Week 5 - Curve fitting

Objectives:

Writing a PYTHON program code to find the performance of curve fitting for the linear and cubic methods.

1. What does popt and pcov mean?

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

3. What does the * in *popt mean?

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

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

Curve Fitting:

Curve fitting is the process of constructing a curve or mathematical function that has the best fit to a series of data points possibly subject to constraints. Curve fitting can involve either interpolation where an exact fit to the data is required, or smoothing, in which a \"smooth\" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors.Fitted curves can be used to summarise the relationships between two or more variables, to infer values of a function in the absence of data, and as a tool for data visualisation. Extrapolation is the use of a fitted curve outside the range of the observed data, and it involves a certain amount of uncertainty because it may reflect the construction process of the curve as well as the observed data.

• The first-degree polynomial equation

• is a line with slope a. A line will connect any two points, so a first-degree polynomial equation is an exact fit through any two points with distinct x coordinates.

  If the order of the equation is increased to a second-degree polynomial, the following results:

  This will exactly fit a simple curve to three points.

  If the order of the equation is increased to a third-degree polynomial, the following is obtained:

• a,b,c,d are variables used to predict the value of y for each X term.

• 1.what is POP and Pcov ?

  • pcov is a 2d array with the estimated covariance of the parameters in popt. It can be used to calculate the the standard deviation errors of popt.
  • \"popt\" gives optimized curve for the given function, its gives best-fit parameter.
  • The \"popt\" is the co-efficient of values of a,b,c,d.
  • popt, pcov = curve_fit(line, x, y)
    The curve_fit function returns two items, which we can popt and pcov. The popt argument are the best-fit paramters for a and b:

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

    • This command is used to convert all temperatures into the numpy function. It is also used when some math operations cannot be performed on a list.

    3. What does the * in *popt mean?

    It should be noted that there is a solution to simplify the function call with the best-fit parameters, which is:

  • line(*popt, x)
    The * notation converts a list of values into function arguments. If your function has more than one or two parameters, this is beneficial. We also employed a secondary and third array for cubic curve fitting.

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

    MAIN PROGRAM:

    Case 1: Linear Curve Fit.

    import matplotlib.pyplot as plt 
    import numpy as np 
    from scipy.optimize import curve_fit
    
    #curve fir function (linear)
    
    def func(t,a,b):
    	return a*t+b
    
    # Reading file
    
    def read_file():
    	temperature =[]
    	cp = []
    
    	for line in open(\'data\',\'r\'):
    		value = line.split(\',\')
    		temperature.append(float(value[0]))
    		cp.append(float(value[1]))
    
    	return[temperature,cp]
    
    
    #Main program
    temperature,cp =read_file()
    popt,pcov = curve_fit(func,temperature,cp)
    fit_cp = func(np.array(temperature),*popt)
    print(popt)
    
    plt.plot(temperature,cp,color =\"blue\",linewidth=\"3\")
    plt.plot(temperature,fit_cp,color=\"red\",linewidth =\"3\")
    plt.legend([\'Actual Data\',\'Linear curve fit\'])
    plt.xlabel(\'Temperature[K]\')
    plt.ylabel(\'Specific Heat(cp)\')
    plt.savefig(\'Linear curve fit\')
    plt.show()

    case 2:cubic polynomial.

    import matplotlib.pyplot as plt 
    import numpy as np 
    from scipy.optimize import curve_fit
    
    #curve fir function (Cubic)
    
    def func(t,a,b,c,d):
    	return a*pow(t,3)+b*pow(t,2)+c*t+d
    
    # Reading file
    
    def read_file():
    	temperature =[]
    	cp = []
    
    	for line in open(\'data\',\'r\'):
    		value = line.split(\',\')
    		temperature.append(float(value[0]))
    		cp.append(float(value[1]))
    
    	return[temperature,cp]
    
    
    #Main program
    temperature,cp =read_file()
    popt,pcov = curve_fit(func,temperature,cp)
    fit_cp = func(np.array(temperature),*popt)
    print(popt)
    
    plt.plot(temperature,cp,color =\"blue\",linewidth=\"3\")
    plt.plot(temperature,fit_cp,color=\"red\",linewidth =\"3\")
    plt.legend([\'Actual Data\',\'Linear curve fit\'])
    plt.xlabel(\'Temperature[K]\')
    plt.ylabel(\'Specific Heat(cp)\')
    plt.savefig(\'Linear curve fit\')
    plt.show()

    PROCEDURE:

    • Importing the required files for program running like \"matplotlib\",\"numpy\",\"scipy for curve fit\".
    • Defining the Linear order equation(a*t+b) for the running linear polynomial .
    • For reading file we need to define the function for read_file(), its read the data in which we stroed in same folder.
    • Creating a empty array for for the temperature and specific heat ,cp used to stored the values .
    • using for loop command in which it used for line to open (\'data\',\'r\'), which is used to read the content of the data in which we read.
    • using command value =line.split(\',\'), which is used to split the data as we required.
    • Data we used to string value used convert with float and append command used to store the value in empty array we created.
    • By using the \"curve_fit\" function fpr linear order, we get an output for temperature and specific heat cp.
    • After getting the output as an popt , the output value of fit_cp we pass the value of temperature in data as numpy array so that it can be executed.
    • We need to plot the output data using the plt.plot of temperature and specific heat cp & temperature with fit_cp.
    • After finishing the case 1, for case 2 we need to use the cubic order of equation a*pow(t,3)+b*pow(t,2)+c*t+d
    • And rest of the procedure are same as the above method.

    OUTPUT: PLOTS FOR LINEAR, CUBIC.

    1. Curve fitting with a linear polynomial

    2.curve fitting with a cubic polynomial

    Results are good are Bad:

    • From above comparing two cases of graphs shows that linear function curve is not accurate with actual data and curve not matched.
    • while using the cubic equation and its output data similary match with data given so, hence we can declare that second equation results is \"GOOD\".
    • 5. What needs to be done in order to make the curve fit perfect?
    • The preceding two methods illustrate that when we use higher order equations, we receive more accurate results based on the data we have, hence it is recommended to utilise higher order equations for desired results with the least amount of error

    • Conclusion:

      Finally, a program using PYTHON for curve fitting done successfully,