least-squares.org
author Shantanu <shantanu@fossee.in>
Tue, 06 Apr 2010 23:43:58 +0530
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* Least Squares Fit
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*** Outline
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***** Introduction
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******* What do we want to do? Why?
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********* What's a least square fit?
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********* Why is it useful?
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******* How are we doing it?
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******* Arsenal Required
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********* working knowledge of arrays
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********* plotting
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********* file reading
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***** Procedure
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******* The equation (for a single point)
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******* It's matrix form
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******* Getting the required matrices
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******* getting the solution
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******* plotting
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*** Script
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    Welcome. 
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    In this tutorial we shall look at obtaining the least squares fit
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    of a given data-set. For this purpose, we shall use the same
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    pendulum data used in the tutorial on plotting from files.
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    To be able to follow this tutorial comfortably, you should have a
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    working knowledge of arrays, plotting and file reading. 
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    A least squares fit curve is the curve for which the sum of the
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    squares of it's distance from the given set of points is
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    minimum. We shall use the lstsq function to obtain the least
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    squares fit curve. 
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    In our example, we know that the length of the pendulum is
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    proportional to the square of the time-period. Therefore, we
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    expect the least squares fit curve to be a straight line. 
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    The equation of the line is of the form T^2 = mL+c. We have a set
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    of values for L and the corresponding T^2 values. Using this, we
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    wish to obtain the equation of the straight line. 
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    In matrix form...
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    {Show a slide here?}
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    We have already seen (in a previous tutorial), how to read a file
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    and obtain the data set. We shall quickly get the required data
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    from our file. 
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    In []: l = []
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    In []: t = []
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    In []: for line in open('pendulum.txt'):
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    ....     point = line.split()
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    ....     l.append(float(point[0]))
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    ....     t.append(float(point[1]))
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    ....
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    ....
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    Since, we have learnt to use arrays and know that they are more
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    efficient, we shall use them. We convert the lists l and t to
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    arrays and calculate the values of time-period squared. 
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    In []: l = array(l)
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    In []: t = array(t)
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    In []: tsq = t*t
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    Now we shall obtain A, in the desired form using some simple array
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    manipulation 
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    In []: A = array([l, ones_like(l)])
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    In []: A = A.T
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    Type A, to confirm that we have obtained the desired array. 
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    In []: A
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    Also note the shape of A. 
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    In []: A.shape
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    We shall now use the lstsq function, to obtain the coefficients m
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    and c. lstsq returns a lot of things along with these
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    coefficients. Look at the documentation of lstsq, for more
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    information. 
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    In []: result = lstsq(A,tsq)
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    We take put the required coefficients, which are the first thing
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    in the list of things that lstsq returns, into the variable coef. 
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    In []: coef = result[0]
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    To obtain the plot of the line, we simply use the equation of the
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    line, we have noted before. T^2 = mL + c. 
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    In []: Tline = coef[0]*l + coef[1]
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    In []: plot(l, Tline)
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    Also, it would be nice to have a plot of the points. So, 
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    In []: plot(l, tsq, 'o')
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    This brings us to the end of this tutorial. In this tutorial,
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    you've learnt how to obtain a least squares fit curve for a given
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    set of points. 
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    Hope you enjoyed it. Thanks. 
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*** Notes
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