* 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

    In our example, we know that the length of the pendulum is

    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, 

    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

    Let's get started. As you can see, the file pendulum.txt

    is on our Desktop and hence we navigate to the Desktop by typing 

    cd Desktop. Let's now fire up IPython: ipython -pylab

    We have already seen (in a previous tutorial), how to read a file

    and obtain the data set using loadtxt(). Let's quickly get the required data

    from our file. 

    l, t = loadtxt('pendulum.txt', unpack=True)

    loadtxt() directly stores the values in the pendulum.txt into arrays l and t

    Let's now calculate the values of square of the time-period. 

    tsq = t*t

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

    manipulation 

    A = array([l, ones_like(l)])

    As we have seen in a previous tutorial, ones_like() gives an array similar

    Type A, to confirm that we have obtained the desired array. 

    A

    Also note the shape of A. 

    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. We may look at the documentation of lstsq, for more

    information by typing lstsq? 

    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. 

    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. 

    Tline = coef[0]*l + coef[1]

    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

    Hope you enjoyed it. Thanks. 

*** Notes