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diff --git a/least-squares.org b/least-squares.org index 9cb0f1e..e4e9402 100644 --- a/least-squares.org +++ b/least-squares.org @@ -21,26 +21,34 @@ In this tutorial we shall look at obtaining the least squares fit of a given data-set. For this purpose, we shall use the same pendulum data that we used in the tutorial on plotting from files. - Hope you have the file with you. To be able to follow this tutorial comfortably, you should have a working knowledge of arrays, plotting and file reading. A least squares fit curve is the curve for which the sum of the squares of it's distance from the given set of points is - minimum. We shall use the lstsq function to obtain the least - squares fit curve. + minimum. + + Previously, when we plotted the data from pendulum.txt we got a + scatter plot of points as shown. In our example, we know that the length of the pendulum is - proportional to the square of the time-period. Therefore, we - expect the least squares fit curve to be a straight line. + proportional to the square of the time-period. But when we plot + the data using lines we get a distorted line as shown. What + we expect ideally, is something like the redline in this graph. + From the problem we know that L is directly proportional to T^2. + But experimental data invariably contains errors and hence does + not produce an ideal plot. The best fit curve for this data has + to be a linear curve and this can be obtained by performing least + square fit on the data set. We shall use the lstsq function to + obtain the least squares fit curve. The equation of the line is of the form T^2 = mL+c. We have a set of values for L and the corresponding T^2 values. Using this, we wish to obtain the equation of the straight line. In matrix form the equation is represented as shown, - Tsq = A.p where Tsq is an NX1 matrix, and A is an NX2 as shown. + Tsq = A.p where Tsq is an NX1 matrix, and A is an NX2 matrix as shown. And p is a 2X1 matrix of the slope and Y-intercept. In order to obtain the least square fit curve we need to find the matrix p @@ -65,9 +73,14 @@ A = array([l, ones_like(l)]) As we have seen in a previous tutorial, ones_like() gives an array similar - in shape to the given array, in this case l, with all the elements as 1. + in shape to the given array, in this case l, with all the elements as 1. + Please note, this is how we create an array from an existing array. + + Let's now look at the shape of A. + A.shape + This is an 2X90 matrix. But we need a 90X2 matrix, so we shall transpose it. - A = A.T to get the transpose of the given array. + A = A.T Type A, to confirm that we have obtained the desired array. A @@ -88,14 +101,15 @@ line, we have noted before. T^2 = mL + c. Tline = coef[0]*l + coef[1] - plot(l, Tline) + plot(l, Tline, 'r') Also, it would be nice to have a plot of the points. So, plot(l, tsq, 'o') This brings us to the end of this tutorial. In this tutorial, you've learnt how to obtain a least squares fit curve for a given - set of points. + set of points using lstsq. There are other curve fitting functions + available in Pylab such as polyfit. Hope you enjoyed it. Thanks. |