Question

Overview: In this exercise, you will be writing a function that finds what spring constant k is required in order to minimise the RMSE of a spring stiffness test. To find the spring constant k_best that minimizes RMSE, you will need to apply MATLAB's in-built function fminsearch and the function rmse (data,k) defined in Exercise 3. You do not have to code this function yourself here. This function is provided to you in AMS and can be called as rmse (data,k). This function was prepared by us as a working solution to Exercise 3. You can execute it in your code by calling it like any other Matlab built-in function. In AMS, this happens automatically. In Matlab, this requires you to put the file rmse.p (available from Blackboard with the template m-files) in the working directory, or to use your solution file rmse.m if you implemented it correctly in Exercice 3. fminsearch is a function in MATLAB that numerically solves a local minimum of a function. x = fminsearch(Func,x0) will numerically approximate the x value that minimizes the function Func(x) and uses x as a starting point. Inputs: data - this is a matrix containing data from a spring stiffness test. Column 1 is the force data and column 2 is the displacement data. Your code should work regardless of how many rows data has. Ik_guess - this is an initial guess for the spring stiffness. Outputs: k_best - the value for k that minimises root mean square error between the displacement data, and the displacement predicted from x= Fik. Process 1. Create an anonymous function k rmse (data, k) that represents the RMSE of the data and the fit line, for any value of k and that can be used as input to fminsearch (remember the creation of anymous functions uses the "@(var)" notation) 2. Apply fminsearch to solve for the k value that minimizes RMSE. Use k_guess as the starting point for the approximation. Function Template function k_best = rmseMin2(data, k_guess) %INSERT CODE end Test your function in MATLAB You can test your function on the data matrix you were provided in Exercise 1 this week, and then redo Exercice 1 with the value of best_k returned by rmseMin2 to see if you get a better line of fit: >> data = [0.1, 2.26; 0.2, 4.43; 0.3, 5.85; 0.4, 8.42; 0.5, 9.91; 0.6, 11.97]; >> rmseMin2(data, 0.058) = the value of k that would provide the line of best fit in Exercice 1 (least square error).

Answer 1

MATLAB Script:

close all
clear
clc

data = [0.1, 2.26; 0.2, 4.43; 0.3, 5.85; 0.4, 8.42; 0.5, 9.91; 0.6, 11.97];
guess_k = 0.058;
RMSE = rmse(data, guess_k); % RMSE before applying rmseMin2()
fprintf('RMSE (with guessed value of k = %.4f) = %.4f\n', guess_k, RMSE)

best_k = rmseMin2(data, guess_k);
RMSE = rmse(data, best_k); % RMSE after applying rmseMin2()
fprintf('RMSE (with best value of k = %.4f) = %.4f\n', best_k, RMSE)
disp('Hence, there''s clearly an improvement as can be seen from reduced RMSE.')

function best_k = rmseMin2(data, guess_k)
Func = @(k) rmse(data, k);
best_k = fminsearch(Func, guess_k);
end

function RMSE = rmse(data, k)
% x = F/k
% where F = force (column 1), x = displacement (column 2)
  
x_measured = data(:,2); % second column
x_theoretical = data(:,1)/k; % F/k
RMSE = sqrt(sum((x_measured - x_theoretical).^2)/length(x_measured));
end

Output:

RMSE (with guessed value of k = 0.0580) = 1.1786
RMSE (with best value of k = 0.0496) = 0.2671
Hence, there's clearly an improvement as can be seen from reduced RMSE.