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Two MATLAB® functions can model your data with a polynomial.
Polynomial Fit Functions
polyfit(x,y,n) finds the coefficients of a polynomial p(x) of degree n that fits the y data by minimizing the sum of the squares of the deviations of the data from the model (least-squares fit).
polyval(p,x) returns the value of a polynomial of degree n that was determined by polyfit, evaluated at x.
For example, suppose you measure a quantity y at several values of time t:
t = [0 0.3 0.8 1.1 1.6 2.3]; y = [0.6 0.67 1.01 1.35 1.47 1.25]; plot(t,y,'o')
Plot of y Versus t
You can try modeling this data using a second-degree polynomial function:
The unknown coefficients a0, a1, and a2 are computed by minimizing the sum of the squares of the deviations of the data from the model (least-squares fit).
To find the polynomial coefficients, type the following at the MATLAB prompt:
MATLAB calculates the polynomial coefficients in descending powers:
p = -0.2942 1.0231 0.4981
The second-degree polynomial model of the data is given by the following equation:
To plot the model with the data, evaluate the polynomial at uniformly spaced times t2 and overlay the original data on a plot:
t2 = 0:0.1:2.8; % Define a uniformly spaced time vector y2=polyval(p,t2); % Evaluate the polynomial at t2 figure plot(t,y,'o',t2,y2) % Plot the fit on top of the data % in a new Figure window
Plot of Data (Points) and Model (Line)
Use the following syntax to calculate the residuals:
y2=polyval(p,t); % Evaluate model at the data time vector res=y-y2; % Calculate the residuals by subtracting figure, plot(t,res,'+') % Plot the residuals
Plot of the Residuals
Notice that the second-degree fit roughly follows the basic shape of the data, but does not capture the smooth curve on which the data seems to lie. There appears to be a pattern in the residuals, which indicates that a different model might be necessary. A fifth-degree polynomial (shown next) does a better job of following the fluctuations in the data.
Repeat the exercise, this time using a fifth-degree polynomial from polyfit:
p5= polyfit(t,y,5) p5 = 0.7303 -3.5892 5.4281 -2.5175 0.5910 0.6000 y3 = polyval(p5,t2); % Evaluate the polynomial at t2 figure plot(t,y,'o',t2,y3) % Plot the fit on top of the data % in a new Figure window
Fifth-Degree Polynomial Fit
When a polynomial function does not produce a satisfactory model of your data, you can try using a linear model with nonpolynomial terms. For example, consider the following function that is linear in the parameters a0, a1, and a2, but nonlinear in the t data:
You can compute the unknown coefficients a0, a1, and a2 by constructing and solving a set of simultaneous equations and solving for the parameters. The following syntax accomplishes this by forming a design matrix, where each column represents a variable used to predict the response (a term in the model) and each row corresponds to one observation of those variables:
% Enter t and y as columnwise vectors t = [0 0.3 0.8 1.1 1.6 2.3]'; y = [0.6 0.67 1.01 1.35 1.47 1.25]'; % Form the design matrix X = [ones(size(t)) exp(-t) t.*exp(-t)]; % Calculate model coefficients a = X\y a = 1.3983 - 0.8860 0.3085
Therefore, the model of the data is given by
Now evaluate the model at regularly spaced points and plot the model with the original data, as follows:
T = (0:0.1:2.5)'; Y = [ones(size(T)) exp(-T) T.*exp(-T)]*a; plot(T,Y,'-',t,y,'o'), grid on
When y is a function of more than one predictor variable, the matrix equations that express the relationships among the variables must be expanded to accommodate the additional data. This is called multiple regression.
Suppose you measure a quantity y for several values of x1 and x2. Enter these variables in the MATLAB Command Window, as follows:
x1 = [.2 .5 .6 .8 1.0 1.1]'; x2 = [.1 .3 .4 .9 1.1 1.4]'; y = [.17 .26 .28 .23 .27 .24]';
A model of this data is of the form
Multiple regression solves for unknown coefficients a0, a1, and a2 by minimizing the sum of the squares of the deviations of the data from the model (least-squares fit).
Construct and solve the set of simultaneous equations by forming a design matrix, X, and solving for the parameters by using the backslash operator:
X = [ones(size(x1)) x1 x2]; a = X\y a = 0.1018 0.4844 -0.2847
The least-squares fit model of the data is
To validate the model, find the maximum of the absolute value of the deviation of the data from the model:
Y = X*a; MaxErr = max(abs(Y - y)) MaxErr = 0.0038
This value is much smaller than any of the data values, indicating that this model accurately follows the data.
In this example, you use MATLAB functions to accomplish the following:
This example uses the data in census.mat, which contains U.S. population data for the years 1790 to 1990.
To load and plot the data, type the following commands at the MATLAB prompt:
load census plot(cdate,pop,'ro')
This adds the following two variables to the MATLAB workspace:
cdate is a column vector containing the years 1790 to 1990 in increments of 10.
pop is a column vector with the U.S. population numbers corresponding to each year in cdate.
The following plot of the data shows a strong pattern, which indicates a high correlation between the variables.
U.S. Population from 1790 to 1990
In this portion of the example, you determine the statistical correlation between the variables cdate and pop to justify modeling the data. For more information about correlation coefficients, see Linear Correlation.
Type the following syntax at the MATLAB prompt:
MATLAB calculates the following correlation-coefficient matrix:
ans = 1.0000 0.9597 0.9597 1.0000
The diagonal matrix elements represent the perfect correlation of each variable with itself and are equal to 1. The off-diagonal elements are very close to 1, indicating that there is a strong statistical correlation between the variables cdate and pop.
% Calculate fit parameters [p,ErrorEst] = polyfit(cdate,pop,2); % Evaluate the fit pop_fit = polyval(p,cdate,ErrorEst); % Plot the data and the fit plot(cdate,pop_fit,'-',cdate,pop,'+'); % Annotate the plot legend('Polynomial Model','Data','Location','NorthWest'); xlabel('Census Year'); ylabel('Population (millions)');
The following figure shows that the quadratic-polynomial fit provides a good approximation to the data:
To calculate the residuals for this fit, type the following syntax at the MATLAB prompt:
res = pop - pop_fit; figure, plot(cdate,res,'+') title('Residuals for the Quadratic Polynomial Model')
Notice that the plot of the residuals exhibits a pattern, which indicates that a second-degree polynomial might not be appropriate for modeling this data.
Confidence bounds are confidence intervals for a predicted response. The width of the interval indicates the degree of certainty of the fit.
The following syntax uses an interval of , which corresponds to a 95% confidence interval for large samples:
% Evaluate the fit and the prediction error estimate (delta) [pop_fit,delta] = polyval(p,cdate,ErrorEst); % Plot the data, the fit, and the confidence bounds plot(cdate,pop,'+',... cdate,pop_fit,'g-',... cdate,pop_fit+2*delta,'r:',... cdate,pop_fit-2*delta,'r:'); % Annotate the plot xlabel('Census Year'); ylabel('Population (millions)'); title('Quadratic Polynomial Fit with Confidence Bounds') grid on
The 95% interval indicates that you have a 95% chance that a new observation will fall within the bounds.