Method and apparatus for analysis of variables

[0001] This application is a divisional application of U.S. patent application Ser. No. 09/921,524 filed on Aug. 3, 2001, which claimed the benefit of U.S. Provisional Patent Application No. 60/223,206 filed on Aug. 04, 2000.

[0002] Portions of this patent application contain materials that are subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever.

[0003] The present invention relates to methods, processes and apparatus for measuring and analysis of variables, provided that the definitions of the terms variable and measuring are adopted from the 7th edition of the International Patent Classification (IPC). This invention also relates to generic measurement systems and processes, that is, the proposed measuring arrangements are not specially adapted for any specific variables, or to one particular environment. This invention also relates to methods and corresponding apparatus for measuring which extend to different applications and provide results other than instantaneous values of variables. The invention further relates to post-processing analysis of measured variables and to statistical analysis.

[0004] In a broad sense, the primary goal of a measurement can be defined as making a phenomenon available for human perception. Even when the results of measurements are used to automatically control machines and processes, the results of such control need to be meaningful, and thus the narrower technical meanings of a measurement still fall under the more general definition. In a technical sense, measurement often means finding a numerical expression of the value of a variable in relation to a unit or datum or to another variable of the same nature. This is normally accomplished by practical implementation of an idealized data acquisition system. The idealization is understood as a (simplified) model of such measuring process, which can be analyzed and comprehended by individuals. This analysis can either be performed directly through senses, or employ additional tools such as computers. When the measurement is reduced to a record, such record is normally expressed in discrete values in order to reduce the amount of information, and to enable storage of this record and its processing by digital machines. The reduction to a finite set of values is also essential for human comprehension. However, a physical embodiment of an idealized data acquisition system is usually an analog machine. That is, it is a machine with continuous action, where the components (mechanical apparatus, electrical circuits, optical devices, and so forth) respond to the input through the continuously changing parameters (displacements, angles of rotation, currents, voltages, and so forth). When the results of such implementation are reduced to numerical values, the uncertainties due to either limitations of the data acquisition techniques, or to the physical nature of the measured phenomenon, are often detached from these numerical values, or from each other. Ignoring the interdependence of different variables in the analyzed system, either intrinsic (due to their physical nature), or introduced by measuring equipment, can lead to misleading conclusions. An example of such idealization of a measurement is its digital record, where the measurement is represented by a finite set of numbers. It needs to be pointed out that the digital nature of a record is preserved even if such record were made continuous in time, that is, available as a (finite) set of instantaneous values.

[0005] Generally, measurement can be viewed as transformation of the input variable into another variable such that it can be eventually perceived, or utilized in some other manner. Measurement may consist of many intermediate steps, or stages between the incoming variable and the output of the acquisition system. For example, a TV broadcast can simplistically be viewed as (optical) measurement of the intensity of the light (incoming variable). where the output (image) is displayed on a TV screen. The same collectively would be true for a recorded TV program, although the intermediate steps of such a measurement will be different.

[0006] Regardless of the physical nature of measuring processes, they all exhibit many common features. Namely, they all involve transformation and comparison of variables at any stage. Transformation may or may not involve conversion of the nature of signals (for instance, conversion of pressure variations into electric signals by a microphone in acoustic measurements), and transformation can be either linear or nonlinear. Most transformations of variables in an acquisition system involve comparison as the basis for such transformations. Comparison can be made in relation to any external or internal reference, including the input variable itself. For example, simple linear filtering of a variable transforms the input variable into another variable, which is a weighted mean of the input variable either in time, space, or both. Here the comparison is made with the sample of the input variable, and the transformation satisfies a certain relation, that is, the output is the weighted average of this sample. An example of such filtering would be the computation of the Dow Jones Industrial Average.

[0007] In measurements of discrete events, a particular nonlinear filtering technique stands out due to its important role in many applications. This technique uses the relative positions, or rank, of the data as a basis for transformation. For example, the salaries and the family incomes are commonly reported as percentiles such as current median salary for a certain profession. The rationale for reporting the median rather than the mean income can be illustrated as follows. Consider some residential neighborhood generating ten million dollars annually. Now, if someone from this neighborhood wins twenty millions in a lottery, this will triple the total as well as the mean income of the neighborhood. Thus reporting the mean family income will create an illusion of a significant increase in the wealth of individual families. The median income, however, will remain unchanged and will reflect the economic conditions of the neighborhood more accurately. As another simple example, consider the way in which a student's performance on a standardized test such as the SAT (Scholastic Aptitude Test) or GRE (Graduate Record Examination) is measured. The results are provided as a cumulative distribution function, that is, are quoted both as a score and as the percentile. The passing criterion would be the score for a certain percentile. This passing score can be viewed as the output of the admission filter.

[0008] In digital signal processing, a similar filtering technique is commonly used and is referred to as rank order or order statistic filtering. Unlike a smoothing filter which outputs a weighted mean of the elements in a sliding window, a rank order filter picks up an output according to the order statistic of elements in this window. See, for example, Arnold et al., 1992, and Sarhan and Greenberg, 1962, for the definitions and theory of order statistics. Maximum, minimum, and median filters are some frequently used examples. Median filters are robust, and can remove impulse noise while preserving essential features. The discussion of this robustness and usefulness of median filters can be found in, for example, Arce et al., 1986. These filters are widely used in many signal and image processing applications. See, for example, Bovik et al., 1983; Huang, 1981; Lee and Fam, 1987. Many examples can be found in fields such as seismic analysis Bednar, 1983, for example, biological signal processing Fiore et al., 1996, for example, medical imaging Ritenour et al., 1984, for example, or video processing Wischermann, 1991, for example. Maximum and minimum selections are also quite common in various applications Haralick et al., 1987, for example.

[0009] Rank order filtering is only one of the applications of order statistic methods. In a simple definition, the phrase order statistic methods refers to methods for combining a large amount of data (such as the scores of the whole class on a homework) into a single number or small set of numbers that give an overall flavor of the data. See, for example, Nevzorov, 2001, for further discussion of different applications of order statistics. The main limitations of these methods arise from the explicitly discrete nature of their definition (see, for example, the definitions in Sarhan and Greenberg, 1962, and Nevzorov, 2001), which is in striking dissonance with the continuous nature of measurements. The discrete approach imposes the usage of algebraic rather than geometric tools in order statistics, and thus limits both the perception of the results through the geometric interpretation and the applicability of differential methods of analysis.

[0010] Order statistics of a sample of a variable is most naturally defined in terms of the cumulative distribution function of the elements composing this sample see David, 1970, for example, which is a monotonic function. Thus computation of an order statistic should be equivalent to a simple task of finding a root of a monotonic function. However, the cumulative distribution of a discrete set is a discontinuous function, since it is composed of a finite number of step functions (see Scott, 1992, for example). As a result, its derivative (the density function) is singular, that is, composed of a finite number of impulse functions such as Dirac -function (see, for example, Dirac, 1958, p. 58-61, or Davydov, 1988, p. 609-612, for the definition and properties of the Dirac -function). When implementing rank order methods in software, this discontinuity of the distribution function prevents us from using efficient methods of root finding involving derivatives, such as the Newton-Raphson method (see Press et al., 1992, and the references therein for a discussion of root finding methods). In hardware, the inability to evaluate the derivatives of the distribution function disallows analog implementation. Even though for a continuous-time signal the distribution function may be continuous in special cases (since now it is an average of an infinitely large number of step functions), the density function is still only piecewise continuous, since every extremum in the sample produces singularity in the density function (Nikitin, 1998, Chapter 4, for example). In fact, the nature of a continuous-time signal is still discrete, since its instantaneous and even time averaged densities are still represented by impulse functions (Nikitin, 1998, for example). Thus the time continuity of a signal does not automatically lead to the continuity of the distribution and the density, functions of a sample of this signal.

[0011] Following from their discrete nature, the limitations of the existing rank order methods (rank order filtering as well as other methods based on order statistics) can roughly be divided into two categories. The first category deals with the issues of the implementation of these methods, and the second one addresses the limitations in the applicability. The implementation of the order statistics methods can in turn be divided into two groups. The first group realizes these methods in software on sequential or parallel computers (see Juhola et al., 1991, for example). The second one implements them on hardware such as Very Large Scale Integration (VLSI) circuits (see Murthy and Swamy, 1992, for example).

[0012] In software implementation, the basic procedure for order statistics calculation is comparison and sorting. Since sorting can be constructed by selection, which is an operation linear in complexity, the algorithms for finding only a specific rank (such as median) are more effective than the algorithms for computation of arbitrary statistics (Pasian, 1988, for example). In addition, the performance of rank order calculations can be improved by taking advantage of the running window where only a minor portion of the elements are deleted and replaced by the same number of new elements (Astola and Campbell, 1989, for example). Regardless of the efficiency of particular algorithms, however, all of them quickly become impractical when the size of the sample grows, due to the increase all both computational intensity and memory requirements.

[0013] The hardware implementation of rank order processing has several main approaches, such as systolic algorithms (Fisher, 1984, for example), sorting networks (Shi and Ward, 1993, and Opris, 1996, for example), and radix (binary partition) methods (Lee and Jen, 1993, for example). The various hardware embodiments of the order statistics methods, however, do not overcome the intrinsic limitations of the digital approach arising from the discontinuous nature of the distribution function, such as inefficient rank finding, difficulties with processing large samples of data, and inability to fully explore differential techniques of analysis. It needs to be pointed out that the differential methods allow studying the properties at a point, that is, the properties which depend on an arbitrary small neighborhood of the point rather than on a total set of the discrete data. This offers more effective technical solutions. Several so-called analog solutions to order statistic filtering have been proposed see Jarske and Vainio, 1993, for example, where the term analog refers to the continuous (as opposed to quantized) amplitude values, while the time remains discrete. Although a definition of the continuous-time analog median filter has been known since the 1980's (see Fitch et al., 1986), no electronic implementations of this filter have been introduced. Perhaps the closest approximation of the continuous-time analog median filter known to us is the linear median hybrid (LMH) filter with active RC linear subfilters and a diode network (Jarske and Vainio, 1993, for example).

[0014] The singular nature of the density functions of discrete variables does not only impede both software and hardware implementations of rank order methods, but also constrains the applicability of these methods (for example, their geometric extension) to signal analysis. The origin of these constraints lies in the contrast between the discrete and the continuous: The mathematical model of a separate object is the unit, and the mathematical model of a collection of discrete objects is a sum of units, which is, so to speak, the image of pure discreteness, purified of all other qualities. On the other hand, the fundamental, original mathematical model of continuity is the geometric figure; . . . (Aleksandrov et al., 1999, v. I, p. 32). Even simple time continuity of the incoming variable enables differentiation with respect to time, and thus expands such applicability to studying distributions of local extrema and crossing rates of signals (Nikitin et al., 1998, for example), which can be extremely useful characteristics of a dynamic system. However, these distributions are still discontinuous (singular) with respect to the displacement coordinates (thresholds). Normally, this discontinuity does not restrain us from computing certain integral characteristics of these distributions, such as their different moments. However, many useful tools otherwise applicable to characterization of distributions and densities are unavailable. For instance, in studies of experimentally acquired distributions the standard and absolute deviations are not reliable indicators of the overall widths of density functions, especially when these densities are multimodal, or the data contain so-called outliers. A well-known quantity Full Width at Half Maximum (FWHM) (e.g., Zaidel' et al., 1976, p.18), can characterize the width of a distribution much more reliably, even when neither standard nor absolute deviation exists. The definition of FWHM, however, requires that the density function be continuous and finite. One can introduce a variety of other useful characteristics of distributions and density functions with clearly identifiable geometrical and physical meaning, which would be unavailable for a singular density function. An additional example would be an -level contour surface (Scott, 1992, p. 22), which requires both the continuity and the existence of the maximum or modal value of the density function.

[0015] Discontinuity of the data (and thus singularity of density functions) is not a direct result of measurements but rather an artifact of idealization of the measurements, and thus a digital record should be treated simply as a sample of a continuous variable. For example, the threshold discontinuity of digital data can be handled by convolution of the density function of the discrete sample with a continuous kernel. Such approximation of the true density is well known as Kernel Density Estimates (KDE) (Silverman, 1986, for example), or the Parzen method (Parzen, 1967, for example). This method effectively transforms a digital set into a threshold continuous function and allows successful inference of true distributions from observed samples. See Lucy, 1974, for the example of the rectification of observed distributions in statistical astronomy. The main limitation of the KDE is that the method primarily deals with samples of finite size and does not allow treatment of spatially and temporally continuous data. For example, KDE does not address the time dependent issues such as order statistic filtering, and does not allow extension of the continuous density analysis to intrinsically time dependent quantities such as counting densities. Another important limitation of KDE is that it fails to recognize the importance of and to utilize the cumulative distribution function for analysis of multidimensional variables. According to David W. Scott (Scott, 1992, page 35), . . . The multivariate distribution function is of little interest for either graphical or data analytical purposes. Furthermore, ubiquitous multivariate statistical applications such as regression and classification rely on direct manipulation of the density function and not the distribution function. Some other weaknesses of KDE with respect to the present invention will become apparent from the further disclosure.

[0016] Threshold, spatial, and temporal continuity are closely related to our inability to conduct exact measurements, for a variety of reasons ranging from random noise and fluctuations to the Heisenberg uncertainty. Sometimes the exact measurements are unavailable even when the measured quantities are discrete. An example can be the pregnant chad problem in counting election votes. As another example, consider the measurement of the energy of a charged particle. Such measurement is normally carried out by means of discriminators. An ideal discriminator will register only particles with energies larger than its threshold. In reality, however, a discriminator will register particles with smaller energies as well, and will not detect some of the particles with larger energies. Thus there will be uncertainties in our measurements. Such uncertainties can be expressed in terms of the response function of the discriminator. Then the results of our measurements can be expressed through the convolution of the ideal measurements with the response function of the discriminator (Nikitin, 1998, Chapter 7, for example). Even for a monoenergetic particle beam, our measurements will be represented by a continuous curve. Since deconvolution is at least an impractical, if not impossible, way of restoring the original signal, the numerical value for the energy of the incoming particles will be deduced from the measured density curve as, for example, its first moment (Zaidel' et al., 1976, pp. 11-24, for example).

[0017] A methodological basis for treatment of an incoming variable in terms of its continuous densities can be found in fields where the measurements are taken by an analog action machine, that is, by a probe with continuous (spatial as well as temporal) impulse response. such as optical spectroscopy (see Zaidel' et al., 1976. for example). The output of such a measuring system is described by the convolution of the impulse response of the probe with the incoming signal, and is continuous even for a discrete incoming signal. For instance, the position of a certain spectral line measured by a monochromator is represented by a smooth curve rather than by a number. If the reduction of the line's position to a number is needed, this reduction is usually done by replacing the density curve by its modal, median, or average value.

[0018] The measurement of variables and analysis of signals often go hand-in-land, and the distinction between the two is sometimes minimal and normally well understood from the context. One needs to understand, however, that a signal, commonly, is already a result of a measurement. That is, a signal is already a result of a transformation (by an acquisition system) of one or many variables into another variable (electrical, optical, acoustic, chemical, tactile, etc.) for some purpose, such as further analysis, transmission, directing, warning, indicating, etc. The relationship between a variable and a signal can be of a simple type, such that an instantaneous value of the variable can be readily deduced from the signal. Commonly, however, this relationship is less easily decipherable. For example, a signal from a charged particle detector is influenced by both the energies of the particles and the times of their arrival at the sensor. In order to discriminate between these two variables, one either needs to use an additional detector (or change the acquisition parameters of the detector), or to employ additional transformation (such as differentiation) of the acquired signal. The analysis of the signal is thus a means for gathering information about the variables generating this signal, and ultimately making the phenomenon available for perception, which is the goal. The artificial division of this integral process into the acquisition and the analysis parts can be a serious obstacle in achieving this goal.

[0019] In the existing art, the measurement is understood as reduction to numbers, and such reduction normally takes place before the analysis. Such premature digitization often unnecessary complicates the analysis. The very essence of the above discussion can be revealed by the old joke that it might be hard to divide three potatoes between two children unless you make mashed potatoes. Thus we recognize that the nature of the difficulties with implementation and applicability of order statistics methods in analysis of variables lies in the digital approach to the problem. By digitizing, we lose continuity. Continuity does not only naturally occur in measurements conducted by analog machines, or arise from consideration of uncertainty of measurements. It is also important for perception and analysis of the results of complex measurements, and essential for geometrical and physical interpretation of the observed phenomena. The geometric representation makes many facts of analysis intuitive by analogy with the ordinary space. By losing continuity, we also lose differentiability, which is an indispensable analytical tool since it allows us to set up differential equations describing the studied system: . . . In order to determine the function that represents a given physical process, we try first of all to set up an equation that connects this function in some definite way with its derivatives of various orders (Aleksanidrov et al., 1999, v. I, p. 119).

[0020] The origin of the limitations of the existing art can thus be identified as relying on the digital record in the analysis of the measurements, which impedes the geometrical interpretation of the measurements and leads to usage of algebraic rather than differential means of analysis.

[0021] As was stated in the description of the existing art, the digital approach limits both the geometrical interpretation of the measurements and prevents usage of the differential means of analysis. In this invention, we present an analog solution to what is usually handled by digital means. We overcome the deficiencies of the prior approach by considering, instead of the values of the variables, such geometrical objects as the densities of these variables in their threshold space. The applicability of the differential analysis is achieved byt either (1) preserving, whenever possible, the original continuity of the measurement in the analysis, or (2) restoring continuity of discrete data through convolution with a continuous kernel which represents the essential qualities of the measuring apparatus. Our approach offers (a) improved perception of the measurements through geometrical analogies, (b) effective solutions of the existing computational problems of the order statistic methods, and (c) extended applicability of these methods to analysis of variables. In the subsequent disclosure we will demonstrate the particular advantages of the invention with respect to the known art.

[0022] In this invention, we address the problem of measuring and analysis of variables, on a consistent general basis, by introducing threshold densities of these variables. As will be described further in detail, the threshold densities result from averaging of instantaneous densities with respect to thresholds, space, and time. Since this averaging is performed by a continuous kernel (test function), it can be interpreted as analog representation, and thus we adopt the collective designation Analysis of Variables Through Analog Representation (AVATAR) for various components of the invention. The interdependence of the variables in the measurement system is addressed by introducing the modulated threshold densities of these variables, which result from the consideration of the joint densities of the inter-dependent variables in their combined threshold space. The particular way in which these densities are introduced leads to the densities being continuous in thresholds, space, and time, even if the incoming variables are of discrete nature. This approach allows us to successfully address the limitations of the prior art identified earlier, opens up many new opportunities for expanding the applicability of rank order analysis of variables, and provides a means for efficient implementation of this analysis in both hardware and software.

[0023] In order to convey the inventive ideas clearly, we adopt the simplified model for measurements as follows (see the analogous ideal system in Nikitin et al., 1998, for example). A variable is described in terms of displacement coordinates, or thresholds, as well as in terms of some other coordinates such as spatial coordinates and physical time. The values of these coordinates are measured by means of discriminators and/or differential discriminators (probes). An ideal discriminator with threshold D returns the value 1 if the measured coordinate exceeds D, if it is equal to D, and it returns zero otherwise. Thus the mathematical expression for an ideal discriminator is the Heaviside unit step function of the difference between the threshold and the input coordinate (see Nikitin et al., 1998, and Nikitin, 1998, for example). Although ideal discriminators can be a useful tool for analysis of a measurement process, different functions of thresholds need to be employed to reflect the vagaries of real measurements (see discussion in Nikitin et al., 1998, for example). In this invention, the peculiarities of real discriminators are reflected by introducing uncertainty into the mathematical description of the discriminator, such that the returned value is in the range zero to one, depending on the values of the input coordinate. As described further in this disclosure, the introduction of such uncertainty can be interpreted as averaging with respect to a threshold test function, or threshold averaging. When the input-output characteristic of the discriminator is a continuous function, then differentiability of the output with respect to threshold is enabled. In addition, if the characteristic of the discriminator is a monotonic function, the rank relations of the input signal are preserved. If the original input signal is not differentiable with respect to time (e.g., the input signal is discontinuous in time), differentiability with respect to time can always be enabled by introducing time averaging into the acquisition system, where under time averaging we understand a suitable convolution transform with a continuous-time kernel. Likewise, differentiability with respect to spatial coordinates can be enabled by spatial averaging.

[0024] The mathematical expression for the response of an ideal probe, or differential discriminator, is the Dirac -function of the difference between the displacement and the input variable, which is the instantaneous density of the input variable. As follows from the properties of the Dirac -function (see, for example, Dirac, 1958, p. 58-61. and Davydov, 1988, p. 609-612, for the definition and properties of the Dirac -function), the output of the real probe is thus the convolution of the instantaneous density with the out-output of characteristic of the differential discriminator, which is equivalent to the threshold averaging of the instantaneous density. When this output is subsequently averaged with respect to space and time, the result is the Threshold-Space-Time Averaged Density.

[0025] Notice that the transition from the ideal to real probes and discriminators preserves the interpretation of their responses as the threshold density and cumulative distribution, respectively. For example, the spectrum acquired by an optical spectrograph can be considered the energy density regardless the width of the slits of its monochromator. Thus a particular way of utilizing the discriminators and probes in this invention is essentially a method of transformation of discrete or continuous variables, and/or ensembles of variables into normalized continuous scalar fields, that is, into objects with mathematical properties of density and cumulative distribution functions. In addition to dependence on the displacement coordinates (thresholds), however, these objects can also depend on other parameters, including spatial coordinates (e.g., if the incoming variables are themselves scalar or vector fields), and/or time (if the variables depend on time). For the purpose of this disclosure, the terms space and time are used to cover considerably more than their primary or basic meaning. Time should be understood as a monotonic scalar, continuous or discrete, common to all other analyzed variables, which can be used for sequential ordering of the measurements. Space is thus all the remaining coordinates which are employed (as opposed to sufficient) to govern the values of the input variables. It is important to note that the use of the invented transformation makes all existing techniques for statistical processing of data, and for studying probability fluxes, applicable to analysis of these variables. Moreover, the transformations of the measured variables can be implemented with respect to any reference variable, or ensemble of reference variables. In this disclosure, we consider two basic transformations with respect to the reference variable, which we refer to as normalization and modulation. The definitions of these transformations will be given later in the disclosure. In both of these transformations, the behavior of the input variable is represented in terms of behavior (and units) of the reference variable. Thus, the values of the reference variable provide a common unit, or standard, for measurement and comparison of variables of different nature, for assessment of the mutual dependence of these variables, and for evaluation of the changes in the variables and their dependence with time. For example, dependence of economic indicators on social indicators, and vice versa, can be analyzed, and the historical changes in this dependence can be monitored. When the reference variable is related in a definite way to the input variable itself, these additional transformations (that is, normalization and modulation) provide a tool for analysis of the interdependence of various properties of the input variable.

[0026] Among various embodiments of the invention, several are of particular importance for analysis of variables. These are the ability to measure (or compute from digital data) (1) quantile density, (2) quantile domain, and (3) quantile volume for a variable. Quantile density indicates the value of the density likely to be exceeded, quantile domain contains the regions of the highest density, and quantile volume gives the (total) volume of the quantile domain. The definitions of these quantities and a means of their implementation are unavailable in the existing art. Detailed definitions and description of these quantities will be given later in the disclosure. As another consequence of the proposed transformation of variables into density functions, the invention enables measurements of currents, or fluxes of these densities, providing a valuable tool for analysis of the variables.

[0027] Another important embodiment of AVATAR is rank normalization of a variable with respect to a cumulative distribution function, generated by another variable or ensemble of variables. The rank normalization of variables can be used for processing and analysis of different time ordered series, ensembles, scalar or vector fields, and time independent sets of variables, especially when the investigated characteristics of the variable are invariant to a monotonic transformation of its values, that is, to a monotonic transformation of the thresholds. This normalization can be performed with respect to a reference distribution of an arbitrary origin, such as the distribution provided by an external reference variable, or by the input variable itself. For example, the reference variable can be a random process with the parameters determined by the input variable. In this case, the reference variable provides a container in the threshold space where the input variable is likely to be found. More importantly, the rank normalization allows computation or measurement of integral estimators of differences between two distributions (or densities) as simple time and/or space averages.

[0028] Another important usage of the rank normalization is as part of preprocessing of the input variable, where under preprocessing we understand a series of steps (e.g., smoothing) in the analysis prior to applying other transformations. Since in AVATAR the extent of the threshold space is determined by the reference variable, the rank normalization allows as to adjust the resolution of the acquisition system according to the changes in the threshold space, as the reference variable changes in time. Such adjustment of the resolution is the key to a high precision of analog processing.

[0029] While rank normalization reflects the rank relations between the input and the reference variables, the modulated threshold density describes the input variable in terms of the rate of change of the reference variable at a certain threshold. As will be clarified further in the disclosure, the modulated threshold densities arise from the consideration of the joint densities of the input and the reference variables in their combined threshold space. Instead of analyzing such joint variable in its total threshold space, however, we consider the behavior of the input variable in the threshold space of the reference variable only. The modulated densities allow us to investigate the interdependence of the input and the reference variable by comparison of the time averages of the input variable at the thresholds of the reference variable with the simple time average of the input variable. As has been mentioned earlier, the important special case of modulated densities arises when there is a definite relation between the input and the reference variables. In this disclosure, we will primarily focus on the densities arising from the two particular instances of this relation, to which we will further refer as the amplitude and the counting densities. Since the definiteness in the relation eliminates the distinction between the input and the reference variables, such special cases of the modulated densities will be regarded simply as modulated threshold densities of the input variable.

[0030] The invention also allows the transformation which can be interpreted as rank filtering of variables. That is, it enables the transformation of a variable into another variable, the value of which at any given space and time is a certain quantile of a modulated cumulative distribution function generated by the input and reference variables. Thus in AVATAR the output of such a filter has simple and clear interpretation as a level line of such a distribution in the time-threshold plane. One needs to notice that the output of a rank filter as defined in the existing digital signal processing methods, will correspond to the discrete points on a level line drawn for an amplitude distribution only. Thus the rank filtering defined in AVATAR extends beyond the known applicability of rank filtering. It is also important that, in this invention, such filtering process is implemented by differential means and thus conducted without sorting. The invention also provides a means for finding (selecting) the rank of a time dependent or static variable or ensemble of variables without sorting. Such analog rank selection permits analog emulation of digital rank filters in an arbitrary window, of either finite or infinite type, with any degree of precision. Moreover, the rank selection is defined for modulated distributions as well, which extends the applicability of rank filtering. The continuous nature of the definitions and the absence of sorting allows easy implementation and incorporation of rank selecting and filtering in analog devices. Rank filtering and selecting also provide alternative embodiments for comparison of variables with respect to a common reference variable, and for detection and quantification of changes in variables.

[0031] Based on the embodiments of AVATAR discussed above, we can define and implement a variety of new techniques for comparison of variables and for quantification of changes in variables. By determining distributions from the signal itself, and/or by providing a common reference system for variables of different natures, the invention provides a robust and efficiently applied solution to the problem of comparing variables. It is important to note that the invention enables comparison of one-dimensional as well as multivariate densities and distributions by simple analog machines rather than through extensive numerical computations.

[0032] The particular advantages of AVATAR stem from the fact that the invention is based on the consideration of real acquisition systems while most of previous techniques assume idealized measurement processes. Even if the incoming variable is already a digital record (such as the result of an idealized measurement process), we restore this record to a continuous form by a convolution with a probe continuous in thresholds, space, and time. It is important to realize that the invention is also applicable to measurements of discrete data. The main distinction between such measurements and the restoration of a digital record lies in the realization that the former are normally taken by continuous action machines. and thus the results of such measurements need to be reformulated, that is, mapped into analog domain. As an example, consider the task of measuring the amplitude (energy) distribution of a train of charged particles. Since this measurement is usually taken by means of sensors with finite time response, the problem of measuring the amplitude density is naturally restated as the problem of finding the distribution of local extrema of a continuous-time signal (Nikitin et al., 1998). This distribution can in turn be found through the derivative of the crossing rates with respect to threshold (Nikitin et al., 1998).

[0033] The invention can be implemented in hardware devices as well as in computer codes (software). The applications of the invention include, but are not limited to, analysis of a large variety of technical, social, and biologic measurements, traffic analysis and control, speech and pattern recognition, image processing and analysis, agriculture, and telecommunications. Both digital and analog implementations of the methods can be used in various systems for data acquisition and analysis. All the above techniques, processes and apparatus are applicable to analysis of continuous (analog) variables as well as discrete (digital). By analyzing the variables through their continuous distributions and the density functions, the invention overcomes the limitations of the prior art by (a) improving perception of the measurements through geometrical analogies, (b) providing effective solutions to the existing computational problems of the order statistic methods, and (c) extending the applicability of these methods to analysis of variables.

[0034] Further scope of the applicability of the invention will be clarified through the detailed description given hereinafter. It should be understood, however, that the specific examples, while indicating preferred embodiments of the invention, are presented for illustration only. Various changes and modifications within the spirit and scope of the invention should become apparent to those skilled in the art from this detailed description. Furthermore, all the mathematical expressions and the examples of hardware implementations are used only as a descriptive language to convey the inventive ideas clearly, and are not limitative of the claimed invention.

[0035] For convenience, the essential terms used in the subsequent detailed description of the invention are provided below. These terms are listed along with their definitions adopted for the purpose of this disclosure. Examples clarifying and illustrating the meaning of the definitions are also provided. Note that the equations in this section are numbered separately from the rest of this disclosure.

[0036] For the purpose of this disclosure, we define a variable as an entity x which, when expressed in figures (numbers), can be represented by one of the mathematical expressions listed below. Throughout this disclosure, the standard mathematical terms such as vector, scalar, or field mostly preserve their commonly acceptable mathematical and/or physical interpretation. The specific meaning of most of the common terms will be clarified through their usage in the subsequent detailed description of the invention. Notice that the most general form of a variable adopted in this disclosure is an Ensemble of Vector Fields. All other types of variables in this disclosure can be expressed through various simplifications of this general representation. For example, setting the ensemble weight n() in the expression for an ensemble of vector fields to the Dirac -function () reduces said expression to a single Vector Field variable. Further, by eliminating the dependence of the latter on spatial coordinates (that is, by setting the position vector aconstant), said single vector field variable reduces to a single Vector variable. Notice also that while ensembles of variables are expressed as integrals/sums of the components of an ensemble, it should be understood that individual components of an ensemble are separately available for analysis.

[0037] 1. Single Variable.

[0038] A single variable in this disclosure can be a vector/scalar variable, or a vector/scalar field variable.

[0039] (a) A elector Field variable can be expressed as

[0040] where a is the vector of spatial coordinates, and t is the time coordinate.

[0041] Representative Examples: (1) A truecolor image can be expressed by a vector field xx(a,t), where the color is described by its coordinates in the three-dimensional color space (red, green, and blue) at the position a. (2) The combination of both the color intensity of a monochrome image and the rate of change of said intensity can be expressed by a vector field x(x(a,t),{dot over (x)}(a,t)).

[0042] (b) A Scalar Field variable can be expressed as

[0043] where a is the vector of spatial coordinates, and t is the time coordinate.

[0044] Representative Example: A monochrome image at a given time is determined by the intensity of the color at location a, and thus it is conveniently described by a scalar field xx(a,t).

[0045] (c) A Vector variable can be represented by the expression

[0046] where t is the time coordinate. Notice that the components of a vector variable do not have to be of the same units or of the same physical nature.

[0047] Representative Examples: (1) The position of a vehicle in a traffic control problem can be described by a single vector variable xx(t). (2) The position and the velocity of a vehicle together can be described by a single vector variable x(x(t),{dot over (x)}(t)).

[0048] (d) A Scalar variable can be represented by the expression

[0049] where t is the time coordinate.

[0050] Representative Example: The current through an element of an electrical circuit can be expressed as a single scalar variable xx(t).

[0051] 2. Ensemble of Variables. Several different variables of the same nature can be considered as a (composite) single entity designated as an ensemble of variables. An individual variable in an ensemble is a component of the ensemble. The relative contribution of a component in the ensemble is quantified by a weight n() of the component.

[0052] (a) Ensemble of Vector Fields:

[0053] where n() dt is the weight of the th component of the ensemble such that dn()N, a is the vector of spatial coordinates, and t is the time coordinate.

[0054] Representative Example: A truecolor image can be expressed by a vector field xx(a,t), where the color is described by its coordinates in the three-dimensional color space (red, green, and blue), at the position a. A compound image consisting of a finite or infinite set of such truecolor images, weighted by the weights n(), can be viewed as an ensemble of vector fields. For example, such a compound image can be thought of as a statistical average of the video recordings taken by several different cameras.

[0055] (b) Ensemble of Scalar Fields:

[0056] where n() d is the weight of the th component of the ensemble such that dn()N, a is the vector spatial coordinates, and t is the time coordinate.

[0057] Representative Example: A monochrome image at a given time is determined by the intensity of the color at location a, and thus it is conveniently described by a scalar field xx(a,t). A compound image consisting of a finite or infinite set of such monochrome images, weighted by the weights n(), can be viewed as an ensemble of scalar fields. For example, such a compound image can be thought of as a statistical average of the video recordings taken by several different cameras.

[0058] (c) Ensemble of Vectors:

[0059] where n() d is the weight of the th component of the ensemble such that dn()N, and t is the time coordinate.

[0060] Representative Example: A variable expressing the average position of N different vehicles in a traffic control problem can be described by an ensemble of vector variables xdn()x(t), where n()n()_i1^N(i).

[0061] (d) Ensemble of Scalars:

[0062] where n() d is the weight of the th component of the ensemble such that dn()N, and t is the time coordinate.

[0063] Representative Example: The total current through N elements of an electrical circuit can be expressed as an ensemble of scalar variables xdn()x(t), where n()n()_i1^N(i).

[0064] Notice that the most general form of a variable among those listed above is the Ensemble of Vector Fields. All other types of variables in this disclosure can be expressed through various simplifications of this general representation. For example, setting the ensemble weight n() in the expression for an ensemble of vector fields to the Dirac -function () reduces said expression to a single Vector Field variable. Further, by eliminating the dependence of the latter on spatial coordinates (that is, by setting the position vector aconstant), said single vector field variable reduces to a single Vector variable. Notice also that while ensembles of variables are expressed as integrals/sums of the components of an ensemble, it should be understood that individual components of an ensemble are separately available for analysis.

[0065] In this disclosure, we define a Threshold Filter as a continuous action machine (a physical device, mathematical function, or a computer program) which can operate on the difference between a Displacement Variable D and the input variable x, and the result of such operation can be expressed as a scalar function of THE Displacement Variable, that is, as a value at a given displacement D. The dependence of the output of a Threshold Filter on the input is equivalent to those of a probe (smoothing threshold filter) or a discriminator (integrating threshold filter) as specified below.

[0066] 1. Probe. A continuous action machine (a physical device, mathematical function, or a computer program) which can operate on the difference between a Displacement Variable D and the input variable x, and the result of such operation can be expressed as a scalar function of the Displacement Variable, that is, as a value at a given displacement D. The dependence of the output y of a probe on its input r is equivalent to the following expression:

[0067] where y is a scalar, r is a vector or a scalar, R is a scalar width parameter of the probe, and the test function f_R is a continuous function satisfying the conditions $\begin{array}{cc}\{\begin{array}{c}{}_{-}^{}{}^{n}r{f}_R\left(r\right)=1,\mathrm{and}\\ \underset{R0}{\lim }{f}_R\left(r\right)=\left(r\right),\end{array}& \left(D\text{-}10\right)\end{array}$

[0068] where (r) is the Dirac -function.

[0069] Representative Examples: (1) The Gaussian test function $\begin{array}{cc}{f}_R\left(r\right)=\underset{i=1}{\overset{n}{}}{}_{D_i}{}_{D_i}\left(r_i\right)=\frac{{}^{-\frac{n}{2}}}{\underset{i=1}{\overset{n}{}}{D}_i}\exp \left[-\underset{i=1}{\overset{n}{}}{\left(\frac{r_i}{D_i}\right)}^2\right]& \left(D\text{-}11\right)\end{array}$

[0070] can act as a probe. In Eq. (D-11), the response of the probe _R(r) to the vector input r(r₁ . . . , r_n) is a product of the responses of the probes ${}_{D_i}{}_{D_i}\left(r_i\right)=\frac{1}{D_i\sqrt{}}{}^{-{\left(\frac{r_i}{D_i}\right)}^2}$

[0071] the components r_i of the input vector. (2) FIG. 6 illustrates an optical threshold smoothing filter (probe). This probe consists of a point light source S and a thin lens with the focal length . The lens is combined with a gray optical filter with transparency described by ₂(x). Both the lens and the filter are placed in an XOY plane at a distance 2 from the source S. The lens-filter combination can be moved in the XOY plane by the incoming signal r so that the center of the combination is located at $\frac{2\mathrm{fr}}{4f-R}$

[0072] in this plane. Then the output of the filter is proportional to the intensity of the light measured at the location D(D_x, D_y) in the D_x-O-D_y plane parallel to the XOY plane and located at the distance R from the image S of the source S (toward the source). That is, the output of this filter can be described by _R(Dr).

[0073] 2. Discriminator. A continuous action machine (a physical device, mathematical function, or a computer program) which can operate on the difference between a Displacement Variable D and the input variable x, and the result of such operation can be expressed as a scalar function of the Displacement Variable, that is, as a value at a given displacement D. The dependence of the output y of a discriminator on the input r is obtained by integrating the response of the respective probe, that is, it is related to the input-output characteristic of the respective probe as

[0074] Representative Examples: (1) The integral of a Gaussian test function $\begin{array}{cc}{}_R\left(x\right)={}_{-}^x{}^n{\mathrm{rf}}_R\left(r\right)=\underset{i=1}{\overset{n}{}}{}_{D_i}\left(x_i\right)=2^{-n}\underset{i=1}{\overset{n}{}}\mathrm{erfc}\left(\frac{-x_i}{D_i}\right),& \left(D\text{-}13\right)\end{array}$

[0075] where erfc(x) is the complementary error function, can act as a discriminator. In Eq. (D-13), the response of the discriminator F_R(x) to the vector input x(x₁ . . . , x_n) is a product of the responses of the discriminators ${}_{D_i}\left(x_i\right)=\frac{1}{2}\mathrm{erfc}\left(\frac{-x_i}{D_i}\right)$

[0076] to the components x_i of the input vector. (2) By replacing the transparency function ₂(x) of the gray filter with F₂(x), the optical probe shown in FIG. 6 is converted into a discriminator with the output F_R(Dr).

[0077] A Displacement Variable is the argument of a function describing the output of a Threshold Filter. For example, if the Threshold Filter is an amplifier operating on the difference between two electrical signals D and x(t), and an output of the amplifier is described as a function of the input signal D, this signal D is a Displacement Variable.

[0078] A Modulating Variable is a unipolar scalar field variable KK(a,t) which can be applied to the output of a Threshold Filter in a manner equivalent to multiplication of said output by the Modulating Variable.

[0079] Representative Example: Imagine that the point light source S in FIG. 6 is produced by an incandescent lamp powered by a unit current. If we now power the lamp by the current K(t), the output of the threshold filter will be modulated by the modulating Variable K(t).

[0080] An Averaging Filter is a continuous action machine (a physical device, mathematical function, or a computer program) which can operate on a variable x(a,t), and the result of such operation can be expressed as convolution with a test function _R(a) and a time impulse response function h(t;T), namely as $\begin{array}{cc}\begin{array}{c}{x\left(r,s\right)}_{T,R}^{h,f}={{x\left(r,s\right)}_R^f}_T^h={{x\left(r,s\right)}_T^h}_R^f=\\ ={}_{-}^{}s{}_{-}^{}{}^{n}rh\left(t-s;T\right)f_R\left(a-r\right)x\left(r,s\right),\end{array}& \left(D\text{-}14\right)\end{array}$

[0081] where a is the position vector (vector of spatial coordinates), and t is the time coordinate. Thus an averaging filter performs both spatial and time averaging. We shall call the product h(t;T) _R(a) the impulse response of the Averaging Filter.

[0082] 1. Time Averaging Filter. An averaging filter which performs only time averaging is obtained by setting the spatial impulse response (test function) of the averaging filter to be equal to the Dirac -function, _R(a)(a). The result of the operation of a time averaging filter with an impulse response h(t;T) on a variable x(a,t) can be expressed by the convolution integral

[0083] where T is the width (time scale) parameter of the filter. For two filters with the width parameters T and T such that T>>T, the former filter is designated as wide, and the latter as narrow.

[0084] Representative Example: An image formed on a luminescent screen coated with luminophor with the afterglow half-time T_1/2T ln(2) is time averaged by an exponentially forgetting Time Averaging Filter $h\left(t;T\right)={}^{-\frac{t}{T}}\left(t\right)/T.$

[0085] 2. Spatial Averaging Filter. An averaging filter which performs only spatial averaging is obtained by setting the time impulse response of the averaging filter to be equal to the Dirac -function, h(t;T)(t). The result of the operation of a spatial averaging filter with an impulse response (test function) _R(a) on a variable x(a,t) can be expressed by the convolution integral

[0086] where R is the width parameter of the filter. For two filters with the width parameters R and R such that R>>R, the former filter is designated as wide, and the latter as narrow.

[0087] Representative Example: A monochrome image given by the matrix ZZ_ij(t) can be spatially averaged by a smoothing filter w_mn, _m,nw_mn1, as $Z_{M,N}^w=\underset{m=-M}{\overset{M}{}}\underset{n=-N}{\overset{N}{}}{}_{\mathrm{mn}}{Z}_{i+m,j+n}\left(t\right).$

[0088] Some other terms and their definitions which appear in this disclosure will be provided in the detailed description of the invention.

[0089] The Detailed Description of the Invention is organized as follows.

[0090] In Section 1 (p. 34) we identify the general form of a variable which is subject to analysis by this invention, and provide several examples demonstrating the convenience of such a general representation.

[0091] In Section 2 (p. 35) we use the example of a single scalar variable to describe the basic elements of the analysis system adopted in this disclosure, and introduce the discriminators and probes as the sensors of such a system. The example of a single scalar variable is used to illustrate that the use of discriminators and probes enables us to reformulate many algebraic problems of the conventional analysis of variables as geometrical problems in the threshold space. The particular continuous fashion in which this geometrical extension is performed enables the solution of these problems by methods of differential geometry.

[0092] In Section 3 (p. 39) we describe some of the exemplary discriminators and the respective probes.

[0093] In Section 4 (p. 40) we introduce the normalized scalar fields in the meaning adopted for the purpose of this disclosure as the density and cumulative distribution functions in the threshold space. We provide a tangible example of how the usefulness of these objects extends beyond making details of the analysis intuitive and more available for human perception by analogy with the ordinary space.

[0094] In Section 5 (p. 42) we provide several examples of equations which reflect the geometrical properties of the threshold distributions, and are later used for development of various practical embodiments of AVATAR. In particular, the definitions of the quantile density, domain, and volume are given along with the explanatory examples.

[0095] Section 6 (p. 47) contains a brief additional discussion of possible relationships between the input and the reference variables.

[0096] In Section 7 (p. 49) we give an introduction to a more general definition of the modulated threshold densities by analyzing an example of the threshold crossing density, a quantity which cannot be defined for digitized data.

[0097] In Section 8 (p. 51) we generalize the result of Section 7 by introducing the modulated threshold densities and the weighted means at thresholds. Along with explanatory examples, we show that the weighted mean at reference threshold is indeed a measurement of the input variable in terms of the reference variable. We also outline an approach to computation of the mean at reference threshold by analog machines.

[0098] In Section 9 (p. 55) we interpret the process of measurement by real analog machines as a transition from the microscopic to macroscopic threshold densities, or as threshold averaging by a probe. Thus we introduce the main practical embodiments of AVATAR as the modulated threshold density (Eq. (52)) and the mean at reference threshold (Eq. (53)), along with the specific embodiments of the amplitude (Eq. (54)) and counting (Eq. (55)) densities, and the counting rates (Eq. (56)). We also provide a simplified diagram of a continuous action machine implementing the transformation of the multivariate input variable(s) into the modulated threshold densities.

[0099] In Section 10 (p. 58) we consider a specific type of weighting function, which is a convenient choice for time averaging in various embodiments of AVATAR.

[0100] In Section 11 (p. 59) we focus on some of the applications of AVATAR for enhancement of analysis through geometric interpretation of the results. We give several examples of displaying the modulated threshold densities and provide illustrative interpretation of the observed results. Among various examples of this section, there are examples of displaying the time evolution of the quantile density, domain, and volume. In this section, we introduce such practical embodiments of AVATAR as the phase space amplitude density (Eq. (60)), the phase space counting density (Eq. (61)), and the phase space counting rates (Eq. (62)). We also provide the illustrative examples of displaying these densities and the rates. In Subsection 11.1 we give some illustrative examples of continuous action machines for displaying the modulated threshold densities and their time evolution.

[0101] In Section 12 (p. 65) we provide a practical embodiment (Eq. (63)) of an estimator of differences in the quantile domain between the mean at reference threshold and the time average of a variable.

[0102] In Section 13 (p. 66) we provide a practical embodiment of the modulated cumulative distribution (Eq. (64)) and describe how the transition from the densities to the cumulative distribution functions in various practical embodiments is formally done by replacing the probes by their respective discriminators. Even though the multivariate cumulative distribution function is often disregarded as a useful tool for either graphical or data analytical purposes (Scott, 1992, page 35, for example), it is an important integral component of AVATAR and is used in its various embodiments.

[0103] In Section 14 (p. 67) we develop simple unimodal approximations for an ideal density function, that is, the density function resulting from the measurements by all ideal probe. Although these approximations are of limited usage by themselves, they provide a convenient choice of approximations for the rank normalization.

[0104] In Section 15 (p. 71) we introduce several practical embodiments of rank normalization, such as the general formula for the rank normalization with respect to the reference distribution C_K,r(D,t) (Eq. (86)), normalization by a discriminator with an arbitrary input-output response (Eq. (88)), and normalization of a scalar variable by a discriminator with an arbitrary input-output response (Eq. (89)).

[0105] In Section 16 (p. 74) we discuss the usage of the rank normalization for comparison of variables and for detection and quantification of changes in variables. We provide several simplified examples of such usage and describe a practical embodiment of a simple estimator of differences between two distributions (Eq. (92)). In Subsection 16.1 we provide additional exemplary practical embodiments of the estimators of differences between two time dependent distributions, Eqs. (95) and (97). In Subsection 16.2 we provide an example of the usage of these estimators for comparing phase space densities and for addressing an exemplary speech recognition problem. We also give an outline of an approach to implementation of such comparison in an analog device. In Subsection 16.3 we provide an embodiment for a time dependent probabilistic comparison of the amplitudes of two signals (Eq. (102)).

[0106] In Section 17 (p. 81) we discuss the usage of AVATAR for analog implementation of rank filtering.

[0107] In Section 18 (p. 82) we discuss the two principal approaches to analog rank filtering of a single scalar variable enabled by AVATAR: (1) an explicit expression for the output of a rank filter (Subsection 18.1), and (2) a differential equation for the output of such a filter (Subsection 18.2). In Subsection 18.1, we also describe a practical embodiment (Eq. (105)) for the explicit analog rank filter.

[0108] In Section 19 (p. 84) we briefly discuss the usage of a particular choice of a time weighting function in analog rank filters.

[0109] In Section 20 (p. 84) we describe the main embodiment (Eq. (113)) of an adaptive analog rank filter. In Subsection 20.1, we also provide an alternative embodiment (Eq. (117)) of this filter.

[0110] In Section 21 (p. 87) we extend the definitions of the modulated threshold densities and cumulative distributions to include ensembles of variables. We provide the expressions for the threshold averaged instantaneous density and cumulative distribution of a continuous ensemble, Eqs. (120) and (121), and for the modulated density and cumulative distribution of a continuous ensemble of variables, Eqs. (122) and (123).

[0111] In Section 22 (p. 88) we introduce the analog rank selectors, and provide the equations for the analog rank selectors for continuous (Eq. (126)) and discrete (Eq. (129)) ensembles.

[0112] In Section 23 (p. 90) we describe the embodiment of an adaptive analog rank filter for an ensemble of variables, Eq. (130).

[0113] In Section 24 (p. 90) we introduce the modulated threshold densities for scalar fields, Eq. (131).

[0114] In Section 25 (p. 91) we describe the analog rank selectors and analog rank filters for scalar fields, Eqs. (133), (134), and (135). In Subsection 25.1, we provide an example of filtering monochrome images using a simple numerical algorithm (Eq. (136)), implementing an analog rank selector for scalar fields.

[0115] In Section 26 (p. 93) we complete the description of the primary embodiment of the AVATAR by generalizing the modulated threshold densities to include vector fields, Eq. (138), and ensembles of vector fields, Eq. (139).

[0116] In Section 27 (p. 94)) we provide the description of the mean at reference threshold for a vector field input variable, Eq. (140).

[0117] In Section 28 (p. 95) we describe such important embodiments of AVATAR as the analog filters for the quantile density, domain, and volume. These quantities are defined in AVATAR for multivariate densities, and thus they are equally applicable to the description of the scalar variables and fields as well as to the ensembles of vector fields.

[0118] In Section 29 (p. 97) we provide several additional examples of performance of analog rank filters and selectors.

[0119] In Section 30 (p. 102) we provide a summary of some of the main transformations of variables employed in this disclosure.

[0120] In order to simplify dividing the problem of measurement and analysis of different variables into specific practical problems, let us assume that a variable x can be presented as an ensemble of vector fields, that is, it can be written as

[0121] where n() d is the weight of the th component of the ensemble such that dn()N, a is the spatial coordinate, and t is the time coordinate.

[0122] Convenience of the general representation of a variable by Eq. (1) can be illustrated by the following simple examples. As the first example, consider a vehicle in a traffic control problem. The position of this vehicle can be described by a single vector variable xx(t). If, however, we are measuring the weight of the vehicle's cargo, it might be more convenient to describe it as a scalar field xx(a,t), since the weight x at a given time depends on the position vector a. If we now extend our interest to the total weight of the cargo carried by N different vehicles, this weight is conveniently expressed by an ensemble of scalar fields xdn()x(a,t), where a is the position vector, n()n() _i1^N(i), x_i is the cargo capacity (volume of the cargo space) of the i th vehicle, and n(i) is the density (weight per unit volume) of the cargo in the i th vehicle. Notice that all the different variables in this example can be written in the general form of Eq. (1).

[0123] As a second example, a monochrome image at a given time is determined by the intensity of the color at location a, and thus it is conveniently described by a scalar field xx(a,t). A truecolor image is then naturally expressed by a vector field xx(a,t), where the color is described by its coordinates in the three-dimensional color space (red, green, and blue), at the position a. We can also consider a compound image as a finite or infinite set of such images, weighted by the weights n(). For example, such a compound image can be thought of as a statistical average of the video recordings, taken by several different cameras.

[0124] Additional particular examples of analysis of the variables satisfying the general form of Eq. (1) will be provided later in the disclosure.

[0125] A system for analysis of variables adopted in this disclosure comprises such basis elements as a Threshold Filter, which can be either a Discriminator or a Probe, and an Averaging Filter operable to perform either time or spatial average or both time and spatial average. This system can also include optional modulation and normalization by a Modulating Variable. A simplified schematic of such a basic system for analysis of variables is shown in FIG. 1a. This system is operable to transform an input variable into an output variable having mathematical properties of a scalar field of the Displacement Variable. The Threshold Filter (a Discriminator or a Probe) is applied to a difference of the Displacement Variable and the input variable, producing the first scalar field of the Displacement Variable. This first scalar field is then filtered with a first Averaging Filter, producing the second scalar field of the Displacement Variable. Without optional modulation, this second scalar field is also the output variable of the system, and has a physical meaning of either an Amplitude Density (when the Threshold Filter is a Probe), or a Cumulative Amplitude Distribution (when the Threshold Filter is a Discriminator) of the input variable.

[0126] A Modulating Variable can be used to modify the system as follows. First, the output of the Threshold Filter (that is, the first scalar field) can be multiplied (modulated) by the Modulating Variable, and thus the first Averaging Filter is applied to the resulting modulated first scalar field. For example, when the Threshold Filter is a Probe and the Modulating Variable is a norm of the first time derivative of the input variable, the output variable has an interpretation of a Counting (or Threshold Crossing) Rate. The Modulating Variable can also be filtered with a second Averaging Filter having the same impulse response as the first Averaging Filter, and the output of the first Averaging Filter (that is, the second scalar field) can be divided (normalized) by the filtered Modulating Variable. As will be discussed further in the disclosure, the resulting output variable will then have a physical interpretation of either a Modulated Threshold Density (when the Threshold Filter is a Probe), or a Modulated Cumulative Threshold Distribution (when the Threshold Filter is a Discriminator). For example, when the Threshold Filter is a Probe and the Modulating variable is a norm of the first time derivative of the input variable, the output variable will have an interpretation of a Counting (or Threshold Crossing) Density.

[0127] Let us now describe the basic elements of the analysis system adopted in this disclosure in more details, using the measurement a single scalar variable xx(t) as all example.

[0128] We assume the data acquisition and analysis system which comprises the elements schematically shown in FIG. 1b. Let the input signal be a scalar function of time x(t). This input signal (Panel I) is transformed by the discriminator (Panel IIa) into a function of two variables, the time t and the displacement D. The latter will also be called threshold in the subsequent mathematical treatment. The result of such transformation of the input signal by the discriminator is illustrated in Panel IIIa. We will provide illustrative examples of such a measuring system below.

[0129] The input-output characteristic of the discriminator is described by the continuous monotonic function F_D(x) and is illustrated in Panel IIa. We shall agree, without loss of generality, that $\begin{array}{cc}\{\begin{array}{c}\underset{x-}{\lim }{}_D\left(x\right)=0\\ {}_D\left(0\right)=\frac{1}{2}\\ \underset{x}{\lim }{}_D\left(x\right)=1\end{array}.& \left(2\right)\end{array}$

[0130] Although the convention of Eq. (2) is not necessary, it is convenient for the subsequent mathematical treatment. One skilled in the art will now recognize that a discriminator can thus be interpreted as a threshold integrating filter. The dependence of F_D on the width parameter D then can be chosen in such way that F_D approaches the Heaviside unit step function (Arfken, 1985, p. 490, for example) as D approaches a suitable limit, namely

[0131] $\begin{array}{cc}\underset{D0}{\lim }{}_D\left(D-x\right)=\left(D-x\right),& \left(3\right)\end{array}$

[0132] where (x) is defined as $\begin{array}{cc}\left(x\right)={}_{-}^{x}s\left(s\right)=\{\begin{array}{cc}0& \mathrm{for}x<0\\ \frac{1}{2}& \mathrm{for}x=0\\ 1& \mathrm{for}x>0\end{array},& \left(4\right)\end{array}$

[0133] and (x) is the Dirac -function. When the functional form of the discriminator is the Heaviside unit step function (x), such discriminator will be called an ideal discriminator. Some other exemplary functional choices for discriminators will be discussed further.

[0134] As an illustration, consider the following example. Imagine that the signal x(t) in Panel I is an electrical current which we measure with the ammeter of Panel IIa. The scale of the ammeter is calibrated in units of current, and D is our reading of this scale. Then F_D(Dx) can be interpreted as the probability that our measurement (reading) D exceeds the true value of the input current x, and D is indicative of the precision of the instrument (ammeter). Thus F_DDx(t) (Panel IIIa) is just such probability with respect to a time-varying input signal, and this probability is now a function of both threshold and time. Notice that this function

[0135] is represented in a three-dimensional rectangular coordinate system by a surface, which is the geometric locus of the points whose coordinates t, D, and z satisfy Eq. (5). Thus the methodological purpose of the measurements by a means of discriminators and probes can be phrased as raising the flat problem of the analysis of the curve on a plane into the embossed surface problem of a three-dimensional space. This alone enables new methods of analysis of the signal x(t), and allows more effective solutions of the existing problems of the prior methods. As a simple analogy, consider the problem of constructing four equilateral triangles out of six wooden matches. This task cannot be achieved on a plane, but can be easily accomplished by constructing a tetrahedron in a three-dimensional space.

[0136] The output of the discriminator can be differentiated with respect to the displacement (threshold). The same can be achieved by a means of transforming the input signal by a differential discriminator, or probe, as illustrated in Panels IIb and IIIb. The input-output characteristic of a probe is coupled with the one of the discriminator by the relation $\begin{array}{cc}{}_D{}_D\left(D-x\right)=\frac{}{D}{}_D\left(D-x\right),& \left(6\right)\end{array}$

[0137] where _D denotes differentiation with respect to the threshold D. As follows from the description of a discriminator, it is convenient, though not necessary, to imagine _DF_D to be nonnegative. It is simplest to assume that _DF_D(x) has a single extremum at x0, and vanishes at x.

[0138] As another example. consider a voltage x(t) (Panel I) applied to the vertical deflecting plates of an oscilloscope (Panel IIb). If the vertical scale of the graticule is calibrated in units of voltage, we can imagine _DF_D(Dx) to be the brightness (in the vertical direction) of the horizontal line displayed for the constant voltage x, with D indicative of the width of this line. Then _DF_DDx(t) (Panel IIIb) describes the vertical profile of the brightness of the displayed line for the time-varying input signal x(t).

[0139] As will be discussed subsequently, the input-output characteristic of a probe call be called the threshold impulse response function of the detection system. One skilled in the art will recognize that a probe can thus be interpreted as a threshold smoothing filter. The functional form of the probe will also be called the (threshold) test function in the subsequent mathematical treatment. Clearly, as follows from Eqs. (4) and (6), the threshold impulse response of an ideal detection system, that is, a system employing ideal discriminators, is described by the Dirac -function. Some other exemplary functional choices for probes will be discussed further.

[0140] For the purpose of this disclosure, we will further refer to the measuring system comprising the non-ideal discriminators and probes as a real measuring system. The output of such a system, and thus the starting point of the subsequent analysis, is no longer a line in the time-threshold plane (as in the case of an ideal system), but a continuous surface (see Eq. (5), for example). Based on the described properties of the discriminators and probes, one skilled in the art will now recognize that discriminators and differential discriminators effectively transform the input signal into objects with mathematical properties of cumulative distribution and density functions, respectively. The main purpose of such transformation is to enable differentiation with respect to displacement (threshold). while preserving, if originally present, differentiability with respect to space and time. If the original input signal is not time-differentiable (e.g., the input signal is time-sampled), differentiability with respect to time can always be enabled by introducing time averaging into the acquisition system. Likewise, differentiability with respect to spatial coordinates can be enabled by spatial averaging.

[0141] Particular practical embodiments of the discriminators and probes will depend on the physical nature of the analyzed signal(s). For example, for a scalar signal of electrical nature, the discriminator can be viewed as a nonlinear amplifier, and the threshold as a displacement voltage (or current, or charge). If the incoming signal describes the intensity of light, then the displacement can be a spatial coordinate z, and the discriminator can be an optical filter with the transparency profile described by F_D(z). The differential discriminator (probe) can then be implemented through the techniques of modulation spectroscopy (see Cardona, 1969, for example, for a comprehensive discussion of modulation spectroscopy). As an additional example, consider the modification of the previously discussed current measurement as follows. Imagine that a gray optical filter is attached to the needle of the ammeter, and the white scale is observed through this filter. Assume that when no current flows through the ammeter, the blackness observed at the position D on the scale is F_D(D), with zero corresponding to the maximum intensity of the white color, and 1 representing the maximum blackness. Then F_DDx(t) will describe the observed blackness of the scale for the time-varying signal x(t) (see Panel IIIa in FIG. 1b). If the profile of the filter were changed into _DF_D(D), then the observed darkness of the scale will correspond to the output of a probe rather than a discriminator (see Panel IIIb in FIG. 1b). Since the mathematical description of any practical embodiment will vary little. if at all, with the physical nature of the measuring system and analyzed signal, we will further use the mathematical language without references to any specific physical (hardware) implementation of the invention. It is also understood that all the formulae manifestations of the embodiments immediately allow software implementation.

[0142] Given the input x, the value of (Dx) is interpreted as the output of an ideal discriminator set at threshold D (see Nikitin et al., 1998, and Nikitin, 1998, for example). Thus the value of $\left(D-x\right)=\frac{}{D}\left(D-x\right)$

[0143] is the output of an ideal probe.

[0144] Input-output characteristics of some exemplary discriminators and the respective probes are shown in FIG. 2. Notice that although we show only symmetric discriminators, asymmetric ones can be successfully applied for particular tasks, as well as any linear combination of the discriminators. A particular mathematical expression describing the input-output characteristic of a discriminator is important only for mathematical computations and/or computer emulations. Any physical device can serve as a discriminator, as long as its input-output characteristic and the characteristic of the respective probe satisfy the requirements for a test function, e.g., Eqs. (2), (3), and (6).

[0145] For those shown in FIG. 2, the mathematical expressions are as follows: $\begin{array}{cc}\begin{array}{ccc}\mathrm{Gaussian}\text{:}& {}_D{}_D\left(D-x\right)=\frac{1}{D\sqrt{}}{}^{-{\left(\frac{D}{D}\right)}^2},& {}_D\left(D\right)=\frac{1}{2}\mathrm{erfc}\left(\frac{-D}{D}\right);\\ \mathrm{Cauchy}\text{:}& {}_D{}_D\left(D\right)={\frac{1}{D}\left[1+{\left(\frac{D}{D}\right)}^2\right]}^{-1},& {}_D\left(D\right)=\frac{1}{2}+\frac{1}{}\mathrm{arc}\tan \left(\frac{D}{D}\right);\\ L\mathrm{aplace}\text{:}& {}_D{}_D\left(D\right)=\frac{1}{2D}{}^{-\frac{D}{D}},& {}_D\left(D\right)=\frac{1}{2}\left[1+\frac{D{}_1-D_1}{D}-\frac{-D(D_1}{D}\right];\\ \mathrm{Hyperbolic}\text{:}& {}_D{}_D\left(D\right)=\frac{2}{D}{\left({}^{\frac{D}{D}}+{}^{-\frac{D}{D}}\right)}^{-2},& {}_D\left(D\right)=\frac{1}{2}\left[1+\tanh \left(\frac{D}{D}\right)\right].\end{array}& \left(7\right)\end{array}$

[0146] Since the invention employs transformation of discrete or continuous variables into objects with mathematical properties of space and time dependent density or cumulative distribution functions, these properties and their consequent utilization need to be briefly discussed. We will further also use the collective term normalized scalar fields to denote the density and cumulative distribution functions. Note that the term time is used as a designation for any monotonic variable, continuous or discrete, common to all other analyzed variables, which can be used for sequential ordering of the measurements. Thus space is all the remaining coordinates which are employed to govern the values of the input variables. The term threshold space will be used for the coordinates describing the values of the variables. For the purpose of this disclosure only, we will further also use the term phase space, which will be understood in a very narrow meaning as the threshold space employed for measuring the values of the variable together with the values of the first time derivative of this variable.

[0147] Let us further use the notation for a volume integral as follows:

[0148] where x(x₁ . . . , x_n) and r(r₁ . . . r_n) are n-dimensional vectors. This definition implies cartesian coordinates, which we also assume in further presentation. If the subsequent equations need to be re-written in curvilinear coordinates (e.g., for the purpose of separation of variables), this can be done by the standard transformation techniques. Refer to Arfken, 1985, Margenau and Murphy, 1956, or Morse and Feshbach, 1953, for example, for a detailed discussion of such techniques.

[0149] Now let F_K(x;a,t) be a space and time dependent cumulative distribution function, i.e.,

[0150] where _K(x;a,t) is a density function, i.e, _K(x;a,t)0, and

[0151] Phase space density (see Nicholson, 1983, for example) in plasma physics and probability density (see Davydov, 1988, and Sakurai, 1985, for example) in wave mechanics are textbook examples of time dependent density functions. Another common example would be the spectral density acquired by a spectrometer with spatial and temporal resolution (see Zaidel' et al., 1976, for example). In Eqs. (9) and (10), the subscript K denotes functional dependence of F_K and _K on some space and time dependent quantity (variable) K. That is although these equations hold for any given space and time, the shape of _K (and, as a result, the shape of F_K) might depend on K. The particular wave in which such dependence is introduced and utilized in this invention will be discussed further in the disclosure.

[0152] The usefulness of density and cumulative distribution functions for analysis of variables extends beyond the fact that the geometric representation makes details of the analysis intuitive and more available for human perception by analogy with the ordinary space. It also lies in one's ability to set up equations better describing the behavior of the variables than the algebraic equations of the prior art. As has been discussed previously, for example, a level line of the cumulative distribution function of a scalar variable in the time-threshold plane corresponds to the output of an order statistic filter. It is an easy-to-envision simple geometric image, having many analogies in our everyday experience (e.g., topographical maps showing elevations). Since it is a curve of the plane, it is completely determined by an algebraic equation with two variables, F(x,y)0, with the easiest transition from the implicit to the explicit form as a well-known differential equation (see Bronshtein and Semendiaev, 1986, p. 405, Eq. (4.50), for example): $\begin{array}{cc}{y}^{}\left(x\right)=-\frac{F_x^{}\left(x,y\left(x\right)\right)}{F_y^{}\left(x,y\left(x\right)\right)}.& \left(11\right)\end{array}$

[0153] As another example, imagine that the function _K(x;a,t) in a traffic control problem describes the density of cars (number of cars per unit length of the road). Then the properties of this density function might be analogous to the properties of the density of fluid. They will, for example, satisfy the continuity equation (Arfken, 1985, p. 40, for example). Then fluid equations will be the most appropriate for the description of the properties of the density function of the traffic problem. Specific applications of the density functions will, of course, depend on the physical nature (or applicable physical analogy) of the variables involved, that is, on the nature of t, a, and x, and on the constraints imposed on these variables. In the next subsection, we will provide several examples of the equations involving the threshold distribution and density functions, which might be of general usage for analysis of variables.

[0154] Using the above definitions for the normalized scalar fields, we can now introduce several examples of additional transformations of variables. Some of these equations will be used further in the disclosure for development of various practical embodiments of AVATAR. Let us first identify rank normalization and filtering of variables as follows.

[0155] Let us consider a new (dimensionless scalar) variable (field) y(a,t) defined as

[0156] where x(a,t) is an arbitrary variable. Apparently, 0y(a,t)1. Since _xiF_K(x;a,t)0, where _xi denotes a partial derivative with respect to x_i, Eq. (12) defines rank normalization of the variable x(a,t) with respect to the reference distribution F_K. Rank normalization transforms a variable into a scalar variable (or scalar field, if the transformed variable is a field variable), the magnitude of which at any given time equals to the value of the reference distribution evaluated at the value of the input variable at a this time. Thus F_K provides a (nonlinear) scale for measurement of x(a,t). We will discuss rank normalization in more details later in the disclosure.

[0157] Differentiating Eq. (12) with respect to time leads to yet another defining equation for an analog rank normalizer (ARN) as follows:

[0158] where _t denotes a partial derivative with respect to t, and {dot over (x)}_x in cartesian coordinates is simply $\begin{array}{cc}\stackrel{.}{x}{}_x=\underset{i=1}{\overset{n}{}}{\stackrel{.}{x}}_i{}_{x_i}.& \left(14\right)\end{array}$

[0159] Let us now introduce another transformation of variables, which can be interpreted as rank filtering. By definition, x_q is the qth quantile of F_K(x;a,t) when

[0160] where 0q1 is the quantile value. Note that Eq. (15) describes a simple surface in the threshold space. When the variable x is a scalar, that is, x(a,t)x(a,t), this surface is a point on the threshold line, and thus (as has been previously discussed) F_K(x_q;a,t)q describes a level line in the time-threshold plane. Taking the full time derivative of F_K(x_q;a,t) allows us to rewrite Eq. (15) in differential form for a family of equiquantile surfaces in the threshold space as

[0161] We can further introduce some constraints on x_q, such as constraints on the direction of {dot over (x)}_q, or other subsidiary conditions. For example, if we allow only the nth component of x_q to depend on time, the time derivative of this component can be written as $\begin{array}{cc}{\stackrel{.}{x}}_{q,n}=-\frac{{}_t{F}_K\left(x_q;a,t\right)}{{}_{x_{q,n}}{F}_K\left(x_q;a,t\right)},& \left(17\right)\end{array}$

[0162] assuming that all derivatives in this equation exist. Eq. (17) thus defines an analog rank filter (ARF). The latter can also be called analog order statistic filter (AOSF), or analog quantile filter (AQF). Note that even though we adopt the existing digital signal processing terminology such as order statistic filter, this is done by a simple analogy between the geometric extension of the AVATAR and the definitions in the discrete domain. Our definitions cannot be derived from the algebraic equations defining order statistic filtering in digital signal processing. Note also that a particular form of Eq. (17) depends on the nature of constraints imposed on x_q. For the important special case of the input variable as an m-dimensional surface, that is, a scalar field xz(a,t), where a(a₁ . . . , a_m), Eq. (17) reads as $\begin{array}{cc}{\stackrel{.}{z}}_q\left(a,t\right)=-\frac{{}_t{F}_K\left(z_q\left(a,t\right);a,t\right]}{F_K\left[z_q\left(a,t\right);a,t\right]}.& \left(18\right)\end{array}$

[0163] In numerical computations, Eqs. (17) and (18) can be considered to be modifications of the Newton-Raphson method of root finding (see Press et al., 1992, for example, and the references therein for the discussion of the Newton-Raphson method).

[0164] When the distribution function does not depend on time explicitly, F_KF_K(z;a), we can introduce an explicit parametric dependence of the density function on some , for example, through the convolution transform

[0165] such that g(z;a, ) approaches _K(z;a) as approaches a suitable limit, for example, when () approaches the Dirac -function (). Then the equality

[0166] leads to the equation for an analog rank finder, or analog rank selector, as follows: $\begin{array}{cc}\frac{}{}{z}_q\left(a,\right)=-\frac{{}_{-}^{z_q\left(a,\right)}{}_{}{g}_{}\left(;a,\right)}{g_{}\left[z_q\left(a,\right);a,\right]}.& \left(21\right)\end{array}$

[0167] Introducing parametric dependence through the convolution transform will later be shown to be convenient in rank selectors for an ensemble of time dependent variables, since the parameter can be chosen to be the time itself. Clearly, there are plenty of alternatives for introduction of such parametric dependence. For example, one can choose _K(;a, )()_K(;a) with () such that lim()1. As an illustration, the choice ()1(1q)e leads to $\begin{array}{cc}\frac{}{}{z}_q=\frac{q-F_K\left(z_q;a\right)}{\left[1-\left(1-q\right){}^{-}\right]f_K\left(z_q;a\right)},& \left(22\right)\end{array}$

[0168] where z_q(a, ) rapidly converges to the true value of z_q(a). Notice again that the definitions of the analog rank selectors require the existence of _K, that is, the threshold continuity of F_K, and thus cannot be introduced in the digital domain.

[0169] Finally, the threshold continuity of the distribution function allows us to write, for both time dependent and time independent scalar fields, an explicit expression for the qth quantile of F_K(x;a,t) as

[0170] The derivation and properties of this equation for rank filtering will be discussed later in the disclosure. Further, we will also provide a means of evaluating this expression by analog machines.

[0171] Let us also define another type of rank filtering, which. unlike the rank filtering defined earlier, is applicable to multivariate variables and does not (and cannot) have a digital counterpart, as follows:

[0172] where _q(a,t) is quantile density. Since the density function _K(x;a,t) vanishes at x_n, the surface in the threshold space defined by Eq. (24) encloses the series of volumes (regions in the threshold space) such that _K(x;a,t)>_q(a,t), and the integral of _K(x;a,t) over these volumes is equal to q. We shall call this series of regions in the threshold space the quantile domain. Notice that the left-hand side of Eq. (24) is non-increasing function of _q(a,t)0. Later in the disclosure, we will provide a means of finding _q(a,t) by continuous action machines.

[0173] We shall designate the total volume enclosed by the surface defined by Eq. (24) as quantile volume R_q, which can be computed as

[0174] Notice that the quantile density indicates the value of the density likely to be exceeded, and the quantile volume gives the total volume of the highest density regions in the threshold space. As an example, consider the density of the cars in a city. The median density will indicate the degree of congestion on the roads, providing the number of cars per unit length of a road (or, inversely, bumper-to-bumper distance) such that half of the traffic is equally or more dense. Then the median domain will indicate the regions (stop lights and intersections) of such congested traffic, and the median volume will give the total length of the congested roads. As another simple example, consider the price (amount per area) of the land in some geographic region. The median density will be the price such that the total amount paid for the land equally or more expensive will be half of the total cost of the land in the region. Then the median domain will map out these more expensive regions, and the median volume will give the total area of this expensive land. Notice that even though in the latter example the median density is price (it has the units of amount per area), it is not the median price in its normal definition as the price such that half of the area is less (or more) expensive. Later in the disclosure, we will provide a means of computation of the quantile domain and volume by continuous action machines.

[0175] Since the distribution and density functions F_K and _K depend on the quantity K, comparison of these functions with respective F_K and _K for a different quantity K will provide a means for assessment of K and K in their relation to the reference variable. that is, to the variable for which the distribution and density were computed. For example, the equality F_KF_K when KK will indicate that even though K and K are not equal, they are equivalent to each other in their relation to the reference variable, at least under the conditions of the conducted measurement. When the quantity K represents the reference variable in some way so that the behavior of K reflects some aspects of the behavior of the reference variable, the reference variable should be considered a component of the measured signal or process rather than a part of the acquisition system. In this case, we shall designate the reference variable as the input variable and call the quantity K an associated variable, or an associated signal. When such interdependence between K and x is not only implied or suspected, but defined in some particular way, we will also call x the input variable, and K a property of the input variable.

[0176] The particular way in which the dependence of the density and the cumulative distribution functions on K are introduced in this invention can be interpreted as measuring the input variable K in terms of the rate of change of the reference variable x at a certain threshold D. The details of this interpretation will be provided later in the disclosure. In the next subsection, we will briefly discuss some aspects of the relationship between the input and the associated variables.

[0177] In order to implement comparison of variables of different natures, it is important to have a reference system, common for all variables. Naturally, time (t) is one of the coordinates in this reference system, since it equally describes evolution of all measured variables. Time can also serve as a proxy for any monotonically ordered index or coordinate. For brevity, we shall call the remaining coordinates governing the values of the variables the spatial coordinates (a). Time and space are the third and second arguments, respectively, in the dimensionless object, cumulative distribution function, we are to define. We will call the first argument of the cumulative distribution the threshold, or displacement (D), and the units of measurements of this argument will be the same as the units of the input (reference) variable.

[0178] There is plenty of latitude for a particular choice of an associated variable K, and a variety of ways to introduce the coupling between K and the input variable. For the purpose of this disclosure, it would be of little interest to us to consider an a priori known K other than Kconstant. Thus K must not be confused with the part of the acquisition system and rather should be considered a component of the measured phenomenon. The relationship between K and the input variable can be of a deterministic nature, such as mathematical transformation, or a result of physical dependence. For example, the reference variable can be the instantaneous air temperature, and the associated variable can be the instantaneous humidity. Then we might be interested in measuring the dependence of the humidity on the temperature variations at a given temperature. Or, the reference variable can be the total population of the psychiatric wards in the United States, and the associated variable the Dow Jones Industrial Average. Then one might try to investigate how the rate of change in the mental health of the nation affects the economic indicators. One skilled in the art will recognize that such dependence between the input and the reference variable is most naturally described in terms of their joint distribution. However, such a joint distribution will be a function of the threshold coordinates of both the input and the reference variables. Thus a different input variable will require a different threshold space for the description of its dependence on the reference variable. In order to enable comparison between input variables of different natures, we would desire to use the reference system common to both input variables, that is, the threshold coordinates of the reference variable.

[0179] As we will try to illustrate further, the choice of both reference and associated variables should be based on the simplicity of treatment and interpretation of the results. For our illustrative purposes, pursuant to easy and useful interpretation, we introduce nonconstant associated variables only as norms of the first two time derivatives of the input signal, {dot over (x)} and {umlaut over (x)}. Our choice of coupling as modulation, as further described in detail, is based solely on the immediate availability of physical interpretation of the results in the case of K{dot over (x)}. For example, the cases Kconstant and K{dot over (x)} relate to each other as the charge and the absolute current in electric phenomena. This coupling (as modulation) allows the input (reference) variable to provide a common unit, or standard, for measuring and comparison of variables of different nature. This coupling also enables assessment of mutual dependence of numerous variables, and for evaluation of changes in the variables and in their dependence with time. For example, dependence of economic indicators on social indicators, and vice versa, can be analyzed, and the historical changes in this dependence can be monitored. Different choices of associated variables, however, may benefit from different ways of coupling.

[0180] For the purpose of this disclosure, we assume that continuous is synonymous to differentiable. Whenever necessary, we assume a continuous input variable x(a,t). When the definition of a particular property requires continuity of derivatives of the input variable, such continuity will also be assumed. For instance, the definitions of the densities for the threshold accelerations and for the phase space threshold crossing rates of a scalar variable x(t) will require continuity of the first time derivative {dot over (x)}(t) of the input variable. Of course, all the resulting equations are applicable to digital analysis as well, provided that they are re-written in finite differences.

[0181] As an introduction to a more general definition. let us consider a single scalar continuous-time variable (signal) x(t), and define a time dependent threshold density for this signal's counting (threshold crossing) rates. First, we notice that the total number of counts, i.e., the total number of crossings of the threshold D by the signal x(t) in the time internal 0tT. can be written as (see Nikitin et al., 1998, and Nikitin, 1998, for example) $\begin{array}{cc}N\left(D\right)=\underset{i}{}{}_0^{T}t\left(t-t_i\right),& \left(26\right)\end{array}$

[0182] where (x) is the Dirac -function, and t_i are such that x(t_i)D for all i. Using the identity (Rumer and Ryvkin, 1977, p. 543, for example) $\begin{array}{cc}\left[a-f\left(x\right)\right]=\underset{i}{}\frac{\left(x-x_i\right)}{f^{}\left(x_i\right)},& \left(27\right)\end{array}$

[0183] we can rewrite Eq. (26) as

[0184] where the dot over x denotes the time derivative. In Eq. (27), (x) denotes the absolute value of the function derivative with respect to x and the sum goes over all x_i such that (x_i)a. Thus the expression $\begin{array}{cc}\begin{array}{c}\left(D\right)=\frac{1}{T}{}_0^{T}t\stackrel{.}{x}\left(t\right)\left[D-x\left(t\right)\right]\\ ={}_{-}^{}s\frac{\left(s\right)\left(T-s\right)}{T}\stackrel{.}{x}\left(s\right)\left[D-x\left(s\right)\right]\end{array}& \left(29\right)\end{array}$

[0185] defines the counting, or threshold crossing, rate.

[0186] FIG. 3 illustrates the counting process for a continuous signal. This illustration utilizes the fact that $\left(x\right)=\frac{}{x}\left(x\right),$

[0187] where (x) is the Heaviside unit step function, and thus {dot over ()}x(t)D{dot over (x)}(t)x(t)D by differentiation chain rule.

[0188] Replacing the rectangular weighting function in Eq. (29) by an arbitrary time window h(t), the rate of crossing of the threshold D by the signal x(t) can be written as the convolution integral

[0189] where the time weighting function h(t) is such that

[0190] and is interpreted as a moving (or sliding) window. It is simplest, though not necessary, to imagine h(t) to be nonnegative. Notice that now the threshold crossing rate (Eq. (30)) depends on time explicitly. Note also that in Eq. (30) this rate is measured by means of an ideal probe _DF_D(Dx)(Dx) with time impulse response h(t).

[0191] If T is a characteristic time, or duration of h(t), we will use shorthand notations for the integral $\begin{array}{cc}{}_{-}^{}sh\left(t-s;T\right)={}_T^h& \left(32\right)\\ ={}_T.& \left(33\right)\end{array}$

[0192] These equations define a time average on a time scale T. We will consequently use the notations of Eqs. (32) and (33) for both continuous and discrete time averages. The notation of Eq. (32) will be used whenever the particular choice of the weighting function is important. Noticing that dDR_h(D,t){dot over (x)}_T, we can now define the counting (threshold crossing) density as $\begin{array}{cc}r\left(D,t\right)=\frac{{\stackrel{.}{x}\left(D-x\right)}_T}{{\stackrel{.}{x}}_T},& \left(34\right)\end{array}$

[0193] where we used the shorthand notation of Eq. (33).

[0194] The meaning of the above equation can be clarified by its derivation from another simple reasoning as follows. Note that a threshold crossing occurs whenever the variable has the value D, and its first time derivative has a non-zero value. Then the density of such events is expressed in terms of the joint density of the amplitudes of the variable and its time derivative as $\begin{array}{cc}\begin{array}{c}r\left(D,t\right)=\frac{{}_{-}^{}{D}_{\stackrel{.}{x}}{D}_{\stackrel{.}{x}}{\left(D_{\stackrel{.}{x}}-\stackrel{.}{x}\right)\left(D-x\right)}_T}{{}_{-}^{}D{}_{-}^{}{D}_{\stackrel{.}{x}}{D}_{\stackrel{.}{x}}{\left(D_{\stackrel{.}{x}}-\stackrel{.}{x}\right)\left(D-x\right)}_T}\\ =\frac{{{}_{-}^{}{D}_{\stackrel{.}{x}}{D}_{\stackrel{.}{x}}\left(D_{\stackrel{.}{x}}-\stackrel{.}{x}\right)\left(D-x\right)}_T}{{{}_{-}^{}D{}_{-}^{}{D}_{\stackrel{.}{x}}{D}_{\stackrel{.}{x}}\left(D_{\stackrel{.}{x}}-\stackrel{.}{x}\right)\left(D-x\right)}_T}\\ =\frac{{\stackrel{.}{x}\left(D-x\right)}_T}{{\stackrel{.}{x}}_T}.\end{array}& \left(35\right)\end{array}$

[0195] The significance of the definition of the time dependent counting (threshold crossing) density, Eq. (34), stems from the importance of zero-crossings. or, more generally. threshold crossings, and zero/threshold crossing rates for many signal processing applications. These quantities characterize the rate of change in the analyzed signal. which is of the most important characteristics of a dynamic system. The importance of threshold crossing rates can be illustrated by the following simple physical analogy: If x(t) describes the location of a unit point charge, then (Dx) is the charge density, and thus {dot over (x)}(Dx) is the absolute current density at the point D. In the next subsection, we generalize the above result and provide its additional interpretation.

[0196] In order to generalize the above result, let us first analyze the example shown in FIG. 4. Consider intersections of a scalar variable x(t) in the interval 0, T with the thresholds D_j. where D_j1D_jD. The instances of these crossings are labeled as t_i t_i1>t_i. The thresholds D_j and the crossing times t_i define a grid. We shall name a rectangle of this grid with the lower left coordinates (t_i, D_j) as a s_ij box. We will now identify the time interval t_ij as t_i1t_i if the box s_ij covers the signal (as shown in FIG. 4), and zero otherwise.

[0197] We can thus define the threshold density, modulated (weighted) by the associated variable K, or simply Modulated Threshold Density (MTD), as $\begin{array}{cc}{c}_K\left(D_j,t\right)=\underset{D->0}{\lim }\frac{1}{D}\frac{\underset{i}{}{t}_{\mathrm{ij}}K\left(t_i\right)}{\underset{i,j}{}{t}_{\mathrm{ij}}K\left(t_i\right)}.& \left(36\right)\end{array}$

[0198] Utilizing Eq. (27) we can rewrite Eq. (36) as $\begin{array}{cc}{c}_K\left(D,t\right)=\frac{\frac{1}{T}{}_0^{T}tK\left(t\right)\left(D-x\left(t\right)\right)}{\frac{1}{T}{}_0^{T}tK\left(t\right)}=\frac{{K\left(D-x\right)}_T}{K_T}.& \left(37\right)\end{array}$

[0199] For example, K(t){dot over (x)}(t) leads to the previously described result, that is, to the counting density. For Kconstant, Eq. (37) reduces to the amplitude density (Nikitin. 1998), namely

[0200] Notice that the modulated threshold density also formally reduces to the amplitude density whenever

[0201] that is, when K(t) and the pulse train Dx(t){dot over (x)}(t)¹_i(tt_i) are uncorrelated.

[0202] To further clarify the physical meaning of MTD, let us first use Eq. (27) and rewrite the numerator of Eq. (37) as $\begin{array}{cc}{K\left(D-x\right)}_T^h=\underset{i}{}h\left(t-t_i\right)\frac{K_i}{\stackrel{.}{x}\left(t_i\right)},& \left(40\right)\end{array}$

[0203] which reveals that it is just a weighted sum of the ratios K_i/{dot over (x)}(t_i), evaluated at the intersections of x with the threshold D. Noticing that the ratio dD/{dot over (x)}(t_i) is equal to the time interval the variable r spends between D and DdD at the ith intersection, we shall realize that $\begin{array}{cc}{\left\{M_{x}K\right\}}_T^h\left(D,t\right)=\frac{{K\left(D-x\right)}_T^h}{{\left(D-x\right)}_T^h}=\frac{\underset{i}{}{K}_i\frac{h\left(t-t_i\right)}{\stackrel{.}{x}\left(t_i\right)}}{\underset{i}{}\frac{h\left(t-t_i\right)}{\stackrel{.}{x}\left(t_i\right)}}& \left(41\right)\end{array}$

[0204] is the time weighted mean of K with respect to x at the threshold D, or simply Mean at Reference Threshold (MRT). Using this designation (MRT) can further be justified by rewriting the middle term of Eq. (41) as $\begin{array}{cc}\begin{array}{c}\frac{{K\left(D-x\right)}_T^h}{{\left(D-x\right)}_T^h}=\frac{{\left[{}_{-}^{}{D}_K{D}_K\left(D_K-K\right)\right]\left(D-x\right)}_T^h}{{\left(D-x\right)}_T^h}\\ ={}_{-}^{}{D}_K{D}_K\frac{{\left(D_K-K\right)\left(D-x\right)}_T^h}{{\left(D-x\right)}_T^h}\\ ={}_{-}^{}{D}_K{D}_{K}f\left(D_K;D,t\right),\end{array}& \left(42\right)\end{array}$

[0205] which demonstrates that MRT is indeed the first moment of the density function (D_K;D,t). Notice that here the threshold coordinates D of the variable x are the spatial coordinates of the density function . Later in the disclosure, we will generalize this result to include the multivariate modulated threshold densities. Then, for example, if the reference variable x(t) is the positions of the particles in an ensemble, and K(t){dot over (x)}(t)v(t) is their velocities. then M_xK_T^h(D, t) will be the average velocity of the particles at the location D as a function of time t. Using Eq. (41), Eq. (39) can be rewritten as

[0206] and understood as the equality, at any given threshold, between the weighted mean of K with respect to x and the simple time average of K. That is, if Eq. (43) holds for any threshold, the weighted mean of K with respect to x is a function of time only. Obviously, Eq. (43) always holds for Kconstant.

[0207] It is very important to notice that although the modulated threshold density given by Eq. (37) implies that K never changes the sign from plus to minus (or vice versa), the weighted mean at threshold of a reference variable defined by Eq. (41) is meaningful for an arbitrary K, and thus the comparison of Eq. (43) can always be implemented. For example, this comparison is implementable for K_T^h0, when the modulated density does not exist.

[0208] Eq. (43) signifies that a simple comparison of the weighted mean at threshold M_xK_T^h. with the simple time average K_T^h, that is, comparison of a modulated density with unmodulated (amplitude), will indicate the presence of correlation between the reference and the associated variables as a function of threshold (and time) on a given time scale. In other words, the equality of Eq. (43) holds when, at a given threshold D, the values of the variable K are uncorrelated with the time intervals the variable x spends between D and DdD. As a simplified example, consider K(t) as a clipped x(t), that is, as $\begin{array}{cc}K\left(t\right)=\{\begin{array}{cc}{D}_0& \mathrm{for}x\left(t\right)<D_0\\ x\left(t\right)& \mathrm{for}x\left(t\right)D_0\end{array}.& \left(44\right)\end{array}$

[0209] In this case, Eq. (43) will hold for DD₀, and will generally fail for D>D₀.

[0210] As another example, let the signal x(t) represent a response of a detector system to a train of pulses with high incoming rate, Poisson distributed in time. The high rate might cause high order pileup effects in some low energy channels of the detector. Then, as follows from (Nikitin, 1998), the amplitude and the threshold crossing densities for such a signal, measured in these channels, will be identical. Thus the channels afflicted by the pileup effects can be identified by comparing the counting rate with the amplitude distribution in different channels.

[0211] As an opposite extreme of Eq. (43), the mean at reference threshold can be a function of threshold only, and thus K would be completely determined by the reference variable. As an illustration, consider a linear reference variable xat, a>0, and thus $\left(D-x\right)=\frac{1}{a}\left(\frac{D}{a}-t\right),$

[0212] which leads to the MRT as $\begin{array}{cc}{\left\{M_{\mathrm{at}}K\right\}}_T^h\left(D,t\right)=\frac{\frac{1}{a}{}_{-}^{}sh\left(t-s\right)K\left(s\right)\left(\frac{D}{a}-s\right)}{\frac{1}{a}{}_{-}^{}sh\left(t-s\right)\left(\frac{D}{a}-s\right)}=K\left(\frac{D}{a}\right),& \left(45\right)\end{array}$

[0213] which depends only on the threshold. One skilled in the art will now recognize that a simple reincarnation of Eq. (45) in a physical device would be an ideal oscilloscope (that is, a precise oscilloscope projecting an infinitesimally small spot on the screen) where the reference variable xat is the voltage across the horizontal deflecting plates, and the MRT is the vertical position of the luminescent spot on the screen at xD. Thus the measurement of the MRT is indeed measuring the input variable K in terms of the reference variable x. For an arbitrary reference variable, the MRT is the average of these vertical positions weighted by the time intervals the reference variable spends at the horizontal position D. Since the afterglow of the luminophor coating of the screen conveniently provides the (exponentially forgetting) time averaging, the MRT can be measured as the average vertical deflection, weighted by the brightness of the vertical line at the horizontal deflection D. In the next three subsections we will clarify how this idealized example relates to the real measurements.

[0214] FIG. 5 provides another example of using modulated densities for measuring the input variable K in terms of the reference variable x. Notice that the amplitude densities (center panels) of the fragments of the signals x₁(t) and x₂(t) shown in the left-hand panels of the figure are identical. Notice also that the modulating signals K₁(t), K₂(2), and K₃(t) are identical for the respective modulated densities of the signals x₁(t) and x₂(t), while the modulated densities are clearly different. Thus even though the amplitude densities and the modulating signals are identical, different reference signals still result in different modulated densities.

[0215] Although the first argument in the density function c_K(D,t) is always a threshold value. we will call the modulated densities for Kconstant, K{dot over (x)}, and K{umlaut over (x)}, for brevity, the amplitude, counting (or threshold crossing), and acceleration densities, respectively. We now proceed with the general definitions of the multivariate modulated threshold densities and the weighted means at thresholds.

[0216] Time averaging with a time weighting function h(t) signifies transition from microscopic (instantaneous) densities (Dx) to macroscopic (time scale T) densities (Dx)_T. Carrying out the same transition from microscopic to macroscopic threshold domain can be done by a means of averaging with respect to a test function _R(x), a standard approach in such fields as electrodynamics (Jackson, 1975, Section 6.7, for example) or plasma physics (Nicholson, 1983, Chapter 3, for example). We thus can define a (multivariate) macroscopic threshold density as a threshold average on a scale R, namely as $\begin{array}{cc}{\left(D-r\right)}_R^f={}_{-}^{}{}^{n}r{f}_R\left(x-r\right)\left(D-r\right)=f_R\left(D-x\right),& \left(46\right)\end{array}$

[0217] where R is a characteristic volume element. We will assume _R(x) to be real, nonzero in some neighborhood of x0, and normalized to unity over all space. It is simplest, though not necessary, to imagine _R(x) to be nonnegative. Such threshold averaging with respect to a test function reflects finite amplitude resolution of data acquisition systems. For digitally recorded data, the lower limit for the characteristic volume is the element of the threshold grid. We will further use the shorthand notation $\begin{array}{cc}{}_{-}^{}{}^{n}r{f}_R\left(x-r\right)={}_R^f& \left(47\right)\end{array}$

[0218] to denote the spatial averaging with the test function _R(x).

[0219] For hardware devices, the choice of _R(x) is dictated by the threshold impulse response of the discriminators (probes) (Nikitin, 1998, Chapter 7, for example). In software, this choice is guided by computational considerations. For the purpose of this disclosure, it is convenient to assume that the total threshold impulse response function is the product of the component impulse responses, that is, it can be written as $\begin{array}{cc}{f}_R\left(x\right)=\underset{i=1}{\overset{n}{}}{}_{D_i}{}_{D_i}\left(x_i\right);& \left(48\right)\end{array}$

[0220] where capital pi denotes product, as capital sigma indicates a sum, that is, $\begin{array}{cc}\underset{i=1}{\overset{n}{}}{f}_i=f_1{f}_2{f}_3{f}_n.& \left(49\right)\end{array}$

[0221] Unless otherwise noted, the subsequent computational examples employ Gaussian test function, namely $\begin{array}{cc}{f}_R\left(x\right)=\underset{i=1}{\overset{n}{}}{}_{D_i}{}_{D_i}\left(x_i\right)=\frac{{}^{-\frac{n}{2}}}{\underset{i=1}{\overset{n}{}}{D}_i}\exp \left[-\underset{i=1}{\overset{n}{}}{\left(\frac{x_i}{D_i}\right)}^2\right].& \left(50\right)\end{array}$

[0222] Notice that, for the Gaussian test function, $\begin{array}{cc}{}_{-}^x{}^{n}r{f}_R\left(r\right)={}_R\left(x\right)=\underset{i=1}{\overset{n}{}}{}_{D_i}\left(x_i\right)=2^{-n}\underset{i=1}{\overset{n}{}}\mathrm{erfc}\left(\frac{-x_i}{D_i}\right),& \left(51\right)\end{array}$

[0223] where erfc(x) is the complementary error function (Abramowitz and Stegun, 1964, for example). F_R(X) should be interpreted as threshold step response.

[0224] FIG. 6 illustrates an optical threshold smoothing filter (probe). This probe consists of a point light source S and a thin lens with the focal length . The lens is combined with a gray optical filter with transparency described by ₂(x). Both the lens and the filter are placed in a XOY plane at a distance 2 from the source S. The lens-filter combination can be moved in the XOY plane by the incoming signal r so that the center of the combination is located at $\frac{2fr}{4f-R}$

[0225] in this plane. Then the output of the filter is proportional to the intensity of the light measured at the location D(D_x,D_y) in the D_x-O-D_y plane parallel to the XOY plane and located at the distance R from the image S of the source S (toward the source). That is, the output of this filter can be described by _R(Dr).

[0226] When a test function is employed for threshold averaging, Eq. (37) can be rewritten for the multivariate modulated threshold densities as Threshold-Time Averaged Density, namely as $\begin{array}{cc}\begin{array}{c}{c}_K\left(D,t\right)=\frac{{K\left(s\right)f_R\left[D-x\left(s\right)\right]}_T}{{K\left(s\right)}_T}\\ =\frac{{K\left(s\right)\underset{i=1}{\overset{n}{}}{}_{D_i}{}_{D_i}\left[D_i-x_i\left(s\right)\right]}_T}{{K\left(s\right)}_T}.\end{array}& \left(52\right)\end{array}$

[0227] As for a scalar variable, in this disclosure we shall call the multivariate modulated threshold densities for Kconstant the amplitude densities, and the densities for $K={[\underset{i=1}{\overset{n}{}}{\left({\stackrel{.}{x}}_i/D_i\right)}^2]}^{\frac{1}{2}}$

[0228] the (multivariate) counting densities. The amplitude density thus indicates for how long the signal occupies an infinitesimal volume in the threshold space, and the counting density indicates how often the signal visits this volume. For example, the amplitude density will indicate the number of cars at a certain intersection (that is, at the intersection positioned at D) at a certain time, and the counting density will describe the total traffic flow through this intersection at a given time. Carrying out the threshold averaging in Eq. (41), we can write the equation for the mean at reference threshold as $\begin{array}{cc}\begin{array}{c}{\left\{M_{x}K\right\}}_T\left(D,t\right)=\frac{{K\left(s\right)f_R\left[D-x\left(s\right)\right]}_T}{{f_R\left[D-x\left(s\right)\right]}_T}\\ =\frac{{K\left(s\right)\underset{i=1}{\overset{n}{}}{}_{D_i}{}_{D_i}\left[D_i-x_i\left(s\right)\right]}_T}{{\underset{i=1}{\overset{n}{}}{}_{D_i}{}_{D_i}\left[D_i-x_i\left(s\right)\right]}_T}.\end{array}& \left(53\right)\end{array}$

[0229] Notice that, unlike in the definition of the modulated threshold density, the variable K no longer has to be a scalar. In the traffic example above, it is obvious that the same traffic flow (number of cars per second) can be achieved either by low density high speed traffic, or by high density low speed traffic. The ratio of the counting rates and the amplitude density will thus give us the average speed at the intersection. If the variable K is different from the speed-for example, it is the rate of carbon monoxide emission by a carthen the mean at reference threshold will indicate this emission rate at location D, and it may (for example, because of the terrain or speed limit) or may not depend on the location.

[0230] In this disclosure, we assume the validity of Eq. (48), and thus the explicit expressions for the amplitude density b(D,t) and the counting density r(D,t) are as follows: $\begin{array}{cc}b\left(D,t\right)={\underset{i=1}{\overset{n}{}}{}_{D_i}{}_{D_i}\left[D_i-x_i\left(s\right)\right]}_T& \left(54\right)\end{array}$

[0231] for the amplitude density, and $\begin{array}{cc}r\left(D,t\right)=\frac{{\sqrt{\underset{i=1}{\overset{n}{}}{\left[\frac{{\stackrel{.}{x}}_i\left(s\right)}{D_i}\right]}^2}\underset{i=1}{\overset{n}{}}{}_{D_i}{}_{D_i}\left[D_i-x_i\left(s\right)\right]}_T}{{\sqrt{\underset{i=1}{\overset{n}{}}{\left[\frac{{\stackrel{.}{x}}_i\left(s\right)}{D_i}\right]}^2}}_T}& \left(55\right)\end{array}$

[0232] for the counting density. In Eq. (55), the numerator in the right-hand side is proportional to the counting rate. Explicitly, the expression for the counting rates reads as $\begin{array}{cc}\left(D,t\right)={\sqrt{{\underset{i=1}{\overset{n}{}}\left[\frac{{\stackrel{.}{x}}_i\left(s\right)}{D_i}\right]}^2}\underset{i=1}{\overset{n}{}}{D}_i{}_{D_i}{}_{D_i}\left[D_i-x_i\left(s\right)\right]}_T.& \left(56\right)\end{array}$

[0233] A type of problem addressed by Eq. (56) might be as follows: How often does a flying object cross a radar beam

[0234] FIG. 7 shows a simplified diagram illustrating the transformation of an input variable into a modulated threshold density according to Eq. (52). The sensor (probe) of the acquisition system has the input-output characteristic _R of a differential discriminator. The width of this characteristic is determined (and may be controlled) by the width, or resolution, parameter R. The threshold parameter of the probe D signifies another variable serving as the unit, or datum. In FIG. 7, the input variable x(t) is a scalar or vector, or a component of an ensemble. For example, a discrete surface (such as an image given by a matrix) can be viewed as a discrete ensemble, being scalar for a monochrome image, and a 3D-vector for a truecolor image. The output of the probe then can be modulated by the variable K(t). which can be of a different nature than the input variable. For example, K(t)constant will lead to the MTD as an amplitude density, and K(t){dot over (x)}(t) will lead to the MTD as a counting density/rate. Both the modulating variable K and its product with the output of the probe K_R can then be time-averaged by a convolution with the time weighting function h(t;T), leading to the averages K_R(Dx)_T^h and K_T^h. The result of a division of the latter average by the former will be the modulated threshold density c_K(D,t). Notice that all the steps of this transformation can be implemented by continuous action machines.

[0235] DENSITIES BY RC_ln IMPULSE RESPONSE FUNCTIONS

[0236] Let us consider a specific choice of a time weighting function as follows: $\begin{array}{cc}{h}_n\left(t\right)=\frac{1}{n!T^{n+1}}{t}^n{}^{-\frac{t}{T}}\left(t\right).& \left(57\right)\end{array}$

[0237] This is a response of a circuit consisting of one RC differentiator and n RC integrators (all time constants RCT) to a unit step of voltage (t). Thus we shall call such weighting function an RC_ln impulse response. FIG. 8 shows the RC_ln time weighting functions for n0 (exponential forgetting), n1, and n2.

[0238] Differentiation of h_n(t) leads to $\begin{array}{cc}{\stackrel{.}{h}}_n\left(t\right)=\frac{1}{T}\left[h_{n-1}\left(t\right)-h_n\left(t\right)\right]\mathrm{for}n1,\mathrm{and}& \left(58\right)\end{array}$ $\begin{array}{cc}{\stackrel{.}{h}}_0\left(t\right)=\frac{1}{T}\left[\left(t\right)-h_0\left(t\right)\right]\mathrm{for}n=0.& \left(59\right)\end{array}$

[0239] An exponential factor in time weighting functions is ubiquitous in nature as well as in technical-solutions. In particular, RC_ln impulse response time averaging functions are quite common, and easily implementable in software as well as in various devices. Although normally the time weighting function does not need to be specified in detail, an exponential factor in the time weighting function allows us to utilize the fact that (e^x)e^x. In particular, the relations of Eqs. (58) and (59) allow us to simplify various practical embodiments of AVATAR. This will become apparent from further disclosure.

[0240] As has been mentioned earlier in this disclosure, the main purpose of the analysis of variables through their continuous density and distribution functions is twofold: (1) to facilitate the perception through geometric interpretation of the results, and (2) to enable the analytical description by differential methods. Let us first address the former part of this goal, that is, the visual presentation of the densities and the interpretation of the underlying qualities of the variable based on these observations.

[0241] Let us first notice that the amplitude density at a given threshold D is proportional to the time the variable spends around this threshold, and thus is proportional to the average inverted (absolute) slope of the variable at this threshold. One might say that the amplitude density is a measure of flatness of the signal. For example, the amplitude density generally increases with the increase in the number of extrema at the threshold D. The counting density is proportional to the number of crossings, or visits, of this threshold by the variable, and the acceleration density is generally proportional to the density of sharp turns (such as extrema and inflection points) of the variable at this threshold. Thus the acceleration (K{umlaut over (x)}), amplitude (K1), and counting (K{dot over (x)}) densities complement each other in a manner necessary for selective shape recognition of a signal, as illustrated in FIGS. 9a and 9b. The left columns of the panels in these figures show the fragments of three different signals in rectangular windows. The second columns of the panels show the amplitude densities, the third columns show the counting densities, and the right columns show the acceleration densities for these fragments. These figures illustrate that the acceleration and counting densities generally reveal different features of the signal than do the amplitude densities. For the fragment x₁(t) in FIG. 9a (the upper row of the panels). {dot over (x)}(t)constant, and thus the counting and the amplitude densities are identical. For the fragment x₂(t) in FIG. 9a (the middle row of the panels), {umlaut over (x)}(t)constant, and thus the acceleration and the amplitude densities are identical.

[0242] The example in FIG. 10 shows time dependent acceleration densities, threshold crossing rates, and amplitude densities computed in a 1-second rectangular moving time window. for two computer generated non-stationary signals (Panels 1a and 1b). Panels 2a and 2b show the acceleration densities, Panels 3a and 3b show the threshold crossing rates, and Panels 4a and 4b show the amplitude densities. The signals represent sequences of (nonlinearly) interacting unipolar Poisson-distributed random pulses, recorded by an acquisition system with an antialiasing bandpass RC-filter with nominal passbands 0.5-70 Hz at 3 dB level. The sequences of the pulses before interaction are identical in both examples, but the rules of the interaction of the pulses are slightly different. These differences are reflected in the shape of the resulting signals, which can in turn be quantified through the displayed densities and the rates.

[0243] As an example of the interpretation of the displayed densities, consider the stretch of the signals in the interval 45 through 70 seconds. For both signals, the amplitude density (Panels 4a and 4b) is highest at the lowest amplitude, and is approximately uniform at other thresholds. This is likely to indicate that the signals in this time interval consist of relatively narrow tall pulses of comparable amplitude, originating from a flat background. The approximate uniformity of the counting rates (Panels 3a and 3b) between the lowest and the highest thresholds confirms the absence of the secondary extrema, that is, these are single pulses. The increased rates in the intervals 50 to 60 seconds and 65 to 70 seconds indicate that there are more pulses per unit time in these intervals than in the intervals 45 to 50 and 60 to 65 seconds. An approximate equality of the acceleration densities (Panels 2a and 2b) at the highest and lowest thresholds is likely to indicate that these pulses might have sharp onsets and sharp tails, in order for the sharpness of the peaks to be equal to the combined sharpness of the onsets and the tails.

[0244] Earlier in this disclosure, we introduced a new type of rank filtering, which is applicable to analysis of scalar as well as multivariate densities. In Eqs. (24) and (25), we introduced the quantile density, domain, and volume. Let us now illustrate how these quantities are applicable to the analysis of scalar variables. Panel I of FIG. 11 shows the fragment of the signal from Panel 1a of FIG. 10 in the time interval between 7 and 29 seconds. The amplitude density of this fragment is plotted in Panels II through IV. In these panels, the quartile densities _1/4 (Panel II), _1/2 (Panel III), and _3/4 (Panel IV) are shown by the horizontal lines. These lines intersect the density in such a way that the shaded areas are equal to , , and , respectively. Then the respective quartile domains will be represented by the intervals on the threshold axis confined between the left and the right edges of these areas, and the respective quartile volumes will be the sums of the lengths of these intervals.

[0245] In FIGS. 12a and 12b, the quantile densities, volumes, and domains are displayed as time dependent quantities computed in a 1-second rectangular sliding window. Panels 1a and 4aof FIG. 12a show the median densities, computed for the amplitude and the counting densities of the signal from Panel 1a of FIG. 10. Panels 1b and 4b of FIG. 12b show the respective median densities for the signal from Panel 1b of FIG. 10. Panels 2a and 5a of FIG. 12a, and Panels 2b and 5b of FIG. 12b, show the median volumes of the amplitude and the counting densities of the respective signals. As can be seen from these examples, both quantile densities and quantile volumes characterize the total width of the densities, that is, the total size of high density regions in the threshold space. Panels 3a and 6a of FIG. 12a, and Panels 3b and 6b of FIG. 12b, display the quartile domains, with the median domain shaded by the gray color, the q domain shaded by the light gray, and the first quartile domain shaded black. These examples illustrate how the quantile domain reveals the location of the high density regions in the threshold space.

[0246] Earlier in this disclosure, we adopted the restricted definition of a phase space as the threshold space of the values of the variable, complemented by the threshold space of the first time derivative of this variable. Thus for a scalar variable the modulated threshold densities in such phase space are the two-variate densities. The introduction of the phase space densities expands the applicability of the density analysis, and allows more detailed study of the changes in the variables. For example, FIG. 13 provides an illustration of the sensitivity of the phase space threshold densities to the signal's shape. The first column of the panels in the figure shows the fragments of three different signals in rectangular windows. The second column of the panels shows the phase space amplitude densities

[0247] and the third column displays the phase space counting densities $\begin{array}{cc}r\left(D_x,D_{\stackrel{.}{x}},t\right)=\frac{\begin{array}{c}\sqrt{{\left(\frac{\stackrel{.}{x}}{D_x}\right)}^2+{\left(\frac{\stackrel{}{x}}{D_{\stackrel{.}{x}}}\right)}^2}\\ {{}_{D_x}{}_{D_x}\left[D_x-x\left(s\right)\right]{}_{D_{\stackrel{.}{x}}}{}_{D_{\stackrel{.}{x}}}\left[D_{\stackrel{.}{x}}-\stackrel{.}{x}\left(s\right)\right]}_T\end{array}}{{\sqrt{{\left(\frac{\stackrel{.}{x}}{D_x}\right)}^2+{\left(\frac{\stackrel{}{x}}{D_{\stackrel{.}{x}}}\right)}^2}}_T}.& \left(61\right)\end{array}$

[0248] This figure also illustrates that while the amplitude density is indicative of the occupancy (that is, the time the variable occupies the infinitesimal volume in the phase space), the counting density reveals the traffic in the phase space, that is, it is indicative of the rates of visiting a small volume in the phase space.

[0249] The example in FIG. 14 shows time dependent phase space amplitude densities computed according to Eq. (60) in a 1-second rectangular moving time window for two computer generated non-stationary signals shown in Panels 1a and 1b of FIG. 10. The figure plots the level lines of the phase space amplitude densities (Panels 1a and 2a), at times indicated by the time ticks. Panels 1b and 2b show the time slices of these densities at time tt₀.

[0250] FIG. 15 shows time dependent phase space counting rates

[0251] computed in a 1-second rectangular moving window for the two signals shown in Panels 1a and 1b of FIG. 10. The figure plots the level lines of the phase space counting rates (Panels 1a and 2a) at times indicated by the time ticks. Panels 1b and 2b show the time slices of these rates at time tt₀. As was discussed in the explanation of FIG. 10, the general shape of the pulses around this time for both signals is similar. Thus the difference in the phase space crossing rates apparent from these time slices results mostly from the small differences in the shape of the tops of these pulses.

[0252] FIGS. 16 and 17 display the boundaries of the median domains for the phase space amplitude and counting densities, respectively. The upper panels of these figures short the respective boundaries for the signal of Panel 1aof FIG. 10, and the lower panels show the median domain boundaries for the signal of Panel 1b of FIG. 10.

[0253] The examples in FIGS. 9 through 17 illustrate the usefulness of AVATAR for visual assessment of various features of a signal, and of the evolution of these features in time. Although FIGS. 13 through 17 deal with the phase space densities of a scalar variable, it should be obvious that densities of any two-dimensional variable can be treated in the same way. For instance, the same technique will apply for describing the time evolution of the population in a geographic region (amplitude density) and for mapping out the routes of migration of this population (counting density). FIG. 16, for example, can represent the time evolution of a quantile domain of the population of a biologic species, that is, the locations of the largest number of specimens per unit area in a region. Then FIG. 17 will represent the time evolution of the respective quantile domain of the traffic of the species, that is, the regions of the most active movement of the species.

[0254] Since many physical sensors have input-output characteristics equivalent to those of the probes in this disclosure, there is a large number of feasible physical embodiments of AVATAR for displaying the modulated threshold densities and their time evolution by continuous action machines. In this subsection, we provide several illustrative examples of such embodiments. The underlying motivation behind constructing an analog machine directly displaying the densities is schematically stated in FIG. 18.

[0255] As the first example, FIG. 19 outlines a conceptual schematic of a simple device for displaying time dependent amplitude densities of a single scalar signal. An electron gun in a cathode-ray tube produces a beam of fast electrons. The tube contains a pair of vertical deflecting plates. By feeding a voltage to this pair of plates, we can produce a proportional displacement of the electron beam in the vertical direction. The screen of the tube is coated with luminophor with the afterglow half-time T_1/2T ln 2. We assume that the brightness of the luminescent spot on the screen is proportional to the intensity of the electron beam, and is described by _YF_Y(Y) when the voltage across the deflecting plates is zero. Then the brightness of the displayed band on the screen, at any given time, will correspond to the amplitude density of the input signal x(t), computed in the exponential moving window of time (RC₁₀) with the time constant T. This band can then be projected on a screen by, for example, a (concave) mirror M. By rotating this mirror, we can display the time evolution of the amplitude density of x(t). If we now modulate the intensity of the electron beam by the signal K(t), then the brightness of the displayed picture will be proportional to K_YF_Y(Yx)_T. For example, when K(t){dot over (x)}(t), the screen will display the threshold crossing rates. A simple conceptual schematic of such a device for displaying time dependent threshold crossing rates of a signal is illustrated in FIG. 20. Note that by displaying only the lines of equal intensity, or by thresholding the intensity, we will reveal the boundaries of the respective quantile domains.

[0256] FIG. 21 provides an illustration for possible hardware implementation of a device for displaying time slices of the phase space amplitude densities. An electron gun in a cathoderay tube of an oscilloscope produces a beam of fast electrons. The tube contains two pairs of mutually perpendicular deflecting plates. By feeding a voltage to any pair of plates, we can produce a proportional displacement of the electron beam in a direction normal to the given plates. The screen of the tube is coated with luminophor with the afterglow half-time T_1/2T ln 2. We assume that the brightness of the luminescent spot on the screen is proportional to the intensity of the electron beam, and is described by _XF_X(x)_YF_Y(Y) when the voltage across the deflecting plates is zero. If the input signals are x(t) and {dot over (x)}(t), respectively, then the displayed picture on the screen, at any given time, will correspond to the phase space amplitude density of the input signal x(t), computed in the exponential moving window of time (RC₁₀) with the time constant T. Thus, the screen will display figures similar to those shown in the second column of the panels in FIG. 13, and in Panels 1b and 2b of FIG. 14. If we now modulate the intensity of the electron beam by the signal K(t), then the brightness of the displayed picture will be proportional to _NF_N(Xx)_YF_Y(Yx)_T. For example, when K(t){square root}{square root over ((x)}Y)²({umlaut over (x)}X)², the screen will display the time slices of the phase space threshold crossing rates of the input signal x(t). computed in the exponential moving window of time (RC₁₀) with the time constant T. Thus, the screen will display figures similar to those shown in the third column of the panels in FIG. 13 and in Panels 1b and 2b of FIG. 15. A simple conceptual schematic of such a device for displaying time slices of the phase space threshold crossing rates is illustrated in FIG. 22.

[0257] One skilled in the art will now recognize that the task of displaying the densities can also be achieved by a variety of other physical devices. In addition to displaying the modulated densities and their time evolution, these devices can also be modified to display the quantile domain, density, and volume, the means at reference thresholds, various other quantities of the MTDs such as their level lines, and to accomplish other tasks of AVATAR. Some of the additional embodiments of such devices will be described later in this disclosure.

[0258] As has been indicated earlier in this disclosure, a comparison of the mean at reference threshold with the simple time average will indicate the interdependence of the input and the reference variables. For the purpose of this disclosure, we will adopt one of many possible ways to measure such dependence within the quantile domain as follows: $\begin{array}{cc}{q}_{}\left(D,t\right)=\frac{{\left\{M_{x}K\right\}}_T\left(D,t\right)-K_T}{K_T}\left[{f_R\left(D-x\right)}_T-f_q\left(t\right)\right],& \left(63\right)\end{array}$

[0259] where _q(t) is the quantile density defined by Eq. (24), and we will assume that the norm is computed simply as the distance in the Euclidean space. Eq. (63) represents an estimator of differences in the quantile domain between the mean at reference threshold and the time average.

[0260] FIG. 23 displays the values of the estimator _q(D,t) of Eq. (63) in q{fraction (9/10)} quantile domain, computed for the two computer generated nonstationary scalar signals (Panels 1a and 1b), used in a number of our previous examples. Panels 2a and 2b display the values of the estimator for K{dot over (x)}, and Panels 3a and 3b display these values for K{umlaut over (x)}.

[0261] As follows from Eq. (9), the time dependent Modulated Cumulative Threshold Distribution (MCTD) C_K(D,t), or Threshold-Time Averaged Cumulative Distribution, is defined as $\begin{array}{cc}\begin{array}{c}{C}_K\left(D,t\right)={}_{-}^D{}^{n}r{c}_K\left(r,t\right)\\ =\frac{{K\left(s\right){}_R\left[D-x\left(s\right)\right]}_T}{{K\left(s\right)}_T}\\ =\frac{{K\left(s\right)\underset{i=1}{\overset{n}{}}{}_{D_i}\left[D_i-x_i\left(s\right)\right]}_T}{{K\left(s\right)}_T},\end{array}& \left(64\right)\end{array}$

[0262] where C_K(D,t) is the modulated threshold density given by Eq. (52). It is easy to see from Eq. (64) and from the definition of the Heaviside unit step function, Eq. (4), that all equations for the counting, amplitude, and acceleration densities (for example, Eqs. (54), (55), (60), and (61)) are valid for the cumulative distributions as well, provided that the symbols b, r, c, , and _D in those equations are replaced by B, R, C, , and F, respectively.

[0263] Note that the transition from the densities to the cumulative distribution functions is equivalent to the threshold integration of the former, and thus the principal examples of the embodiments for the densities can be easily modified for handling the respective cumulative distributions. For instance, the embodiments of FIGS. 19 and 20 can be easily converted to display the cumulative distributions instead of the densities. Then, for example, the lines of equal intensity on the screen will correspond to the level lines of these cumulative distributions, and thus will be equivalent to the outputs of the respective rank filters.

[0264] Note also that the ability to compute or measure the time dependent threshold densities and cumulative distributions for a signal gives access to a full range of time dependent statistical measures and estimates, such as different threshold moments (e.g., mean and median, skew, kurtosis, and so on) with respect to these distributions. Although the defining equations for the threshold densities and cumulative distributions are given for a continuous variable or signal, in numerical computations these quantities can be calculated in finite differences. Clearly, the introduction of threshold averaging alleviates the computational problems caused by the singularity of the delta function.

[0265] For time independent thresholds, numerical (or hardware) computation of densities and cumulative distributions according to Eqs. (52) and (64) should not cause any difficulties. However, in the equations for rank normalization. Eqs. (12) and (13), and for filtering. Eqs. (17) and (18), densities and cumulative distributions appear with thresholds dependent on time, and their evaluation may present a significant computational challenge.

[0266] This challenge is greatly reduced when the thresholds vary slowly, which is usually the case in rank filtering. In such cases, it might be sufficient to approximate (replace) _RD(t)x(s)_T by the first term in its Taylor expansion, namely by _RD(t)x(s)_T, and higher order terms can be retained when necessary. This approximation will be our prime choice in most embodiments of AVATAR discussed further in this disclosure. For rank normalization, however, this approach is only adequate when certain relations between the input and reference signals are held, and thus different approximations should be developed. Let us first develop a unimodal approximation for the ideal modulated density function (K_T)¹K(Dx)_T, that is, the density function resulting from the measurements by an ideal probe. Although we will later present more accurate approximations for rank normalization, the unimodal approximation has certain merits of its own, e.g., simplicity of implementation and analytical treatment of the results. To develop such unimodal approximation, we can use, for example, the integral representation of the delta function as (Arfken, 1985, p.799, for example) $\begin{array}{cc}\left(D-x\right)=\frac{1}{2}{}_{-}^{}u{}^{u\left(D-x\right)}.& \left(65\right)\end{array}$