Writing codes to perform curve fitting using MATLAB

AIM: CURVE FITTING USING MATLAB

OBJECTIVE:

1. Linear and cubic polynomial curve fitting to a set of data points.

2. Plotting predicted curves against raw data points.

3. Studding parameters that measure the goodness of a fit

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)= ax + 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 prospective line is. A line that provides

us a concrete number that expresses how far from 'best' the 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 .

PROGRAM

clear all
close all
clc
%prepare the data

 cp_data = load('data');

temperature = cp_data(:,1);
cp = cp_data(:,2);

co_effs = polyfit(temperature,cp,1);
predicted_cp = polyval(co_effs,temperature);

figure(1)
plot(temperature, cp,'linewidth',3)
hold on
plot(temperature,predicted_cp,'linewidth',3)
xlabel('temperature [k]')
ylabel('specific heat [kj/kmol-k]')
title('linear')
legend('original dataset','curve fit')

co_effs = polyfit(temperature,cp,3);
predicted_cp = polyval(co_effs,temperature);

figure(2)
plot (temperature,cp,'linewidth',3)
hold on
plot(temperature,predicted_cp,'linewidth',3)
xlabel('temperature [k]')
ylabel('specific heat [kj/kmol-k]')
title('cubic')
legend('original dataset','curve fit')

ERROR FACED : no error encounterd

RESULT :

Plot for linear

Plot for cubic

COCULUSION :

from the above plot, we can clearly say that the cubic curve is clearly fits better compare than the linear curve.

But if we increase the order 4 5 6 we might get a much better fit than cubic and to know which order of polynomial fits the best we have to error check by RMSE lower the RMSE the better the curved fit.