Hi, thanks for the great tutorial! I noticed that the example only covers the case with a single feature. Could you please explain how to adapt the code for multiple features? I'd love to understand how to handle the design matrix and update the regression model in that scenario. Thanks!
Hi Henry! The only difference will be in the way you build the design matrix (the rest is the same). First, start with your original features, x1, x2, ..., xd. These represent the dimensions of your data. When you apply an RBF kernel, you map the data into a higher-dimensional space using the kernel function. For example. the RBF kernel is given by the formula: K(xi, xj) = exp(-gamma * ||xi - xj||^2) In this formula, xi and xj are two data points (represented as vectors of features), gamma is a parameter controlling the width of the kernel, and ||xi - xj||^2 is the squared distance between these two points. The kernel computes a similarity measure between data points based on this distance. To incorporate this into Bayesian linear regression, you need to build a design matrix (also called the kernel matrix) where each element is the kernel value between two data points. The matrix will contain the similarities between every pair of data points. For example, if you have three data points (x1, x2, x3), the design matrix (Phi(X)) will look like this: Phi(X) = [ K(x1, x1), K(x1, x2), K(x1, x3) ] [ K(x2, x1), K(x2, x2), K(x2, x3) ] [ K(x3, x1), K(x3, x2), K(x3, x3) ] (That will be a nxn matrix, where n is the number of training points.) Each element K(xi, xj) is calculated using the RBF kernel formula from above. Once you have this design matrix, you can use it in your Bayesian linear regression model. This allows you to handle non-linear relationships between the data points while still applying linear regression techniques. The most important equations in python would be: # Compute pairwise squared Euclidean distances between all points squared_distances = np.sum(X**2, axis=1).reshape(-1, 1) + np.sum(X**2, axis=1) - 2 * np.dot(X, X.T) # Apply the RBF kernel formula to the squared distances design_matrix = np.exp(-gamma * squared_distances)
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Hi, thanks for the great tutorial! I noticed that the example only covers the case with a single feature. Could you please explain how to adapt the code for multiple features? I'd love to understand how to handle the design matrix and update the regression model in that scenario. Thanks!
Hi Henry! The only difference will be in the way you build the design matrix (the rest is the same).
First, start with your original features, x1, x2, ..., xd. These represent the dimensions of your data. When you apply an RBF kernel, you map the data into a higher-dimensional space using the kernel function. For example. the RBF kernel is given by the formula:
K(xi, xj) = exp(-gamma * ||xi - xj||^2)
In this formula, xi and xj are two data points (represented as vectors of features), gamma is a parameter controlling the width of the kernel, and ||xi - xj||^2 is the squared distance between these two points. The kernel computes a similarity measure between data points based on this distance.
To incorporate this into Bayesian linear regression, you need to build a design matrix (also called the kernel matrix) where each element is the kernel value between two data points. The matrix will contain the similarities between every pair of data points.
For example, if you have three data points (x1, x2, x3), the design matrix (Phi(X)) will look like this:
Phi(X) = [ K(x1, x1), K(x1, x2), K(x1, x3) ] [ K(x2, x1), K(x2, x2), K(x2, x3) ] [ K(x3, x1), K(x3, x2), K(x3, x3) ]
(That will be a nxn matrix, where n is the number of training points.)
Each element K(xi, xj) is calculated using the RBF kernel formula from above.
Once you have this design matrix, you can use it in your Bayesian linear regression model. This allows you to handle non-linear relationships between the data points while still applying linear regression techniques.
The most important equations in python would be:
# Compute pairwise squared Euclidean distances between all points
squared_distances = np.sum(X**2, axis=1).reshape(-1, 1) + np.sum(X**2, axis=1) - 2 * np.dot(X, X.T)
# Apply the RBF kernel formula to the squared distances
design_matrix = np.exp(-gamma * squared_distances)