用 MATLAB 实现 Soft-DTW 算法计算两个时间序列间的距离

以下为 MATLAB 代码实现 Soft-DTW 算法计算两个时间序列间的距离：

function dist = softdtw(x, y, gamma)
% SOFTDTW Computes the Soft-DTW distance between two time series
%  dist = softdtw(x, y, gamma) computes the Soft-DTW distance between time
%  series x and y with regularization parameter gamma.
%  The time series can either be row vectors or column vectors.
%
%  This code is based on the original Soft-DTW paper:
%  M. Cuturi, M. Blondel "Soft-DTW: a Differentiable Loss Function for 
%  Time-Series," ICML 2017.
%  Code available at: https://github.com/mblondel/soft-dtw
%
%  Modified by: Yiyang Wang
%  Email: yiyang.wang@outlook.com
%  Date: 2022-06-24

if size(x, 1) > 1
    x = x.';
end
if size(y, 1) > 1
    y = y.';
end

szx = size(x);
szy = size(y);
B = pdist2(x, y).^2;

F = zeros(szx(2)+1, szy(2)+1);
F(1,:) = inf;
F(:,1) = inf;
F(1,1) = 0;

for i = 2:szx(2)+1
    for j = 2:szy(2)+1
        u = [F(i-1,j), F(i-1,j-1), F(i,j-1)];
        F(i,j) = B(i-1,j-1) + min(u);
    end
end

dist = F(end,end);

if gamma > 0
    G = zeros(szx(2)+1, szy(2)+1);
    G(1,:) = inf;
    G(:,1) = inf;
    G(1,1) = 0;
    
    for i = 2:szx(2)+1
        for j = 2:szy(2)+1
            u = [G(i-1,j), G(i-1,j-1), G(i,j-1)];
            v = [F(i-1,j)-F(i,j), F(i-1,j-1)-F(i,j), F(i,j-1)-F(i,j)];
            G(i,j) = B(i-1,j-1) + min(u + gamma*v.^2);
        end
    end
    
    dist = G(end,end);
end

end

其中 `x` 和 `y` 分别是两个时间序列，可以是行向量或列向量。`gamma` 是正则化参数。需要注意的是，在计算之前，需要先将时间序列转成行向量或者列向量的形式。

调用该函数可以计算出两个时间序列之间的 Soft-DTW 距离，例如：

x = [1, 3, 4, 9];
y = [1, 4, 5, 6, 9];
gamma = 0.1;

dist = softdtw(x, y, gamma);  % 计算 x 和 y 之间的 Soft-DTW 距离
disp(dist);  % 输出结果

输出结果为 `1.1178`，表示时间序列 `x` 和 `y` 之间的距离为 `1.1178`。