Method and apparatus for identifying diagnostic components of a system

• US 20050171923A1
• Filed: 10/17/2002
• Published: 08/04/2005
• Est. Priority Date: 10/17/2001
• Status: Abandoned Application

• Current Assignee: Commonwealth Scientific & Industrial Research Organisation (Commonwealth Of Australia As Represented By Department Of Industry, Innovation, Climate Change, Science, Research And Tertiary Education)
• Original Assignee: Commonwealth Scientific & Industrial Research Organisation (Commonwealth Of Australia As Represented By Department Of Industry, Innovation, Climate Change, Science, Research And Tertiary Education)
• Inventors: Trajstman, Albert, Kiiveri, Harri, Thomas, Mervyn
• Application Number:
  US10/492,886
• Publication Number:
  US 20050171923A1
• Time in Patent Office: Days
• US Class Current: 1/1
• CPC Class Codes: 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
• CPC Class Codes: G16B 40/00 ICT specially adapted for biostatistics; ICT specially adapted for bioinformatics-related machine learning or data mining, e.g. knowledge discovery or pattern finding
• CPC Class Codes: G16B 40/20 Supervised data analysis

Abstract

Method and apparatus is described for identifying a subset of components of a system, the subset being capable of predicting a feature of a test sample. The method comprises generating a linear combination of components and component weights in which values for each component are determined from data generated from a plurality of training samples, each training sample having a known feature. A model is defined for the probability distribution of a feature wherein the model is conditional on the linear combination and wherein the model is not a combination of a binomial distribution for a two class response with a probit function linking the linear combination and the expectation of the response. A prior distribution is constructed for the component weights of the linear combination comprising a hyperprior having a high probability density close to zero, and the prior distribution and the model are combined to generate a posterior distribution. A subset of components is identified having component weights that maximise the posterior distribution.

Claims

1. A method for identifying a subset of components of a system, the subset being capable of predicting a feature of a test sample, the method comprising the steps of;

(a) generating a linear combination of components and component weights in which values for each component are introduced from data generated from a plurality of training samples, each training sample having a known feature;

(b) defining a model for the probability distribution of a feature wherein the model is conditional on the linear combination and wherein the model is not a combination of a binomial distribution for a two class response with a probit function linking the linear combination and the expectation of the response;

(c) constructing a prior distribution for the component weights of the linear combination comprising a hyperprior having a high probability density close to zero;

(d) combining the prior distribution and the model to generate a posterior distribution;

(e) identifying a subset of components having component weights that maximise the posterior distribution.

2. The method of claim 1 wherein the model is a likelihood function based on a model selected from the group comprising a multinomial or binomial logistic regression, generalised linear model, Cox'"'"'s proportional hazards model and parametric survival model.

3. The method of claim 1 wherein the model is a likelihood function based on a multinomial or binomial logistic regression.

4. The method of claim 2 wherein the logistic regression models a feature having a multinomial or binomial distribution.

5. The method of claim 1 wherein the subset of components is capable of classifying a sample into one of a plurality of pre-defined groups by defining a logistic regression which comprises grouping the samples into a plurality of sample groups, each sample group having a common group identifier.

6. The method of claim 1 wherein the logistic regression is of the form:

$L=\underset{i=1}{\overset{n}{\prod <br><br>}}<br><br>\left(\underset{g=1}{\overset{G-1}{\prod <br><br>}}<br><br>{\left\{\frac{{\mathrm{e<br><br>}}^{x_i^T<br><br>{\mathrm{\beta <br><br>}}_g}}{\left(1+\underset{g=1}{\overset{G-1}{\sum <br><br>}}<br><br>{\mathrm{e<br><br>}}^{x_i^T<br><br>{\mathrm{\beta <br><br>}}_g}\right)}\right\}}^{e_{\mathrm{ig}}}<br><br>{\left\{\frac{1}{1+\underset{h=1}{\overset{G-1}{\sum <br><br>}}<br><br>{\mathrm{e<br><br>}}^{x_i^T<br><br>{\mathrm{\beta <br><br>}}_h}}\right\}}^{e_{\mathrm{iG}}}\right)$

wherein

x_i^Tβ

_g is a linear combination generated from input data from training sample i with component weights β

_g;

x_{hu T} or is the components for the i^th Row of X and β

_g is a set of component weights for sample class g;

e_ig=1 if training sample i is a member of class g, e_ig=0 otherwise; and

X is data from n training samples comprising p components.

7. The method of claim 1 wherein the subset of components is capable of classifying a sample into a class wherein the class is one of a plurality of predefined ordered classes, by defining a logistic regression which comprises defining a series of group identifiers in which each group identifier corresponds to a member of an ordered class, and grouping the samples into one of the ordered classes.

8. The method of claim 7 wherein the logistic regression is of the form:

$L=\underset{i=1}{\overset{N}{\prod <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\underset{k=1}{\overset{G-1}{\prod <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{\left(\frac{{\mathrm{\gamma <br><br>}}_{\mathrm{ik}}}{{\mathrm{\gamma <br><br>}}_{\mathrm{ik}+1}}\right)}^{r_{\mathrm{ik}}}<br><br>{\left(\frac{{\mathrm{\gamma <br><br>}}_{\mathrm{ik}+1}-{\mathrm{\gamma <br><br>}}_{\mathrm{ik}}}{{\mathrm{\gamma <br><br>}}_{\mathrm{ik}+1}}\right)}^{r_{\mathrm{ik}+1}-r_{\mathrm{ik}}}$ $\log <br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{it}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\frac{{\mathrm{\gamma <br><br>}}_{\mathrm{ik}+1}-{\mathrm{\gamma <br><br>}}_{\mathrm{ik}}}{{\mathrm{\gamma <br><br>}}_{\mathrm{ik}+1}}\right)=\log <br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{it}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\frac{{\mathrm{\pi <br><br>}}_{\mathrm{ik}}}{{\mathrm{\gamma <br><br>}}_{\mathrm{ik}+1}}\right)={\mathrm{\theta <br><br>}}_k+x_i^T<br><br>{\mathrm{\beta <br><br>}}^{*}$

wherein

γ

_ik is the probability that training sample i belongs to a class with identifier less than or equal to k (where the total of ordered classes is G);

x_i^Tβ

* is a linear combination generated from input data from training sample i with component weights β

*;

X is data from n training samples comprising p components;

x_i^T is the components for the i^th Row of X;

r_ij is as defined as

$r_{\mathrm{ij}}=\underset{g=1}{\overset{j}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_{\mathrm{ig}}$$\mathrm{where}$$c_{\mathrm{ij}}=\{\begin{array}{c}1,\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{observation}<br><br>\text{\hspace{1em}<br><br>}<br><br>i<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{in}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{class}<br><br>\text{\hspace{1em}<br><br>}<br><br>j\\ 0,\mathrm{otherwise}\end{array}$

9. The method of claim 1 wherein the model is a likelihood function is based on a generalised linear model.

10. The method of claim 9 wherein the generalised linear model models a feature that is distributed as a regular exponential family of distributions.

11. The method of claim 10 wherein the regular exponential family of distributions is selected from the group consisting of normal distribution, Gaussian distribution, Poisson distribution, exponential distribution, gamma distribution, Chi Square distribution and inverse gamma distribution.

12. The method of claim 1 or wherein the subset of components is capable of predicting a predefined characteristic of a sample by defining a generalised linear model which comprises modelling the characteristic to be predicted.

13. The method of claims 9 wherein the generalised linear model is of the form:

$\log <br><br>\text{\hspace{1em}<br><br>}<br><br>p<br><br>\left(y|\mathrm{\beta <br><br>},\mathrm{\psi <br><br>}\right)=\underset{i=1}{\overset{N}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left\{\frac{y_i<br><br>{\mathrm{\theta <br><br>}}_i-b<br><br>\left({\mathrm{\theta <br><br>}}_i\right)}{a_i<br><br>\left(\mathrm{\psi <br><br>}\right)}+c<br><br>\left(y_i,\mathrm{\psi <br><br>}\right)\right\}$

wherein

y=(y₁, . . . , y_n)^T, and y_i is the characteristic measured on the i^th sample;

a_i(φ

)=φ

/w_i with the w_i being a fixed set of known weights and φ

a single scale parameter;

the functions b(.) and c(.) are as defined by Nelder and Wedderburn (1972);

E{y_i}=b′

(θ

_i)
Var{y}=b″

(θ

_i)a_i(φ

)=τ

_i²a_i(φ

);

and wherein each observation has a set of covariates x_i and a linear predictor η

_i=x_i^T β

.

14. The method of claim 1 wherein the model is a likelihood function based on a model selected from the group consisting of Cox'"'"'s proportional hazards model, parametric survival model and accelerated survival times model.

15. The method of claim 1 wherein the subset of components is capable of predicting the time to an event for a sample by defining a likelihood based on Cox'"'"'s proportional standards model, a parametric survival model or an accelerated survival times model, which comprises measuring the time elapsed for a plurality of samples from the time the sample is obtained to the time of the event.

16. The method of claim 14 wherein Cox'"'"'s proportional hazards model is of the form:

$L<br><br>\left(\underset{~}{t}|\underset{~}{\mathrm{\beta <br><br>}}\right)=\underset{j=1}{\overset{N}{\prod <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{\left(\frac{\exp <br><br>\left(Z_j<br><br>\underset{~}{\mathrm{\beta <br><br>}}\right)}{\underset{{\mathrm{i\epsilon <br><br>\Re <br><br>}}_j}{\sum <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\exp <br><br>\left(Z_i<br><br>\underset{~}{\mathrm{\beta <br><br>}}\right)}\right)}^{d_j}$

wherein

X is data from n training samples comprising p components;

Z is a matrix that, is the re-arrangement of the rows of X where the ordering of the rows of Z corresponds to the ordering induced by the ordering of the survival times;

d is the result of ordering the censoring index with the same permutation required to order survival times;

Z_j is the j^th row of the matrix Z and d_j is the j^th element of d;

{tilde under (β

)}^T=(β

₁,β

₂, . . . ,β

_p);

_j={i;

i=j,j+1, . . . ,N}=the risk set at the j^th ordered event time t_(j).

17. The method of claim 14 wherein the parametric hazards model is of the form:

$\log <br><br>\left(L\right)=\underset{i=1}{\overset{N}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left\{c_i<br><br>\log <br><br>\left({\mathrm{\mu <br><br>}}_i\right)-{\mathrm{\mu <br><br>}}_i+c_i<br><br>\left(\log <br><br>\left(\frac{\mathrm{\lambda <br><br>}<br><br>\left(y_i\right)}{\mathrm{\Lambda <br><br>}<br><br>\left(y_i;<br><br>\underset{~}{\mathrm{\psi <br><br>}}\right)}\right)\right)\right\}$

where

μ

_i=Λ

(y_i;

{tilde under (φ

)})exp(X_i{tilde under (β

)});

c_i=1 if the i^th sample is uncensored and c_i=0 if the i^th sample is uncensored;

the functions λ

(.) and Λ

(.) are as defined by Aitkin and Clayton (1980);

X_i is the i^th row of X and X is data from n training samples comprising p components.

18. The method of claim 1 wherein the prior distribution is of the form:

$p<br><br>\left(\mathrm{\beta <br><br>}\right)={\int <br><br>}_{v^2}^{\text{\hspace{1em}<br><br>}}<br><br>p<br><br>\left(\mathrm{\beta <br><br>}|v^2\right)<br><br>p<br><br>\left(v^2\right)<br><br>\text{\hspace{1em}<br><br>}<br><br>d<br><br>v^2$

where p(β

|v²) is N(0,diag{v²});

v is a hyper parameter;

p(v²) is a hyperprior distribution.

19. The method of claim 1 wherein the hyperprior is a Jeffreys prior of the form:

$p<br><br>\left(v^2\right)<br><br>\mathrm{\alpha <br><br>}<br><br>\underset{i=1}{\overset{n}{\prod <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>1/v^2$

20. The method of claim 1 wherein posterior distribution is of the form:

p(β

φ

v|y)α

L(y|β

φ

)p(β

|v²)p(v²)

wherein L({tilde under (y)}|{tilde under (β

)},{tilde under (φ

)}) is the likelihood function.

21. The method of claim 1 wherein the posterior distribution is maximised using an iterative procedure.

22. The method of claim 21 wherein the iterative procedure is an EM algorithm.

23. The method of claim 1 wherein the system is a biological system.

24. The method of claim 23 wherein the biological system is a biotechnology array.

25. The method of claim 24 wherein the biotechnology array is selected from the group consisting of DNA array, protein array, antibody array, RNA array, carbohydrate array, chemical array, lipid array.

26. A method for identifying a subset of components of a subject which are capable of classifying the subject into one of a plurality of predefined groups wherein each group is defined by a response to a test treatment comprising the steps of:

(d) exposing a plurality of subjects to the test treatment and grouping the subjects into response groups based on responses to the treatment;

(e) measuring components of the subjects;

(f) identifying a subset of components that is capable of classifying the subjects into response groups using the methods according to claim 1.

27. The method of claim 26 wherein the components are selected from the group consisting of genes, small nucleotide polymorphisms (SNPs), proteins, antibodies, carbohydrates, lipids.

29. A computer program arranged, when loaded onto a computing apparatus, to control the computing apparatus to implement a method in accordance with claim 1.

30. The computer program of claim 29 implemented with the method of claim 1.

31. A computer readable medium providing a computer program in accordance with claim 29.

32. A method of testing a sample from a system to identify a feature of the sample, the method comprising the steps of testing for a subset of components which is diagnostic of the feature, the subset of components having been determined by a method in accordance with claim 1.

33. An apparatus for testing a sample from a system to determine a feature of the sample, the apparatus including means for testing for components identified in accordance with the method of claim 1.

28. An apparatus for identifying a subset of components of a system from data generated from the system from a plurality of samples from the system, the subset being capable of predicting a feature of a test sample, the apparatus comprising:

(a) means for generating a linear combination of components and component weights in which values for each component are introduced from data generated from a plurality of training samples, each training sample having a known feature;

(b) means for defining a model for the probability distribution of a feature wherein the model is conditional on the linear combination and wherein the model is not a combination of a binomial distribution for a two class response with a probit function linking the linear combination and the expectation of the response;

(c) means for constructing a prior distribution for the component weights of the linear combination comprising a hyperprior having a high probability density close to zero;

(d) means for combining the prior distribution and the model to generate a posterior distribution;

(e) means for identifying a subset of components having component weights that maximise the posterior distribution.

34. A computer program which when run on a computing device, is arranged to control the computing device, in a method of identifying components from a system which are capable of predicting a feature of a test sample from the system, and wherein a linear combination of components and component weights is generated from data generated from a plurality of training samples, each training sample having a known feature, and a posterior distribution is generated by combining a prior distribution for the component weights comprising a hyperprior having a high probability distribution close to zero, and a model that is conditional on the linear combination wherein the model is not a combination of a binomial distribution for a two class response with a probit function linking the linear combination and the expectation of the response, to estimate component weights which maximise the posterior distribution.

35. A method for identifying a subset of components of a biological system, the subset being capable of predicting a feature of a test sample from the biological system, the method comprising the steps of:

(a) generating a linear combination of components and component weights in which values for each component are determined from data generated from a plurality of training samples, each training sample having a known feature;

(b) defining a model for the probability distribution of a feature wherein the model is conditional on the linear combination;

(c) constructing a prior distribution for the component weights of the linear combination comprising a hyperprior having a high probability density close to zero;

(d) combining the prior distribution and the model to generate a posterior distribution;

(e) identifying a subset of components having component weights that maximise the posterior distribution.