Non Linear Model

Let us consider an equation Y = 10 + 5{X^2}

By putting the values of X = 0,1,2,3,4 in this equation, we find the values of Y as given in the table below. The first and second differences are calculated in that given table.

X
Y
First differences \Delta Y
Second differences {\Delta ^2}Y
\begin{array}{*{20}{c}} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{array}
\begin{array}{*{20}{c}} {10} \\ {15} \\ {30} \\ {55} \\ {90} \end{array}
\begin{array}{*{20}{c}} {15 - 10 = 5} \\ {30 - 15 = 15} \\ {55 - 30 = 25} \\ {90 - 55 = 35} \end{array}
\begin{array}{*{20}{c}} {15 - 5 = 10} \\ {25 - 15 = 10} \\ {35 - 25 = 10} \end{array}

 

The second differences are exactly constant. The general quadratic equation or non linear model is written as

Y = a + bX + c{X^2}

       \left( {c \ne 0} \right)

It is also called second degree parabola or second degree curve. The graph of the data is shown in the figure given below:


non-linear-model

This figure is not a straight line. It is a curve or we say that the model Y = 10 + 5{X^2} in non-linear. The readers are advised to remember that if in a certain observed data, the second differences are constant or almost constants, we find the second degree curve close to the observed data. We shall face this type of situation in time series.