# US 2017/0286593 A1 · Computationally efficient correlation of genetic effects with function-valued traits

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- Number: US 2017/0286593 A1
- Document ID: US20170286593A1
- Status: Published application
- Kind: A1
- Application number: 15/084,951
- Filed: 2016-03-30
- Published: 2017-10-05
- Earliest claimed: 2016-03-30
- Assignees: Microsoft Technology Licensing, LLC
- Applicants: Microsoft Technology Licensing, LLC
- Inventors: Nicolo Fusi; Jennifer Listgarten
- Web page: https://patentgrep.com/patent/US20170286593A1

## CPC

- G16B 20/20: Allele or variant detection, e.g. single nucleotide polymorphism [SNP] detection
- G16B 20/00: ICT specially adapted for functional genomics or proteomics, e.g. genotype-phenotype associations
- 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
- G16B 20/40: Population genetics; Linkage disequilibrium
- G16B: BIOINFORMATICS, i.e. INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR GENETIC OR PROTEIN-RELATED DATA PROCESSING IN COMPUTATIONAL MOLECULAR BIOLOGY

## Abstract

This disclosure presents a model for identifying correlations in genome-wide association studies (GWAS) with function-valued traits that provides increased power and computational efficiency by use of a Gaussian process regression with radial basis function (RBF) kernels to model the function-valued traits and specialized factorizations to achieve speed. A Gaussian Process is assigned to each partition for each allele of a given single nucleotide polymorphism (SNP) which yields flexible alternative models and handles a large number of data points in a way that is statistically and computationally efficient. This model provides techniques for handling missing and unaligned function values such as would occur when not all individuals are measured at the same time points. If the data is complete algebraic re-factorization by decomposition into Kronecker products reduces the time complexity of this model thereby increasing processing speed and reducing memory usage as compared to a naive implementation.

## Claims

### Claim 1

A computationally-efficient method for identifying a correlation between a single-nucleotide polymorphism (SNP) and a function-valued trait in a genome-wide association study (GWAS), the method comprising:
for a plurality of subjects, populating a trait matrix with values of the function-valued trait, a domain matrix with values of a plurality of function domain points, and a genotype matrix with allele values of the SNP;
linking the trait matrix, the domain matrix, and the genotype matrix;
generating a null model comprising a Gaussian distribution in the trait matrix with mean zero and a first covariance term, the first covariance term being determined in part by a first radial basis function (RBF) kernel applied to the domain matrix and in part by a first length-scale parameter;
representing the first covariance term as a first Kronecker product;
calculating a first maximum likelihood for the null model;
partitioning the matrix into a first partition, a second partition, and a third partition based at least in part on alleles of the SNP;
generating an alternative model that sums together Gaussian distributions of the first partition, the second partition, and the third partition of the matrix, each with mean value and a second, third, and fourth covariance term determined in part by a second, third, and fourth RBF kernel applied to first data from the domain matrix in the first partition, second data from the domain matrix in the second partition, and third data from the domain matrix in the third partition of the matrix and a second set of one or more length-scale parameters;
representing the second, third and fourth covariance terms as a second, third, and fourth Kronecker product;
calculating a second maximum likelihood for the alternative model;
calculating a difference between the first maximum likelihood for the null model and the second maximum likelihood for the alternative model;
calculating a test statistic based at least in part on the difference; and
identifying existence of the correlation between the SNP and the function-valued trait based at least in part on the test statistic.

### Claim 2 (depends on 1)

The method of claim 1, wherein the second set of one or more length-scale parameters comprises a single length-scale parameter shared across the first partition, the second partition, and the third partition.

### Claim 3 (depends on 1)

The method of claim 1, wherein the second set of length-scale parameters comprises a first length-scale parameter associated with the first partition, a second length-scale parameter associated with the second partition, and a third length-scale parameter associated with the third partition.

### Claim 4 (depends on 1)

The method of claim 1, wherein the first covariance term of the Gaussian distribution includes an NT×NT identity matrix, wherein N is a number of the plurality of subjects and T is a number of the plurality of function domain points.

### Claim 5 (depends on 1)

The method of claim 1, wherein the first covariance term is defined in part by a first covariance matrix, the second covariance term is defined in part by a second covariance matrix, the third covariance term is defined in part by a third covariance matrix, and the fourth covariance term is defined in part by a fourth covariance matrix, all of which are at least partially defined by similarities across subjects based in part on population structures, on family structures, or on other forms of similarity between the subjects.

### Claim 6

A computing device for identifying correlations in genome-wide association studies (GWAS), the computing device comprising:
a processing unit;
a memory;
a network connector;
input/output components;
a null model of data encoding absence of a correlation between a single-nucleotide polymorphism (SNP) and function-valued trait, the null model comprising a Gaussian distribution with a first covariance term determined in part by a non-linear kernel having a kernel-associated parameter;
an alternative model of the data encoding the correlation between the SNP and the function-valued trait, the alternative model comprising summation of separate Gaussian distributions each representing an effect of a single allele of the SNP or an effect of interaction between the SNP and at least one other SNP on the function-valued trait;
a model modification module, stored in the memory and implemented by the processing unit, configured to modify the null model and the alternative model by addition of pseudo-observations that make observations within the data conditionally independent; and
a model comparison module, stored in the memory and implemented by the processing unit, configured to:
find, by way of an optimization procedure, a first likelihood of the data fitting the null model;
find, by way of the optimization procedure, a second likelihood of the data fitting the alternative model;
determine that the second likelihood of the data fitting the alternative model is higher than the first likelihood of the data fitting the null model; and
send an indication to at least one of the network connector or the input/output components that the function-valued trait is correlated with the SNP.

### Claim 7 (depends on 6)

The computing device of claim 6, wherein the null model and the alternative model enforce smooth variation of the function-valued trait across a function domain.

### Claim 8 (depends on 7)

The computing device of claim 7, wherein the kernel-associated parameter determines an overall scale on which the function-valued trait varies, where the scale is defined with respect to the function domain.

### Claim 9 (depends on 6)

The computing device of claim 6, wherein the non-linear kernel comprises a radial basis function (RFB) kernel and the kernel-associated parameter comprises a length-scale parameter.

### Claim 10 (depends on 6)

The computing device of claim 6, wherein the optimization procedure maximizes the lower bound of the null likelihood and the lower bound of the alternative likelihood, by way of variational inference.

### Claim 11 (depends on 6)

The computing device of claim 6, wherein the model comparison module is further configured to:
specify a number of the pseudo-observations and not specify locations of the pseudo-observations; and
learn the position of pseudo-observations as part of the optimization procedure.

### Claim 12 (depends on 6)

The computing device of claim 6, further comprising obtaining a value of the kernel-associated parameter by a quasi-Newton method of optimization.

### Claim 13 (depends on 6)

The computing device of claim 6, wherein the pseudo-observations are shared across the multiple separate models in the alternative model.

### Claim 14 (depends on 6)

The computing device of claim 6, wherein the data is provided to the Gaussian distribution as a vector derived from unrolling a matrix containing data for the traits of the subjects at function domain points.

### Claim 15

A computationally-efficient method for identifying a correlation between a single-nucleotide polymorphism (SNP) and a function-valued trait in genome-wide association study (GWAS), the method comprising:
creating a linked matrix having values of the function-valued trait, values of function domain points, and phenotype values for each allele combination of the SNP;
generating an alternative model having a single Gaussian process of a vector derived from the linked matrix and a covariance term, the covariance term based upon at least one scalar and a non-linear kernel;
generating a null model by setting a value of the at least one scalar to zero;
comparing a first maximum likelihood for the alternative model with a second maximum likelihood for the null model; and
determining that the SNP is correlated with the function-valued trait.

### Claim 16 (depends on 15)

The method of claim 15, wherein the non-linear kernel is a radial basis function (RBF) kernel or a Matern kernel.

### Claim 17 (depends on 15)

The method of claim 15, wherein the at least one scalar comprises a correlation parameter for each pair of alleles of the SNP indicating a degree of trait similarity between the data with each allele in the pair.

### Claim 18 (depends on 15)

The method of claim 15, wherein the alternative model comprises a parameter that determines an overall scale on which the function-valued trait varies as a function varies.

### Claim 19 (depends on 15)

The method of claim 15, further comprising:
determining that there are no missing data in the values of the function-valued trait and that the function domain points for each of the plurality of subjects align; and
decomposing at least part of the covariance term to a Kronecker product thereby reducing computational time and memory needed to calculate the first maximum likelihood for the alternative model.

### Claim 20 (depends on 15)

The method of claim 15, further comprising:
marginalizing over the missing values of the function-valued trait;
adding pseudo-observations as inputs to the non-linear kernel thereby making the values of the function-valued trait conditionally independent and reducing a time complexity for maximizing a lower bound of the first likelihood and a lower bound of the second likelihood, wherein variational inference maximizes a lower bound of the first likelihood and maximizes a lower bound of the second likelihood.