import numpy as np def referenceRbfKernel( X, lengthScale, signalVariance ): (nDataPoints, inputDimensions ) = X.shape kernel = np.zeros( (nDataPoints, nDataPoints ) ) for row_index in range( nDataPoints ): for column_index in range( nDataPoints ): vecA = X[row_index,:] vecB = X[column_index,:] delta = vecA - vecB distanceSquared = np.dot( delta.T, delta ) kernel[row_index, column_index ] = signalVariance * np.exp( -0.5*distanceSquared / lengthScale** 2) return kernel def referenceArcCosineKernel( X, order, weightVariances, biasVariance, signalVariance ): num_points = X.shape[0] kernel = np.empty((num_points, num_points)) for row in range(num_points): for col in range(num_points): x = X[row] y = X[col] numerator = (weightVariances * x).dot(y) + biasVariance x_denominator = np.sqrt((weightVariances * x).dot(x) + biasVariance) y_denominator = np.sqrt((weightVariances * y).dot(y) + biasVariance) denominator = x_denominator * y_denominator theta = np.arccos(np.clip(numerator / denominator, -1., 1.)) if order == 0: J = np.pi - theta elif order == 1: J = np.sin(theta) + (np.pi - theta) * np.cos(theta) elif order == 2: J = 3. * np.sin(theta) * np.cos(theta) + (np.pi - theta) * (1. + 2. * np.cos(theta) ** 2) kernel[row, col] = signalVariance * (1. / np.pi) * J * \ x_denominator ** order * \ y_denominator ** order return kernel def referencePeriodicKernel( X, lengthScale, signalVariance, period ): # Based on the GPy implementation of standard_period kernel base = np.pi * (X[:, None, :] - X[None, :, :]) / period exp_dist = np.exp( -0.5* np.sum( np.square( np.sin( base ) / lengthScale ), axis = -1 ) ) return signalVariance * exp_dist