* Least Squares Fit
*** Outline
***** Introduction
******* What do we want to do? Why?
********* What's a least square fit?
********* Why is it useful?
******* How are we doing it?
******* Arsenal Required
********* working knowledge of arrays
********* plotting
********* file reading
***** Procedure
******* The equation (for a single point)
******* It's matrix form
******* Getting the required matrices
******* getting the solution
******* plotting
*** Script
    Welcome. 
    
    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.

    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. 

    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. 

    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...
    {Show a slide here?}
    
    We have already seen (in a previous tutorial), how to read a file
    and obtain the data set. Let's quickly get the required data
    from our file. 

    In []: l = []
    In []: t = []
    In []: for line in open('pendulum.txt'):
    ....     point = line.split()
    ....     l.append(float(point[0]))
    ....     t.append(float(point[1]))
    ....
    ....

    Since, we have learnt to use arrays and know that they are more
    efficient, we shall use them. We convert the lists l and t to
    arrays and calculate the values of time-period squared. 

    In []: l = array(l)
    In []: t = array(t)
    In []: tsq = t*t

    Now we shall obtain A, in the desired form using some simple array
    manipulation 

    In []: A = array([l, ones_like(l)])
    In []: A = A.T
    
    Type A, to confirm that we have obtained the desired array. 
    In []: A
    Also note the shape of A. 
    In []: A.shape

    We shall now use the lstsq function, to obtain the coefficients m
    and c. lstsq returns a lot of things along with these
    coefficients. Look at the documentation of lstsq, for more
    information. 
    In []: result = lstsq(A,tsq)

    We extract the required coefficients, which is the first element
    in the list of things that lstsq returns, and store them into the variable coef. 
    In []: coef = result[0]

    To obtain the plot of the line, we simply use the equation of the
    line, we have noted before. T^2 = mL + c. 

    In []: Tline = coef[0]*l + coef[1]
    In []: plot(l, Tline)

    Also, it would be nice to have a plot of the points. So, 
    In []: 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. 

    Hope you enjoyed it. Thanks. 

*** Notes