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Cubic Regression Calculator + Online Solver With Free Steps
The Cubic Regression Calculator performs the cubic regression calculation using the least-squares method. In reality, the model matrix X, including the independent variable, and the vector y, containing the values of the dependent variable, employ the normal equation.
This equation enables us to determine the cubic regression coefficients using a sequence of matrix operations.
What Is a Cubic Regression Calculator?
The Cubic Regression Calculator is an online tool that uses a statistical method that identifies the cubic polynomial (a polynomial of degree 3) that best fits our sample.
This is a particular type of polynomial regression, which also has quadratic and simple linear versions.
Regression is a statistical method that, in general, enables us to model the connection between two variables by identifying the curve that most closely matches the observed samples.
We deal with cubic functions, or polynomials of degree 3, in the cubic regression model.
The concept is the same in all regression models, whether it be quadratic regression or linear regression, where we deal with parabolas instead of trying to fit a straight line to data points.
Polynomial regression is illustrated by these three types of regression.
How To Use a Cubic Regression Calculator
You can use the Cubic Regression Calculator by following the given instructions to receive the desired results.
Step 1
Enter the data points in the respective input field
Step 2
Click on the “SUBMIT” button to determine the Cubic Regression and also the whole step-by-step solution for the Cubic Regression will be displayed.
When the scatter plot indicates that the data follows a cubic curve, we use a cubic equation. We always strive to fit a simpler model, such as basic linear or quadratic. Keep in mind that we want our models to be as straightforward as possible.
How Does a Cubic Regression Calculator Work?
The Cubic Regression Calculator works by using the least-squares method to compute cubic regression.
In real-world applications, we use the normal equation, which makes use of the model matrix X, which involves the independent variable, and the vector y, which holds the values of the dependent variable.
This equation enables us to determine the cubic regression coefficients using a sequence of matrix operations.
The Formula for Cubic Regression
We need to introduce some notation to discuss the cubic regression formula more formally in the following data points:
(x1, y1), … , (xn, yn)
The cubic regression function takes the form:
y = a + b.x + c.$x^2$ + d.$x^3$
where a, b, c, and d are real integers that represent the cubic regression model’s coefficients. As you can see, we simulate the impact of a change in x on the value of y.
In other words, we assume that y is the dependent (response) variable and that x is the independent (explanatory) variable in this situation.
- We get quadratic regression if d = 0.
- A straightforward linear regression model results if c = d = 0.
The primary difficulty right now is figuring out what the four coefficients’ real values are. In most cases, we use the least-squares method to determine the coefficients of the cubic regression model.
Specifically, we seek a, b, c, and d values that reduce the squared distance between each data point (x$_\mathsf{i}$, y$_\mathsf{i}$) and the equivalent point that the equation for cubic regression predicts as:
\[ (x_i\,,\, a + bx_i + c(x_i)^2 + d(x_i)^3) \]
Solved Examples
Let’s explore some examples to better understand the working of the Cubic Regression Calculator.
Example 1
Let us find the cubic regression function for the following dataset:
(0, 1), (2, 0), (3, 3), (4, 5), (5, 4)
Solution
Here are our matrices:
- The matrix X:
\[ \begin{bmatrix} 1 & 0 & 0 & 0\\ 1 & 2 & 4 & 8\\ 1 & 3 & 9 & 27\\ 1 & 4 & 16 & 64\\ 1 & 5 & 25 & 125\\ \end{bmatrix} \]
- The vector y:
\[\begin{bmatrix} 1 \\ 0 \\ 3 \\ 5 \\ 4 \\ \end{bmatrix}\]
We apply the formula step-by-step:
- First, we determine X$^\mathsf{T}$:
\[\begin{bmatrix} 1 & 1 & 1 & 1 & 1\\ 0 & 2 & 3 & 4 & 5\\ 0 & 4 & 9 & 16 & 25\\ 0 & 8 & 27 & 64 & 125\\ \end{bmatrix}\]
- Next, we compute X$^\mathsf{T} \cdot$ X:
\[\begin{bmatrix} 5 & 14 & 54 & 224 \\ 14 & 54 & 224 & 978 \\ 54 & 224 & 978 & 4424 \\ 224 & 978 & 4424 & 20514 \\ \end{bmatrix}\]
- Then, we find (X$^\mathsf{T} \cdot$ X)$^\mathsf{-1}$:
\[\begin{bmatrix} 0.9987 & -0.9544 & 0.2844 & -0.0267 \\ -0.9544 & 5.5128 & -2.7877 & 0.3488 \\ 0.2844 & -2.7877 & 1.4987 & -0.1934 \\ -0.0267 & 0.3488 & -0.1934 & 0.0254 \\ \end{bmatrix}\]
- Finally, we perform the matrix multiplication (X$^\mathsf{T}\cdot$ X)$^\mathsf{-1}\,\cdot$ X$^\mathsf{T}\cdot$ X. The linear regression coefficients we wanted to find are:
\[\begin{bmatrix} 0.9973 \\
-5.0755 \\ 3.0687 \\ -0.3868 \\ \end{bmatrix}\]
- Therefore, the cubic regression function that best fits our data is:
y = 0.9973-5.0755.x + 3.0687.$x^2$-0.3868.$x^3$
Example 2
Let us find the cubic regression function for the following dataset:
(10, 15), (11, 5), (3, 4), (8, 8), (10, 12)
Solution
Fitted coefficients of the dataset:
a = 129.1429
b = -69.7429
c = 10.8536
d = -0.5036
Cubic model:
y = 129.1429 – 69.7429.x + 10.8536.$x^2$-0.5036.$x^3$
The goodness of fit:
Standard Error of Regression: 2.1213
Coefficient of determination R$^\mathsf{2}$: 0.9482