Little Bits of Artificial Intelligence

Mathematical Explanation

Let’s go from the farmer’s story to the real math.

1. Linear Regression Recap:

We model the prediction as:

y = w^T x + b

where:

• x: input features (e.g., fertilizer),
• w: weights,
• b: bias/intercept.

We try to minimize the error between predicted and actual values.

2. Nonlinear Relationship:

When the relation is not linear, we map input x to a higher-dimensional space using a function ϕ(x):

y=w^Tϕ(x)+b

But computing ϕ(x) for all x is expensive.

3. Kernel Function:

A kernel function K(xi,xj) computes:

K(xi,xj)=⟨ϕ(xi),ϕ(xj)⟩

This gives us the dot product in high-dimensional space without computing ϕ(x) directly!

Some popular kernels:

• Linear Kernel: K(x,x′)=x^Tx′
• Polynomial Kernel: K(x,x′)=(x^Tx′+c)^d
• RBF / Gaussian Kernel:

Basic Math Knowledge Required

To understand and implement kernel regression, here’s what you should know: