Method and apparatus for mapping components of descriptor vectors to a space that discriminates between groups

[0002] The present application is related to U.S. Patent Applications (Attorney Docket YO998-112) and (Attorney Docket No. Y0999-150), herein incorporated by reference in their entirety.

[0003] 1. Technical Field

[0004] The invention relates to the field of statistical analysis, and, more specifically, to principle component analysis of descriptor vectors.

[0005] 2. Description of the Related Art

[0006] Areas of inquiry in computationally biology and chemistry typically characterize an item of interest (e.g., molecular complex such as a molecule, compound or portion of one or more molecules/compounds, or property of one or more molecules/compounds) with a vector of descriptors (i.e., descriptor vector) that represent one or more characteristics of the item. For example, the descriptor vector for a molecular complex may characterize structure, charge distribution, biological activity, and/or other property of the molecular complex.

[0007] Principal Component Analysis is often applied to select a subset of components of the descriptor vectors associated with a set of items that approximates the data within the set. The selected subset of components is typically used to perform analysis of regression and/or correlation on the set of items. Generally, such analysis of regression and correlation both concern the following questions:

[0008] 1) does a statistical relation affording some predictability appear between the set of items!

[0009] 2) how strong is the apparent statistical relation, in the sense of the possible predictive ability that the statistical relation affords!

[0010] 3) can a rule be formulated for predicting relations among the set of items, and, if so, how good is this rule!

[0011] A more detailed description of Principal Component Analysis together with regression analysis and/or correlation analysis may be found in I. T. Jolliffe, “Principal Component Analysis”, Springer Verlag, New York, 1986, herein incorporated by reference in its entirety.

[0012] Typically, Principal Component Analysis generates a correlation matrix based upon the descriptor vectors for a given set of items, identifies the largest principal values of such correlation matrix, and selects those components of the descriptor vector that correspond to the identified principle values.

[0013] Although traditional Principle Component Analysis selects components that approximate both the items of interest and correlation matrix, it may reject information that discriminates between groups of items, thus leaving a need in the art for an improved method for capturing information that optimally discriminates between groups of items.

[0014] The problems stated above and the related problems of the prior art are solved with the principles of the present invention, method and apparatus for mapping components of descriptor vectors to a space that discriminates between groups. The present invention transforms descriptor vectors that characterize items partitioned into groups into a space that discriminates between those groups in a well defined optimal sense. First data is generated that represents a differences between the groups of descriptor vectors. Second data is generated representing variation within the groups of descriptor vectors. A set of component vectors is then identified that maximizes an F distributed criterion function that measures differences of descriptor vectors between groups relative to variations of descriptor vectors within groups. A statistic is generated for subsets of the component vectors. For each particular subset of component vectors, a probability value for the statistic associated with the particular subset is calculated. The subset with the minimum probability value is selected. Finally, one or more of the descriptor vectors for the items are mapped to a space corresponding to the selected subset of component vectors.

[0015] FIGS. 1(A) and 1(B) are block diagrams of computer processing systems wherein the methods of the present invention may be embodied.

[0016] FIGS. 2(A) and (B) is a flow chart illustrating the method of the present invention in mapping a descriptor vector for an item to a space that optimally discriminates between groups of items in accordance with the present invention;

[0017] FIG. 3 is a flow chart illustrating operations of step 205 of FIG. 2 in generating a set of component vectors that maximize an F distributed criterion function in accordance with the present invention.

[0018] FIG. 4 is a table illustrating a multi-factor design with multi-way analysis of variance.

[0019] FIG. 5 is a flow chart illustrating the training phase of a system that identifies molecules within a database of molecules that has similar structure to a query molecule in accordance with the present invention.

[0020] FIG. 6 is a flow chart illustrating the acquisition phase of a system that identifies molecules within a database of molecules that has similar structure to a query molecule in accordance with the present invention.

[0021] FIGS. 7(A) and (B) is a pictorial illustration of a multi-map, which is part of the system that identifies molecules within a database of molecules that has similar structure to a query molecule.

[0022] FIGS. 8(A) and (B) is a flow chart illustrating the recognition phase of a system identifies molecules within a database of molecules that has similar structure to a query molecule in accordance with the present invention.

[0023] FIG. 9 is a flow chart illustrating an exemplary embodiment of the operations in the recognition phase of the present invention.

[0024] The present invention may be implemented on any computer processing system including, for example, a personal computer or a workstation. As shown in FIG. 1(A), a computer processing system 100 as may be utilized by the present invention generally comprises memory 101, at least one central processing unit (CPU) 103 (one shown), and at least one input device 107 (such as a keyboard, mouse, joystick, voice recognition system, or handwriting recognition system). In addition, the computer processing system includes a nonvolatile storage device 108, such as a ROM or fixed disk drive, that stores an operating system and one or more application programs that are loaded into the memory 101 and executed by the CPU 103. In the execution of the operating system and application program(s), the CPU may use data stored in the nonvolatile storage device 108 and/or memory 101. In addition, the computer processing system includes a graphics adapter 104 coupled between the CPU 103 and a display device 105 such as a CRT display or LCD display. In addition, the computer processing system may include a communication link 109 (such as a network adapter, RF link, or modem) coupled to the CPU 103 that allows the CPU 103 to communicate with other computer processing systems over the communication link, for example over the Internet. The CPU 103 may receive portions of the operating system, portions of the application program(s), or portions of the data used by the CPU 103 in executing the operating system and application program(s).

[0025] It should be noted that the application program(s) executed by the CPU 103 may perform the methods of the present invention described below. Alternatively, portions or all of the methods described below may be embodied in hardware that works in conjunction with the application program executed by the CPU 103.

[0026] According to the present invention, a computer implemented method provides a mapping (i.e., a transformation) for the components of the descriptor vectors for a series of items to a space that optimally discriminates between groups of items. With reference to FIGS. 2(A) and (B), the operation begins in step 201 wherein a series of items are classified into N groups (N is an integer greater than 1), wherein each group is identified by an identifier i ranging from 1 to N. For the sake of description, consider groups that contain n_iitems and descriptor vectors; a however, the present invention is not limited in this respect and can be used for groups that contain a non-uniform number of items and descriptor vectors. In addition, m_ijidentifies item j belonging to group i where j ranges from 1 to n_i, and {right arrow over (x_iy+L )} is the descriptor vector corresponding to the item m_ij. Preferably, the descriptor vectors are stored as part of a file in persistent storage and loaded into non-persistent storage for use by the CPU 103 as needed.

[0027] In step 203, first data representing covariance between the groups of the items, denoted ε_b, are generated. In addition, second data representing covariance within the items belonging to the groups, denoted ε_w, are generated. Note that both the first data and second data follow a chi-square distribution. An example of the operations in determining ε_band ε_wis provided below. In this example, the first data (ε_b) have a chi-square distribution with N−1 degrees of freedom (where N represents the number of groups of items); and the second data (ε_w) have a chi-square distribution with Σn_i−N degrees of freedom (where N represents the number of groups of items, n_irepresents the number of items in group i, and Σn_irepresents the sum of n_ifor the N groups).

[0028] An example of the operations in determining ε_band ε_wis now provided. One may break any given descriptor vector {right arrow over (x_ij+L )} into variation between groups and variations within a group as follows:

[0029] {right arrow over (x_ij+L )}={right arrow over (α)}+{right arrow over (α_i+L )}+{right arrow over (α_ij+L )},

[0030] where

[0031] {right arrow over (α)} represents the mean of all the items (of all the groups);

[0032] {right arrow over (α_i+L )} represents deviation from mean of all the items ({right arrow over (α)}) to the mean of group i; and

[0033] {right arrow over (α_ij+L )} represents deviation from the mean of group i ({right arrow over (α_i+L )}) to the descriptor vector {right arrow over (x_ij+L )}.

[0034] For each group i of items (i.e., i ranging from 1 to N), the sample mean, denoted {right arrow over (X_i+L )}., of the descriptor vectors within group i is calculated. For example, the sample mean of the descriptor vectors in a given group i may be calculated as follows:

[0035] $\overrightarrow{X_{i.}}=\frac{1}{n_i}\Sigma \overrightarrow{x_{\mathrm{ij}}},$

[0036] where the Σ operator sums over the range of j=1 to n_i

[0037] The sample mean {right arrow over (α)} of all the items is calculated. For example, the sample mean {right arrow over (α+L )} may be calculated as follows:

[0038] {right arrow over (α)}={fraction (1+L /N)}Σ{right arrow over (X_i.+L )},

[0039] where the Σ operator sums over the range of i=1 to N

[0040] For each group i of items (i.e., i ranging from 1 to N), the deviation {right arrow over (α_i+L )} is calculated. For example, the deviation {right arrow over (α_i+L )} for a given group i may be calculated as follows:

[0041] {right arrow over (α_i+L )}={right arrow over (X_i.+L )}−{right arrow over (α)}

[0042] For each descriptor vector {right arrow over (x_ij+L )}, the deviation {right arrow over (α_ij+L )} is calculated. For example, the deviation {right arrow over (α_ij+L )} may be calculated for a given descriptor vector {right arrow over (x_ij+L )} as follows:

[0043] {right arrow over (α_ij+L )}={right arrow over (x_ij+L )}−{right arrow over (X_i.+L )}

[0044] The covariance between groups, denoted ε_b, may be calculated as follows:

[0045] ε_b=Σn_i{right arrow over (α_i+L )}{right arrow over (α_i+L )}^T

[0046] where the Σ operator sums over the range of i=1 to N

[0047] And the covariance within groups, denoted ε_w, may be calculated as follows:

[0048] ε_w=ΣΣ{right arrow over (α_ij+L )}{right arrow over (α_ij+L )}^T

[0049] where the ΣΣ operator sums over the range j=1 to n_ifor each group i in the range 1 to N

[0050] Note that the covariance between groups ε_b has a chi-square distribution with N−1 degrees of freedom (where N represents the number of groups of items); and the covariance within groups ε_w, has a chi-square distribution with Σn_i−N degrees of freedom (where N represents the number of groups of items, n_irepresents the number of items in group i, and Σn_irepresents the sum of n_ifor the N groups). The total covariance, denoted ε, is represented as follows:

[0051] ε=ΣΣ({right arrow over (x_ij+L )}−{right arrow over (α)})({right arrow over (x_ij+L )}−{right arrow over (α)})^T=ε_b+ε_w

[0052] where the ΣΣ operator sums over the range j=1 to n_ifor each group i in the range 1 to N

[0053] In step 205, an F distributed criterion function is used to determine a set of component vectors that maximize the criterion function. The criterion function has a numerator and a denominator, whereby the numerator is based upon the first data (ε_b) generated in step 203 and the denominator is based upon the second data (ε_w) generated in step 203. The criterion function has the general form: $f\left(\hat{w}\right)=C\left(\frac{{\hat{w}}^T{ϵ}_b\hat{w}}{{\hat{w}}^T{ϵ}_w\hat{w}}\right)$

[0054] where ŵ is a vector in the direction whose variation is being tested, and C is a constant based upon the degrees of freedom in ε_band ε_w.

[0055] For example, the constant C may be determined as follows: $C=\frac{1/\mathrm{degrees}\mathrm{of}\mathrm{freedom}\mathrm{in}{ϵ}_b}{1/\mathrm{degrees}\mathrm{of}\mathrm{freedom}\mathrm{in}{ϵ}_w}=\frac{1/\left(N-1\right)}{1/\left(\sum n_i-N\right)}$

[0056] where N represents the number of groups of items,

[0057] n_irepresents the number of items in a group, and Σn_irepresents the sum of n_ifor the N groups, and

[0058] The set of component vectors is determined by solving for those ŵ that maximize the criterion function 𝑓(ŵ) An example of the operation of the computer processing system 100 in determining the set of component vectors is set forth below with respect to FIG. 3. For the sake of description, the number of component vectors belonging to this set is denoted D.

[0059] In step 207, a loop is performed over the set of component vectors generated in step 205 (i.e., k=1 . . . D) to calculate the value, denoted 𝑓_k, of the criterion function 𝑓(ŵ) at the given component vector ŵ. If the operation set forth below with respect to FIG. 3 is used, the loop of step 207 may be performed over the set of vectors {right arrow over (γ)}, and the value 𝑓_kfor a given vector {right arrow over (γ)} may be calculated as follows:

[0060] 𝑓({circumflex over (γ)})={circumflex over (γ)}^T(ε_w)^−½ε_b(ε_w)^−½{right arrow over (γ)}

[0061] where {right arrow over (γ)}=(ε_w)^½ŵ, and

[0062] where {right arrow over (γ)} is the unit vector corresponding to the vector {right arrow over (γ)}.

[0063] In addition, in step 207, the value 𝑓_kof the criterion function 𝑓(ŵ) at the given component vector ŵ and the associated component vector ŵ (or the value 𝑓_kof the criterion function 𝑓({circumflex over (γ)}) at the given vector {circumflex over (γ)} and associated vector {circumflex over (γ)}) are stored by the processing system 100.

[0064] In step 209, an F distributed statistic is generated for subsets of the set of component vectors generated in step 205. The F distributed statistic for a given subset of component vectors preferably represents a ratio of the variance between groups of items to the variance within groups of items along the given subset of component vectors. In this case, the statistic, denoted ψ_s, characterizing a given subset of component vectors preferably has following form: ${\psi }_S=C\left(\frac{1}{L_S}\right)\sum f_k$

[0065] where 𝑓_krepresents the value of the criterion function 𝑓({right arrow over (w)}) at a given component vector in the subset,

[0066] C is a constant,

[0067] L_srepresents the number of 𝑓_kvalues in the given subset of component vectors, and

[0068] the Σ operation sums over the L_S𝑓_kvalues in the given subset of component vectors.

[0069] Note that in the example above where $C=\frac{1/\left(N-1\right)}{1/\left(\sum n_i-N\right)},$

[0070] the ψ_s' s are F distributed with the ((N−1)L_S) degrees of freedom in the numerator and (Σn_i−N) degrees of freedom in the denominator.

[0071] An optimal subset S of the component vectors is then selected by identifying a subset of component vectors such that a probability value for the statistic ψ_sassociated with the subset (preferably, the probability value represents that the probability that the statistic ψ_sfor the subset could have been larger by chance) satisfies a predetermined criterion (significance level). This significance level represents a threshold at which an hypothesis that the aggregate F-distributed ratios (for the subset) representing discrimination between groups of items is smaller than that within groups of items can be rejected. A small significance level implies a very significant rejection of this hypothesis, in turn implying high confidence in the hypothesis that the aggregate F-distributed ratios (for the subset) representing discrimination between groups of items is greater than or equal to that within groups of items. In this case, the optimal subset S of the component vectors is selected by identifying the subset of component vectors whose probability value for the statistic ψ_sassociated with the subset is a minimum. This minimum probability value implies a maximum confidence in the hypothesis that the aggregate F-distributed ratios (for the subset) representing discrimination between groups of items is greater than or equal to that within groups of items. Selection of the optimal subset S is preferably accomplished as follows.

[0072] In step 211, the D 𝑓_kvalues generated in step 207 are ranked in descending order (i.e., from largest to smallest).

[0073] In step 213, a loop is performed over subsets of component vectors whereby, for each subset, the statistic ψ_sassociated with the subset is generated and a probability value for statistic ψ_sis calculated. Preferably, the loop is performed over a counter X ranging from 1 to D. In each iteration of the loop, the following operations are performed. First, the largest X 𝑓_kvalues identified in step 211 are added together as follows

[0074] (F_X=ψ_s=C*(1/L_S)*(𝑓_k+𝑓_k+ . . . 𝑓_k)), and the resultant sum is normalized by a division by X (i.e., F_X=F_X|X). Second, a probability value, denoted ρ_x, for the normalized sum F_Xis calculated. Third, the probability value ρ_x, and associated counter X are stored.

[0075] The probability value ρ_xfor a normalized sum F_xmay be calculated in step 213 as follows:

[0076] ρ_x=Q(F_x|N−1,X)

[0077] A more detailed description of the calculation of the probability value ρ_xfor a subset of component vectors may be found in Press et al., “Numerical Recipes”, Cambridge University Press, 1986, pp. 175-190, herein incorporated by reference in its entirety. And a more detailed description of the F distribution function is set forth in Freund, “Mathematical Statistics”, 5th Ed., Prentice Hall, 1992, pp. 314-315, herein incorporated by reference in its entirety.

[0078] In step 215, one or more probability values ρ_xgenerated in step 213 that satisfy a predetermined criterion are selected, and the corresponding subset of component vectors corresponding to selected probability value(s) is identified as the optimal subset S of component vectors. Preferably, in step 215, the minimum probability value ρ_xgenerated in step 213 is selected, and the corresponding subset of component vectors corresponding to selected probability value is identified as the optimal subset S of component vectors.

[0079] Finally, in step 217, one or more of the descriptor vectors for the series of items are mapped to a space corresponding to the optimal subset S of component vectors. This mapping is preferably accomplished for a given descriptor vector {right arrow over (X_ij+L )} by performing a loop over each component vector ŵ belonging to the optimal subset S of component vectors whereby, in each iteration of the loop, the contribution ŵ^T{right arrow over (X_ij+L )} (where ŵ^Tis the transpose of the unit vector ŵ for the given component vector {right arrow over (w)}) is added to a running sum. Consider an example wherein the optimal subset S includes three component vectors ({right arrow over (w₁+L )}, {right arrow over (w₂+L )}, {right arrow over (w₃+L )}). In this example, the mapping, denoted M, may be represented by the following:

[0080] M({right arrow over (x_ij+L )})=(^T{right arrow over (x_ij+L )})+(^T{right arrow over (x_ij+L )})+(^T{right arrow over (x_ij+L )})

[0081] (where ^Tis the transpose of the unit vector for the component vector ,

[0082] where ^Tis the transpose of the unit vector for the component vector ,

[0083] where ^Tis the transpose of the unit vector for the component vector ).

[0084] FIG. 3 illustrates an example of the operation of the computer processing system 100 in determining the set of component vectors (i.e., those ŵ) that maximize the criterion function 𝑓(ŵ). The approach defines a vector {circumflex over (γ)} that is a function of ŵ along which the contribution from the denominator of the criterion function 𝑓(ŵ) is independent.

[0085] In step 301, the set of eigenvalue/eigenvector pairs of the matrix ε_ware calculated. This may be accomplished, for example, with the techniques set forth in Press et al., “Numerical Recipes”, Cambridge University Press, 1986, pp. 349-363, herein incorporated by reference in its entirety. The set of eigenvalue/eigenvector pairs for which the eigenvalue is non-zero, denoted set E_w, is then stored by the computer processing system 100. For the sake of description, the number of eigenvalue/eigenvector pairs belonging to set E_wis denoted K, and a given eigenvalue/ eigenvector pair belonging to set E. is denoted (e_k, {right arrow over (ν_k+L )}).

[0086] The operations may then define a vector, denoted {circumflex over (γ)}, as a function of ŵ as follows:

[0087] {right arrow over (γ)}=ε_w)^½{right arrow over (w)}

[0088] where

[0089] (ε_w)^½=Σ^T(e_k)^½

[0090] where is the unit vector corresponding to eigenvector {right arrow over (ν_k+L )} and the Σ operation sums over the K eigenvalue/eigenvector pair belonging to set E_w.

[0091] The criterion function 𝑓(ŵ) may be rewritten as 𝑓({circumflex over (γ)}) as follows:

[0092] 𝑓({circumflex over (γ)})={circumflex over (γ)}^T(ε_w)^−½ε_b(ε_w)^−½{circumflex over (γ)}

[0093] where {circumflex over (γ)} is the unit vector corresponding to the vector {right arrow over (γ)}.

[0094] The set of vectors {circumflex over (γ)} that maximize the criterion function 𝑓({circumflex over (γ)}) may be generated by solving the following eigenvalue equation:

[0095] (ε_w)^−½ε_b(ε_w)^−½ =𝑓_k

[0096] Finally, the set of vectors {circumflex over (γ)} that maximize the criterion function 𝑓({circumflex over (γ)}) may then be transformed to generate the corresponding set of component vectors ŵ that maximize the criterion function 𝑓(ŵ) as follows:

[0097] ŵ=(ε_w)^−½{circumflex over (γ)}

[0098] In step 303, the computer processing system 100 calculates the matrix (ε_w)^−½ε_b(ε_w)^−½ utilizing the eigenvalue/eigenvector pairs belonging to set E_wcalculated in step 301. Note that (ε_w)^½ may be calculated as follows:

[0099] (ε_w)^½=Σ^T(e_k)^½

[0100] where {circumflex over (ν_k+L )} is the unit vector corresponding to eigenvector {right arrow over (ν_k+L )} and the Σ operation sums over the K eigenvalue/eigenvector pair belonging to set E_w.

[0101] In addition, in step 303, the computer processing system 100 calculates the set of eigenvalue/eigenvector pairs for the matrix (ε_w)^−½ ε_b(ε_w)^−½. This may be accomplished, for example, with the techniques set forth in Press et al., “Numerical Recipes”, Cambridge University Press, 1986, pp. 349-363, incorporated by reference above in its entirety. The set of eigenvalue/eigenvector pairs for which the eigenvalue is non-zero, denoted set E_wb, is then stored by the computer processing system 100. For the sake of description, the number of eigenvalue/eigenvector pairs belonging to set E_wbis denoted K′, and a given eigenvalue/eigenvector pair belonging to set E_wbis denoted (e_k′, {right arrow over (ν_k′+L )}). The K′ eigenvectors (each denoted {right arrow over (ν_k′+L )}) of the eigenvalue/eigenvector pairs belonging to set E_wbrepresent the set of vectors {circumflex over (γ)} that maximize the criterion function 𝑓({right arrow over (γ)}).

[0102] In step 305, the computer processing system 100 transforms the K′ eigenvectors (each denoted {right arrow over (ν_k′+L )}) of the eigenvalue/eigenvector pairs belonging to set E_wbto a set of corresponding component vectors in {right arrow over (w)} space (each denoted {right arrow over (w_k+L )}) as follows:

[0103] {right arrow over (W_k+L )}=(ε_w)^−½{right arrow over (ν_k′+L )}

[0104] The K′ set of vectors (each denoted {right arrow over (w_k+L )}) represent the set of component vectors that maximize the criterion functions 𝑓({right arrow over (w)}).

[0105] As described above, the computer implemented method of the present invention provides a mapping (i.e., a transformation) for the descriptor vectors for a series of items to a space that optimally discriminates between groups of items. The method may be used in many domains.

[0106] For example, in the domain of bio-informatics, the items of interest may be genotypes that are partitioned into groups based upon phenotypes exhibited by such genotypes; and the descriptor vectors associated with such genotypes may represent biological, chemical, and/or physical properties of such genotypes. Typically, a candidate genotype and associated descriptor vector is identified and one wishes to ascertain to which group the candidate genotype belongs. The method discussed above may be used to provide a suggestion as to which group the candidate genotype belongs.

[0107] More specifically, the method described above is used to map the descriptor vector for each genotype to a space that optimal discriminates between groups. In addition, the statistical mean of the mapped descriptor vectors for each group [or some other statistical variable (such as the covariance about the mean) based upon the mapped descriptor vectors] is calculated. In addition, the mapping function that optimally discriminates between groups is applied to the descriptor vector for the candidate genotype. A suggestion as to which group the candidate genotype belongs is then determined based upon differences between the mapped descriptor vector for the candidate genotype and the statistical mean of the mapped descriptor vectors for the groups.

[0108] In order to illustrate this example in more detail, consider the case where the genotypes are a population of male patients partitioned into two groups (A,B) based upon prostate cancer activity experienced by the patients (i.e., patients belonging to group A have not experienced any prostate cancer activity, and patients belonging to group B have experienced prostate cancer activity). In addition, the descriptor vector associated with each male patient is expression data acquired from a gene probe array system, such as the GeneChipsystems developed by Affymetrix, Inc. (information is available at http://www.affymetrix.com/products/). In addition, a candidate and associated descriptor vector has been identified. Again, the descriptor vector associated with the candidate is expression data acquired from a gene probe array system, and one wishes to ascertain to which group the candidate belongs. The method discussed above may be used to provide a suggestion as to which group the candidate belongs. More specifically, the method described above is used to map the descriptor vector for each patient to a space that optimal discriminates between groups. In addition, the statistical mean of the mapped descriptor vectors for each group (or some other statistical variable, such as the covariance about the mean, based upon the mapped descriptor vectors) is calculated. In addition, the same mapping function is applied to the descriptor vector for the candidate. Finally, a suggestion as to which group the candidate belongs is determined based upon differences between the mapped descriptor vector for the candidate and the statistical mean of the mapped descriptor vectors for the groups A,B.

[0109] In another example in the domain of bio-informatics, the items of interest may be gene sequences that are partitioned into groups; and the descriptor vectors associated with such gene sequences may represent expression activities of such gene sequences. The method discussed above may be used to derive a probability value for the statistic ψ_sthat represents a significance level at which an hypothesis that the aggregate F-distributed ratios representing discrimination between groups of expression activities of gene sequences is smaller than that within groups of expression activities of gene sequences can be rejected. A small significance level implies a very significant rejection of this hypothesis, in turn implying high confidence in the hypothesis that the aggregate F-distributed ratios representing discrimination between groups of expression activities of gene sequences is greater than or equal to that within groups of expression activities of gene sequences. In the event that this probability value is less than a threshold value, there is a suggestion that the gene sequences express differences in activities.

[0110] In the domain of computational biology and chemistry, for example, the items of interest may be molecular complexes (a molecule or portion of a molecule) that are partitioned into groups based upon a biological activity (for example, molecular complexes belonging to a group all bind to another molecular complex); and the descriptor vectors associated with such molecular complexes may represent the physical property (structure, charge distribution), biological property and/or chemical property of such molecular complexes. For instance, the descriptor vectors associated with a molecule may be based upon data representing the structure and/or charge distribution of molecule as outlined in i) U.S. Pat. No. 5,784,294 to Platt et al., which is commonly assigned to the assignee of the present invention, ii) R. D. Cramer et al., “Comparative Molecular Field Analysis (CoMFA). Effect of Shape on Binding of Steroids to Carrier Proteins, J. Am. Chem. Soc. Vol. 110, 1988 pp. 5959-5967, iii) A. C. Good et al., “Structure-Activity Relationships from Molecular Similarity Matrices,” J. Med. Chem., Vol. 36, 1993, pp. 433-438, iv) A. Jain et al., “Predicting Biological Activities from Molecular Surface Properties. Performance Comparisons on a Steroid Benchmark,” J. Med. Chem., Vol. 37, 1994, pp. 2315-2327, v) W. Fisanick et al., “Similarity Searching on CAS Registry Substances, 1: Global Molecular Property and Generic Atom Triangle Geometric Searching,” Journal of Chemical Information and Computer Sciences, Vol. 32, No. 6, 1992, pp. 664-674, and W. Fisanick et al., “Similarity Searching on CAS Registry Substances, 2: 2D Structural Similarity,” Journal of Chemical Information and Computer Sciences, Vol. 34, No. 1, 1994, pp. 130-140, hereinafter incorporated by reference in their entirety. Similarly, descriptor vectors may be associated with a portion of a molecule (i.e., a subset of the atoms that make up a molecule) and represent the structure and/or charge distribution of the portion of the molecule. In such domains, typically, a candidate molecular complex and associated descriptor vector is identified and one wishes to ascertain to which group the candidate molecular complex belongs. The computer implemented method of the present invention described above may be used to provide a suggestion as to which group the candidate molecular complex belongs. More specifically, the method described above is used to map the descriptor vector for each molecular complex to a space that optimal discriminates between groups. In addition, the statistical mean of the mapped descriptor vectors for each group [or some other statistical variable (such as the covariance about the mean) based upon the mapped descriptor vectors] is calculated. In addition, the mapping function that optimally discriminates between groups is applied to the descriptor vector for the candidate molecular complex. A suggestion as to which group the candidate molecular complex belongs is then determined based upon differences between the mapped descriptor vector for the candidate molecular complex and the statistical mean of the mapped descriptor vectors for the groups.

[0111] The description above illustrates application of the present invention in one-way analysis of variance. However, the present invention is not limited in this respect, and can be applied to multi-factor designs with multi-way analysis of variance. An example of such a design for an item of interest is illustrated in FIG. 4. In this design, a series of items are classified into N groups (N is an integer greater than 1), wherein each group is identified by an identifier i ranging from 1 to N represented by the columns of the matrix. A series of factors (typically each factor represents one or more experimental treatments) are attributed to the items belonging to the groups, wherein each factor is identified by an identifiers ranging from 1 to M represented by the rows of the matrix. An identifier m_ijkidentifies item k that is attributed to the group/factor pair i,j , and vector {right arrow over (x_ijk+L )} identified a descriptor vector attributed to the item m_ijk. Preferably, the descriptor vectors are stored as part of a file in persistent storage and loaded into non-persistent storage for use by the CPU 103 as needed.

[0112] In such a system, one may break any given descriptor vector {right arrow over (x_ijk+L )} into the following components:

[0113] {right arrow over (x_iyk+L )}={right arrow over (α)}+{right arrow over (β)}+{right arrow over (γ_j+L )}+{right arrow over (δ_ij+L )}+{right arrow over (ε_ijk+L )},

[0114] where

[0115] {right arrow over (α)} represents the mean of the descriptor vectors for all the items (of all the groups and all the factors);

[0116] {right arrow over (β_i+L )} represents the mean of the descriptor vectors for all the items in group i (of all the factors);

[0117] {right arrow over (γ_j+L )} represents the mean of the descriptor vectors for all the items in factorj (of all the groups);

[0118] {right arrow over (δ_ij+L )} by represents the mean of the deviation of the descriptor vectors for all the items in group/factor pair i,j from the sum ({right arrow over (α)}+{right arrow over (β_i+L )}+{right arrow over (γ_j+L )}); and

[0119] {right arrow over (_ijk+L )} represents deviation from the mean δ_ijto the descriptor vector {right arrow over (x_ijk+L )}.

[0120] These values may be calculated as follows. For each group/factor pair i,j of items (i.e., i ranging from 1 to N, and j ranging from 1 to M), the mean of the descriptor vectors within group/factor pair i,j , denoted {right arrow over (x_ji.+L )}, may be calculated as follows:

[0121] $\overrightarrow{x_{\mathrm{ij}.}}=\frac{1}{n_{\mathrm{ij}}}\Sigma \overrightarrow{x_{\mathrm{ijk}}}$

[0122] where n_ijrepresents the number of items in the group/factor pair i,j , and the Σ operator sums over the range (k=1 . . . n_ij) of items in the group/factor pair i,j

[0123] Then, for each group i of items (i.e., i ranging from 1 to N), the mean {overscore (x_i..+L )} may be calculated as follows:

[0124] $\overrightarrow{x_{i..}}=\frac{1}{n_i}\Sigma \overrightarrow{x_{\mathrm{ij}.}}$

[0125] where ⁿ_irepresents the number of items in the group i , and the Σ operator sums over the range (1 . . . M) of factors

[0126] Then, for each factorj (i.e., i ranging from 1 to M), the mean {overscore (x_.j.+L )} may be calculated as follows:

[0127] $\overrightarrow{x_{.j.}}=\frac{1}{n_{j.}}\Sigma \overrightarrow{x_{\mathrm{ij}.}}$

[0128] where n_jrepresents the number of items in the factorj , and the Σ operator sums over the range (1 . . . N) of groups

[0129] Then, the mean {right arrow over (α)} may be calculated as follows:

[0130] {right arrow over (α)}={fraction (1+L /N)}Σ{overscore (x_i..+L )}

[0131] where N represents the number of groups , and the Σ operator sums over the range (1 . . . N) of groups; or equivalently

[0132] {right arrow over (α)}={fraction (1+L /M)}Σ{overscore (x_i..+L )}

[0133] where M represents the number of factors, and the Σ operator sums over the range (1 . . . M) of factors

[0134] For each group i of items (i.e., i ranging from 1 to N), the mean {right arrow over (β_i+L )} may be calculated as follows:

[0135] {right arrow over (β_i+L )}={overscore (x_i..+L )}−{right arrow over (α)}

[0136] For each factor j of items (i.e., j ranging from 1 to M), the mean {right arrow over (γ_j+L )} may be calculated as follows:

[0137] {right arrow over (γ_j+L )}={overscore (x_.j.+L )}−{right arrow over (α)}

[0138] For each group/factor pair i,j of items (i.e., i ranging from 1 to N, andj ranging from 1 to M), the mean {right arrow over (_ij+L )} may be calculated as follows:

[0139] {right arrow over (δ_ij+L )}={right arrow over (x_ij.+L )}−{right arrow over (x_.j.+L )}

[0140] Finally, for each descriptor vector {right arrow over (x_ijk+L )} in the group/factor pair i,j, the deviation {right arrow over (ε_ijk+L )} may be calculated as follows:

[0141] {right arrow over (ε_ijk+L )}={right arrow over (x_ijk+L )}−{right arrow over (δ_ij+L )}

[0142] The total covariance, denoted ε, may be represented by the sum of the following four (4) terms:

[0143] ΣM_i{right arrow over (β_i+L )}{right arrow over (β_i+L )}^T,   (1)

[0144] where the Σ operator sums over the range of groups i=1 to N and M_iis the number of items in group i.

[0145] ΣN_j{right arrow over (γ_j+L )}{right arrow over (γ_j+L )}^T,   (2)

[0146] where the Σ operator sums over the range of factors and Nj is the number of items in factor i.

[0147] ΣN_ij{right arrow over (δ_i+L )}{right arrow over (δ_i+L )}^T,   (3)

[0148] where the Σ operator sums over the range of group/factor pairs i,j and N_ijis the number if items belonging to the group/factor pair i,j.

[0149] Σε_ijk{right arrow over (ε_ijk+L )}^T,   (4)

[0150] where the Σ operator sums over the range of group/factor pairs i,j and k items in each group/factor pair

[0151] Note the first term represents the covariance between groups of items; the second term represents the covariance between factors; and the third term represents the covariance of the interaction between the groups and factors.

[0152] Generally, the principles of the present invention as described above may then be used to map the components of the descriptors vectors of the items to a space that:

[0153] 1) optimally discriminates between the groups of items;

[0154] 2) optimally discriminates between factors; or

[0155] 3) optimally discriminates interactions between the groups and factors.

[0156] More specifically, in order to map the components of the descriptors vectors of the items to a space that optimally discriminates between groups of items (case 1 above), an F distributed criterion function is used to determine a set of component vectors that maximize the criterion function. The criterion function has a numerator and a denominator, whereby the numerator is based upon the covariance between groups of items (the first term described above) and the denominator is based upon the fourth term described above. The criterion function has the general form: $f\left(\overrightarrow{w}\right)=C\left(\frac{{\overrightarrow{w}}^T\left(\sum M_i\overrightarrow{{\beta }_i}{\overrightarrow{\beta }}_i^T\right)\overrightarrow{w}}{{\overrightarrow{w}}^T\left(\sum \overrightarrow{{ϵ}_{\mathrm{ijk}}}{\overrightarrow{{ϵ}_{\mathrm{ijk}}}}^T\right)\overrightarrow{w}}\right)$

[0157] where {right arrow over (w)} is some vector

[0158] The set of component vectors is determined by solving for those ŵ that maximize the criterion function 𝑓({right arrow over (w)}), and an optimal subset of the component vectors is identified. A more detailed description of this operation is described above with respect to steps 205-215 of FIG. 2. Finally, the components of the descriptor vectors for the items are mapped to a space corresponding to the optimal subset of the component vectors. The resultant data optimally discriminates between groups of items. A more detailed description of this operation is described above with respect to step 217 of FIG. 2.

[0159] In order to map the components of the descriptors vectors of the items to a space that optimally discriminates between factors (case 2 above), an F distributed criterion function is used to determine a set of component vectors that maximize the criterion function . The criterion function has a numerator and a denominator, whereby the numerator is based upon the covariance between factors (the second term described above) and the denominator is based upon the fourth term described above. The criterion function has the general form: $f\left(\overrightarrow{w}\right)=C\left(\frac{{\overrightarrow{w}}^T\left(\sum N_j\overrightarrow{{\gamma }_j}{\overrightarrow{\gamma }}_j^T\right)\overrightarrow{w}}{{\overrightarrow{w}}^T\left(\sum \overrightarrow{{ϵ}_{\mathrm{ijk}}}{\overrightarrow{{ϵ}_{\mathrm{ijk}}}}^T\right)\overrightarrow{w}}\right)$

[0160] where {right arrow over (w)} is some vector

[0161] The set of component vectors is determined by solving for those ŵ that maximize the criterion function 𝑓({right arrow over (w)}), and an optimal subset of the component vectors is identified. A more detailed description of this operation is described above with respect to steps 205-215 of FIG. 2. Finally, the components of the descriptor vectors for the items are mapped to a space corresponding to the optimal subset of the component vectors. The resultant data optimally discriminates between factors. A more detailed description of this operation is described above with respect to step 217 of FIG. 2.

[0162] In order to map the components of the descriptors vectors of the items to a space that optimally discriminates interactions between the groups and factors (case 3 above), an F distributed criterion function is used to determine a set of component vectors that maximize the criterion function. The criterion function has a numerator and a denominator, whereby the numerator is based upon the covariance of the interaction between the groups and factors (the third term described above) and the denominator is based upon the fourth term described above. The criterion function has the general form:

[0163] $f\left(\overrightarrow{w}\right)=C\left(\frac{{\overrightarrow{w}}^T\left(\sum N_{\mathrm{ij}}\overrightarrow{{\delta }_i}{\overrightarrow{\delta }}_i^T\right)\overrightarrow{w}}{{\overrightarrow{w}}^T\left(\sum \overrightarrow{{ϵ}_{\mathrm{ijk}}}{\overrightarrow{{ϵ}_{\mathrm{ijk}}}}^T\right)\overrightarrow{w}}\right)$

[0164] where {right arrow over (w)} is some vector

[0165] The set of component vectors is determined by solving for those ŵ that maximize the criterion function 𝑓(ŵ), and an optimal subset of the component vectors is identified. A more detailed description of this operation is described above with respect to steps 205-215 of FIG. 2. Finally, the components of the descriptor vectors for the items are mapped to a space corresponding to the optimal subset of the component vectors. The resultant data optimally discriminates interaction between the groups and factors. A more detailed description of this operation is described above with respect to step 217 of FIG. 2.

[0166] The application of the present invention to multi-factor designs may be used in many domains.

[0167] In the domain of insurance, for example, the items of interest may be individuals that are partitioned into groups based upon certain characteristics such as age, gender, etc. Categories of auto insurance polices (for example, categorized based upon the limits/deductable amount for liability coverage and/or collision coverage) attributed to the groups of individual may represent the factors of the design. The descriptor vectors associated with such individuals may represent risk of the individual (for example, may be a dollar amount of the moneys paid to the individual arising from auto insurance claims during a predetermined period of time). In this example, the computer implemented method discussed above may be used to provide a mapping of the components of the descriptor vectors to a space that optimal discriminates between groups of individuals; or optimally discriminates between the policies, or optimally discriminates interaction between the groups and policies.

[0168] In the domain of agriculture, for example, the items of interest may be plant species that are partitioned into groups based upon certain characteristics such as genetic makeup of the plant species. Categories of fertilizers (or pesticides) that are applied to the groups of plant species may represent the factors of the design. The descriptor vectors associated with such plant species may represent a characteristic of the plant species such as yield and/or drought resistance. In this example, the computer implemented method discussed above may be used to provide a mapping of the components of the descriptor vectors to a space that optimal discriminates between groups of plant species; or optimally discriminates between the fertilizers (or pesticides), or optimally discriminates interaction between the groups and fertilizers (or pesticides).

[0169] It should be noted that the information encoded by the descriptor vectors may cause membership in a group associated with an item of interest. For example, the behavior of a molecular complex, which is encoded by a descriptor vector associated with the molecular complex, may cause membership in a group (or category) of molecular complexes (for example, hydrophilic, polar, “active” with respect to a class of reactions, etc.). In the alternative, the information encoded by the descriptor vector associated with an item of interest may be a response to group membership. For example, the risks associated with an insurance policy may be encoded by a descriptor vector associated with the insurance policy, whereby the risks are dictated by the group (i.e., type) membership of the policy.

[0170] In addition, the present invention may be used to identify molecules within a large database of molecules that are similar to a query molecule. The methodology may be conceptually divided into three phases: a training phase, an acquisition phase, and a recognition phase.

[0171] In the training phase, for each molecule in a series of molecules (denoted the training set), the following steps are performed for a given molecule in the training set. Data that represents the atomic structure of the given molecule in the training set is stored in persistent memory. In addition, data that represents one or more properties of the given molecule is stored in persistent memory. The data that represents the atomic structure of the given molecule is read from persistent memory and used to define a set of three-dimensional regions of space that contain portions of the given molecule. For each region of the given molecule, one or more properties of the given molecule are generated and mapped to a one or more property values for the grid points of the region; and the properties values for the grid points of the region are used to determine a descriptor vector associated with the region that characterizes the region. Preferably, one or more components of the descriptor vector associated with a given region represent position and/or orientation of axes derived from the property distribution of the region, wherein the axes are invariant with respect to translation and rotation of the region. In addition, a group identifier is assigned to the region. The group identifier identifies the group to which the region belongs. For example, the regions may be partitioned into groups based upon the charge distribution of the region (e.g., charged, neutral, polar, non-polar, etc.) or other property or behavior of the region. A mapping of one or more components of the descriptor vector for the regions is then calculated (preferably by applying the method discussed above is used to generate a mapping of components of the descriptor vectors to a space that optimally discriminates between the groups of regions—step 217 of FIG. 2). The mapping generated in the training phase is stored in persistent storage and used in the acquisition phase and recognition phase. A more detailed description of the training phase for a particular region is illustrated in FIG. 5.

[0172] In the acquisition phase, for each molecule in a series of molecules (which is typically distinct from the series of molecules processed in the training phase), the following steps are performed for a given molecule. Similar to the training phase, data that represents the atomic structure of the given molecule is used to define a set of regions that contain portions of the given molecule, and descriptor vectors are generated for such regions of the given molecule. Similar to the training phase, one or more items of the descriptor vector associated with a given region preferably represents position and/or orientation of axes derived from the property distribution of the region, wherein the axes are invariant with respect to translation and rotation of the region. The mapping generated in the training phase is used to map one or more components of the descriptor vector for each given region (preferably to a space that optimally discriminates between groups of regions), and a key is generated based upon the mapping of the component(s) of the associated descriptor vector. The key identifies an entry in a multi-map (described below) stored in persistent memory. Data identifying the given region, data (or a pointer to such data) characterizing a set of axes derived from the property distribution of the region, and preferably other data (for example, data identifying the molecule to which the region belongs, and data representing the geometric center of the molecule to which the region belongs) are then stored in the multi-map at a location identified by the key. As described below in more detail, the data characterizing a set of axes derived from the property distribution of a given region preferably characterizes transformation between an input reference frame and the inertial reference frame for the given region. A more detailed description of an exemplary embodiment of the acquisition phase for a particular region is illustrated in FIG. 6. Note that if there are common molecules (or regions) in the series of molecules processed in the training phase and acquisition phase, data generated in the training phase may be used in the acquisition phase, and the operations that use and/or generate such data may be bypassed accordingly.

[0173] In the recognition phase, a query molecule is provided. Similar to the training and acquisition phase, data that represents the atomic structure of the query molecule is used to define a set of regions that contain portions of the query molecule, and descriptor vectors are generated for such regions of the query molecule. Similar to the training phase, one or more items of the descriptor vector associated with a given region preferably represents position and/or orientation of axes derived from the property distribution of the region, wherein the axes are invariant with respect to translation and rotation of the region. For each query region, the mapping generated in the training phase is used to map one or components of the descriptor vector for the given query region (preferably to a space that optimally discriminates between groups of regions), and a key is generated based upon the mapping of the component(s) of the associated descriptor vector. For each key, a multi-map entry identified by the key is retrieved and data stored therein [i.e., data identifying one or more regions associated therewith, data characterizing a set of axes derived from the property distribution of each region associated therewith, data identifying the molecule to which each region associated therewith belongs, and data representing the geometric center of the molecule to which each region associated therewith belongs] are read from the table. For each region identified by the retrieved table entry, an hypothesized match is constructed and added to a vote table. After processing all of the regions identified by the retrieved table entry, the vote table is sorted to determine a set of potential matching regions, and the set of potential matching regions (and/or the molecules to which the set of potential matching regions belong) is made accessible to the user via an I/O device. A more detailed description of an exemplary embodiment of the recognition phase for a region of the query molecule is illustrated in FIGS. 7(A) and (B).

[0174] In the training phase, for each molecule in a training set, data that represents the structure of the given molecule is stored in persistent memory. Preferably, the data represents the atomic structure of the given molecule in an arbitrary three-dimensional reference frame, which is referred to below as the “input reference frame”. The data may be obtained from a database (public or private) or be derived by traditional molecular modeling techniques.

[0175] In addition, for each molecule in the training set, data that represents the property distribution of one or more properties of the given molecule is generated, for example by reading such data from persistent memory. The property distribution for the property of a given molecule may be an atomic property distribution (i.e., data representing a heuristic, such as electronegativity, hydrophobicity or polarity, that characterizes the behavior or property of an atom in the given molecule), a surface property distribution (i.e., data representing a heuristic that characterizes the behavior or property of a surface on or within the molecule), or a volumetric property distribution (i.e., data characterizing the behavior or property of a region of the molecule—for example, electron density).

[0176] In addition, for each given molecule in the training set, the structure data is used to generate a set of three-dimensional regions of space, hereinafter referred to “scoops”, in the input reference frame. Preferably, each scoop is a spherical region in the input reference frame having a center (i.e., a point in the input reference frame) and a radius. Preferably, the center of each scoop corresponds to one or more heuristics of the given molecule. For example, the heuristic may correspond to the coordinate of i) the nucleus of one or more atoms of the given molecule, ii) the center of one or more bonds between atoms of the given molecule, iii) the end point of an extension to a bond (typically the length of the extension is a factor of bond length) between atoms of the given molecule, iv) a grid points in the given molecule, v) the geometric center of the molecule, vi) ring centers in the given molecule, or vii) lone pairs of electrons in the given molecule. In addition, the radius of each scoop may be set to a predetermined value (for example, 3 angstroms, or 5 angstroms) or may be based upon heuristics of the given molecule. It should be noted that a scoop can be any arbitrary three-dimensional region in the input reference frame.

[0177] As illustrated in FIG. 5, the training phase preferably performs a nested loop over the molecules in the training set; wherein, for a given molecule, each scoop in the set of scoops for the given molecule is processed as follows.

[0178] In step 501, the property distribution of one or more properties of the given molecule that is relevant to the scoop are identified, and the relevant property distribution is mapped to a property field that represents the value of the property at points in the input reference frame. There are many well-known techniques to map a property distribution (atomic, surface, volumetric) to a property field, the specifics of which is not relevant to the present invention. Examples of such techniques may be found in U.S. Pat. No. 5,025,388 to Cramer, III et al., hereinafter incorporated by reference in its entirety. With regard to atomic properties for the given molecule, preferably the properties of atoms that are contained within the scoop are identified as relevant properties for the scoop in step 501. With regard to surface properties for the given molecule, preferably the properties of surfaces that are contained (partially or fully) within the scoop are identified as relevant properties for the scoop in step 501. With regard to volumetric properties for the given molecule, preferably the properties of volumes that are contained (partially or fully) within the scoop are identified as relevant properties for the scoop in step 501.

[0179] In step 503, the property field generated in step 501 is mapped to grid points contained in the scoop to determine contribution of property field at the grid points. Preferably, the grid points are evenly spaced within the given scoop. In addition, the contribution of the property field at a given grid point preferably includes two values: the first value is a positive value, which for the sake of description is referred to below as a mass value; the second value is a real value, which for the sake of description is referred to below as a charge value. There are many possible techniques to calculate the contribution of a property field at a given grid point, the specifics of which is not relevant to the present invention. For example, the following “smearing function” can be used in step 503 to map values of the points of the property field to a first value μ_i(mass value) at grid point i:

[0180] μ_i=ΣA_j*e −R_ij/L_o*(1/R_ij+B)

[0181] where

[0182] A_jrepresents the value of the property field at a point j in the property field,

[0183] R_ijrepresents the distance between the point j in the property field and the point i in the scoop;

[0184] L₀is a damping factor, which typically ranges between 0.5 and 2.5;

[0185] B is an arbitrary constant, which typically is 1; and

[0186] Σ sums over all the points in the property field for the scoop.

[0187] Alternate smearing functions may be gaussian forms. In addition, the following equation may be used to map values of the points of the property field to a second value χ_i(charge value) at grid point i:

[0188] χ_i=μ_i−{overscore (μ)}

[0189] where {overscore (μ)} represents the mean first value (mass value) for all the grid points of the scoop

[0190] In step 505, a descriptor vector for the scoop is generated based upon the property values of the grid points of the scoop. Preferably, the components of the descriptor vector for the scoop includes one or more of the following data values, details of which are set forth below:

[0191] ID data value identifying the molecule from which the scoop is derived

[0192] N an integer value representing the number of grid points for the scoop

[0193] M total mass for the scoop, which is a sum of the first values (i.e., “mass values” or μ_i) for the grid points of the scoop

[0194] Q total charge for the scoop, which is the sum of the second values (i.e., “charge values” or χ_i) for the grid points of the scoop

[0195] S_x, S_y, S_zcomponents of a vector describing translation between origin of the input reference frame and center of mass in the input reference frame

[0196] I_x, I_y, I_z, principal values of the moment of inertia tensor of the scoop with respect to the center of mass of the scoop

[0197] S_θ, S_Φ, S_ΩS_Φ, S_θ are polar angles that describe an axis of rotation. S_Ω represents an angle of rotation about that angle. Together, the angles represent a rotation transformation between principal axes of the moment of inertia tensor and axes of the input reference frame

[0198] v_x, v_y, v_z, components of a vector describing translation between the center of scoop in the inertial reference frame and center of mass in the inertial reference frame

[0199] d_x, d_y, d_z, components of a vector describing translation between center of mass in the inertial reference frame and center of dipole in the inertial reference frame

[0200] Ci_x, Ci_y, Ci_zcomponents of a vector describing third order moment of mass about a center of expansion in the internal reference frame

[0201] Cq_x, Cq_y, Cq_z, components of a vector describing third order moment of charge about a center of expansion in the internal reference frame

[0202] Qx_x, Qx_y, Qx_zcomponents of a tensor characterizing quadrupolar moment about a

[0203] Qy_y, Qy_z, Qz_zcenter of expansion in the internal reference frame

[0204] Note that the components ID, N, M, Q, I_x, I_y, I_z, v_x, v_y, v_z, d_x, d_y, d_z, Ci_x, Ci_y, Ci_z, Cq_x, Cq_y, Cq_z, Q_xx, Q_xy, Q_xz, Q_yy, Q_yz, Q_zzcharacterize the property distribution with respect to invariant axes (i.e., axes that are invariant with respect to translation and rotation of the given scoop—for example the sensed inertial axes of the given scoop). Also note that the components S_x, S_y, S_z, S_θ, S_Φ, S_Ω, are variant with respect to translation and rotation of the scoop, and characterize orientation of the invariant axes with respect to the input reference frame of the scoop. A more detailed description of the calculation of the components of the descriptor vector for a given scoop is set forth below.

[0205] In step 507, a group identifier is assigned to the scoop. The group identifier identifies the group to which the region belongs. For example, the scoops may be partitioned into groups based upon the charge distribution of the scoop (e.g., charged, neutral, polar, non-polar, etc.) or other property or behavior of the scoops.

[0206] In step 509, statistical analysis is used to generate a mapping of one or more components of the descriptor vector for the scoop. Preferably, the computer implemented method of the present invention discussed above is used to generate a mapping of the components of the descriptor vector for the scoop that characterize position and/or orientation of invariant axes derived from the property distribution of the given scoop (i.e., ID, N, M, Q, I_x, I_y, I_z, v_x, v_y, v_z, d_x, d_y, d_z, Ci_x, Ci_y, Ci_z, Cq_x, Cq_y, Cq_z, Q_xx, Q_xy, Q_xz, Q_yy, Q_yz, Q_zz) to a space that optimally discriminates between the groups of scoops (described above with respect to step 217 of FIG. 2). As described above with respect to step 217 of FIG. 2, the mapping is based upon the transpose of unit vectors for the component vectors {right arrow over (w₁+L )}, {right arrow over (w₂+L )}, {right arrow over (w₃+L )}. The mapping generated in step 509 is preferably stored in persistent storage for subsequent use in the acquisition phase and recognition phase.

[0207] In the acquisition phase, for each molecule in a series of molecules, data that represents the structure of the given molecule is stored in persistent memory. Preferably, the data represents the atomic structure of the given molecule in an arbitrary three-dimensional reference frame, which is referred to below as the “input reference frame”. The data may be obtained from a database (public or private) or be derived by traditional molecular modeling techniques.

[0208] In addition, for each molecule in a series of molecules, data that represents the property distribution of one or more properties of the given molecule is generated, for example, by reading such data from persistent memory. The property distribution for a property of the given molecule may be an atomic property distribution, a surface property distribution, or a volumetric property distribution as described above.

[0209] In addition, for each given molecule in the series of molecules, the structure data is used to define a set of scoops in the input reference frame. Preferably, each scoop is a spherical region in the input reference frame having a center and radius as described above.

[0210] As illustrated in FIG. 6, the acquisition phase preferably performs a nested loop over the series of molecules, wherein, for a given molecule, each scoop in the set of scoops for the given molecule is processed as follows.

[0211] In step 601, properties of the given molecule that are relevant to the scoop are identified, and the relevant properties are mapped to a property field that represents the value of the property at points in the input reference frame. This operation is similar to the processing described above with respect to step 501 of the training phase.

[0212] In step 603, the property field generated in step 601 is mapped to grid points contained in the scoop to determine contribution of property field at the grid points. Preferably, the contribution of property field at a given grid point includes two values: a positive mass value and a real charge value. This operation is similar to the processing described above with respect to step 503 of the training phase.

[0213] In step 605, a descriptor vector for the scoop is generated based upon the property values of the gird points of the scoop generated in step 603. Preferably, the components of the descriptor vector for the scoop includes one or more of the data values as described above with respect to step 505 of the training phase.

[0214] In step 607, the mapping generated in step 509 of the training phase is used to map one or more components of the descriptor vector for the scoop (preferably, the component(s) of the descriptor vector is mapped to a space that optimally discriminates between the groups of scoops as described above with respect to step 217 of FIG. 2).

[0215] In step 609, a key is generated based upon the mapping of the component(s) of the descriptor vector for the scoop in step 607. The key identifies an entry in a multi-map stored in persistent memory. The multi-map is an associative memory which permits more than one entry stored in the memory to be associated with the same key. A detailed description of a multi-map is set forth in D. R. Musser and Atul Saini, STL Tutorial and Reference Guide: C++ Programming with the Standard Template Library (Addison-Wesley 1996), herein incorporated by reference in its entirety. Preferably, the multi-map is formed from a hash table. A more detailed description of a hash table may be found in R. Sedgewick, Algorithms in C++ (Addison-Wesley 1992), herein incorporated by reference in its entirety. In the alternative, the multi-map container may be formed from a linked list data structure, or a tree structure such as an AVL-tree or B* tree as described in D. R. Musser and Atul Saini, herein incorporated by reference in its entirety. One skilled in the art will realize that there are many possible underlying implementations for the multi-map data structure.

[0216] Finally, in step 611, data identifying the given scoop, data (or a pointer to such data) characterizing a set of axes derived from the property distribution of the scoop, and preferably other data (for example, data identifying the molecule to which the scoop belongs, and data representing the geometric center in the input reference frame of the molecule to which the scoop belongs) are then stored in the multi-map at a location identified by the key generated in step 609. Preferably, the data characterizing a set of axes derived from the property distribution of a given scoop characterizes transformation between an input reference frame and the inertial reference frame for the given scoop—the components S_x, S_y, S_z, S_θ, S_Φ, S_Ω of the descriptor vector for the scoop as discussed above with respect to step 605.

[0217] An exemplary multi-map entry is illustrated in FIGS. 7(A) and (B). The entry 701 includes a series of segments 703-1 and 703-2 (two shown) coupled via a link-list data structure. Each segment includes an first ID field 711 that stores an identifier for a scoop, a pointer 713 to data (or the data itself) characterizing a set of axes derived from the property distribution of the scoop, a second ID field 715 identifying the molecule to which the scoop belongs, a pointer 717 to data (or the data itself) representing the geometric center of the molecule to which the scoop belongs, and a pointer 719 to the next segment in the table entry.

[0218] The operations of step 601-611 are applied for each scoop in a given molecule. At the end of the acquisition phase, the multi-map stores entries each corresponding to one or more scoops for the series of molecules being studied.

[0219] FIGS. 8(A) and (B) illustrates the recognition phase for a query molecule. In step 801, data that represents the structure of the query molecule in the input reference frame is stored in persistent memory. In step 803, data that represents the property distribution of one or more properties of the query molecule is generated, for example by reading such data from persistent memory. The property distribution of the property for a given molecule may be an atomic property distribution, a surface property distribution, or a volumetric property distribution as described above. In step 805, the structure data of the query molecule is used to define a set of scoops (in the input reference frame) in the query molecule. In steps 807-825, a loop is performed over the set of scoops in the query molecule wherein each scoop is processed as follows.

[0220] In step 809, the properties of the query molecule that are relevant to the scoop are identified, and the relevant properties are mapped to a property field that represents the value of the property at points in the input reference frame of the query molecule. This operation is similar to the processing described above with respect to step 501 of the training phase.

[0221] In step 811, the property field generated in step 809 is mapped to grid points contained in the scoop to determine contribution of property field at the grid points. Preferably, the contribution of property field at a given grid point includes two property values as described above: a positive mass value and a real charge value. This operation is similar to the processing described above with respect to step 503 of the training phase.

[0222] In step 813, a descriptor vector for the scoop is generated based upon the property values (mass values and charge values) of the grid points of the scoop generated in step 811. Preferably, the components of the descriptor vector for the scoop includes one or more of the data values as described above with respect to step 505 of the training phase.

[0223] In step 815, the mapping generated in step 509 of the training phase is used to map one or more components of the descriptor vector for the scoop generated in step 813 (preferably, the component(s) of the descriptor vector are mapped to a space that optimally discriminates between the groups of scoops as described above with respect to step 217 of FIG. 2). This operation is similar to the processing described above with respect to step 607 of the acquisition phase.

[0224] In step 817, a key is generated based upon the mapping of the component(s) of the descriptor vector generated in step 815. The key identifies an entry in the multi-map stored in persistent memory. This operation is similar to the processing described above with respect to step 609 of the acquisition phase.

[0225] In step 819, the multi-map entry identified by the key generated in step 817 is retrieved and the data stored therein are read from the retrieved multi-map entry.

[0226] In step 821, for each scoop identified by the retrieved multi-map entry (i.e., each entry segment in FIG. 7), an hypothesized match is constructed to determine a set of transformation parameters whereby a set of axes derived from property distribution of the query scoop is aligned with a set of axes derived from property distribution for each molecular scoop identified by the retrieved multi-map entry, which for the sake of description is referred to below as the stored scoop.

[0227] In step 823, a label corresponding to transformation parameters of the hypothesized match is generated, and the hypothesized match is added to a vote table. The vote table is an associative memory of entries keyed by the label. Each entry of the vote table stores i) an accumulated score of the number of stored scoops whose hypothesized match corresponds to the label, and ii) data identifying those stored scoops whose hypothesized match corresponds to the label. Preferably, the vote table is implemented as a map. A detailed description of a map is set forth in D. R. Musser and Atul Saini, STL Tutorial and Reference Guide: C++ Programming with the Standard Template Library (Addison-Wesley 1996), incorporated by reference above in its entirety. The map may be formed from a hash table. A more detailed description of a hash table may be found in R. Sedgewick, Algorithms in C++ (Addison-Wesley 1992), incorporated by reference above in its entirety. In the alternative, the map may be formed from a linked list data structure, or a tree structure such as an AVL-tree or B* tree as described in D. R. Musser and Atul Saini. One skilled in the art will realize that there are many possible underlying implementations for the map data structure.

[0228] In step 827, after processing all of the stored scoops identified by the retrieved multi-map entry (step 819), the vote table is sorted to determine a set of potential matching scoops. For example, one or more entries with the highest score may be selected, and those scoops identified by the selected entries of the vote table may be selected as the set of potential matching scoops for the query molecule.

[0229] Finally, in step 829, the set of potential matching scoops (and/or the molecules to which the set of potential matching scoops belong) is made accessible to the user, for example, via graphical user interface 907 of the computer processing system.

[0230] In the preferred embodiment of the present invention, the operations of the recognition phase in constructing the hypothesized match, adding the hypothesized match to the vote table, and sorting the vote table to report results to the user are illustrated in the flow chart of FIG. 9. As described above, each entry of the multi-map generated in step 611 of the acquisition phase preferably stores: i) data characterizing transformation between the input reference frame and the sensed inertial reference frame for a given stored scoop—which may include the components S_x, S_y, S_z, S_θ, S_Φ, S_Ω of the descriptor vector for the stored scoop; ii) data identifying the molecule to which the stored scoop belongs, denoted the stored molecule; and iii) data representing the geometric center in the input reference frame of the stored molecule. Similar operations are performed in the recognition phase in constructing the hypothesized match as illustrated in the flow chart of FIG. 9.

[0231] More specifically, in step 901, data characterizing transformation between the input reference frame and the sensed inertial reference frame for the query scoop—which may include the components S_x, S_y, S_z, S_θ, S_Φ, S_Ω of the descriptor vector for the query scoop is generated.

[0232] In step 903, data characterizing transformation between the input reference frame for the stored molecule (denoted “stored frame”) and the sensed inertial reference frame of the stored scoop (, , of the stored scoop as described below) and data characterizing transformation between the input reference frame of the query molecule (denoted “query frame”) and the sensed inertial reference frame of the query scoop (, , of the query scoop as described below) are used to calculate a transformation (a translation and rotation transformation) that aligns the two sensed inertial frames, which also represents a transformation from the stored frame to the query frame. Details of an exemplary technique for calculating this transformation may be found in K. Turkowski, “Graphics Gems,” Academic Press, edited by A. Glassner, pgs. 522-532, herein incorporated by reference in its entirety.

[0233] In step 905, the alignment transformation generated in step 903 is applied to the data representing the geometric center of the stored molecule, which is preferably retrieved from data included in the matching entry of the multi-map, to generate data representing center of the stored molecule in the query frame.

[0234] In step 907, it is determined if the center of the stored molecule in the query frame lies within the volume of the query molecule. The volume of the query molecule may be calculated utilizing various well-known techniques, such as the techniques described in Connelly, M. L., “Computation of Molecular Volume,” JACS, Vol. 107, 1985, pg. 1118-1124, hereinafter incorporated by reference in its entirety.

[0235] If the test of step 907 fails (the center of the stored molecule in the query frame lies outside the volume of the query molecule), in step 909 the construction of the match hypothesis ends without adding an entry to the vote table, and processing continues to step 909

[0236] However, if the comparison of step 907 is successful, in step 911 data representing the rotation of the alignment transformation generated in step 903 and data representing center of the stored molecule in the query frame are preferably quantized to form a integer pair, and a label is generated based upon such data and the identifier of the stored molecule.

[0237] In step 913, if the vote table does not include an entry corresponding to the label generated in step 911, a new entry associated with the corresponding label is added to the vote table. The new entry includes a score field with an initial value (for example, 1), data identifying the given stored scoop, and data identifying the alignment transformation generated in step 903. Otherwise (the vote table does not include an entry corresponding to the label generated in step 911), the score field of the corresponding entry is incremented and possibly data identifying the stored scoop is added to the entry. In addition, the alignment transformation data may be updated, for example, to represent the cumulative average alignment transformation. The operation then continues to step 909.

[0238] In step 909, one or more entries of the vote table is selected, and the alignment transformation corresponding to the selected entry (ies) is applied to the corresponding stored molecule (which is identified by the label associated with the vote table entry) to generate an alignment of the stored molecule in the query frame.

[0239] Finally, in step 915, the alignment of the stored molecule in the query frame is reported to the user via an I/O device and operation of the recognition phase ends.

[0240] It should be noted that the operations described above with respect to FIG. 9 represent a preferred embodiment of the present invention. One skilled in the art will realize that the operations similar to those described above can be used to construct an hypothesized match between a stored scoop and a query scoop based upon any data characterizing a set of axes derived from the property distribution of the stored scoop and query scoop, respectively.

[0241] As described above, the training phase is used to define the association criteria between query scoops and stored scoops, and keys and the corresponding multi-map data structure capture the associations. In an alternate embodiment, any arbitrary selection scheme (for example, one based on a distance metric) can be used to associate a query scoop with a stored scoop, which leads to construction of a match hypothesis between the query scoop and the associated stored scoop.

[0242] The computer processing system 100 that implements the present invention may be distributed in nature as shown in FIG. 1(B). More specifically, a distributed computer processing system comprises more than one CPU 103 (three shown 103-1, 103-2,103-3) with each of these CPUs communicating with one another via message passing utility 113. The message passing utility 113 may be implemented via shared memory, a network connection, a high speed switch or some other method that allows data to be passed from CPU to CPU. A distributed computer processing system is preferably used for the recognition phase of the present invention because of the inherently parallel nature of the algorithm. More specifically, the multi-map data structure generated in the acquisition mode as described below is preferably partitioned amongst the CPUs of the distributed system. The multi-map data structure may also be partitioned amongst the various nonvolatile storage devices 108 associated with a given CPU 103. For example, the multi-map may be partitioned into nine portions MM11, MM12, MM13, MM21, MM22, MM23, MM31, MM32, MM33 among three CPUs 103-1,103-2 and 103-3 and their associated nine storage devices 108-11, 108-12 ,108-13, 108-21, 108-22,108-23, 108-31, 108-32, 108-33 shown. In addition, the vote table data structure may be similarly partitioned amongst the CPUs of the distributed system. When such a system is used in the recognition phase described above, as table entries are retrieved from the multi-map, such table entries are routed via the message passing utility 113 to the appropriate CPU for accumulation in the proper segment of the vote table. In the end, a distributed merge sort is preferably used to collate all of the resulting hypothesized matches on a single CPU.

[0243] As described above, the components of the descriptor vector characterizing a given scoop preferably includes S_x, S_y, S_z, which represent the components of a vector describing translation between the origin of input reference frame and center of mass in the input reference frame. These components may be calculated utilizing well known techniques, including the operations described in column 6 of U.S. Pat. No. 5,784,294 to Platt et al., incorporated by reference above in its entirety.

[0244] In addition, the components of the descriptor vector characterizing a given scoop preferably includes the principal values of the moment of inertia tensor I for the scoop, which are denoted I_x, I_y, I_z. The inertial reference frame for the scoop is preferably characterized by an origin at the center of mass in the input reference frame, and a set of three axes (denoted , , ) in the input reference frame. The axes , , are unit vectors that point in the same direction as corresponding eigenvectors (principal axes) of the diagonalized moment of inertia tensor I of the scoop. The principal values of the moment of inertia tensor I of the scoop (I_x, I_y, I_z) are represented by the eigenvalues of the diagonalized moment of inertia tensor I of the scoop. The eigenvectors (principal axes) and the corresponding eigenvalues of the diagonalized moment of inertia tensor I for the scoop may be calculated utilizing the operations described in columns 6-8 of U.S. Pat. No. 5,784,294 to Platt et al., incorporated by reference above in its entirety.

[0245] Importantly, the three axes , , do not sufficiently denote an internal frame of reference for coordinate transformation because the signs of the corresponding eigenvectors are ambiguous. The signs must be determined from other information.

[0246] Accordingly, the present invention provides a procedure for sensing the axes , , with a third order moment vector, which is denoted the asymmetric vector, in order to denote an internal frame of reference sufficient for coordinate transformation. The asymmetric vector has a general form {right arrow over (C)}=(C_x, C_y, C_z). This general form may characterize a third order moment of mass about a center of expansion, denoted above as {right arrow over (C_i+L )}=(Ci_x, Ci_y, Ci_z). In the alternative, the general form may characterize a third order moment of charge about center of expansion, denoted above as {right arrow over (C_q+L )}=(Cq_x, Cq_y, Cq_z).

[0247] In the preferred embodiment of the present invention, the components of the asymmetric vector {right arrow over (C)}=(C_x, C_y, C_z) are derived as follows:

[0248] {right arrow over (C)}∫ρ({right arrow over (x)})|{right arrow over (x)}|²{right arrow over (x)}d³x

[0249] where

[0250] {right arrow over (x)} represents a vector between the center of expansion and any arbitrary point in the input reference frame, and

[0251] ρ({right arrow over (x)}) represents the relevant distribution (i.e., represents the mass distribution for the asymmetric vector characterizing third order moment of mass, or represents the charge distribution for asymmetric vector characterizing third order moment of charge) of the scoop at points in the input reference frame corresponding to the vector {right arrow over (x)}.

[0252] In systems where the distribution ρ({right arrow over (x)}) of the scoop has discrete values over points in the input reference frame, the components of the asymmetric vector {right arrow over (C)}=(C_x, C_y, C_z) may be derived as follows:

[0253] {right arrow over (C)}=Σρ_i|{right arrow over (x_i+L )}|²{right arrow over (x_i+L )}

[0254] where

[0255] {right arrow over (x_i+L )} represents a vector between the center of expansion in the input reference frame and a grid point i in the input reference frame,

[0256] p_iis the mass/charge property at the grid point i (when generating the asymmetric vector characterizing third order moment of mass, p_iis the mass value μ_ifor the grid point i as described above with respect to step 503; and when generating the asymmetric vector characterizing third order moment of charge, p_iis the charge value χ_ifor the grid point i as described above with respect to step 503); and

[0257] Σ sums over the grid points of the scoop.

[0258] With respect to the asymmetric vector {right arrow over (C_i+L )} characterizing third order moment of mass for the scoop, preferably the center of expansion is the center of mass of the scoop. With respect to the asymmetric vector {right arrow over (Cq)} characterizing third order moment of charge for the scoop, the center of expansion may be the center of charge, center of dipole, or center of quadrupole for the scoop.

[0259] As described above, the components of asymmetric vector {right arrow over (C)} (such as the asymmetric vector {right arrow over (C_i+L )} characterizing third order moment of mass for the scoop, or the asymmetric vector {right arrow over (Cq)} characterizing third order moment of charge for the scoop) are used to sense the axes , , in order to denote an internal frame of reference sufficient for coordinate transformation. Preferably, the axes , , are sensed by looping over a countery ranging from 1 to 2 with an increment of 1. In each iteration of the loop, the following conditional operation is performed: if the dot product of the axis and the asymmetric vector is less than zero, then the sign of is swapped (i.e., =−). In the first iteration of the loop, the countery is 1 and the conditional operation tests whether the dot product of the axis and the asymmetric vector is less than zero. If so, the sign of is swapped. Otherwise, the sign of remains unchanged. In the second iteration of the loop, the counter y is 2 and the conditional operation tests whether the dot product of the axis and the asymmetric is less than zero. If so, the sign of is swapped. Otherwise, the sign of remains unchanged. The loop then terminates, and the third axis is constructed as the cross product of and .

[0260] Importantly, the sensed axes , , denote a reference frame suitable for coordinate transformation. More specifically, this reference frame is characterized by an origin at the center of mass in the input reference frame, and the sensed axes (, , in the input reference frame). This reference frame is preferably used to generate components of the descriptor vector that characterizes a scoop.

[0261] In addition, the components of the descriptor vector characterizing a given scoop includes the components of a transformation matrix representing rotation between th e principal axes of moment of inertia tensor I for the scoop and the axes of input reference frame. Preferably, this transformation matrix, which for the sake of description is denoted R, is derived from the sensed axes (=u_1x{circumflex over (x)}+u_1yŷ+u_1z{circumflex over (z)}), (=u_2x{circumflex over (x)}+u_2yŷ+u_2z{circumflex over (z)}), and (=u_3x{circumflex over (x)}+u_3yŷ+u_3z{circumflex over (z)}) as follows: $R=\begin{array}{ccc}{u}_{1x}& u_{1y}& u_{1z}\\ u_{2x}& u_{2y}& u_{2z}\\ \mathrm{u3x}& u_{3y}& u_{3z}\end{array}$

[0262] Note that this representation has nine terms. Preferably, the transformation matrix R may be represented by three angles: S₇₄, S_Φ, S_Ω, S_Φ, S_θ are polar angles that describe an axis of rotation. S_Ω represents an angle of rotation about that angle. Together, the angles represent the rotation transformation represented by the transformation matrix R. The rotation matrix R and the angles S_θ, S_Φ, S_Ω can be calculated using the technique described in M. Pique, “Graphics Gems,” Academic Press, edited by A. Glassner, pgs. 465-467, herein incorporated by reference in its entirety. Note that this representation of the rotation transformation using the angles S_θ, S_Φ, S_Ω only has three terms, which is advantageous because it lowers the storage allocation requirements for the components of the descriptor vector of the scoop that represent the transformation.

[0263] In addition, the components of the descriptor vector characterizing a given scoop preferably includes the components of a vector describing translation between center of scoop in the inertial reference frame and center of mass in the inertial reference frame, which are denoted v_x, v_y, v_z. Preferably, the components v_x, v_y, and v_zare derived by generating a vector representing translation between center of scoop in the input reference frame and center of mass in the input reference frame, and then applying the rotation transformation matrix R described above to transform the vector from the input reference frame to the inertial reference frame represented by the sensed axes , , .

[0264] In addition, the components of the descriptor vector characterizing a given scoop preferably includes the components of a vector describing translation between center of mass in the inertial reference frame and center of dipole in the inertial reference frame, which are denoted d_x, d_y, d_z. Preferably, the components d_x, d_y, and d_zare derived by generating a vector representing translation between center of mass in the input reference frame and center of dipole mass in the input reference frame, and then applying the rotation transformation matrix R described above to transform the vector from the input reference frame to the inertial reference frame represented by the sensed axes , , . The center of dipole in the inertial reference frame may calculated utilizing the operations described in columns 10-11 of U.S. Pat. No. 5,784,294 to Platt et. al., incorporated by reference above in its entirety.

[0265] In addition, the components of the descriptor vector characterizing a given scoop preferably includes one or more components of a tensor characterizing the quadrupolar moment about a center of expansion, which are denoted Q_xx, Q_xy, Q_xz, Q_yy, Q_yz, Q_zz. Preferably, the components Q_xx, Q_xy, Q_xz, Q_yy, Q_yz, Q_zzcharacterize contribution of such quadrupolar moment along the sensed axes , , . In this case, components of a tensor Q characterizing quadrupolar moment about a center of expansion are generated. The components of the tensor Q may be calculated utilizing the operations described in columns 10-11 of U.S. Pat. No. 5,784,294 to Platt et. al., incorporated by reference above in its entirety. Finally, the following operations are performed with respect to the tensor Q to calculate contribution of the tensor Q along the sensed axes , , :

[0266] Q_xx=•Q•

[0267] Q_xy=•Q•

[0268] Q_xz=•Q•

[0269] Q_yy=•Q•

[0270] Q_yz=•Q•

[0271] Q_zz=•Q•

[0272] While the invention has been described in connection with specific embodiments, it will be understood that those with skill in the art may develop variations of the disclosed embodiments without departing from the spirit and scope of the following claims.