### Function Reference: `regress_gp`

statistics: [Yfit, Yint, m, K] = regress_gp (X, Y, Xfit)
statistics: [Yfit, Yint, m, K] = regress_gp (X, Y, Xfit, `"linear"`)
statistics: [Yfit, Yint, Ysd] = regress_gp (X, Y, Xfit, `"rbf"`)
statistics: […] = regress_gp (X, Y, Xfit, `"linear"`, Sp)
statistics: […] = regress_gp (X, Y, Xfit, Sp)
statistics: […] = regress_gp (X, Y, Xfit, `"rbf"`, theta)
statistics: […] = regress_gp (X, Y, Xfit, `"rbf"`, theta, g)
statistics: […] = regress_gp (X, Y, Xfit, `"rbf"`, theta, g, alpha)
statistics: […] = regress_gp (X, Y, Xfit, theta)
statistics: […] = regress_gp (X, Y, Xfit, theta, g)
statistics: […] = regress_gp (X, Y, Xfit, theta, g, alpha)

Regression using Gaussian Processes.

`[Yfit, Yint, m, K] = regress_gp (X, Y, Xfit)` will estimate a linear Gaussian Process model m in the form `Y = X' * m`, where X is an N×P matrix with N observations in P dimensional space and Y is an N×1 column vector as the dependent variable. The information about errors of the predictions (interpolation/extrapolation) is given by the covarianve matrix K. By default, the linear model defines the prior covariance of m as `Sp = 100 * eye (size (X, 2) + 1)`. A custom prior covariance matrix can be passed as Sp, which must be a P+1×P+1 positive definite matrix. The model is evaluated for input Xfit, which must have the same columns as X, and the estimates are returned in Yfit along with the estimated variation in Yint. `Yint(:,1)` contains the upper boundary estimate and `Yint(:,1)` contains the upper boundary estimate with respect to Yfit.

`[Yfit, Yint, Ysd, K] = regress_gp (X, Y, Xfit, `"rbf"`)` will estimate a Gaussian Process model with a Radial Basis Function (RBF) kernel with default parameters `theta = 5`, which corresponds to the characteristic lengthscale, and `g = 0.01`, which corresponds to the nugget effect, and `alpha = 0.05` which defines the confidence level for the estimated intervals returned in Yint. The function also returns the predictive covariance matrix in Ysd. For multidimensional predictors X the function will automatically normalize each column to a zero mean and a standard deviation to one.

Run `demo regress_gp` to see examples.

See also: regress, regression_ftest, regression_ttest

Source Code: regress_gp

### Example: 1

     ## Linear fitting of 1D Data
     rand ("seed", 125);
     X = 2 * rand (5, 1) - 1;
     randn ("seed", 25);
     Y = 2 * X - 1 + 0.3 * randn (5, 1);

     ## Points for interpolation/extrapolation
     Xfit = linspace (-2, 2, 10)';

     ## Fit regression model
     [Yfit, Yint, m] = regress_gp (X, Y, Xfit);

     ## Plot fitted data
     plot (X, Y, "xk", Xfit, Yfit, "r-", Xfit, Yint, "b-");
     title ("Gaussian process regression with linear kernel");

### Example: 2

     ## Linear fitting of 2D Data
     rand ("seed", 135);
     X = 2 * rand (4, 2) - 1;
     randn ("seed", 35);
     Y = 2 * X(:,1) - 3 * X(:,2) - 1 + 1 * randn (4, 1);

     ## Mesh for interpolation/extrapolation
     [x1, x2] = meshgrid (linspace (-1, 1, 10));
     Xfit = [x1(:), x2(:)];

     ## Fit regression model
     [Ypred, Yint, Ysd] = regress_gp (X, Y, Xfit);
     Ypred = reshape (Ypred, 10, 10);
     YintU = reshape (Yint(:,1), 10, 10);
     YintL = reshape (Yint(:,2), 10, 10);

     ## Plot fitted data
     plot3 (X(:,1), X(:,2), Y, ".k", "markersize", 16);
     hold on;
     h = mesh (x1, x2, Ypred, zeros (10, 10));
     set (h, "facecolor", "none", "edgecolor", "yellow");
     h = mesh (x1, x2, YintU, ones (10, 10));
     set (h, "facecolor", "none", "edgecolor", "cyan");
     h = mesh (x1, x2, YintL, ones (10, 10));
     set (h, "facecolor", "none", "edgecolor", "cyan");
     hold off
     axis tight
     view (75, 25)
     title ("Gaussian process regression with linear kernel");

### Example: 3

     ## Projection over basis function with linear kernel
     pp = [2, 2, 0.3, 1];
     n = 10;
     rand ("seed", 145);
     X = 2 * rand (n, 1) - 1;
     randn ("seed", 45);
     Y = polyval (pp, X) + 0.3 * randn (n, 1);

     ## Powers
     px = [sqrt(abs(X)), X, X.^2, X.^3];

     ## Points for interpolation/extrapolation
     Xfit = linspace (-1, 1, 100)';
     pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3];

     ## Define a prior covariance assuming that the sqrt component is not present
     Sp = 100 * eye (size (px, 2) + 1);
     Sp(2,2) = 1; # We don't believe the sqrt(abs(X)) is present

     ## Fit regression model
     [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, Sp);

     ## Plot fitted data
     plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                             Xfit, polyval (pp, Xfit), "g-;True;");
     axis tight
     axis manual
     hold on
     plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
     hold off
     title ("Linear kernel over basis function with prior covariance");

### Example: 4

     ## Projection over basis function with linear kernel
     pp = [2, 2, 0.3, 1];
     n = 10;
     rand ("seed", 145);
     X = 2 * rand (n, 1) - 1;
     randn ("seed", 45);
     Y = polyval (pp, X) + 0.3 * randn (n, 1);

     ## Powers
     px = [sqrt(abs(X)), X, X.^2, X.^3];

     ## Points for interpolation/extrapolation
     Xfit = linspace (-1, 1, 100)';
     pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3];

     ## Fit regression model without any assumption on prior covariance
     [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi);

     ## Plot fitted data
     plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                             Xfit, polyval (pp, Xfit), "g-;True;");
     axis tight
     axis manual
     hold on
     plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
     hold off
     title ("Linear kernel over basis function without prior covariance");

### Example: 5

     ## Projection over basis function with rbf kernel
     pp = [2, 2, 0.3, 1];
     n = 10;
     rand ("seed", 145);
     X = 2 * rand (n, 1) - 1;
     randn ("seed", 45);
     Y = polyval (pp, X) + 0.3 * randn (n, 1);

     ## Powers
     px = [sqrt(abs(X)), X, X.^2, X.^3];

     ## Points for interpolation/extrapolation
     Xfit = linspace (-1, 1, 100)';
     pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3];

     ## Fit regression model with RBF kernel (standard parameters)
     [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf");

     ## Plot fitted data
     plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                             Xfit, polyval (pp, Xfit), "g-;True;");
     axis tight
     axis manual
     hold on
     plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
     hold off
     title ("RBF kernel over basis function with standard parameters");
     text (-0.5, 4, "theta = 5\n g = 0.01");

### Example: 6

     ## Projection over basis function with rbf kernel
     pp = [2, 2, 0.3, 1];
     n = 10;
     rand ("seed", 145);
     X = 2 * rand (n, 1) - 1;
     randn ("seed", 45);
     Y = polyval (pp, X) + 0.3 * randn (n, 1);

     ## Powers
     px = [sqrt(abs(X)), X, X.^2, X.^3];

     ## Points for interpolation/extrapolation
     Xfit = linspace (-1, 1, 100)';
     pxi = [sqrt(abs(Xfit)), Xfit, Xfit.^2, Xfit.^3];

     ## Fit regression model with RBF kernel with different parameters
     [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 10, 0.01);

     ## Plot fitted data
     plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                             Xfit, polyval (pp, Xfit), "g-;True;");
     axis tight
     axis manual
     hold on
     plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
     hold off
     title ("GP regression with RBF kernel and non default parameters");
     text (-0.5, 4, "theta = 10\n g = 0.01");

     ## Fit regression model with RBF kernel with different parameters
     [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.01);

     ## Plot fitted data
     figure
     plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                             Xfit, polyval (pp, Xfit), "g-;True;");
     axis tight
     axis manual
     hold on
     plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
     hold off
     title ("GP regression with RBF kernel and non default parameters");
     text (-0.5, 4, "theta = 50\n g = 0.01");

     ## Fit regression model with RBF kernel with different parameters
     [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.001);

     ## Plot fitted data
     figure
     plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                             Xfit, polyval (pp, Xfit), "g-;True;");
     axis tight
     axis manual
     hold on
     plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
     hold off
     title ("GP regression with RBF kernel and non default parameters");
     text (-0.5, 4, "theta = 50\n g = 0.001");

     ## Fit regression model with RBF kernel with different parameters
     [Yfit, Yint, Ysd] = regress_gp (px, Y, pxi, "rbf", 50, 0.05);

     ## Plot fitted data
     figure
     plot (X, Y, "xk;Data;", Xfit, Yfit, "r-;Estimation;", ...
                             Xfit, polyval (pp, Xfit), "g-;True;");
     axis tight
     axis manual
     hold on
     plot (Xfit, Yint(:,1), "m-;Upper bound;", Xfit, Yint(:,2), "b-;Lower bound;");
     hold off
     title ("GP regression with RBF kernel and non default parameters");
     text (-0.5, 4, "theta = 50\n g = 0.05");

### Example: 7

     ## RBF fitting on noiseless 1D Data
     x = [0:2*pi/7:2*pi]';
     y = 5 * sin (x);

     ## Predictive grid of 500 equally spaced locations
     xi = [-0.5:(2*pi+1)/499:2*pi+0.5]';

     ## Fit regression model with RBF kernel
     [Yfit, Yint, Ysd] = regress_gp (x, y, xi, "rbf");

     ## Plot fitted data
     r = mvnrnd (Yfit, diag (Ysd)', 50);
     plot (xi, r', "c-");
     hold on
     plot (xi, Yfit, "r-;Estimation;", xi, Yint, "b-;Confidence interval;");
     plot (x, y, ".k;Predictor points;", "markersize", 20)
     plot (xi, 5 * sin (xi), "-y;True Function;");
     xlim ([-0.5,2*pi+0.5]);
     ylim ([-10,10]);
     hold off
     title ("GP regression with RBF kernel on noiseless 1D data");
     text (0, -7, "theta = 5\n g = 0.01");

### Example: 8

     ## RBF fitting on noisy 1D Data
     x = [0:2*pi/7:2*pi]';
     x = [x; x];
     y = 5 * sin (x) + randn (size (x));

     ## Predictive grid of 500 equally spaced locations
     xi = [-0.5:(2*pi+1)/499:2*pi+0.5]';

     ## Fit regression model with RBF kernel
     [Yfit, Yint, Ysd] = regress_gp (x, y, xi, "rbf");

     ## Plot fitted data
     r = mvnrnd (Yfit, diag (Ysd)', 50);
     plot (xi, r', "c-");
     hold on
     plot (xi, Yfit, "r-;Estimation;", xi, Yint, "b-;Confidence interval;");
     plot (x, y, ".k;Predictor points;", "markersize", 20)
     plot (xi, 5 * sin (xi), "-y;True Function;");
     xlim ([-0.5,2*pi+0.5]);
     ylim ([-10,10]);
     hold off
     title ("GP regression with RBF kernel on noisy 1D data");
     text (0, -7, "theta = 5\n g = 0.01");