Week 5 - Curve fitting

• Aim:- To write codes to perform curve fitting.
• Objectives of the project:

1. Write code to fit a linear and cubic polynomial for the Cp data.

2.Plot the linear and cubic fit curves along with the raw data point.

3.Explain the parameters used to measure the fitness characteristics for the curves

Solution:

Answers of questions

popt: array stores the values of coefficients that are given in a function

1.pcov :two dimensional array that stores the values of estimated covariance of the coefficients i.e popt

2.np.array(temperature) used to pass the values of temperature as np array

3.* in * popt helps to refer coefficient of given equation

• Governing equations:-

Method used to solve this problem is polynominal curve fitting

1.Linear curve fitting (linear regression)-f(x)=ax+b">f(x)=ax+bf(x)=ax+b

2.Quadratic curve fitting-f(x)=ax2+bx+c">f(x)=ax^2+bx+cf(x)=ax^2+bx+c

3.Cubic curve fitting-f(x)=ax3+bx2+cx+d">f(x)=ax^3+bx^2+cx+df(x)=ax3+bx2+cx+d

4.4th order polynominal-f(x)=ax4+bx3+cx2+dx+f">f(x)=ax^4+bx^3+cx^2+dx+df(x)=ax4+bx3+cx2+dx+f

Incaseofourprob≤mcalca̲tiondo≠→calcua<ebestfitforspecificheat(cp)consider∈gtemperature.">I

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

To find the the goodness of fit method of least square is used. Errors calculation helps to find best fits.

`Notation used:

Error=e,

Sum of squares of error=sse, sum of square of regression=ssr,

total sum of squares=sst e =(cp-predicted_cp1);

sse =sum(e.^2);

ssr=sum((cp-mean(cp)).^2);

sst =(sse +ssr);

r_square=(ssr/sst);

root mean square error= rmse=sqrt(sse/n); where, ‘n’ defines no. of datapoints`

4.Body of the content:

Introduction:

Steps in code:

1. First the file of data named 'curve_it_data' get loaded in the programme
2. The 1st column of data specifies Temperature,temp (k) and 2nd column specifies Specific heat, cp-(Kj/kmol-k)
3. Program is written in two parts such as: part 1 is for linear curve fit and part 2 Cubic curve fit.
4. After running the code coefficients are get calculated for each type of fit and Plots are get plotted for Curve fit .

   # -*- coding: utf-8 -*-
   """
   Created on Tue Jul 14 17:15:54 2020
   
   @author: ASUS
   """
   ''' Program for linear curve fit'''
   
   import numpy as np
   from scipy.optimize import curve_fit
   import matplotlib.pyplot as plt
   
   def function(t,a,b):    ## defining function
       return a*t+b        ## function for linear fit
   def read_file():
       temp=[]        ### empty arrays to store values of temperature
       cp=[]
       for line in open('curve_fit_data','r'):   ## ro read each line in data
           values=line.split(',')   ## splitting data in  line using comma
           temp.append(float(values[0]))   ## temperature values assigned to first column
           cp.append(float(values[1]))     ## cp values assigned to first column
       return[temp,cp]
       
   
   temp,cp=read_file()     ##calling function read-file and store value in temp and cp
   popt,pcov=curve_fit(function,temp,cp)
   fit_cp=function(np.array(temp),*popt)
   
   plt.plot(temp,cp,'r')   
   plt.plot(temp,fit_cp,'b') 
   plt.legend(['actual data', 'curve fit data'])
   plt.xlabel('temp')
   plt.ylabel('cp')
   plt.show()
   
   """Program for cubic curve fit"""
   
   def function_2(t,a,b,c,d):
       return a*pow(t,3)+b*pow(t,2)+c*t+d    ## function for cubic fit
   
   temp,cp=read_file()     ##calling function read-file and store value in temp and cp
   popt,pcov=curve_fit(function_2,temp,cp)
   fit_cp_2=function_2(np.array(temp),*popt)
   
   plt.plot(temp,cp,'g')   
   plt.plot(temp,fit_cp_2,'y') 
   plt.legend(['actual data', 'curve fit data_2'])
   plt.xlabel('temp')
   plt.ylabel('cp_2')
   plt.show()​

   Window screenshot:

linear fit graph

Cubic fit

From the graphs it is concluded cubic fit is good than linear fit.