# US 7,610,169 B2 · Regression analysis apparatus and method

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- Number: US 7,610,169 B2
- Document ID: US7610169B2
- Status: Granted
- Kind: B2
- Application number: 11/757,545
- Filed: 2007-06-04
- Published: 2007-12-13
- Granted: 2009-10-27
- Earliest claimed: 2006-06-09
- Assignees: Canon Kabushiki Kaisha
- Inventors: Teruyoshi Washizawa
- Web page: https://patentgrep.com/patent/US7610169B2

## CPC

- G06F 17/18: for evaluating statistical data {, e.g. average values, frequency distributions, probability functions, regression analysis (forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04)}
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

A variance-covariance matrix of a matrix having a combination of multivariate data and objective variables is obtained, and multiple eigenvalues and their corresponding eigenvectors are calculated by eigenvalue decomposition of the variance-covariance matrix. Accumulated contributions are calculated from the multiple eigenvalues in descending order of absolute value of the eigenvalues. Regression coefficients are calculated from eigenvalues and eigenvectors that correspond to accumulated contributions that exceed a predetermined value.

## Claims

### Claim 1

A regression analysis method for obtaining an objective variable from multivariate data by performing, with a computer program stored in a computer readable medium, for controlling a computer, multiple linear regression analysis to obtain regression coefficient, comprising:
a first input step of inputting a combination of the objective variable and the multivariate data;
an obtaining step of obtaining a variance-covariance matrix having the combination of the multivariate data and the objective variable;
a vector calculation step of calculating a plurality of eigenvalues and corresponding eigenvectors by eigenvalue decomposition of the variance-covariance matrix;
an accumulated contributions calculation step of calculating accumulated contributions from the plurality of eigenvalues in descending order of absolute value of the plurality of eigenvalues;
a regression coefficients calculation step of calculating regression coefficients from the eigenvalues and eigenvectors corresponding to the accumulated contributions that exceed a predetermined value;
a second input step of inputting multivariate data that is not used in said regression coefficients calculation step;
a calculation step of calculating, by the computer, the objective variable with respect to the multivariate data input in said second input step by using the regression coefficients calculated in said regression coefficients calculation step; and
an output step of outputting the objective variable calculated in said calculation step.

### Claim 2 (depends on 1)

The regression analysis method according to claim 1, wherein ordinal numbers j are assigned to the eigenvalues and eigenvectors corresponding to the accumulated contributions that exceed the predetermined value in descending order of absolute value of the eigenvalues and the (j+1)-th eigenvector is ζ1(j+1),
the regression coefficients (θ1 and θ2) are calculated in said regression coefficients calculation step by using equations (A) and (B)
θ1=(ζ1(2)/ζ1(1), ζ1(3)/ζ1(1), . . . , ζ1(d+1)/ζ1(1))T (A)
and
θ2=<y>−θ1T<x> (B)
where <x> is the average of samples of x and <y> is the average of samples of y.

### Claim 3 (depends on 2)

The regression analysis method according to claim 2, further comprising a step of comparing the ζ1(1) with a preset positive real number and, if the ζ1(1) is smaller than the positive real number, determining that the regression coefficient cannot be calculated.

### Claim 4 (depends on 1)

The regression analysis method according to claim 1, wherein the regression coefficients have a relation with a set x={x1, x2, . . . , xN} of d-dimensional variable vectors xj=(xj(1), xj(2), . . . , xj(d))T and a set Y={y1, y2, . . . , yN} of one-dimensional variables yj such that y =θ1Tx+θ2.

### Claim 5 (depends on 1)

The regression analysis method according to claim 1, wherein said accumulated contributions calculation step calculates the accumulated contributions from the eigenvalues {λ1, λ2, . . . , λ(d+1)} and the eigenvectors {q1, q2, . . . qd+1} by using the following equation (C)
η
⁢
⁢
k
=
(
∑
j
=
1
j
=
k
⁢
λ
⁢
⁢
j
)
/
(
∑
j
=
1
j
=
d
+
1
⁢
λ
⁢
⁢
j
)
(
C
)
wherein ηk is an accumulated contribution corresponding to each eigenvalue λj.

### Claim 6 (depends on 5)

The regression analysis method according to claim 5, wherein said regression coefficients calculation step obtains k that causes the accumulated contribution to exceed the predetermined value and substitutes the k for ζ1(j) in the equations (A) and (B) to calculate the regression coefficients.

### Claim 7

A computer program stored in a computer readable medium, for controlling a computer to determine a regression coefficient by multiple linear regression analysis for obtaining an objective variable from multivariate data, the program comprising code for implementing the following steps:
a first input step of inputting a combination of the objective variable and the multivariate data;
an obtaining step of obtaining a variance-covariance matrix having a combination of the multivariate data and the objective variable;
a vector calculation step of calculating a plurality of eigenvalues and corresponding eigenvectors by eigenvalue decomposition of the variance-covariance matrix;
an accumulated contributions calculation step of calculating accumulated contributions from the plurality of eigenvalues in descending order of absolute value of the plurality of eigenvalues;
a regression coefficients calculation step of calculating regression coefficients from the eigenvalues and eigenvectors corresponding to the accumulated contributions that exceed a predetermined value;
a second input step of inputting multivariate data that is not used in said regression coefficients calculation step;
a calculation step of calculating the objective variable with respect to the multivariate data input in said second input step by using the regression coefficients calculated in said regression coefficients calculation step; and
an output step of outputting the objective variable calculated in said calculation step.

### Claim 8

A regression analysis method for obtaining a height of a person as an objective variable from multivariate data including weights of a plurality of persons by performing, with a computer program stored in a computer readable medium, for controlling a computer, multiple linear regression analysis to obtain a regression coefficient, comprising:
a first input step of inputting a combination of the objective variable and the multivariate data;
an obtaining step of obtaining a variance-covariance matrix having the combination of the multivariate data and the objective variable;
a vector calculation step of calculating a plurality of eigenvalues and corresponding eigenvectors by eigenvalue decomposition of the variance-covariance matrix;
an accumulated contributions calculation step of calculating accumulated contributions from the plurality of eigenvalues in descending order of absolute value of the plurality of eigenvalues;
a regression coefficients calculation step of calculating regression coefficients from the eigenvalues and eigenvectors corresponding to the accumulated contributions that exceed a predetermined value;
a second input step of inputting a weight of a person;
a calculation step of calculating, by the computer, a height of the person whose weight was inputted in the second input step by using the regression coefficients calculated in said regression coefficients calculation step; and
an output step of outputting the height calculated in said calculation step.