ensemble {CVEK}R Documentation

Estimating Ensemble Kernel Matrices

Description

Give the ensemble projection matrix and weights of the kernels in the library.

Usage

ensemble(strategy, beta_exp, error_mat, A_hat)

Arguments

strategy

(character) A character string indicating which ensemble strategy is to be used.

beta_exp

(numeric/character) A numeric value specifying the parameter when strategy = "exp" ensemble_exp.

error_mat

(matrix, n*K) A n\*K matrix indicating errors.

A_hat

(list of length K) A list of projection matrices to kernel space for each kernel in the kernel library.

Details

There are three ensemble strategies available here:

Empirical Risk Minimization (Stacking)

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

\hat{u}={argmin}_{u \in \Delta}\parallel \sum_{d=1}^Du_d\hat{\epsilon}_d\parallel^2 \quad where\; \Delta=\{u | u \geq 0, \parallel u \parallel_1=1\}

Then produce the final ensemble prediction:

\hat{h}=\sum_{d=1}^D \hat{u}_d h_d=\sum_{d=1}^D \hat{u}_d A_{d,\hat{\lambda}_d}y=\hat{A}y

where \hat{A}=\sum_{d=1}^D \hat{u}_d A_{d,\hat{\lambda}_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{\epsilon}_d\}_{d=1}^D.

u_d(\beta)=\frac{exp(-\parallel \hat{\epsilon}_d \parallel_2^2/\beta)}{\sum_{d=1}^Dexp(-\parallel \hat{\epsilon}_d \parallel_2^2/\beta)}

Value

A_est

(matrix, n*n) The ensemble projection matrix.

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 Gaussian Processes. October 2017.

Xiang Zhan, Anna Plantinga, Ni Zhao, and Michael C. Wu. A fast small-sample kernel independence 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


[Package CVEK version 0.1-2 Index]