ensemble: Estimating Ensemble Kernel Matrices

In IrisTeng/CVEK: Cross-Validated Kernel Ensemble

Description

Give a list of estimated kernel matrices and their weights.

Usage

ensemble(n, kern_size, strategy, beta, error_mat, A_hat)

Arguments

n	(integer) A numeric number specifying the number of observations.
kern_size	(integer, =K) A numeric number specifying the number of kernels in the kernel library.
strategy	(character) A character string indicating which ensemble strategy is to be used.
beta	(numeric/character) A numeric value specifying the parameter when strategy = "exp" ensemble_exp.
error_mat	(matrix, n*K) A n\*kern_size matrix indicating errors.
A_hat	(list of length K) A list of projection matrices for every kernels in the kernel library.

Details

There are three ensemble strategies available here:

Empirical Risk Minimization

After obtaining the estimated errors \{\hat{ε}_d\}_{d=1}^D, we estimate the ensemble weights u=\{u_d\}_{d=1}^D such that it minimizes the overall error

\hat{u}=\underset{u \in Δ}{argmin}\parallel ∑_{d=1}^Du_d\hat{ε}_d\parallel^2 \quad where\; Δ=\{u | u ≥q 0, \parallel u \parallel_1=1\}

Then produce the final ensemble prediction:

\hat{h}=∑_{d=1}^D \hat{u}_d h_d=∑_{d=1}^D \hat{u}_d A_{d,\hat{λ}_d}y=\hat{A}y

where \hat{A}=∑_{d=1}^D \hat{u}_d A_{d,\hat{λ}_d} is the ensemble matrix.

Simple Averaging

Motivated by existing literature in omnibus kernel, we propose another way to obtain the ensemble matrix by simply choosing unsupervised weights u_d=1/D for d=1,2,...D.

Exponential Weighting

Additionally, another scholar gives a new strategy to calculate weights based on the estimated errors \{\hat{ε}_d\}_{d=1}^D.

u_d(β)=\frac{exp(-\parallel \hat{ε}_d \parallel_2^2/β)}{∑_{d=1}^Dexp(-\parallel \hat{ε}_d \parallel_2^2/β)}

Value

A_est	(matrix, n*n) A list of estimated kernel matrices.
u_hat	(vector of length K) A vector of weights of the kernels in the library.

Author(s)

Wenying Deng

References

Jeremiah Zhe Liu and Brent Coull. Robust Hypothesis Test for Nonlinear Effect with Gaus- sian Processes. October 2017.

Xiang Zhan, Anna Plantinga, Ni Zhao, and Michael C. Wu. A fast small-sample kernel inde- pendence test for microbiome community-level association analysis. December 2017.

Arnak S. Dalalyan and Alexandre B. Tsybakov. Aggregation by Exponential Weighting and Sharp Oracle Inequalities. In Learning Theory, Lecture Notes in Computer Science, pages 97– 111. Springer, Berlin, Heidelberg, June 2007.

See Also

mode: tuning

Examples

ensemble(n = 100, kern_size = 3, strategy = "erm", beta = 1, 
error_mat, A_hat)

IrisTeng/CVEK documentation built on May 31, 2019, 4:50 p.m.