Week 5 - Curve fitting

Aim: curve fitting

• What do popt and pcov mean?
  =The popt argument are the best-fit parameters for the coefficients
  The pcov variable contains the covariance matrix, which indicates the uncertainties and correlations between parameters. This is most useful when the data has uncertainties.
• What does np.array(temperature) do?
  =np.array(temperature) complies the temperature list from the data as an array since operation like numpy are allowed on numpy array and not on python list
• What does the * in *popt mean?
  =It refers to the coefficient values stored in the popt
• Write code to fit a linear and cubic polynomial for the Cp data. Explain if your results are good or bad.

• #Curve fitting
  
  import matplotlib.pyplot as plt
  import numpy as np
  from scipy.optimize import curve_fit
  
  
  #define the function to read the 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]
  
  
  
  #define the function for curve fittng
  def fun1(t,a,b,c,d,e):
      return a*t + b
  
  def fun2(t,a,b,c,d,e):
      return a*t**2 + b*t + c
  
  def fun3(t,a,b,c,d,e):
      return a*t**3 + b*t**2 + c*t + d
  
  def fun4(t,a,b,c,d,e):
      return a*t**4 + b*t**3 + c*t**2 + d*t + e
  
  #main program
  i = -1
  temperature,cp = read_file()
  for fun in (fun1,fun2,fun3,fun4):
      popt, pcov = curve_fit(fun, temperature, cp)
      cp_fit_data = fun(np.array(temperature), *popt)
      i = i+1
      #plotting the real data and the curve fitting data
      plt.plot(temperature, cp_fit_data,color = 'red',label='build curve')
      plt.plot(temperature,cp, label = 'real curve')
      plt.grid(True)
      plt.xlabel('Temperature (kelvin)')
      plt.ylabel('C_p')
      plt.title('curve_fitting_practice')
      plt.legend(loc= 'lower center')
      plt.savefig('{}'.format(i))
      plt.show()

• Explanation: With cubic polynomial, the curve fit tend to fit the better because with higher polynomial the curve fit tend to better and has lower errors
  
  
• What needs to be done in order to make the curve fit perfectly? = To make the curve fit perfectly we have to minimize the error to zero which is hard to achieve so we are trying to best fit the curve by increasing from linear polynomial to cubic polynomial and check the fitness of the curve by using

  The sum of squares of regression is found by using the formula SSR=n∑i=1(f(xi)−mean)2 where f(xi)f(xi) is the predicted Cp values and then

  Then the sum of squared errors is found by using the formula SSE=n∑i=1[y(i)−f(xi)]2 then he R^2 is calculated by SSR/SST where SST is SSR+SSE

  when the R^2 value is greater than 0.95 then the fit is assumed to be good and we can check the values of the value of fitness