Curve fitting (Python)

AIM: To write codes in Python to perform curve fitting.

OBJECTIVE : 

1. To wite codes to fit a linear and cubic polynomial for the Cp data.
2. To plot the linear and cubic fit curves along with the raw data points.
3. To measure the fitness characteristics for both the curves.

THEORY: Curve fitting is the way we model or represent a data spread by assigning the best fit function (curve) along with the entire range. Ideally, it will capture the trend in the data and allow us to make predictions of how the data series will behave in the future.

Types of curve fitting include:

• Interpolation, where you discover a function that is an exact fit to the data points. Since this assumes no measurement error, it has limited applicability to real-life scenarios.
• Smoothing is when we find a function that is an approximate fit to the data points, but we give room for error and we allow our actual points to be near, but not necessarily on the line; given the error is minimized overall.

Linear and Polynomial Curve fitting :

(i) Linear curve fitting, or linear regression, is when the data is fit to a straight line. Although there might be some curve to your data, a straight line provides a reasonable enough fit to make predictions.

Since the equation of a generic straight line is always given by f(x)= a x + b, the question becomes: what a and b will give us the best fit line for our data?

Considering the vertical distance from each point to a prospective line as an error, and summing them up over our range, gives us a concrete number that expresses how far from ‘best’ the prospective line is.

A line that provides a minimum error can be considered the best straight line.

Since it’s the distance from our points to the line we’re interested in—whether it is positive or negative distance is not relevant—we square the distance in our error calculations. This also allows us to weight greater errors more heavily. So this method is called the least square approach.

(ii) Polynomial curve fitting is when we fit our data to the graph of a polynomial function. The same least-squares method can be used to find the polynomial, of a given degree, that has a minimum total error.

To choose the best fit for the curve the following four parameters help us to measure the goodness of fit criteria or how well the equations are representing the given datapoints:

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

GOVERNING EQUATIONS USED :

• Error = $\mid y\left(i\right))-$value of $f\left(x\right)$at $x\left(i\right)\mid$ = $|y\left(i\right)-f\left(x\left(i\right)\right)|$,
• SSE (sum of squared error) = $\sum _{i=1}^n$Error${\left(i\right)}^2$,
• SSR (sum of square of regression) = $\sum _{i=1}^n{(f\left(x\left(i\right)\right)-Mean)}^2$,
• SST (sum of the squared total) = $SSE+SSR$ ,
• $R^2$ ( R squared) = $\frac{SSR}{SST}$,
• RMSE (root mean squared error) = $\sqrt{\frac{SSE}{n}}$,

          where n is the  total number of datapoints available.

The raw data of Temperature (K) and specific heat (kJ/kcalK) is obtained from here: Data

SOLUTION STEPS :

1. At first, necessary modules such as math,matplotlib,NumPy,scipy are imported to perform their respective functions.
2. Two functions are defined which return linear and cubic equations.
3. The following data is loaded into Python by placing it in the right directory and a function read_file is created. For loop is used to run each line by using 'open' command with 'r' to read it.
4. From the data obtained, two individual column matrices of temperature and specific heat are derived.
5. For curve fitting, 'popt' and 'pcov' are used as output for storing parameters obtained by 'curve_fit' command which takes input as the required function(linear or cubic) , temperature, and Cp. Now, another variable named 'fit_cp' stores the new values of Cp calculated by using temperature array.
6. Plots of both raw dataset and predicted dataset are made and compared to find fitness characteristics for both the curves for both linear and cubic polynomial.
7. To measure the fitness characteristics for both the curves, various parameters such as $SSE$,$SSR$,$SST$,$R^2$,$RMSE$ are calculated. The plot which has highest $R^2$ or closest to 1 is considered to be a good fit.

PYTHON CODE :

# A program to measure fitness characteristics for the linear and cubic polynomial for the Cp data:
import math
import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import curve_fit

# Curve fit function
# Linear function
def function_0(t,a,b):
	return a*t + b 
# Cubic function	
def function_2(t,a,b,c,d):
	return a*pow(t,3) + b*pow(t,2) + c*t +d
	

# Reading 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]

# Main Program
temperature , cp = read_file()

popt, pcov = curve_fit(function_0,temperature,cp)
fit_cp_l = function_0(np.array(temperature), *popt)
plt.figure(1)
plt.plot(temperature,cp,'k--')
plt.plot(temperature,fit_cp_l,color='red',linewidth = 1)
plt.legend(['Actual data','Curve fit'])
plt.xlabel('Temperature (K)')
plt.ylabel('Cp')
plt.title('Linear Curve fitting')
plt.show()

popt, pcov = curve_fit(function_2,temperature,cp)
fit_cp_c = function_2(np.array(temperature), *popt)
plt.figure(3)
plt.plot(temperature,cp,'k--')
plt.plot(temperature,fit_cp_c,color='red',linewidth = 1)
plt.legend(['Actual data','Curve fit'])
plt.xlabel('Temperature (K)')
plt.ylabel('Cp')
plt.title('Cubic Curve fitting')
plt.show()


''' 
Measuring the fitness characetristics of the curves 

 SSE (sum of error squared)
 SSR (sum of squares of the regression)
 SST = SSE + SSR
 R^2 = SSR/SST
 RMSE = (SSE/l)^0.5

 Finding mean of Cp data
 mean = ( sum of all elements)/(total number of elements)'''

s = np.sum(cp)
l = np.size(cp)
m = s/l
print('Mean of all cp :',m)
print('')
# For Linear curve fit

linear_sse = 0
linear_ssr = 0
for i in range(l):
    linear_error = abs((np.sum((cp[i] - fit_cp_l[i]))))
    linear_sse = linear_sse+pow(linear_error,2)
    linear_ssr = linear_ssr+ np.sum(pow((fit_cp_l[i] - m),2))

 
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)
print('')


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

 
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)

ERRORS : 

• The following errors occurred-

• This was caused by the statement "for i in l:" because the for statement operates on iterable data such as a string, list, tuple, or another object container. It was rectified by providing range to it. The range function is a great way to loop for a specified number of times.

• The error says the list is not callable because I was trying to access/use a python list but using parenthesis for it. Parenthesis are used to call a function and because a list is not a function. To access/use a python list, you should use square brackets. 

RESULTS :

• The following are the output of the above program-

• For Linear Curve fitting the plot obtained is-

• For Cubic Curve fitting the plot obtained is-

Based on the above work, we can answer the following questions:

Q.  What do popt and pcov mean?

Ans. 'popt' represents the matrix that stores and extracts the coefficients for the fitting functions according to the general equations defined.

popt : array, Optimal values for the parameters so that the sum of the squared residuals of f(xdata, *popt) - ydata is minimized

       'pcov' represents the square matrix which stores the estimated values of covariance of the coefficients of the above. The diagonal elements of this square matrix also represent the variance of these coefficients. 

pcov : 2d array

The estimated covariance of popt. The diagonals provide the variance of the parameter estimate. To compute one standard deviation errors on the parameters use perr = np.sqrt(np.diag(pcov)).How the sigma parameter affects the estimated covariance depends on absolute_sigma argument, as if True, sigma is used in an absolute sense and the estimated parameter covariance pcov reflects these absolute values. If False, only the relative magnitudes of the sigma values matter. The returned parameter covariance matrix pcov is based on scaling sigma by a constant factor. 

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

Ans This command converts each value of temperature in the data file into an array using the NumPy module.

Q. What does the * in *popt mean?

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

Q. What needs to be done in order to make the curve fit perfectly?

Ans If we increase the order of the polynomial, the error produced will be less and curve fit perfectly. On increasing the order of the polynomial  the value of $R^2$ will be close to 1 and is assumed to a perfect fit.

This method requires defining functions every time for another polynomial and can be considered lengthy. Instead of this method one of the data says temperature can be split into multiple domains say temperature column has 5000 data in it, we can split it into 10 domains of 500 values and if we plot them it will fit perfectly on a curve.

CONCLUSION: Hence we can conclude that to make the curve fit good $R^2$ should be close to 1. It can be found from the results obtained as for linear curve fit $R^2$ is approximately 0.93 and for the cubic curve fit it is nearly 0.99. So it can be said that cubic polynomial fits better than linear polynomial. If the order of the polynomial increases, the value of $R^2$ will inch towards 1 and fit will be assumed to be good.