EXPLANATION NOTE 1

Overview

The code implements a simple linear regression calculation. Linear regression is a statistical method to model the relationship between a dependent variable (output) and one or more independent variables (inputs). In this case, the code calculates the linear regression output for a given set of input features.

Code Explanation

Initializing Variables

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# Linear regression variable
w0 = 7.17
xi = [453, 11, 86]
w = [0.01, 0.04, 0.002]

• w0: This is the intercept or bias term in the linear regression model.
• xi: This is a list of input feature values.
• w: This is a list of weights corresponding to each feature in xi.

Adding the Intercept to the Weight List

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# Adding the w0 to the w list
w_new = [w0] + w

• w_new: This creates a new list of weights that includes the intercept w0 as the first element, followed by the original weights w.

Dot Product Function

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# Linear regression function
def dot(xi, w):
    """
    Calculates the dot product between two vectors.

    Parameters:
    xi (list): The first vector.
    w (list): The second vector.

    Returns:
    float: The dot product of the two vectors.
    """
    n = len(xi)

    res = 0.0
    for j in range(n):
        res = res + xi[j] * w[j]

    return res

• dot(xi, w): This function calculates the dot product between two lists (vectors) xi and w.
• Parameters:
• xi: The first vector (list of input features).
• w: The second vector (list of weights).
• Returns: The dot product of the two vectors.
• Process:
• It initializes a result variable res to 0.
• It iterates over each element in the vectors, multiplies corresponding elements, and adds the result to res.
• Finally, it returns the computed dot product.

Linear Regression Function

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# Linear regression function
def linear_regression(xi):
    xi = [1] + xi
    return dot(xi, w_new)

• linear_regression(xi): This function calculates the linear regression output for a given input vector xi.
• Parameters:
• xi: The input feature vector.
• Returns: The output of the linear regression model.
• Process:
• It adds 1 to the beginning of the input vector xi. This 1 corresponds to the intercept term in the linear regression model.
• It calls the dot function to calculate the dot product of the modified input vector xi and the weight vector w_new.
• It returns the result of the dot product, which is the linear regression output.

Running the Linear Regression Function

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linear_regression(xi)

• This line calls the linear_regression function with xi as the input, and it calculates the linear regression output.

Putting It All Together

1. Initialization:
• w0 is the intercept (7.17).
• xi is the list of input features ([453, 11, 86]).
• w is the list of weights for these features ([0.01, 0.04, 0.002]).
2. Adding Intercept to Weights:
• w_new becomes [7.17, 0.01, 0.04, 0.002].
3. Dot Product Function:
• dot(xi, w) computes the sum of element-wise products of xi and w.
4. Linear Regression Function:
• linear_regression(xi) prepends 1 to xi (to account for the intercept) making it [1, 453, 11, 86].
• It computes the dot product of this new xi with w_new.
5. Execution:
• The function linear_regression(xi) is called, and it returns the calculated linear regression output.

Example Calculation

Let's manually calculate the result of the linear regression function to better understand the output.

1. Modified xi: [1, 453, 11, 86]
2. Weights w_new: [7.17, 0.01, 0.04, 0.002]
3. Dot product calculation:
• 1×7.17+453×0.01+11×0.04+86×0.0021×7.17+453×0.01+11×0.04+86×0.002
• 7.17+4.53+0.44+0.1727.17+4.53+0.44+0.172
• 12.31212.312

So, the output of linear_regression(xi) will be approximately 12.312.