 derive solutions for the parameters a and b of the linear model
y = ax + b and for N training samples {(x1, y1),(x2, y2), . . . ,(xN , yN )}.

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.

derive solutions for the parameters a and b of the linear model y = ax + b and for N training samples {(x1, y1),(x2, y2), . . . ,(xN , yN )}.

Explore how to derive the parameters a and b for the linear model y = ax + b using the least squares method. This detailed guide covers key concepts in linear regression, including essential formulas and theorems. Ideal for undergraduate students studying statistics or machine learning.

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