Week 5 - Curve fitting

THEORY:

Curve fitting is the process of constructing a curve or mathematical function, that has the best fit to a series of data points. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. By curve fitting, we can capture the trend in the data and 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 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:

Linear curve fitting, or linear regression, is when the data is fit to a straight line. 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, equation for linear curve fitting

The goal is to identify the coefficients ‘a’ and ‘b’ such that f(x) ‘fits’ the data well.

How to find coefficients for best fit?

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-squares approach.

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.

GOODNESS OF FIT:

There are four quantities that help us measure the goodness of fit or how well the predicted curve is representing the raw data points:

1. SUM OF SQUARE OF ERRORS(SSE)

Assume data points (x1,y1),(x2,y2)………(x_n,y_n) and assume curve fit for these data points to be y=f(x),

when we differentiate the SSE with each and every coefficient of f(x) and equate to zero, we find the values of coefficients

This method is known as LEAST SQUARE METHOD.

2. R-square

R-square is the square of the correlation between the response values and the predicted values.

3. Adjusted R-square

Adjusted R-square is a normalizing variation of R-square which neglects the variation of R-square due to insignificant independent variable.

4. Root mean square error(RMSE)

n = no. of data points

Code:-

import numpy as np 
import matplotlib.pyplot as plt 
import math
from scipy.optimize import curve_fit

def func_linear(t,a,b):

	return a*t + b


def func_cubic(t,a,b,c,d):

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

#reading data file
def read_file():
	temp=[]
	cp=[]
	for line in open('data','r'):
		line=line.split(',')
		t=float(line[0])
		temp.append(t)
		s=float(line[1])
		cp.append(s)

	return [temp,cp]

#CURVE FITTING Linear

[temp,cp]=read_file()

popt, pcov = curve_fit(func_linear,temp,cp)
fit_cp= func_linear(np.array(temp),*popt)

#calculating goodness of linear fit
#SSE
sse=0
for i in range(0,np.size(cp)):
	sse=sse + pow(cp[i]-fit_cp[i],2)

print('FOR LINEAR CURVE FITTING')
print('SSE:', sse)

#R_sqaure
mean=np.sum(cp)/np.size(cp)
ssr=0
for i in range(0,np.size(cp)):
	ssr= ssr + pow(fit_cp[i] - mean,2)

sst=ssr+sse
R_sqaure=ssr/sst
print('R_sqaure:',R_sqaure)

#RMSE
rmse=pow(sse/np.size(cp),0.5)
print('RMSE:', rmse)

#plotting linear curve fit
plt.figure(1)
plt.plot(temp,cp,label= 'orginal data')
plt.plot(temp,fit_cp,label='linear curve fit')
plt.legend()
plt.xlabel('Temperature (K)')
plt.ylabel('Specific Heat (KJ/K)')
plt.title('Linear Curve Fitting')
plt.grid()
plt.savefig('linear curve fitting')
plt.show()

#cubic curve fitting
popt, pcov = curve_fit(func_cubic,temp,cp)
fit_cp= func_cubic(np.array(temp),*popt)

#calculating goodness of cubic fit
#SSE
sse=0
for i in range(0,np.size(cp)):
	sse=sse + pow(cp[i]-fit_cp[i],2)

print(' ')
print('FOR CUBIC CURVE FITTING')
print('SSE:', sse)

#R_sqaure
mean=np.sum(cp)/np.size(cp)
ssr=0
for i in range(0,np.size(cp)):
	ssr= ssr + pow(fit_cp[i] - mean,2)

sst=ssr+sse
R_sqaure=ssr/sst
print('R_sqaure:',R_sqaure)

#RMSE
rmse=pow(sse/np.size(cp),0.5)
print('RMSE:', rmse)


#plotting cubic curve fit
plt.figure(2)
plt.plot(temp,cp,label='orginal data')
plt.plot(temp,fit_cp,label='cubic curve fit')
plt.legend()
plt.xlabel('Temperature (K)')
plt.ylabel('Specific Heat (KJ/K)')
plt.title('Cubic Curve Fitting')
plt.grid()
plt.savefig('cubic curve fitting')
plt.show()

STEPS:

1. Load data points using open() and read the temp and cp values by splitting every line in text file.
2. Using curve_fit function in scipy, get coefficients of linear and cubic fit and using store the value of predicted linear and cubic polynomials.
3. Plot predicted values of linear and cubic polynomials along with raw data points to compare the best fit.
4. Define formulas of SSE, R-square, RMSE to get the values of parameters which measure the goodness of fit.
5. Compare the goodness of fit of linear and cubic curve fitting.

ERRORS: No error encountered

OUTPUT:

ANSWERS:

1. What does popt and pcov mean?

popt gives an array which contains the coefficients of the fitting function. These coefficients are optimized by minimizing the Sum of Squared Errors.

pcov: gives 2d array

It is 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)).

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

This command converts the temperature values into an array and stores them into it.

3. What does the * in *popt mean?

* refers to each element of popt array. It gives us all the elements which are stored in popt.

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

We can compare the value of R-square in linear and cubic curve fitting. R-square value of Cubic fitting is closer to 1. but still there is some error in both curve fitting i.e. the fit in both linear and cubic fit is not perfect.

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

By minimizing the SSE we can make a curve to perfectly fit data points. With a high degree of the polynomial we get less value off SSE and more value of R-square(closer to 1), this results in high accuracy of curve fitting.

Also to perfectly fit cubic polynomial in data points, we can split the data points and perform cubic curve fitting in two separate regions. this will give us more accuracy in curve fitting.

REFERENCE:

https://projects.skill-lync.com/projects/Curve-Fitting-and-criterion-to-choose-best-fit-41860