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

Step 1: Kernel Matrix (K)

The kernel matrix defines the covariance structure between all training points. It encodes how much each point influences others based on their distance.

Step 2: Inverse of K (K⁻¹)

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.

Step 3: Observations (y)

The observation vector contains the actual function values at the training points. These are the values we want to interpolate between.

Step 4: Multiply K⁻¹ and y

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.

Step 5: Final Posterior Mean
Posterior Mean = K*ᵀ · (K⁻¹ · y)
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.

Data Points
Posterior Mean
Confidence Band
Kernel Correlations

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.

Kernel Parameters

0.5
Controls how far correlations extend
1.0
Controls the amplitude of functions
0.05
Controls observation noise

Data Controls

Animation Settings

1.0x

Kernel Type