Question

Use cubic regression to find a function that fits the following points (1,4),(2,9),(3,34),(-2,-11)

Answer 1

Answer:

The cubic regression that fits the given set of points is $y = 2\cdot x^{3}-2\cdot x^{2} -3\cdot x + 7$.

Step-by-step explanation:

Let the cubic regression be represented by the following third order polynomial:

$y = a\cdot x^{3}+b\cdot x^{2}+c\cdot x + d$ (1)

Where:

$x$ - Independent variable.

$y$ - Dependent variable.

$a,b,c,d$ - Coefficients.

By Algebra, we can determine analytically the values of all coefficients by knowing four distinct points. If we know that $(x_{1}, y_{1}) = (1,4)$, $(x_{2},y_{2}) = (2,9)$, $(x_{3}, y_{3}) = (3, 34)$ and $(x_{4}, y_{4}) = (-2, -11)$, then the system of linear equations is:

$a+b+c+d = 4$ (1)

$8\cdot a + 4\cdot b + 2\cdot c + d = 9$ (2)

$27\cdot a + 9\cdot b + 3\cdot c + d = 34$ (3)

$-8\cdot a + 4\cdot b -2\cdot c + d = -11$ (4)

The solution of the system of linear equations is: $a = 2$, $b = -2$, $c = -3$, $d = 7$

The cubic regression that fits the given set of points is $y = 2\cdot x^{3}-2\cdot x^{2} -3\cdot x + 7$.