ℓ_𝑝-Norm Multikernel Learning Approach for Stock Market Price Forecasting

Algorithm 1

1. Input 𝑓_𝑚,𝑖=𝑓_𝑖=𝛼_𝑖=0, 𝑔_𝑚,𝑖=𝑔_𝑖=𝛼_𝑖=0, ∀𝑖=1,2,…,𝑁; 𝐿=𝑆=−∞, 𝑑_𝑘=(1/𝑘)^𝑝,

∀𝑘=1,2,…,𝑀.

2. Iterate

 (1) Select 2𝑄 variables based on the gradient of (6): 𝜶_𝑄=𝛼_𝑖1,…,𝛼_𝑖𝑄, 𝜶_𝑄=𝛼_𝑖1,…,𝛼_𝑖𝑄.

 (2) Store 𝜶^old_𝑄=𝜶_𝑄, 𝜶^old_𝑄=𝜶_𝑄 and new 𝜶_𝑄, 𝜶_𝑄 can be obtained according to (6) with

 respect to the selected variables.

 (3) Update gradient 𝑓_𝑚,𝑖←𝑓_𝑚,𝑖+∑^𝑄_𝑞=1(𝛼_𝑖𝑞−𝛼^old_𝑖𝑞)𝐾_𝑘(𝐱_𝑖𝑞,𝐱_𝑖), 𝑔_𝑚,𝑖←𝑔_𝑚,𝑖+∑^𝑄_𝑞=1(𝛼_𝑖𝑞−𝛼^old_𝑖𝑞)𝐾_𝑘(𝐱_𝑖𝑞,𝐱_𝑖),

 ∀𝑘=1,2,…,𝑀,𝑖=1,2,…,𝑁.

 (4) Compute the quadratic terms 𝑆_𝑘=(1/2)∑_𝑖(𝑓_𝑚,𝑖−𝑔_𝑚,𝑖)(𝛼_𝑖−𝛼_𝑖), ∀𝑘=1,2,…,𝑀.

 (5) 𝐿_old=𝐿, 𝐿=∑_𝑖𝑦_𝑖(𝛼_𝑖−𝛼_𝑖)−∑_𝑖𝜀(𝛼_𝑖+𝛼_𝑖), 𝑆_old=𝑆, 𝑆=∑_𝑘𝑑_𝑘𝑆_𝑘.

 (6) If |1−(𝐿−𝑆)/(𝐿_old−𝑆_old)|≥𝜀, update 𝑑_𝑘 with (10), ∀𝑘=1,2,…,𝑀,

 else

  break

 endif

3. Output 𝜶,𝜶,𝐝.