Question

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

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

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.

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 ```

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.

Key concepts

Slope Calculation

In the world of linear least-squares fitting, the slope \(m\) is a crucial component. It represents how much the dependent variable \(y\) changes for a unit change in the independent variable \(x\). To calculate this slope effectively, we use a special formula designed for a least-squares approach:\[m = \frac{N\sum xy - \sum x\sum y}{N\sum x^2 - (\sum x)^2}\]Here's a simple breakdown:

• \(\sum x\), \(\sum y\): Sum of all the values in the respective arrays.
• \(\sum xy\): Sum of products of corresponding \(x\) and \(y\) values.
• \(\sum x^2\): Sum of squares of the \(x\) values.
• \(N\): Total number of data points.

Once you plug these values into the equation, solving for \(m\) gives you the slope of the line that best fits your data. Short and simple, this slope is a measure of your data's trend.

Intercept Calculation

The intercept \(b\) in the least-squares line equation tells us where the line crosses the y-axis. It's the value of \(y\) when \(x\) is zero. Calculating the intercept is straightforward once the slope \(m\) is known:\[b = \frac{\sum y - m\sum x}{N}\]Let's unfold this:

• \(\sum y\): Sum of all \(y\) values.
• \(m\sum x\): The product of the slope and the sum of all \(x\) values.
• \(N\): Number of data points, useful for averaging.

By substituting these into the equation, you can quickly find \(b\), painting a complete picture of your least-squares line. Understanding \(b\) helps in interpreting the dataset's initial value or starting point.

MATLAB Function Development

Developing a function to automate calculations can greatly simplify data analysis. In MATLAB, creating a least-squares fitting function involves coding the formula for slope and intercept calculation into a single function. Consider a Python example, which can be easily adapted to MATLAB:```pythondef 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```Here's a closer look at how this works:

• Input Arrays: Provide the x and y data as input arrays.
• Summations: Use loops or array operations to compute necessary summations.
• Calculations: Implement the formulas for \(m\) and \(b\).
• Return Values: Output the calculated slope and intercept.

In MATLAB, you'd leverage similar mathematical constructs with slightly different syntax. This function forms the backbone of many data analysis tasks, giving clear interpretations quickly.