Tutorial: Linear Regression

Least Squares Regression

Part 1: Linear Regression (line)

Problem

Predict the pressure if the temperature is increased to 150C based on Charles's law for ideal gas P=kT, where k is a constant.

Tutorial: Matlab

T=[30	40	50	60	70	80];
P=[1.05	1.07	1.09	1.14	1.17	1.21];

% MATLAB function of curvefitting
% z=[a1, a0]; 
Z=polyfit(T,P,1)

% yopt=Z(1).*x+Z(2);
x=30:10:150;
yopt=polyval(Z,x);  

Exercise: MATLAB

Download the tutorial source file

• TU_Curvefitting_student_2024.mlx

Fill-in the blanks to create function [a0,a1] = linearFit(X, Y)

Exercise: C

Download the tutorial source code

• Assignment_LinearRegression_student.cpp

Fill-in the blanks to create functions that calculates coefficients of least squares regression (line)

void linearRegression (double z_opt[], double xdata[], double ydata[], int dataN)

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Part 2: Higher-order polynomial curve fitting (Optional)

Problem

Find the optimal higher-order polynomial to fit the given dataset. assume the model has n=4 order polynomial form

Exercise: MATLAB

Xdata = 0:0.4:6;
Ydata = [0, 3, 4.5, 5.8, 5.9, 5.8, 6.2, 7.4, 9.6, 15.6, 20.7, 26.7, 31.1, 35.6, 39.3, 41.5];

% MATLAB function of curvefitting
% z=[an,..., a1, a0]; 

% polynomial order n=4
n=4;
Z=polyfit(Xdata,Ydata,n)

% yhat on the fitted curve:
% yopt=Z(5).*(x.^4) +...+ +Z(0);
Yopt=polyval(Z,Xdata);

figure
plot(Xdata,Ydata, '*r')
hold on
plot(Xdata,Yopt, '-b')
xlabel('strain','fontsize',15)
ylabel('stress','fontsize',15)
title('Polyfit')

Exercise: C (optional)

Fill-in the blanks to create functions that calculates coefficients of least squares regression of Nth order polynomial

Download the tutorial source code

• Assignment_CurveFit_student.cpp

void polyFit(double vecZ[], double vecX[], double vecY[], int n);

If you choose to use Matrix structure

Matrix polyFit_mat(Matrix _vecX, Matrix _vecY, int n);

	double T[] = { 30, 40, 50, 60, 70, 80 };
	double P[] = { 1.05, 1.07, 1.09, 1.14, 1.17, 1.21 };
	double Z_Q1[2] = { 0 };
	int n = 1;	// nth order
	int m_Q1 = 6;	// length of dataset

	// Option 1: using 1D array
	n = 1;
	polyFit(Z_Q1, T, P, n);
	
	// Option 2	
	// Delete the below if you selected Option 1
	Matrix matT = arr2Mat(T, m_Q1, 1);
	Matrix matP = arr2Mat(P, m_Q1, 1);
	Matrix vecZ_Q1 = polyFit_mat(matT, matP, n);
	printMat(vecZ_Q1, "Z_Q1");