Fitting a Straight Line and a Parabola

Definition

Curve fitting is the mathematical process of constructing a curve (a mathematical function) that has the best fit to a series of data points. When we fit a straight line ($y = a + bx$) or a parabola ($y = a + bx + cx^2$), we use the Method of Least Squares to minimize the sum of the squares of the vertical deviations between the observed data points and the fitted curve.

Main Content

1. Fitting a Straight Line (Linear Regression)

A straight line represents a linear relationship where a constant change in $x$ leads to a constant change in $y$.
It is defined by the equation $y = a + bx$, where '$a$' is the y-intercept and '$b$' is the slope of the line.

2. Fitting a Parabola (Quadratic Regression)

A parabola represents a non-linear relationship where the rate of change is not constant, typically used when data shows a "U" shape or a curvature.
It is defined by the equation $y = a + bx + cx^2$, involving three constants ($a, b, c$) that determine the shape and position of the curve.

3. The Method of Least Squares

This principle states that the line or curve of best fit is the one where the sum of the squares of the residuals (the difference between actual $y$ and predicted $y$) is minimized.
It translates into "Normal Equations" which allow us to solve for the unknown constants ($a, b, c$) algebraically.

Working / Process

1. Identify Normal Equations

For a straight line ($y = a + bx$): 1) $\sum y = na + b \sum x$ 2) $\sum xy = a \sum x + b \sum x^2$
For a parabola ($y = a + bx + cx^2$): 1) $\sum y = na + b \sum x + c \sum x^2$ 2) $\sum xy = a \sum x + b \sum x^2 + c \sum x^3$ 3) $\sum x^2y = a \sum x^2 + b \sum x^3 + c \sum x^4$

2. Tabulation of Data

Create a table to calculate the required summations ($\sum x, \sum y, \sum xy, \sum x^2$, etc.) based on your data points $(x, y)$.
Ensure the number of data points $n$ is clearly identified to substitute into the equations.

3. Solve for Constants

Substitute the summation values from the table into the Normal Equations.
Use substitution or elimination methods to solve the system of linear equations to find the values of $a, b$, and $c$.

Visual representation of a Linear vs Parabolic fit:

      |      / (Linear)       |     \     / (Parabolic)
      |     /                 |      \   /
      |    /                  |       \ /
      |   /                   |        -
      ------------------      ------------------

Advantages / Applications

Predictive Analysis: Allows researchers to forecast future trends based on existing historical data points.
Scientific Modeling: Helps in understanding physical laws, such as projectile motion (parabolic) or velocity-time relationships (linear).
Data Smoothing: Eliminates "noise" or random fluctuations in data to reveal the underlying trend.

Summary

Curve fitting is a technique used to find the best-fitting mathematical model for a set of observations. By minimizing the sum of the squares of the errors, we can determine the specific parameters of a straight line or a parabola that represent a data set.

Key terms: Residual (error), Normal Equations, Least Squares Method, y-intercept, Slope.