Week 5 - Curve fitting

Aim:- To write codes to perform curve fitting by using python.

Objective:-

1. Writing the codes to fit a linear and cubic polynomial for the Cp data.
2. Plotting the linear and cubic fit curve with help of cp vs temperature data file.
3. Measuring the fitness characteristics for both the curves

Theory:-

Curve fitting is one of the techniques of data analysis to validate and to find the mathematical relationship between experimental data.

• Interpolation is a method where a function is used to exact to the data points.
• Smoothing is a method to finding a function that is an approximate fit to the data points.

The goodness of fit - It's a measure of fitness of curve for given data and there are four parameters to calculate the goodness of fit.

1. The sum of squares due to error (SSE)
2. R-Square (R^2)
3. Adjusted R-Square,
4. Root mean squared error (RMSE)

Input:- inputing file- Cp vs Temperature data file

Program:-

# Program for curve fitting
#Importing module
import math
import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import curve_fit

# reading the thermodynamic data file
def read_file ():
	temperature = []
	cp = []
	for line in open('data','r'):
		values = line.split(',')
		temperature.append(float(values[0]))
		cp.append(float(values[1]))

	return[temperature,cp]

# curve fit function (linear function)
def func(t,a,b):
	return a*t+b

# Main program
temperature,cp = read_file()
#print (temperature,cp)

popt,pcov = curve_fit(func,temperature,cp)
#print(popt)
#print(pcov)
fit_cp = func(np.array(temperature),*popt)

plt.figure()
plt.plot(temperature,cp,color='blue',linewidth=1)
plt.plot(temperature,fit_cp,color='red',linewidth=1)
plt.legend(['Actual data','linear curve fit'])
plt.xlabel('temperature(k)')
plt.ylabel('specific heat (Cp)')
plt.title('linear curve fitting')
plt.grid()
plt.show()


# curve fit function (cubic polynomial function)

def func(t,a,b,c,d):
	return a*pow(t,3)+b*pow(t,2)+c*t+d

popt,pcov = curve_fit(func,temperature,cp)
predicted_cp = func(np.array(temperature),*popt)
plt.figure()
plt.plot(temperature,cp,color='blue',linewidth=1)
plt.plot(temperature,predicted_cp,color='red',linewidth=1)
plt.legend(['Actual data','cubic curve fit'])
plt.xlabel('temperature(k)')
plt.ylabel('specific heat (Cp)')
plt.title('cubic curve fitting')
plt.grid()
plt.show()

# Goodness of fitting
#Measuring the fitness characterstic of the curves
"""SSE = Sum of error squared
SSR = Sum of erroe squared
SST = Sum of the squared total
RMSE = Root mean squared error
SST = SSR/SST
RMSE = (SSE/1)^0.5
"""
s = np.sum(cp)
l = np.size(cp)

# Finding mean of Cp data
m = s / l 
print('Mean of all cp :', m,'\n')

# For Linear curve fit
linear_sse = 0
linear_ssr = 0
for i in range(l):
    linear_error = abs((np.sum((cp[i] - fit_cp[i]))))
   
    linear_sse = linear_sse + pow(linear_error, 2)
    linear_ssr = linear_ssr + np.sum(pow((fit_cp[i] - m), 2))

print('For Linear Curve Fit')
linear_sst = linear_sse + linear_ssr
print('Linear SST :', linear_sst)
linear_R2 = linear_ssr / linear_sst
print('Linear R2 :', linear_R2)
linear_rmse = pow((linear_sse / l), 0.5)
print('Linear RMSE :', linear_rmse ,'\n')

# For Cubic curve fit
cubic_sse = 0
cubic_ssr = 0
for j in range(l):
    cubic_error = abs(np.sum((cp[i] - predicted_cp[i])))
    cubic_sse = cubic_sse + (pow(cubic_error, 2))
    cubic_ssr = cubic_ssr + np.sum(pow((predicted_cp[i] - m), 2))

print('For Cubic Curve Fit')
cubic_sst = cubic_sse + cubic_ssr
print('Cubic SST :', cubic_sst)
cubic_R2 = cubic_ssr / cubic_sst
print('Cubic R2 :', cubic_R2)
cubic_rmse = pow((cubic_sse / l), 0.5)
print('Cubic RMSE :', cubic_rmse)

Output:-

Plot of Linear function

Plot of Cubic polynomial function

Output values

Q&A:-

1. What does popt and pcov mean?

Ans. ‘popt’ : It is an array that stores the values of the coefficients that are being passed in a given function.

‘pcov’ : It is a two-dimensional array that stores the values of the estimated co-variance of ‘popt’ i.e the coefficients.

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

Ans. This is a command which is used to converts each value of temperature in the data file into an array using the NumPy module.

3. What does the * in *popt mean?

Ans. '*' in ‘*popt' indicates and returns each coefficient stored in popt array.

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

Ans . By increasing the order of the polynomial the value of will be close to 1 and is assumed to a perfect fit and also splitting the data set into different groups of certain range or both.

Conclusion:- In linear function, curve fitting is bad and In Cubic function, curve fitting is good. Curve fitting program run successfully and result found satisfactory.