1. What does popt and pcov mean? (Watch the video to get some context)

In the curve fitting code, the variables "popt" and "pcov" are defined to the curve fit function. The first variable, "popt", stores and extracts the coefficients of the curve fitting function into an array, with respect to the general polynomial equation defined in the curve fit function. In this assignment, there are two curve fit functions used for a linear and cubic model. For each mode, the "popt" variable will be an array with the coefficients of the linear and cubic model. The second variable, "pcov", represents a matrix in square form, which stores the covariance values of the coefficients stored in the array defined by the "popt" variable. The diagonal elements of this square matrix will match with the variance of the same coefficients. 

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

The "np.array(temperature)" command creates an array from the temperature values that were retrieved from the thermodynamic data file, using the "numpy" library. Having the temperature data in an array is useful in this code as it helps the code to calculate fitted "Cp" values for each respective temperature value based on the "popt" array. 

3. What does the * in *popt mean?

The "*" in the "*popt* tells the python code to indicate and return the values of the coefficients that are stored in the "popt" array. 

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

import numpy as np

import matplotlib.pyplot as plt

from scipy.optimize import curve_fit

# Linear Fit Function

def linear_func(t, a, b):

          return a*t + b

# Reading Thermodynamic 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(linear_func, temperature, cp)

fit_cp = linear_func(np.array(temperature), *popt)

plt.plot(temperature, cp, color = "blue", linewidth = 3)

plt.plot(temperature, fit_cp, color = "red", linewidth = 3)

plt.legend(["Actual Data", "Curve Fit"])

plt.xlabel("Temperature [K]")

plt.ylabel("Cp")

plt.show()

total_cp = np.sum(cp)

numval_cp = np.size(cp)

avg_cp = total_cp/numval_cp

print('Average Cp is:', avg_cp)

for i in range(numval_cp):

          sst = np.sum(pow((cp[i]-avg_cp),2))

          ssr = np.sum(pow((cp[i]-fit_cp[i]),2))

# R-Squared Value

r_squared = 1-(ssr/sst)

print("R-Squared:", r_squared)

# Root Mean Square Error (RMSE)

rmse = pow((ssr/numval_cp),0.5)

print("RMSE:", rmse)

For the goodness of fit calculations, a for loop was created. In this for loop, two terms were calculated. The first term is the total sum of squares. This term is calculated using the formula below:

TSS =  ∑ (y_i − y¯)²

where,

y_i - Original "Cp" value

y¯- Average "Cp" value

Then, the second term was calculated called the sum of squares regression. This term is calculated using the formula below:

SSR = ∑ (y_i − y_fit)2

where,

y_fit- Fitted "Cp" value from the model

Finally, the R-Squared value and root mean squared error (RMSE) were calculated using the formulas shown below:

R² = 1 − ( RSS )

TSS

RMSE = ( RSS )

n

0.5

where,

n - number of "Cp" values in the thermodynamic data file.

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

To make the curve fit model better, there are two methods that can be followed. The first method involves adjusting the fitted function using a trial and error method and observing the changes in the graphical output to see if the model is a better fit to the actual data. There are unlimited combinations of coefficients to try and observe whether the model is a better fit or not. However, this method has a limitation with respect to the order of the polynomial. This is because no matter how many combinations of coefficients we try, the order of that model will only reach a maximum level of fit that cannot be exceeded. this is when the second method of using higher order polynomials is more useful. When using higher order polynomials, the fitted model will be a better fit to the actual data. This is because higher order polynomials have more coefficients and thus have more coefficients that can be adjusted to be a better fit. From the previous question, it was evident that the cubic fitted model was superior to the linear fitted model. The cubic fitted model had better goodness of fit characteristics and followed the shape of the actual data more closely on the graph. 

6. Show empirically as to how well your curve has been fit.

From the answer to question 4, the goodness of fit for the linear and cubic fitted models was explained based on the parameters of the R-Squared value and the RMSE. In addition, the graphical output for both models was also shown. 

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


# Linear Fit Function
def linear_func(t, a, b):
	return a*t + b


# Reading Thermodynamic 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(linear_func, temperature, cp)
fit_cp = linear_func(np.array(temperature), *popt)


plt.plot(temperature, cp, color = "blue", linewidth = 3)
plt.plot(temperature, fit_cp, color = "red", linewidth = 3)
plt.legend(["Actual Data", "Curve Fit"])
plt.xlabel("Temperature [K]")
plt.ylabel("Cp")
plt.show()


total_cp = np.sum(cp)
numval_cp = np.size(cp)
avg_cp = total_cp/numval_cp
print('Average Cp is:', avg_cp)



for i in range(numval_cp):
	sst = np.sum(pow((cp[i]-avg_cp),2))
	ssr = np.sum(pow((cp[i]-fit_cp[i]),2))

# R-Squared Value
r_squared = 1-(ssr/sst) 
print("R-Squared:", r_squared)

# Root Mean Square Error (RMSE)
rmse = pow((ssr/numval_cp),0.5)
print("RMSE:", rmse)