Method and system for analyzing multi-variate data using canonical decomposition

• US 20050015205A1
• Filed: 07/11/2001
• Published: 01/20/2005
• Est. Priority Date: 07/12/2000
• Status: Active Grant

• Current Assignee: Cornell Research Foundation Incorporated (Cornell University)
• Original Assignee: Cornell Research Foundation Incorporated (Cornell University)
• Inventors: Schiff, Nicholas, Victor, Jonathan, Repucci, Michael
• Granted Patent: US 7,171,339 B2
• Time in Patent Office: Days
• US Class Current: 702/10
• CPC Class Codes: G06F 18/213 Feature extraction, e.g. by transforming the feature space; Summarisation; Mappings, e.g. subspace methods
• CPC Class Codes: G06F 2218/08 Feature extraction
• CPC Class Codes: G06Q 90/00 Systems or methods specially adapted for administrative, commercial, financial, managerial or supervisory purposes, not involving significant data processing

Abstract

A canonical decomposition (CD) method that includes building a multi-variate linear autoregressive (“MLAR”) model from an original data set or from a reduced set derived by data reduction methods from the original data set. The MLAR analysis is followed by seeking a coordinate transformation of the MLAR model to obtain the best possible match with one or more canonical forms representing relationships among components. For multi-variate data with a truly hierarchical structure, CD accurately extracts the underlying sources of the system.

Claims

1. A method of determining dynamic characteristics of data, the data being causal and having translation-invariant statistics with respect to a variable, which method comprises the steps of:

receiving signals representing the data, the signals originating from a set of components, the set of components having dynamic relationships with each other;

calculating a set of multi-linear autoregressive coefficients of the data, the multi-linear autoregressive coefficients yielding an array of square matrices, each square matrix reflecting the dynamic relationships among the components for a certain value of the variable;

transforming each square matrix to a substantially canonical form corresponding to a selected canonical form suitable for analyzing the dynamic relationships, yielding a set of canonical form matrices representing a transformation of the components; and

is analyzing the selected canonical form and transformed components to determine the dynamic characteristics of the data with respect to the variable.

2. A method according to claim 1, wherein the dynamic relationships are hierarchical.

3. A method according to claim 1, wherein the dynamic relationships are dependent.

4. A method according to claim 1, wherein the dynamic relationships are independent.

5. A method according to claim 1, wherein the dynamic relationships are partially dependent and partially independent.

6. A method according to claim 1, wherein the receiving step further comprises applying a data reduction method to the data.

7. A method according to claim 6, wherein the data reduction method comprises Principal Component Analysis.

8. A method according to claim 1, wherein the variable is time.

9. A system for determining dynamic characteristics of data, the data being causal and having translation-invariant statistics with respect to a variable, the system comprising:

means for receiving signals representing the data, the data originating from a set of components, the set of components having dynamic relationships with each other;

means for calculating a set of multi-linear autoregressive coefficients of the data, the multi-linear autoregressive coefficients yielding an array of square matrices, each square matrix reflecting the dynamic relationships among the components for a certain value of the variable;

means for transforming the square matrices to a substantially canonical form corresponding to a selected canonical form suitable for analyzing the dynamic relationships, yielding a set of canonical form matrices representing a transformation of the components; and

means for analyzing the canonical form matrices and transformed components to determine the dynamic characteristics of the data with respect to the variable.

10. A system according to claim 9, wherein the dynamic relationships are hierarchical.

11. A system according to claim 9, wherein the dynamic relationships are dependent.

12. A system according to claim 9, wherein the dynamic relationships are independent.

13. A system according to claim 9, wherein the dynamic relationships are partially dependent and partially independent.

14. A method for determining an orthogonal rotation for analyzing data, the data being causal and having translation-invariant statistics with respect to a variable, the data originating from a set of components, the set of components having dynamic relationships to each other, which method comprises the steps of:

creating a multi-variate linear autoregressive model of the data, the model describing influences of one component on another component with respect to the variable;

decorrelating random terms in the model so that the decorrelated random terms are orthogonalized to produce a new model with autoregression coefficients, the random terms driving channels of the components; and

identifying an orthogonal rotation that preserves the influences from the creating step and the orthogonality of the random terms from the decorrelating step, and that transforms the new model autoregression coefficients into a canonical form.

15. A method according to claim 14, wherein the creating step further comprises performing data reduction on the data.

16. A method according to claim 15, wherein the data reduction performed is Principal Component Analysis decomposition.

17. A method according to claim 14, wherein the components are temporal, T_p,n, and are modeled according to:

$T_{p,n}=R_{p,n}+{\mathrm{\mu <br><br>}}_p+\underset{q=1}{\overset{P}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\underset{l=1}{\overset{L}{\sum <br><br>}}<br><br>A_{q,p,l}<br><br>\text{\hspace{1em}<br><br>}<br><br>T_{q,n-l}$

whereby, P represents principal components;

L represents a predetermined number of lags;

N represents a total number of samples of the data observed at evenly spaced intervals;

R_p,n is a matrix of residual values of dimension P×

N;

μ

_p is a mean value of a p^th component;

A is a three dimensional array of model coefficients of dimension P×

P×

L, and n is a particular sample or interval of time.

18. A method according to claim 14, wherein a set of residuals R_MLAR is calculated according to:

$R_{\mathrm{MLAR}}=\underset{p=1}{\overset{P}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>R_{p,n}^2$

whereby, N represents a total number of samples of the data observed at evenly spaced intervals; and

P represents principal components.

19. A method according to claim 14, wherein the canonical form has a main diagonal populated with zero and nonzero elements, the nonzero elements being located in positions only along the main diagonal and anywhere at locations right of the main diagonal.

20. A method according to claim 14, further comprising predicting future dynamic characteristics of the data using the canonical form.

21. A computer system for analyzing signals originating from a plurality of channels dependent on a variable, the signals being causal and having translation-invariant statistics with respect to the variable, comprising:

an input for receiving the signals from a plurality of components, the signals representing multi-channel data;

a memory device; and

a processor for creating a multi-variate linear autoregressive model of the components, the model representing relationships among the components, the model being stored in the memory device, the processor transforms the model so that the transformed model decorrelates and orthogonalizes random terms within the model, the transformed model having a set of autoregressive coefficients, the processor identifying an orthogonal rotation that preserves the relationships among the components in the model and the orthogonalized random terms in the transformed model, and the processor converting the autoregressive coefficients of the transformed model into a canonical form.

23. A computer system according to claims 21 or 22, wherein the system is used to analyze neural activity.

24. A computer system according to claims 21 or 22, wherein the system is used to analyze polygraphic biomedical time series.

25. A computer system according to claims 21 or 22, wherein the system is used to analyze electrocardiographic records.

26. A computer system according to claims 21 or 22, wherein the system is used to analyze at least one of gene expression and protein expression level time series.

27. A computer system according to claims 21 or 22, wherein the system is used to analyze geophysical data.

28. A computer system according to claims 21 or 22, wherein the system is used for financial modeling and forecasting.

29. A computer system according to claims 21 or 22, wherein the system is used to analyze inventory levels.

30. A computer system according to claims 21 or 22, wherein the system is used to analyze routing data.

31. A computer system according to claims 21 or 22, wherein the system is used to analyze network data transmission.

32. A computer system according to claims 21 or 22, wherein the system is used to analyze a quantity and a type of procedural inputs and procedural outputs.

33. A computer system according to claims 21 or 22, wherein the system is used to analyze user interfaces.

34. A computer system according to claims 21 or 22, wherein the system is used to analyze outputs of an electrical device.

35. A computer system according to claims 21 or 22, wherein the system is used to analyze telemetry signals to determine the behavior of a tracking system.

36. A computer system according to claims 21 or 22, wherein the system is used to analyze data patterns so as to improve information dispersal.

37. A computer system according to claim 36, wherein information dispersal includes marketing materials.

38. A computer system according to claim 36, wherein information dispersal includes surveys.

22. A computer system for predicting future dynamic characteristics of data that is causal and has translation-invariant statistics with respect to a variable, comprising:

an input for receiving signals from a plurality of components, the signals representing the data;

a memory device; and

a processor for calculating a plurality of multi-linear autoregressive coefficients of the data to yield an array of square matrices, each square matrix reflecting relationships among the components for a certain value of the variable, the processor selecting a canonical form suitable for analyzing relationships among the components, and the processor transforming the square matrices to a substantially canonical form corresponding to the selected canonical form, yielding a plurality of canonical form matrices representing a transformation of the components.

39. A method of identifying canonical forms of equally-sized square matrices under orthogonal transformation, comprising the step of:

identifying using orthogonal rotation a canonical form within rotations of the matrices.

40. A method according to claim 39, wherein the canonical form is a triangular canonical form.

41. A method according to claim 39, wherein the canonical form has nonzero elements along a main diagonal and in locations just right of the main diagonal.

42. A method according to claim 39, wherein the canonical form has nonzero elements along a main diagonal and nonzero elements just right of the main diagonal, and one of the nonzero elements just right of the main diagonal is wrapped around to a lower left corner.

43. A method according to claim 39, wherein the canonical form has nonzero elements along a main diagonal and has nonzero elements just right and just left of the main diagonal, and one of the elements right of the main diagonal is wrapped around to a lower left corner and one of the elements left of the main diagonal is wrapped around to an upper right corner.

44. A method according to claim 39, wherein the canonical form is a diagonal canonical form.

45. A method according to claim 39, wherein the canonical form is a block triangular canonical form including at least one of triangular canonical form, restricted triangular canonical form, unidirectional cyclic canonical form, bidirectional cyclic canonical form, and diagonal canonical form.