14.1 Curve of Best Fit

There are many situations in which two continuous random variables, $X$ and $Y$, are related to one another. The question is, “how are they related?” Or another question one could ask is, “if one of the variables changes, how does the other change?” In order to determine a particular relationship, we rely on using a method of regression.

In order to use a particular method of regression, is important to first classify what are the independent and dependent variables.

So our objective to show a relationship between the variables $X$ and $Y$. Ideally, we would like to find a function that maps values data values of $x$ to data values of $y$. In mathematics, such a function is often denoted as $Y(X)$.

Let’s briefly go through a few examples of different methods of regression.

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  Linear Regression: Here the goal is to find a linear relationship between $X$ and $Y$, assuming that $X$ and $Y$ are continuous random variables. That is, we would like to write

  $$Y(X)=mX+b,$$

  where $m$ is the slope and $b$ is the $y$-intercept. Below is an example of pairwise data values from $X$ and $Y$, written as $(x_i,y_i)$, plotted on the cartesian plane ($xy$-plane) that somewhat show a linear relationship. A line is approximated and calculated to best describe the relationship. Can you see the estimated line?

  

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  Quadratic Regression: In quadratic regression, the goal is to find a quadratic relationship between $X$ and $Y$, assuming that $X$ and $Y$ are continuous random variables. That is, we would like to write

  $$Y(X)=a{X}^2+bX+c,$$

  where $a,b$ and $c$ are real numbers. Below is an example of data that exhibits a quadratic regression. Notice the parabolic-like curve that fits the data.

  

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  Sinusoidal Regression: In sinusoidal regression, the objective is to find a sinusoidal (sine function) relationship between $X$ and $Y$, assuming that $X$ and $Y$ are continuous random variables. That is, we would like to write

  $$Y(X)=A\sin (BX+C)+D,$$

  where $A,B,C,$ and $D$ are real numbers. Below is an example of such a sine-like curve that fits the data.

  

There are other types of regression not described here, such as logarithmic regression and logistic regression. For the purposes of a brief introduction, we leave out the details for these types and others. A subsequent regression analysis course would elaborate further on the different types of regressions that are possible. In this chapter, we will focus on linear regression.