Kernel Matrix Formation
Physical Representation in Data Space
The kernel matrix (K) represents the covariance structure between all pairs of data points. Each element K[i][j] shows how much point i influences point j based on their distance and the kernel parameters. This creates a smoothness constraint on possible functions.
Posterior Mean Calculation
The kernel matrix defines the covariance structure between all training points. It encodes how much each point influences others based on their distance.
The inverse matrix (K⁻¹) represents the precision matrix. It determines how much each observation should contribute to the final prediction, accounting for correlations between points.
The observation vector contains the actual function values at the training points. These are the values we want to interpolate between.
K⁻¹y represents the weights for each observation. It's a projection of the data onto the space defined by the kernel, determining how much each point contributes to predictions.
Where K* is the covariance between test points and training points
The posterior mean is a weighted combination of kernel functions centered at each data point. The weights (K⁻¹y) determine how much each point contributes to the prediction at any location.
How Gaussian Processes Work
A Gaussian Process is a powerful non-parametric method that defines a distribution over functions. It's completely specified by its mean function and covariance (kernel) function. As you add data points, the GP updates its posterior distribution to reflect both the observed data and its uncertainty in regions without data.