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ECOR 2606 - Lecture 16

ECOR 2606 - Lecture 16 - Problem(text1329: Year Population...

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3/8/2010 1 Problem (text 13 29): The table below gives the population of a small but growing suburb over a twenty year period. Year 0 5 10 15 20 Population 100 200 450 950 2000 The growth is assumed to be exponential: population = α *exp( β t ), where t is a time in years What values of α and β best fit the data? What can the population be expected to be after 25 years? Ideally a nonlinear regression technique would be used to find the α and β that absolutely minimize the sum of the squares of the errors between the data points and the fitted curve. A good (but not perfect) answer can be obtained more simply by transforming the data and using linear regression. The basic procedure: y = α *exp( β x ) ln( y ) = ln( α ) + β x Let y ’ = ln( y ) Then y ’= ax + b , where a = β and b = ln( α ) Use linear regression to find best a and b . Then find α and β by applying α =exp( b ) and β = a Matlab part 1: x = [0 5 10 15 20]; y = [100 200 450 950 2000]; yt = log(y); % transform the y values p = polyfit (x, yt, 1); % fit a straight line to the transformed data fitt = @(x) p(1) * x + p(2); % function for the fitted line
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3/8/2010 2 Matlab part 2: figure (1) plot (x, yt, 'x', x, fitt(x), 'MarkerSize', 10); grid on; xlabel ('Years'); ylabel ('ln(Population)'); fprintf ('For transformed data a = %f, b = %f, r = %f\n', ... p(1) p(2) orrelate ( t fitt)) p(1), p(2), correlate (x, yt, fitt)); The first data point doesn’t appear. This seems to happen a lot (a Matlab bug?). The best fit line is y = 0.1510 * x + 4.5841 The correlation coefficient for this straight line and the transformed data is 0.999789. Matlab part 3: % calculate alpha and beta alpha = exp(p(2)); beta = p(1); fit = @(x) alpha * exp(beta * x); % function for fitted curve % need lots of x values to get a smooth plot of the fitted curve
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