[Matlab] Support Vector Regression二次规划实现

使用x = quadprog(H,f,A,b,Aeq,beq,lb,ub)训练Lagrange multipliers αi and αi’,并用于回归分析。


Training

function [alphas, alphas_dash, d] = svrTrain( X, y, K, epsilon, C )
% Input
% -----
%
% X          ... Data points.
%                [ x_11, x_12;
%                  x_21, x_22;
%                  x_31, x_32;
%                  ...              ]
%
% y          ... Class labels.
%                [ s_1; s_2; s_3; ... ]
%
% K          ... Kernel.
%                @(x, y) ...
%
% epsilon    ... SVR parameter.
%
%
% C          ... SVR parameter.

% Output
% ------
%
% alphas     ... Lagrange multipliers.
%
% alphas_dash... Lagrange multipliers.
%
% d          ... Distance from the origin.


% Obtain the size of data
n = size(X, 1);
% Initialization quadprog vars
A = [];
b = [];
Aeq = [];
f = [];
ub = []
lb = []
%Aqe,ub,lu,f
for i = 1:n
    Aeq = [Aeq,1,-1]
    ub = [ub,C,C]
    lb = [lb,0,0]
    f = [f,-epsilon + y(i),-epsilon - y(i)];
end
%initialize Q
H = zeros(2*n,2*n)
for i = 1:2*n
    for j = 1:2*n
        if mod(i,2) == 0
            xi = X(i/2,:)
        else
            xi = X((i+1)/2,:)
        end
        if mod(j,2) == 0
            xj = X(j/2,:)
        else
            xj = X((j+1)/2,:)
        end
        if mod((i + j),2) == 0
          H(i,j) = 1 * K(xi,xj)
        else
          H(i,j) = -1 * K(xi,xj)
        end
    end
end
%max -> min
f = -f
beq = 0;
[alphalist] = quadprog(H, f, A, b, Aeq, beq, lb, ub);
%alphalist:[alphas1,alphas1',alphas2,alphas2'...]
za = size(alphalist);

alphas = [];
alphas_dash = [];
%seperate alphas1,alphas1'
for i = 1:za(1)
    if mod(i,2) == 1
        alphas = [alphas;alphalist(i)]
    else
        alphas_dash = [alphas_dash;alphalist(i)]
    end
end
%record j and where it from
j = 0
isDash = 0
for i = 1:n
    if(alphas(i)>0&&alphas(i)0&&alphas_dash(i)

Produce

function f = svrProduce( X, K, alphas, alphas_dash, d, x1, x2 )
% Input
% -----
%
% X          ... Data points.
%                [ x_11, x_12;
%                  x_21, x_22;
%                  x_31, x_32;
%                  ...              ]
%
% K          ... Kernel.
%                @(x, y) ...
%
% alphas     ... Lagrange multipliers.
%
% alphas_dash... Lagrange multipliers.
%
% d          ... Distance from the origin.
%
% x1         ... Domain of x1, e.g. [-3 -2 -1 0 1 2 3].
%
% x2         ... Domain of x2, e.g. [-3 -2 -1 0 1 2 3].

% Output
% ------
%
% f          ... Approximated values of f(x) on the domain(s) of x1 and x2.


% Initialization 
size1 = size(x1,2);
size2 = size(x2,2);
n = size(X, 1);
f = zeros(size1,1);

%calculate the predicted value of 1d or 2d data set
if(size2 == 0)
    for d1 = 1:size1
        temp = 0;
        z = x1(d1) ;
        for i = 1:n
            temp = temp + (alphas(i,1) - alphas_dash(i,1)) * K(X(i,:), z);
        end
        f(d1) = temp + d;
    end 
else
    for d1 = 1:size1
        for d2 = 1:size2
            temp = 0;
            z = [x1(d1),x2(d2)];

            for i = 1:n
                temp = temp + (alphas(i,1) - alphas_dash(i,1)) * K(X(i,:), z);
            end
            f(d1,d2) = temp + d;
        end
    end
end
end

1D data Test

clear all;
close all;
clc;

% --------------------------------------------------
% Test instance 1-D data.
% --------------------------------------------------
nSamples = 11;
X = linspace(-16, 16, nSamples)';
noiseFactor = 20;
y = ((X-5).^2 + 2 + noiseFactor * (rand(size(X)) - 0.5))';

% Set up kernel.
K = @(x, y) (x*y'+1)^2;

% Set up parameters.
epsilon = 2
C = 10;

% Call dual QP with kernel.
[alphas, alphas_dash, d] = svrTrain(X, y, K, epsilon, C);

% Calculate regression line / regression plane.
x1 = -16:0.05:16;
f = svrProduce(X, K, alphas, alphas_dash, d, x1, []);

% Plot results.
svrPlot(x1, [], (x1-5).^2 + 2, X, y, f);

text(5, 450, 'K(x,y) = (x*y+1)^2', 'FontSize', 14, 'FontWeight', 'bold');
text(5, 400, ['Noise factor: ' num2str(noiseFactor)], 'FontSize', 14, 'FontWeight', 'bold');
text(5, 350, ['nSamples: ' num2str(nSamples)], 'FontSize', 14, 'FontWeight', 'bold');
text(5, 300, ['Sampling interval: ' num2str(X(1)) ' - ' num2str(X(end))], 'FontSize', 14, 'FontWeight', 'bold');

print('-dpng', '1D.png', '-r150');

2D data test

clear all;
close all;
clc;

% --------------------------------------------------
% Test instance 2-D data.
% --------------------------------------------------
x1 = -3:0.1:3;
x2 = -3:0.1:3;
[X1, X2] = meshgrid(x1, x2);
mu = [0, 0];
sigma = eye(2);
F = mvnpdf([X1(:) X2(:)], mu, sigma);
F = reshape(F, length(x1), length(x2));

nSamples = 40;
X = zeros(nSamples, 2);
y = zeros(1, nSamples);
for i = 1:nSamples
    xi = ceil(rand() * length(x1));
    yi = ceil(rand() * length(x2));
    X(i, 1) = x1(xi);
    X(i, 2) = x2(yi);
    y(i) = F(xi, yi);
end

% Set up kernel.
K = @(x, y) exp(-norm(x-y)^2);

% Set up parameters.
epsilon = 0.00002;
C = 7;

% Call dual QP with kernel.
[alphas, alphas_dash, d] = svrTrain(X, y, K, epsilon, C);

% Calculate regression line / regression plane.
x1 = -3:0.1:3;
x2 = -3:0.1:3;
f = svrProduce(X, K, alphas, alphas_dash, d, x1, x2);


% Plot results.
svrPlot(x1, x2, F, X, y, f);

text(0, 0, 0.21, 'K(x,y) = exp(-norm(x-y)^2)', 'FontSize', 14, 'FontWeight', 'bold');
text(0, 0, 0.20, ['nSamples: ' num2str(nSamples)], 'FontSize', 14, 'FontWeight', 'bold');

print('-dpng', '2D.png', '-r150');

plot figure tool

function svrPlot( x1, x2, F, X, y, f )

figure();
set(gcf, 'Units', 'normalized', 'OuterPosition', [0.05 0.05 0.9 0.9]);
set(gcf, 'PaperOrientation', 'landscape');
set(gcf, 'PaperUnits', 'centimeters', 'PaperPosition', [0 0 29.7 21]);
set(gcf, 'PaperSize', [29.7 21.0]);
if (isempty(x2))
    plot(x1, F, 'k--', 'LineWidth', 2); hold on;
    plot(X, y, 'k*', 'MarkerSize', 15); 
    plot(x1, f, 'k-', 'LineWidth', 2); hold off;
    xlabel('x', 'FontSize', 14, 'FontWeight', 'bold');
    ylabel('f(x)', 'FontSize', 14, 'FontWeight', 'bold');
    legend('f(x)', 'Sample points', 'Regression line', 'Location', 'SouthEast');
    set(gca, 'FontSize', 14, 'FontWeight', 'bold');
    xlim([-18 18]);
    ylim([-100 500]);
else
    subplot(1, 2, 1);
    surf(x1, x2, F); hold on;
    plot3(X(:, 1), X(:, 2), y, 'r*', 'MarkerSize', 17, 'LineWidth', 2); hold off;
    view([-45 15]);
    xlabel('x_1', 'FontSize', 14, 'FontWeight', 'bold');
    ylabel('x_2', 'FontSize', 14, 'FontWeight', 'bold');
    zlabel('f(x_1, x_2)', 'FontSize', 14, 'FontWeight', 'bold'); 
    legend('f(x_1, x_2)', 'Sample points', 'Location', 'NorthEast');
    set(gca, 'FontSize', 14, 'FontWeight', 'bold');
    xlim([-5 5]);
    ylim([-5 5]);
    zlim([0 0.2]);    
    grid on;
    
    subplot(1, 2, 2);
    surf(x1, x2, f);
    view([-45 15]);
    xlabel('x_1', 'FontSize', 14, 'FontWeight', 'bold');
    ylabel('x_2', 'FontSize', 14, 'FontWeight', 'bold');
    zlabel('f(x_1, x_2)', 'FontSize', 14, 'FontWeight', 'bold'); 
    legend('Regression plane', 'Location', 'NorthEast');
    set(gca, 'FontSize', 14, 'FontWeight', 'bold');
    xlim([-5 5]);
    ylim([-5 5]);
    zlim([0 0.2]);
    grid on;
end
RMS = mean((F(:)-f(:)).^2);
title(['RMS: ' num2str(RMS)], 'FontSize', 14, 'FontWeight', 'bold');

end