17.28 An object is suspended in a wind tunnel and the force measured for various levels of wind velocity. The results are tabulated below.
v, m/s 10 20 30 40 50 60 70 80
F, N 25 70 380 550 610 1220 830 1450
Use least-squares regression to fit this data with (a) a straight line linear regression and (b) a nonlinear polynomial regression. Display the results graphically and compare which technique will give the best fit.
Regression is a modeling of relationship between dependent and independent variables. Besides, it is for finding a curve which represent the best approximation of a series of data points and the curve is the estimate of trend of dependent variables.
For our problems, we want to show the relationship between velocity of the wind and the force exerted. We solve the problem by using linear regression and polynomial regression.
By using linear regression, we are using the formula,
Equation of straight line
Where; ao = y intercept
a1 = slope of linear line
e = error the actual y value and the approximate
value given by equation ao + a1x
Thus the error we rewrite as;
Error Quantification in Linear Regression
Sum of residual square formula
We may then define two coefficients:
Coefficient of determination,
Coefficient of correlation,
In a more convenient form for computation of r is
By using polynomial regression, we are using the formula,
The mth-degree polynomial may be represented as
Where; a0, a1, a2, . . . , and am are unknown constant coefficients.
e is the error between the polynomial value and the actual y value.
The error is then given by
Using the least-squares procedure, the sum of the squares of the residuals is
When we partially differentiate the sum of the squares of the residuals with respect to...