Description Usage Arguments Details Value Author(s) References See Also
Give a list of estimated kernel matrices and their weights using empirical risk minimization.
1 | ensemble_erm(n, kern_size, beta, error_mat, A_hat)
|
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. |
beta |
(numeric/character) A numeric value specifying the parameter when
strategy = "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. |
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.
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. |
Wenying Deng
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.
mode: tuning
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