Week 5 - Curve fitting

Week 5 – Curve Fitting

1. What does popt and pcov mean?

For Eg- def func1(t , a , b, c):

return a*pow(t,2)+b*t+c

popt - The curve_fit function is going to calculate the variables a, b, c values and store the values in popt. Hence, popt is going to be our coefficients.

pcov- It is the covariance matrix of the parameters and can be used to get an estimate of the errors on the parameter and the relation the parameters have to each other.

Eg - ([[ 1 , 2 , 3 ],

[ 4 , 5 , 6 ],

[ 7 , 8 , 9 ]])

Here 1, 5, 6 are related to the errors with a, b, c and the diagonal elements [2,3,5,6,7,8] tell us how the parameters are related

2. What does np.array(temperature) do?

While we pass the temperature we pass it as a numpy array (np.array). The np.array converts the python list into a numpy array. Here an array of values are created which helps the program to select the particular temperature and calculate the specific heat values to obtain the result.

3. What does the * in *popt mean?

A * in the program is a pointer which stores the values or the address of the variables in an array. Here in the program *popt stores in the integer values of the constants and gives them out as well.

4. Write code to fit a linear and cubic polynomial for the Cp data. Explain if your results are good or bad.

for biquadratic polynomial

import numpy as np #used for scientific computing

import matplotlib.pyplot as plt #module used to plot graphs

from scipy.optimize import curve_fit #scipy module provides more utility functions for optimization

#using a four degree quadratic equation

def func(t,a,b,c,d,e): #defining the function

return a*pow(t,4)+b*pow(t,3)+c*pow(t,2)+d*t+e #amthematical model used to represent the data, i.e to represent the relation between temperature and specific heat

def read_file():

temperature=[] #empty array for temperature

spc_ht=[] #empty array for specific heat

for line in open('data','r'): #this loop states that we are opening a file called 'data' and reading the file line by line with he;lp of 'r' parameter which reffers to read mode

values = line.split(',') #splitting the line with a comma operator

temperature.append(float(values[0])) #atter splitting there are gonna be two values in the array. first value is temperature value obtained by [0]

spc_ht.append(float(values[1])) #second value is obtained by index [1]

return [temperature,spc_ht]

#main program

temperature , spc_ht = read_file() #reads the data file which contains specific heat of oxygen

popt, pcov = curve_fit(func, temperature, spc_ht) #using the curve_fit function in the scipy module which contains parameters of func(given mathamtical model), temperature and pecific heat values.here popt contains a,b,c,d variables and pcov is the covariance matrix

fit_spe_ht = func(np.array(temperature), *popt) #after having new coeffecients, the defined function is taken again and the numpy temperature array and *popt are passes. here *popt are the coeeficents a,b,c,d

plt.plot(temperature, spc_ht, color='blue',linewidth=3) #plotting temperature against soecific heat with a blue colour

plt.plot(temperature, fit_spe_ht, color='pink',linewidth=3) #plotting temperature against fit specific heat with a pink line

plt.legend(['actual data','curve fit']) #assigning the meaning of the various plot elements

plt.xlabel('temperature K') #labeling x axis as temperature

plt.ylabel('specific heat') #labeling y axis as speific heat

plt.show() #showing the final graph

Explanation of the result:

By the obtained result we can say that the results are good as the biquadratic polynomial fit is better than a linear fit or a cubic fit. As we increase the power of the polynomial simultaneously the fit gets better and better.

RESULT-

5. What needs to be done in order to make the curve fit perfect?

As we increase the power of the polynomial simultaneously the fit gets better and better.

Fig 1 is a cubic polynomial and Fig 2 is a biquadratic polynomial and Fig 2 has a better curve fit

Fig 1 Fig 2

import numpy as np #used for scientific computing
import matplotlib.pyplot as plt #module used to plot graphs
from scipy.optimize import curve_fit #scipy module provides more utility functions for optimization 


#using a four degree quadratic equation 
def func(t,a,b,c,d,e): #defining the function
	return a*pow(t,3) + b*pow(t,2) + c*t + d #amthematical model used to represent the data, i.e to represent the relation between temperature and specific heat

def read_file():
	temperature=[] #empty array for temperature
	spc_ht=[] #empty array for specific heat
	for line in open('data','r'): #this loop states that we are opening a file called 'data' and reading the file line by line with he;lp of 'r' parameter which reffers to read mode
		values = line.split(',') #splitting the line with a comma operator
		temperature.append(float(values[0])) #atter splitting there are gonna be two values in the array. first value is temperature value obtained by [0] 
		spc_ht.append(float(values[1])) #second value is obtained by index [1]
	return [temperature,spc_ht]

#main program
temperature , spc_ht = read_file() #reads the data file which contains specific heat of oxygen
popt, pcov = curve_fit(func, temperature, spc_ht) #using the curve_fit function in the scipy module which contains parameters of func(given mathamtical model), temperature and pecific heat values.here popt contains a,b,c,d variables and pcov is the covariance matrix
fit_spe_ht = func(np.array(temperature), *popt) #after having new coeffecients, the defined function is taken again and the numpy temperature array and *popt are passes. here *popt are the coeeficents a,b,c,d

plt.plot(temperature, spc_ht, color='blue',linewidth=3) #plotting temperature against soecific heat with a blue colour
plt.plot(temperature, fit_spe_ht, color='pink',linewidth=3) #plotting temperature against fit specific heat with a pink line
plt.legend(['actual data','curve fit']) #assigning the meaning of the various plot elements
plt.xlabel('temperature K') #labeling x axis as temperature 
plt.ylabel('specific heat') #labeling y axis as speific heat
plt.show() #showing the final graph