How do you solve quadratic regression?

Solving quadratic regression involves finding the quadratic equation (y = ax² + bx + c) that best fits a given set of data points. The process, according to the reference, generally follows these steps:

Steps to Solve Quadratic Regression

Here's a breakdown of how to solve quadratic regression:

1. Count the Total Number of Data Points: Determine the total number of data points (n) you have.

2. Find All the Values Needed for the Formula: This typically involves calculating sums and sums of squares and cross-products of your x and y values. Specifically, you'll need to calculate the following sums:

   • ∑x (sum of x values)
   • ∑y (sum of y values)
   • ∑x² (sum of x-squared values)
   • ∑x³ (sum of x-cubed values)
   • ∑x⁴ (sum of x to the power of 4 values)
   • ∑xy (sum of the product of x and y values)
   • ∑x²y (sum of the product of x-squared and y values)

3. Substitute the Values in the Formula: These sums are then substituted into a system of three linear equations. These equations are derived from the normal equations used in least squares regression. The goal is to solve for the coefficients a, b, and c in the quadratic equation. The general form of the system of equations is:

   • Equation 1: a∑x⁴ + b∑x³ + c∑x² = ∑x²y
   • Equation 2: a∑x³ + b∑x² + c∑x = ∑xy
   • Equation 3: a∑x² + b∑x + cn = ∑y

4. Calculate a: Solve the system of equations for 'a'. This often involves using methods like matrix algebra, substitution, or elimination to isolate 'a'.

5. Calculate b: After finding 'a', substitute its value back into the system of equations to solve for 'b'.

6. Calculate c: Once 'a' and 'b' are known, substitute their values into any of the three equations to solve for 'c'.

7. Substitute the value of a, b, c in the Quadratic Regression Equation: Finally, plug the calculated values of a, b, and c into the quadratic regression equation: y = ax² + bx + c.

In summary, the process involves gathering data, calculating necessary sums, setting up and solving a system of equations, and then plugging the resulting coefficients into the quadratic equation. Most statistical software packages and spreadsheet programs can perform quadratic regression automatically, making the process much easier than manual calculation.