Difference between revisions of "Python-3.4.3/C3/Least-square-fit/English"

Objectives

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

System Specifications

• Ubuntu Linux 16.04 operating system
• Python 3.4.3 and
• IPython 5.1.0

Pre-requisites

• 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.

Pendulum.txt

Save it in the current working directory.

Example

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

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.

Open the terminal.

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

from numpy import loadtxt

L, t = loadtxt("pendulum.txt", unpack=True)

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.

L

t

Then type t

We can see that L and t are length and time values.

Press q to exit.

import matplotlib.pyplot as plt

tsq = t * t

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

Highlight bo in the code

Type

plt.show()

Type as shown.

Here bo represents the blue circle marker.

Then type,

plt.show open and close brackets

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

least square fit line

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.

steps for 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.

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.

from numpy import array,ones_like

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

inter_mat

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.

A = inter_mat.T

A

Type,

Capital A is equal to inter underscore mat dot Capital T

Now type capital A

from numpy.linalg import lstsq

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

Type,

result

We only need to use lstsq.

Type as shown.

Now to see the result, type result

The result is a sequence of values.

p = result[0]

m, c = p

m

c

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.

tsq_fit = m * L + c

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

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

Type,

plt.show()

Highlight the output

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.

Summary slide

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

• What does the following function produce?

Solutions

• 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.

Acknowledgement

For more details, visit this website.