Implement a line fitting model using the least squares method to find the best-fit line $y = mx + b$ for a given set of points. This task involves minimizing the sum of squared residuals between observed points and predicted values.

The least squares approach is a fundamental concept in model fitting and optimization, aiming to find the optimal parameters that result in the smallest possible sum of squared errors. In this case, the goal is to determine the slope $m$ and intercept $b$ of the line that minimizes the sum of squared residuals $E = \sum_i (y_i - (mx_i + b))^2$ .

To achieve this, the process involves the following steps:

Calculate the necessary summations of $x$ , $y$ , $x^2$ , and $xy$ for the given points.
Use these summations to derive the parameters $m$ and $b$ . The key to this process lies in the mathematical formulation of the normal equations, which provide a way to solve for $m$ and $b$ .

E = \sum_i (y_i - (mx_i + b))^2

This technique is widely used in computer vision for tasks such as image processing and object detection.

The input points are $(0,0)$ , $(1,1)$ , and $(2,2)$ , which all lie exactly on the line $y = x$ .
For a line $y = mx + b$ , this means choosing $m = 1$ and $b = 0$ makes every residual $y_i - (mx_i + b)$ equal to $0$ , so the total squared error $E$ is minimized.
Therefore, the fitted line parameters are $[m, b] = [1.0, 0.0]$ , matching the sample output.

Return [m, b] rounded to 4 decimal places

Editor

Python 3.13.1

Test Results

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Run code to see test results.

📘

Line Fitting

MediumOptimization

To achieve this, the process involves the following steps:

Calculate the necessary summations of $x$ , $y$ , $x^2$ , and $xy$ for the given points.
Use these summations to derive the parameters $m$ and $b$ . The key to this process lies in the mathematical formulation of the normal equations, which provide a way to solve for $m$ and $b$ .

E = \sum_i (y_i - (mx_i + b))^2

This technique is widely used in computer vision for tasks such as image processing and object detection.

Example:

Input:

fit_line([(0,0), (1,1), (2,2)])

Output:

[1.0, 0.0]

Reasoning:

The input points are $(0,0)$ , $(1,1)$ , and $(2,2)$ , which all lie exactly on the line $y = x$ .
For a line $y = mx + b$ , this means choosing $m = 1$ and $b = 0$ makes every residual $y_i - (mx_i + b)$ equal to $0$ , so the total squared error $E$ is minimized.
Therefore, the fitted line parameters are $[m, b] = [1.0, 0.0]$ , matching the sample output.

Constraints:

Return [m, b] rounded to 4 decimal places

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.