Nancyvarkey at 04:14, 14 November 2018

2018-11-14T04:14:17Z

Priyacst: Created page with "'''Title of script''': '''Least Square Fit''' '''Author: Aditya Palaparthy, Arun KP''' '''Keywords: Python, IPython, Linear Regression, lstsq(), least square fit, video tuto..."

2018-11-01T09:14:07Z

Created page with "'''Title of script''': '''Least Square Fit''' '''Author: Aditya Palaparthy, Arun KP''' '''Keywords: Python, IPython, Linear Regression, lstsq(), least square fit, video tuto..."

New page

'''Title of script''': '''Least Square Fit'''

'''Author: Aditya Palaparthy, Arun KP'''

'''Keywords: Python, IPython, Linear Regression, lstsq(), least square fit, video tutorial'''

{| style="border-spacing:0;"
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| <center>'''Visual Cue'''</center>
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| <center>'''Narration'''</center>

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide title
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Welcome to the spoken tutorial on '''Least Square Fit'''.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide

Objectives
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| In this tutorial, you will learn to,

* Generate the '''least square fit '''line for a given set of '''points.'''

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide

System Specifications
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| To record this tutorial, I am using

* '''Ubuntu Linux 16.04''' operating system
* '''Python 3.4.3 '''and
* '''IPython 5.1.0'''

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide

Pre-requisites
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| To practise this tutorial, you should know about

* using '''plot''' interactively
* '''loading''' '''data''' from '''files'''
* '''using arrays''' and '''matrices '''and
* theoretical knowledge of''' least square method'''

If not, see the relevant '''Python '''tutorials on this website.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Slide:

Pendulum.txt
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Before beginning this tutorial,

Please download the file''' pendulum.txt '''from the '''Code files''' link of this tutorial.

Save it in the current working directory.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide

Example
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Let us start the tutorial with the help of an example.

Generate a least square fit line for''' L '''versus '''t square''' using the data in the file ''''pendulum.txt''''.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Open the file '''pendulum.txt'''
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Let us open the file '''pendulum.txt.'''

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Open the file 'pendulum.txt' and show
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| This is an input '''file''' generated from a simple '''pendulum''' '''experiment'''.

The first column is the '''length''' of the '''pendulum.'''

The second is the corresponding '''time''' '''period''' of the '''pendulum'''.

The '''square''' of '''time''' '''period''' is directly '''proportional''' to its '''length.'''

We shall plot '''L''' versus '''t square''' and verify this.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Open '''terminal'''
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Let us start '''ipython.'''

Open the '''terminal'''.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type '''ipython3'''

| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type '''ipython3 '''and press '''Enter'''.

From here onwards remember to press the Enter key after typing every command on the terminal.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

'''from numpy import loadtxt'''

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

| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| To read the input file and '''parse''' the '''data''', we are going to use the '''loadtxt''' '''function'''.

Type,

'''from numpy import loadtxt'''

Now type,

Capital '''L '''''comma''''' t '''''is equal to''''' loadtxt''''' inside brackets inside double quotes '''''pendulum.txt '''''comma''''' unpack '''''is equal to''''' True'''

'''loadtxt''' is a method available in the '''numpy''' library.

Since '''True''' is passed to '''unpack''' argument, the returned array is transposed.

This means that we will get one array per column in the file.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

'''L'''

'''t'''

| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

Capital '''L'''

Then type,

'''t'''

We can see that''' L '''and '''t''' are '''length''' and '''time''' values.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type, '''loadtxt?'''
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| To know more about '''loadtxt, '''type '''loadtxt '''''question mark''

Press '''q '''to exit.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

'''import matplotlib.pyplot as plt'''

'''tsq = t * t '''

'''plt.plot(L, tsq, 'bo')'''

Highlight bo in the code

Type

'''plt.show()'''
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Let us first '''plot''' '''L '''versus '''t square. '''

Type as shown.

Here '''bo''' represents the blue circle marker.

Then type,

'''plt.show '''''open and close brackets''

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show plot window and highlight the graph
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| We can see that there is a '''visible linear trend''', but we do not get a straight line connecting them.

Looking at the trend, we can now propose a model for the data.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Close the image
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Let me close this image.

|-
| style="background-color:#ffffff;border-top:none;border-bottom:0.5pt solid #000000;border-left:0.5pt solid #000000;border-right:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Slide:

'''least''' '''square fit line'''

| style="background-color:#ffffff;border-top:none;border-bottom:0.5pt solid #000000;border-left:0.5pt solid #000000;border-right:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| We need to '''fit''' a '''line''' through '''points''' for the the '''equation'''

Capital '''T '''''square is''''' '''''equal to''''' m '''''asterisk''''' '''Capital '''L '''''plus''''' c'''

where '''m''' represents the slope of the line and '''c''' represents the intercept of the line.

We will obtain '''m''' and '''c''' using linear regression.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| slide:

'''steps for least''' '''square fit line'''
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Let us see the steps to generate a '''least''' '''square fit line.'''

* First generate the two '''matrices''' '''tsq''' and '''A'''.
* Use the '''lstsq''' '''function''' to find the values of the slope '''m''' and the intercept '''c'''.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Slide Matrix Formulation
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| In matrix form, the equation can be represented as

'''tsq''' ''is equal to'' '''A '''''asterisk '''''p'''

'''tsq''' is a one-dimensional array of size '''n'''.

Each element of this array will contain the square of the time period.

'''A''' is a matrix of size '''n '''by '''2'''.

The first column will contain the length of the pendulum.

The second column will contain the number 1.

'''p''' is a one-dimensional array of size 2.

The first row contains the slope of the line.

The second row contains the intercept of the line.

We need to find '''p''' to plot the line.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

'''from numpy import array,ones_like'''

'''inter_mat = array((L, ones_like(L)))'''

'''inter_mat'''

| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Let us now generate the '''A''' '''matrix''' with '''L''' values.

We shall generate a '''matrix''' with the first '''row''' as '''L''' values and the second '''row''' as ones.

Then take the transpose of it.

Type as shown.

We see that we have intermediate matrix.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

'''A = inter_mat.T '''

'''A'''
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Now we need the transpose.

Type,

Capital '''A '''is equal to''' inter''''' underscore '''''mat '''dot''' '''Capital '''T '''

Now type,

Capital '''A'''

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

'''from numpy.linalg import lstsq'''

'''result = lstsq(A, tsq, rcond=None)'''

Type,

'''result'''
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Now we have both the '''matrices''' '''A''' and '''tsq'''.

We only need to use '''lstsq.'''

Type as shown.

Now to see the result, type

'''result'''

The '''result''' is a sequence of '''values'''.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

'''<nowiki>p = result[0]</nowiki>'''

'''m, c = p '''

'''m'''

'''c'''
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Now we will store result of index 0 in '''m''' and '''c''' respectively.

Type,

'''p '''''is equal to''''' result '''''inside square brackets''''' 0'''

Then type,

'''m '''''comma '''''c '''''is equal to''''' p'''

Now type,

'''m'''

'''c'''

We can see the values for '''m''' and '''c'''.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Type,

'''tsq_fit = m * L + c '''

'''plt.plot(L, tsq, 'bo')'''

'''plt.plot(L, tsq_fit, 'r')'''

Type,

'''plt.show()'''

Highlight the output
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Now we have '''m''' and '''c.'''

We need to generate the fitted values of '''t square'''.

Type as shown.

Then type,

'''plt.show''''' open and close brackets''

We get the least square fit of '''L''' versus '''t square'''.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Close the image.
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Let me close this window.

|-
| style="background-color:#ffffff;border-top:none;border-bottom:0.5pt solid #000000;border-left:0.5pt solid #000000;border-right:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide

Summary slide

| style="background-color:#ffffff;border-top:none;border-bottom:0.5pt solid #000000;border-left:0.5pt solid #000000;border-right:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| This brings us to the end of this tutorial. Let us summarize.

In this tutorial, we have learnt to,''' '''

* Generate a least square fit using matrices
* Use the function '''lstsq()''' to generate a least square fit line

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide Evaluation

| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Here is a self assessment question for you to solve

* What does the following function produce?

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide

Solutions

| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| And the answer,''' '''

* The function '''ones''' underscore '''like '''''inside brackets inside square brackets ''1 ''comma'' 2 ''comma'' 3 will generate '''array''' ''inside brackets inside square brackets '''''1''' ''comma'' '''1''' ''comma'' '''1'''.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide Forum
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Please post your timed queries in this forum.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide Fossee Forum
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Please post your general queries on Python in this forum.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide TBC
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| FOSSEE team coordinates the TBC project.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide

Acknowledgement
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Spoken Tutorial Project is funded by NMEICT, MHRD, Govt. of India.

For more details, visit this website.

|-
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| Show Slide Thank You
| style="background-color:#ffffff;border:0.5pt solid #000000;padding-top:0cm;padding-bottom:0cm;padding-left:0.097cm;padding-right:0.191cm;"| This is Priya from IIT Bombay signing off. Thanks for watching.

|}

Python-3.4.3/C3/Least-square-fit/English - Revision history

Nancyvarkey at 04:14, 14 November 2018

Priyacst: Created page with "'''Title of script''': '''Least Square Fit''' '''Author: Aditya Palaparthy, Arun KP''' '''Keywords: Python, IPython, Linear Regression, lstsq(), least square fit, video tuto..."