.. Author              : Nishanth

   Internal Reviewer 1 : Puneeth

   Internal Reviewer 2 : 

   External Reviewer   :

Hello friends and welcome to the tutorial on Least Square Fit

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In this tutorial, we shall look at generating the least square fit line for a

given set of points.

First let us have a look at the problem.

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We have an input file generated from a simple pendulum experiment.

It contains two columns of data. The first column is the length of the

pendulum and the second is the corresponding time period of the pendulum.

As we know, the square of time period of a pendulum is directly proportional to

its length, we shall plot l vs t^2 and verify this. 

#[Puneeth:] removed the explanation about loadtxt and unpack

 option. It's been done in another LO already. simple dependency 

 should work?

To read the input file and parse the data, we are going to use the

loadtxt function.  Type 

::

    l, t = loadtxt("/home/fossee/pendulum.txt", unpack=True)

    l

    t

We can see that l and t are two sequences containing length and time values

correspondingly.

Let us first plot l vs t^2. Type

::

    tsq = t * t

    plot(l, tsq, 'bo')

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#[Puneeth:] Moved explanation of least square fit here. seems more

apt. 

We can see that there is a visible linear trend, but we do not get a

straight line connecting them. We shall, therefore, generate a least

square fit line.

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As shown in the slide, we are first going to generate the two matrices

tsq and A. Then we are going to use the ``lstsq`` function to find the

values of m and c.

let us now generate the A matrix with l values.

We shall first generate a 2 x 90 matrix with the first row as l values and the

second row as ones. Then take the transpose of it. Type

::

    inter_mat = array((l, ones_like(l)))

    inter_mat

We see that we have intermediate matrix. Now we need the transpose. Type

::

    A = inter_mat.T

    A

Now we have both the matrices A and tsq. We only need to use the ``lstsq``

Type

::

    result = lstsq(A, tsq)

The result is a sequence of values. The first item in this sequence,

is the matrix p i.e., the values of m and c. Hence, 

::

    m, c = result[0]

    m

    c

Now that we have m and c, we need to generate the fitted values of t^2. Type

::

    tsq_fit = m * l + c

    plot(l, tsq, 'bo')

    plot(l, tsq_fit, 'r')

We get the least square fit of l vs t^2

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%% 2 %% change the label on y-axis to "y" and save the lines of code

        accordingly

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This brings us to the end of the tutorial.

we have learnt

 * how to generate a least square fit

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#[Nishanth]: Will add this line after all of us fix on one.

This tutorial was created as a part of FOSSEE project, NME ICT, MHRD India

Hope you have enjoyed and found it useful.

Thank you

Questions

=========

 1. What does ones_like([1, 2, 3]) produce

   a. array([1, 1, 1])

   #. [1, 1, 1]

   #. [1.0, 1.0, 1.0]

   #. Error

 2. What does ones_like([1.2, 3, 4, 5]) produce

   a. [1.2, 3, 4, 5]

   #. array([1.0, 1.0, 1.0, 1.0])

   #. array([1, 1, 1, 1])

   #. array([1.2, 3, 4, 5])

 3. What is the shape of the