Question

Mathematical models of physical phenomena are infinitely precise. However, we are not able to make measurements of physical phenomena with anywhere near that precision. Therefore, we often must “fit” the data we measure to a mathematical model we construct to describe the system. How well the data "fits" helps us understand how good our model is and how well our data collection systems perform.

Using the velocity and position data for a falling object provided by the instructor, do the following in MATLAB:

1. Add in instrumentation noise by introducing a random offset of your design. Explain the rationale for your range of values.
2. Fit the data to polynomial curves using the methods shown in the textbook.
3. Determine which "order" of curve best fits the data. The order refers to the highest power of the polynomial equation.
4. For both velocity and position, plot the curve and the data on the same figure.

velocity = 11, 31, 64, 112, 176, 259, 362
position = 2, 3, 4, 5, 6, 7, 8

Answer 1

Here we are going to demonstrate the effect of 3 polynomial orders (1 2 3) for the relation between velocity and position, obeying the models (between position p and velocity v)

%===================================================

clear all;
clc;

% Assuming position(p) to be a function of velocity(v) as for x=v and y=p giving y=f(x)

x = [11, 31, 64, 112, 176, 259, 362];
y = [2, 3, 4, 5, 6, 7, 8];
y = y + rand(1,length(y)); % Introducing uniform random noise having value between 0 to 1

% Fitting 1,2,3 order polynomials
poly1 = polyfit(x,y,1);
poly2 = polyfit(x,y,2);
poly3 = polyfit(x,y,3);

% Generting fitted values
yfit1 = polyval(poly1,x);
yfit2 = polyval(poly2,x);
yfit3 = polyval(poly3,x);

% Goodness of fit calculation
% == order-1 ==
% Compute the residual values as a vector of signed numbers:

yresid1 = y - yfit1;
yresid2 = y - yfit2;
yresid3 = y - yfit3;

% Square the residuals and total them to obtain the residual sum of squares:

SSresid1 = sum(yresid1.^2);
SSresid2 = sum(yresid2.^2);
SSresid3 = sum(yresid3.^2);

% Compute the total sum of squares of y by multiplying the variance of y by the number of observations minus 1:

SStotal = (length(y)-1) * var(y);

% Compute R2 using the formula given in the introduction of this topic:

rsq1 = 1 - SSresid1/SStotal;
rsq2 = 1 - SSresid2/SStotal;
rsq3 = 1 - SSresid3/SStotal;
fprintf('Goodness of fit for polynomials of order 1 = %f, order 2 = %f, order 3 = %f\n',rsq1,rsq2,rsq3);

% plot
figure
scatter(x,y);
hold on
plot(x,yfit1)
hold on
plot(x,yfit2)
hold on
plot(x,yfit3)
legend('Original data','Polynomial order-1','Polynomial order-2','Polynomial order-3');
xlabel('Velocity');
ylabel('Position');

%===================================================

sample output:

Goodness of fit for polynomials of order 1 = 0.905276, order 2 = 0.943981, order 3 = 0.995069

Note: the output on each run is highly likely to be different that the previous run due to the introduction of random noise in y values.