To derive the solutions for the parameters $a$ and $b$ of the linear model $y = ax + b$ given $N$ training samples $\{(x_1, y_1), (x_2, y_2), \ldots, (x_N, y_N)\}$, we can use the method of least squares. The objective is to minimize the sum of squared errors between the observed values $y_i$ and the predicted values $\hat{y}_i = ax_i + b$.

Step 1: Define the Error Function

The error (or cost) function $E(a, b)$ is given by:

$E(a, b) = \sum_{i=1}^{N} (y_i - (ax_i + b))^2$

Our goal is to find $a$ and $b$ that minimize this error function.

Step 2: Compute Partial Derivatives

To minimize $E(a, b)$, we need to take partial derivatives with respect to $a$ and $b$ and set them to zero.

1. Partial Derivative with respect to $a$:

$\frac{\partial E(a, b)}{\partial a} = -2 \sum_{i=1}^{N} x_i (y_i - (ax_i + b))$

2. Partial Derivative with respect to $b$:

$\frac{\partial E(a, b)}{\partial b} = -2 \sum_{i=1}^{N} (y_i - (ax_i + b))$

Set both partial derivatives to zero:

$\sum_{i=1}^{N} x_i (y_i - (ax_i + b)) = 0$

$\sum_{i=1}^{N} (y_i - (ax_i + b)) = 0$

Step 3: Simplify the Equations

Simplify the equations obtained from the derivatives:

1. Simplifying for $a$:

$\sum_{i=1}^{N} x_i y_i = a \sum_{i=1}^{N} x_i^2 + b \sum_{i=1}^{N} x_i$

2. Simplifying for $b$:

$\sum_{i=1}^{N} y_i = a \sum_{i=1}^{N} x_i + Nb$

Step 4: Solve the System of Equations

Let's denote:

$S_x = \sum_{i=1}^{N} x_i, \quad S_y = \sum_{i=1}^{N} y_i, \quad S_{xy} = \sum_{i=1}^{N} x_i y_i, \quad S_{xx} = \sum_{i=1}^{N} x_i^2$

Substituting these into the equations, we get:

1. Equation for $a$:

$S_{xy} = a S_{xx} + b S_x$

2. Equation for $b$:

$S_y = a S_x + Nb$

Solve these equations for $a$ and $b$:

From the second equation:

$b = \frac{S_y - a S_x}{N}$

Substitute $b$ in the first equation:

$S_{xy} = a S_{xx} + \left(\frac{S_y - a S_x}{N}\right) S_x$

Simplify:

$S_{xy} = a S_{xx} + \frac{S_x S_y - a S_x^2}{N}$

$S_{xy} = a \left(S_{xx} - \frac{S_x^2}{N}\right) + \frac{S_x S_y}{N}$

Solving for $a$:

$a = \frac{S_{xy} - \frac{S_x S_y}{N}}{S_{xx} - \frac{S_x^2}{N}}$

Now, substitute $a$ back into the equation for $b$:

$b = \frac{S_y - a S_x}{N}$

Final Solution

The values for $a$ and $b$ are:

$a = \frac{N \sum_{i=1}^{N} x_i y_i - \sum_{i=1}^{N} x_i \sum_{i=1}^{N} y_i}{N \sum_{i=1}^{N} x_i^2 - (\sum_{i=1}^{N} x_i)^2}$

$b = \frac{\sum_{i=1}^{N} y_i - a \sum_{i=1}^{N} x_i}{N}$

These equations provide the optimal values of $a$ and $b$ for the linear model $y = ax + b$.

Would you like more details or have any questions? Here are some related questions:

1. How does the least squares method handle outliers in the data?
2. Can you provide an example with specific data points to calculate $a$ and $b$?
3. How does this derivation change for multiple linear regression with more than one predictor variable?
4. What are the assumptions underlying the least squares approach?
5. How does adding a regularization term affect the values of $a$ and $b$?

Tip: Always check your data for outliers and linearity before applying linear regression to ensure reliable results.