Quadratic and Cubic Modeling



Quadratic and Cubic Modeling

Polynomials can take the form of powered coefficients in OLS linear equations used for trend analysis. The linear equation of y = a + b1X + b2X2 illustrates the use of powered polynomials, with b1X as the linear term (not shown is X taken to the first power), and the squared term b2X2 producing a quadratic term. When powered polynomials are used for trend analysis, the resulting trends are non orthogonal, meaning non independent. When the trends are created using powered polynomials, the quadratic and higher order terms are built from the linear term (for example, X is used to create X2), and thus are highly correlated.

Find a polynomial equation of degree 2 (quadratic regression) that models the data (add trendline in Excel). Call this equation Y1 and include the value. Print your graph with labeled axes, and include the graph in your report.

Solution

The quadratic regression is the process by which we find the parameters of a parabola that best fit a set of data we possess, whether measurements or other. As a result, we obtain an equation of the form

y = ax2 + bx + c

where a ? 0.

The relative predictive power of a quadratic model is denoted by R 2. The value of R 2 varies between 0 and 1. The closer the value is 1, the more accurate the model. Consider the data set that has been chosen from section I. In this analysis we will determine the quadratic regression for the set.

(-3, 7.5), (-2, 3), (-1, 0.5), (0, 1), (1, 3), (2, 6), (3, 14)

Entering the data set of the x-coordinates and the y coordinates into the Excel sheet in order to perform the quadratic regression. The equation of the parabola that best approximates the point is

y = 1.1071x2 + 0.5714

The graph for the given coordinated is shown in the picture below.

You can see the value of R 2 for the data is 0.9942. Find a polynomial equation of degree 3 (cubic regression in Excel) that models the data and call this equation Y2 and include the value. Print your graph with labeled axes, and include the graph in your report.

Solution

A cubic equation with one unknown is an equation that can be put under the canonical form:

ax³ + bx ² + cx + d = 0,

where a, b, c and d (a ? 0) are numbers that belong to a body, usually R or C . Let K be a commutative, where you can extract square and cube roots. In this body, it is possible to factor all a ? 0, and the following identity holds:

(a- b) 3 = a3 - 3a2 b +3 ab 2 - b 3

Just find a solution, say r, then by factoring ax3 + bx 2 + cx + d for x - r; we obtain a quadratic equation that we solve, giving the other roots. In an algebraically closed, it is known that ...