Treelet Covariance SmoothersEstimation of Genetic Parameters

B. Draves1

1Department of MathematicsLafayette College

Advisor: T. Gaugler

Moravian College, 2017

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 1 / 11

Treelet Covariance Smoothers

Overview

Principle Component Analysis

Motivation in Statistical Genetics

Existing Methods

New Methods & Results

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 2 / 11

Treelet Covariance Smoothers

Overview

Principle Component Analysis

Motivation in Statistical Genetics

Existing Methods

New Methods & Results

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 2 / 11

Treelet Covariance Smoothers

Overview

Principle Component Analysis

Motivation in Statistical Genetics

Existing Methods

New Methods & Results

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 2 / 11

Treelet Covariance Smoothers

Overview

Principle Component Analysis

Motivation in Statistical Genetics

Existing Methods

New Methods & Results

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 2 / 11

Treelet Covariance Smoothers

Principle Component Analysis (PCA)

PCA looks to reduce the number of input variables X1,X2, . . . ,Xk (klarge) to k0 < k variables

Reduce to fewer inputs while preserving as much variability in thedata as possible

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 3 / 11

Treelet Covariance Smoothers

Principle Component Analysis (PCA)

PCA looks to reduce the number of input variables X1,X2, . . . ,Xk (klarge) to k0 < k variables

Reduce to fewer inputs while preserving as much variability in thedata as possible

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 3 / 11

Treelet Covariance Smoothers

Principle Component Analysis (PCA)

PCA looks to reduce the number of input variables X1,X2, . . . ,Xk (klarge) to k0 < k variables

Reduce to fewer inputs while preserving as much variability in thedata as possible

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 3 / 11

Treelet Covariance Smoothers

Principle Component Analysis (PCA)

PCA looks to reduce the number of input variables X1,X2, . . . ,Xk (klarge) to k0 < k variables

Reduce to fewer inputs while preserving as much variability in thedata as possible

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 3 / 11

Treelet Covariance Smoothers

Principle Component Analysis (PCA)

PCA looks to reduce the number of input variables X1,X2, . . . ,Xk (klarge) to k0 < k variables

Reduce to fewer inputs while preserving as much variability in thedata as possible

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 3 / 11

Treelet Covariance Smoothers

Genetics Motivation

Individual’s genetic material can be described by a panel of genotypedSNPs

Using this genetic information, an estimate of the relationship matrix,A, can be calculated

Genetically inferred relationship matrices are typically very noisy

Idea: use PCA to set the data on top of a more “natural” basis

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 4 / 11

Treelet Covariance Smoothers

Genetics Motivation

Individual’s genetic material can be described by a panel of genotypedSNPs

Using this genetic information, an estimate of the relationship matrix,A, can be calculated

Genetically inferred relationship matrices are typically very noisy

Idea: use PCA to set the data on top of a more “natural” basis

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 4 / 11

Treelet Covariance Smoothers

Genetics Motivation

Individual’s genetic material can be described by a panel of genotypedSNPs

Using this genetic information, an estimate of the relationship matrix,A, can be calculated

Genetically inferred relationship matrices are typically very noisy

Idea: use PCA to set the data on top of a more “natural” basis

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 4 / 11

Treelet Covariance Smoothers

Genetics Motivation

Individual’s genetic material can be described by a panel of genotypedSNPs

Using this genetic information, an estimate of the relationship matrix,A, can be calculated

Genetically inferred relationship matrices are typically very noisy

Idea: use PCA to set the data on top of a more “natural” basis

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 4 / 11

Treelet Covariance Smoothers

Treelet Covariance Smoothing

Goal: preserve local structure of data

Iteratively change basis via PCA to preserve the variability betweentwo most closely related individuals

Repeat this process until all individuals are processed

Once all individuals are processed, enforce sparsity to improve theestimator

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 5 / 11

Treelet Covariance Smoothers

Treelet Covariance Smoothing

Goal: preserve local structure of data

Iteratively change basis via PCA to preserve the variability betweentwo most closely related individuals

Repeat this process until all individuals are processed

Once all individuals are processed, enforce sparsity to improve theestimator

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 5 / 11

Treelet Covariance Smoothers

Treelet Covariance Smoothing

Goal: preserve local structure of data

Iteratively change basis via PCA to preserve the variability betweentwo most closely related individuals

Repeat this process until all individuals are processed

Once all individuals are processed, enforce sparsity to improve theestimator

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 5 / 11

Treelet Covariance Smoothers

Treelet Covariance Smoothing

Goal: preserve local structure of data

Iteratively change basis via PCA to preserve the variability betweentwo most closely related individuals

Repeat this process until all individuals are processed

Once all individuals are processed, enforce sparsity to improve theestimator

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 5 / 11

Treelet Covariance Smoothers

TCS Properties

P8

P7

P6

P5

P4

P3

P2

P1

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 6 / 11

Treelet Covariance Smoothers

TCS Properties

P8

P7

P6

P5

P4

P3

P2

P1

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 6 / 11

Treelet Covariance Smoothers

TCS Properties

P8

P7

P6

P5

P4

P3

P2

P1

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 6 / 11

Treelet Covariance Smoothers

TCS Properties

P8

P7

P6

P5

P4

P3

P2

P1

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 6 / 11

Treelet Covariance Smoothers

TCS Properties

P8

P7

P6

P5

P4

P3

P2

P1

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 6 / 11

Treelet Covariance Smoothers

TCS Properties

P8

P7

P6

P5

P4

P3

P2

P1

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 6 / 11

Treelet Covariance Smoothers

TCS Properties

P8

P7

P6

P5

P4

P3

P2

P1

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 6 / 11

Treelet Covariance Smoothers

TCS Properties

P8

P7

P6

P5

P4

P3

P2

P1

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 6 / 11

Treelet Covariance Smoothers

Expanding TCS

Why do we require merging into one cluster?

Idea: Stop merging variables to utilize familiar blocks

Smooth by projecting data onto sum variables

Treelet Covariance Blocking (TCB)

Further enforce sparsity by thresholding estimates

Treelet Covariance Blocked Smoothing (TCBS)

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 7 / 11

Treelet Covariance Smoothers

Expanding TCS

Why do we require merging into one cluster?

Idea: Stop merging variables to utilize familiar blocks

Smooth by projecting data onto sum variables

Treelet Covariance Blocking (TCB)

Further enforce sparsity by thresholding estimates

Treelet Covariance Blocked Smoothing (TCBS)

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 7 / 11

Treelet Covariance Smoothers

Expanding TCS

Why do we require merging into one cluster?

Idea: Stop merging variables to utilize familiar blocks

Smooth by projecting data onto sum variables

Treelet Covariance Blocking (TCB)

Further enforce sparsity by thresholding estimates

Treelet Covariance Blocked Smoothing (TCBS)

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 7 / 11

Treelet Covariance Smoothers

Expanding TCS

Why do we require merging into one cluster?

Idea: Stop merging variables to utilize familiar blocks

Smooth by projecting data onto sum variables

Treelet Covariance Blocking (TCB)

Further enforce sparsity by thresholding estimates

Treelet Covariance Blocked Smoothing (TCBS)

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 7 / 11

Treelet Covariance Smoothers

Treelet Covariance Smoothers

At each level of the tree there are basis vectors {v (`)1 , . . . , v(`)N }

We can then write our estimate of A, Σ, by

Σ =∑i∈S`

γ(`)i ,i v

(`)i

(v(`)i

)t+

∑i ,j∈S`,i 6=j

γ(`)i ,j v

(`)i

(v(`)j

)t

Here γi ,j(`) are the variance - covariance estimates of transformed

relationship variables

We can enforce sparsity by further thresholding these γi ,j(`) values

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 8 / 11

Treelet Covariance Smoothers

Treelet Covariance Smoothers

At each level of the tree there are basis vectors {v (`)1 , . . . , v(`)N }

We can then write our estimate of A, Σ, by

Σ =∑i∈S`

γ(`)i ,i v

(`)i

(v(`)i

)t+

∑i ,j∈S`,i 6=j

γ(`)i ,j v

(`)i

(v(`)j

)t

Here γi ,j(`) are the variance - covariance estimates of transformed

relationship variables

We can enforce sparsity by further thresholding these γi ,j(`) values

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 8 / 11

Treelet Covariance Smoothers

Treelet Covariance Smoothers

At each level of the tree there are basis vectors {v (`)1 , . . . , v(`)N }

We can then write our estimate of A, Σ, by

Σ =∑i∈S`

γ(`)i ,i v

(`)i

(v(`)i

)t+

∑i ,j∈S`,i 6=j

γ(`)i ,j v

(`)i

(v(`)j

)t

Here γi ,j(`) are the variance - covariance estimates of transformed

relationship variables

We can enforce sparsity by further thresholding these γi ,j(`) values

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 8 / 11

Treelet Covariance Smoothers

Treelet Covariance Smoothers

At each level of the tree there are basis vectors {v (`)1 , . . . , v(`)N }

We can then write our estimate of A, Σ, by

Σ =∑i∈S`

γ(`)i ,i v

(`)i

(v(`)i

)t+

∑i ,j∈S`,i 6=j

γ(`)i ,j v

(`)i

(v(`)j

)t

Here γi ,j(`) are the variance - covariance estimates of transformed

relationship variables

We can enforce sparsity by further thresholding these γi ,j(`) values

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 8 / 11

Treelet Covariance Smoothers

Simulation Results

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 9 / 11

Treelet Covariance Smoothers

Conclusion

Treelet Covariance Smoothers is a class of methods which improvethe estimation of distant relationships

Estimating relationships is the centerpiece of successful estimation ofother genetic parameters such as heritability

Understanding of genetic basis of heritability can lead to bettertreatment

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 10 / 11

Treelet Covariance Smoothers

Conclusion

Treelet Covariance Smoothers is a class of methods which improvethe estimation of distant relationships

Estimating relationships is the centerpiece of successful estimation ofother genetic parameters such as heritability

Understanding of genetic basis of heritability can lead to bettertreatment

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 10 / 11

Treelet Covariance Smoothers

Conclusion

Treelet Covariance Smoothers is a class of methods which improvethe estimation of distant relationships

Estimating relationships is the centerpiece of successful estimation ofother genetic parameters such as heritability

Understanding of genetic basis of heritability can lead to bettertreatment

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 10 / 11

Treelet Covariance Smoothers

Thanks for listening

Questions? Comments?

B. Draves (Lafayette College) Treelet Covariance Smoothers Moravian College, 2017 11 / 11