kernelRidge

kernelRidge(ds, yColName, xColNames, [alpha=1.0], [kernel='linear'], [gamma=0], [degree=3], [coef0=1], [swColName])

ds is an in-memory table or a data source usually generated by the sqlDS function.

yColName is a string indicating the column name of the dependent variable in ds, corresponding toyin the objective function.

xColNames is a string scalar/vector indicating the column names of the independent variables in ds, corresponding to X in the objective function.

alpha (optional) is a positive floating-point number indicating the regularization strength, corresponding to alpha in the objective function. The default value is 1.0.

kernel (optional) is a string indicating the kernel, corresponding to Φ in the objective function. It can take one of the following values:

• 'linear' (default): the linear kernel <x, y>
• 'rbf': the RBF (Gaussian) kernel exp(-gamma * ||x - y||²)
• 'poly'/'polynomial': the polynomial kernel (gamma * <x, y> + coef0)^degree
• 'sigmoid': the sigmoid kernel tanh(gamma * <x, y> + coef0)
• 'laplacian': the laplacian kernel exp(-gamma * ||x - y||₁)
• 'cosine': the cosine similarity kernel <x, y> / (||x|| * ||y||)
• 'additive_chi2': the additive chi-squared kernel -∑[(x - y)² / (x + y)]
• 'chi2': the chi-squared kernel exp(-gamma * ∑[(x - y)² / (x + y)])

gamma (optional) is a numeric scalar indicating the gamma parameter for kernels. The default value is 0, indicating that gamma will be automatically determined based on the number of features.

degree (optional) is a numeric scalar indicating the degree of the polynomial kernel.

coef0 (optional) is a numeric scalar indicating the zero coefficient for kernels.

swColName (optional) is a string indicating a column name in ds as the sample weight. If it is not specified, the sample weight is treated as 1.

This function combines ridge regression regularization with the kernel trick to model complex nonlinear relationships in the data and outputs a regression prediction model.

The objective function is:

Return Value:

A dictionary with the following keys:

• modelName: A string indicating the model name, which is ‘kernelRidge’.
• coefficients: A numeric vector indicating the parameters, corresponding to the w in the objective function.
• xFit: A numeric matrix indicating the independent variables.
• predict: The prediction function of the model.
• The other keys, including xColNames, alpha, kernel, gamma, degree and coef0, correspond to the arguments with the same names.

x1 = [-1.5, 2.3, 4.2, 1.6];
x2 = [-2.2, 3.9, 2.8, 0.5];
sw = [2, 5, 8, 1];
y = [0.0, 0.9, 3.2, 3.1];
t = table(y, x1, x2, sw);

m = kernelRidge(t, `y, `x1`x2, 1, 'laplacian', 0, 3, 1, `sw);
m
/*
modelName->kernelRidge
coefficients->#0                
------------------
-0.061436450468451
0.09192514209272  
2.716894143137368 
1.428547958639275 

xColNames->["x1","x2"]
xFit->#0                #1  
----------------- ----
-1.5              -2.2
2.3               3.9 
4.200000000000001 2.8 
1.6               0.5 

alpha->1
kernel->laplacian
gamma->0.5
degree->3
coef0->1
predict->kernelRidgePredict
*/



x1 = [-1.2, 4.6, 2.4];
x2 = [-1.0, 2.8, -4.7];
t_test = table(x1, x2);
predict(m, t_test);
/*
#0               
-----------------
0.166070925673076
2.34188781136904 
0.095783250622374
*/