Corticomuscular Activity Modeling by Combining Partial Least Squares and Canonical Correlation Analysis

Algorithm 1

The combined PLS + CCA method.

Input: two data sets 𝐗 (with size 𝑁×𝑝) and 𝐘 (with size 𝑁×𝑞)

Output: corresponding LVs matrices 𝐔_𝐗, and 𝐔_𝐘

The First Step:

(1) Solve the eigen decomposition problems:

    𝐗^𝑇𝐘𝐘^𝑇𝐗𝐰₁=𝜆₁𝐰₁ and 𝐘^𝑇𝐗𝐗^𝑇𝐘𝐰₂=𝜆₂𝐰₂.

(2) Determine 𝑅₁ and 𝑅₂, the numbers of LVs extracted, corresponding to

    the above two problems by the ratio of explained variance.

(3) Determine the final number of LVs: 𝑅=min(𝑅₁,𝑅₂).

(4) Set 𝑐𝑜𝑢𝑛𝑡=𝑅.

(5) Initialize both LVs matrices to be empty, that is, 𝐓_𝐗=[] and 𝐓_𝐘=[].

(6)        while  𝑐𝑜𝑢𝑛𝑡>0  do

(7)      Set 𝐰₁ and 𝐰₂ to be the largest eigenvectors of the matrices

        𝐗^𝑇𝐘𝐘^𝑇𝐗 and 𝐘^𝑇𝐗𝐗^𝑇𝐘, respectively.

(8)      Calculate the LVs as 𝐭_𝐗=𝐗𝐰₁ and 𝐭_𝐘=𝐘𝐰₂.

(9)      Set 𝐓_𝐗=𝐓_𝐗𝐭_𝐗 and 𝐓_𝐘=𝐓_𝐘𝐭_𝐘.

(10)  Deflate 𝐗 by subtracting the effects of the LV 𝐭_𝐗 from the data space:

        𝐗=𝐗−𝐭_𝐗(𝐭^𝑇_𝐗𝐭_𝐗)⁻¹𝐭^𝑇_𝐗𝐗.

(11)  Deflate 𝐘  by subtracting the effects of the LV 𝐭_𝐘 from the data space:

        𝐘=𝐘−𝐭_𝐘(𝐭^𝑇_𝐘𝐭_𝐘)⁻¹𝐭^𝑇_𝐘𝐘.

(12)  Let 𝑐𝑜𝑢𝑛𝑡=𝑐𝑜𝑢𝑛𝑡−1.

(13)  end while

The Second Step:

(14) Solve the following eigen decomposition problems:

        (𝐓^𝑇_𝐗𝐓_𝐗)⁻¹𝐓^𝑇_𝐗𝐓_𝐘(𝐓^𝑇_𝐘𝐓_𝐘)⁻¹𝐓^𝑇_𝐘𝐓_𝐗𝐯₁=𝜂₁𝐯₁ and

        (𝐓^𝑇_𝐘𝐓_𝐘)⁻¹𝐓^𝑇_𝐘𝐓_𝐗(𝐓^𝑇_𝐗𝐓_𝐗)⁻¹𝐓^𝑇_𝐗𝐓_𝐘𝐯₂=𝜂₂𝐯₂.

(15) Set 𝐕₁ and 𝐕₂ to be the 𝑅 associated eigenvectors, respectively.

(16) The recovered LVs 𝐔_𝐗 and 𝐔_𝐗 can be calculated by

        𝐔_𝐗=𝐓_𝐗𝐕₁ and 𝐔_𝐘=𝐓_𝐘𝐕₂.