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Chapter 5: Problem 18

Linear Least-Squares Fit. Develop a function that will calculate slope \(m\) and intercept \(b\) of the least-squares line that best fits an input data set. The input data points \((x, y)\) will be passed to the function in two input arrays, \(x\) and \(y\). (The equations describing the slope and intercept of the least- squares line are given in Example \(4.7\) in the previous chapter.) Test your function using a test program and the 20-point input data set given in Table 5.2.

Short Answer

Expert verified
To calculate the slope and intercept of the least-squares line that best fits an input data set, we can implement the following function: ```python def linear_least_squares_fit(x, y): N = len(x) sum_x = sum(x) sum_y = sum(y) sum_xy = sum(x[i] * y[i] for i in range(N)) sum_x_sq = sum(x[i] ** 2 for i in range(N)) m = (N * sum_xy - sum_x * sum_y) / (N * sum_x_sq - sum_x ** 2) b = (sum_y - m * sum_x) / N return m, b ``` This function takes two input arrays, `x` and `y`, representing the data points, and returns the slope `m` and intercept `b` of the least-squares line using the mentioned formulas. To test this function, simply call it with desired input arrays and print the obtained slope and intercept values.

Step by step solution

01

Understanding the Slope and Intercept Formulas for Linear Least-Squares Fit

We will use the following equations to calculate the slope \(m\) and intercept \(b\) of the least-squares line that best fits the input data points \((x, y)\). The formulas are: 1. \(m = \frac{N\sum xy - \sum x\sum y}{N\sum x^2 - (\sum x)^2}\) 2. \(b = \frac{\sum y - m\sum x}{N}\) Here, N is the number of data points.
02

Developing the Function to Calculate Slope and Intercept

Now, we will develop the function that takes input arrays for data points (x, y) and calculates the slope 'm' and intercept 'b' using the formulas mentioned in step 1. The function should take the following input arguments: Input: - x - The array containing the x-coordinates of the data points - y - The array containing the y-coordinates of the data points Output: - The function should return the calculated slope 'm' and intercept 'b' Function: ``` def linear_least_squares_fit(x, y): N = len(x) sum_x = sum(x) sum_y = sum(y) sum_xy = sum(x[i] * y[i] for i in range(N)) sum_x_sq = sum(x[i] ** 2 for i in range(N)) m = (N * sum_xy - sum_x * sum_y) / (N * sum_x_sq - sum_x ** 2) b = (sum_y - m * sum_x) / N return m, b ```
03

Testing the Function with the Provided Data Set

We will now test the developed function using the 20-point input data set given in Table 5.2. Assume that we have already stored the data set in arrays 'x_values' and 'y_values'. Then, we can call the 'linear_least_squares_fit' function to calculate the slope and intercept: ``` slope, intercept = linear_least_squares_fit(x_values, y_values) print("Slope (m):", slope) print("Intercept (b):", intercept) ``` After running this test code, we will obtain the slope 'm' and intercept 'b' of the linear least-squares fit for the given input data set.

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Most popular questions from this chapter

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