Question

How many normal equations are in second degree parabola method ?

Answer 1

Answer:

18

Step-by-step explanation:

The second degree parabola = It describes the trend( non linear) in a time series where a change in the amount of change is constant per unit time.

The quadratic (parabolic)trend can be describe by equation

⇒ y = a + bx + cx2

The method of least square gives the normal equation as 

⇒ ∑y = na + b∑x + c∑x2

⇒ ∑xy = a∑x + b∑x2 + c∑x3

⇒ ∑x2y = a∑x2 + b∑x3 + c∑x4

However if ∑x = 0 then, the normal equation reduces to

⇒ ∑y = an + c∑x2

⇒ ∑xy = b∑x2

⇒ ∑x2y = a∑x2 + c∑x4

The values of a, b, and c can be found as

⇒ c = (n∑x2y - ∑x2)(∑y)/[n∑x2 - (∑x2)2]

⇒ a = (∑y - c∑x2)/n

⇒ b = ∑xy/∑x2 

Answer 2

Answer:

There will be three norma equations

Given:

The second-degree parabola describes the trend (non-linear) in a time series where the amount of change is constant per unit of time.

To find: Number normal equation is in second-degree parabola method

Step-by-step explanation:

Let the equation of best fitted second-degree polynomial be,

$Y=a x^{2} +b x+c$

So, $D^2=\sum\left(a x^2+b x+c-y\right)^ 2$

For $D^2$ to be minimum the necessary conditions are given by

1. $\frac{d D^{2}}{d a}=0$

$\sum\left(a x^2+b x+c-y\right)^ 2=0$

$a \sum x^ 4+b \sum x^ 3+c \sum x^ 2=\sum x ^2 y \ldots .... .(1)$

2. $\frac{d D^{2}}{d b}=0$

$\sum\left(a x^ 2+b x+c-y\right)^ 2=0$

$a \sum x^3+b \sum x^ 2+c \sum x=\sum x y \ldots... . .(2)$

3. $\frac{d D^{2}}{d c}=0$

$\begin{aligned}&\sum\left(a x^2+b x+c-y\right)^ 2=0 \\&a \sum x^2+b \sum x+c n=\sum y \text {......(3) }\end{aligned}$

Solving the above equations 1,2, and 3, simultaneously give values of ' $a , b$ ', and 'c' for the best-fitted parabola.