We have demonstrated in this section that constant, linear, and piecewise combinations of these two function types can be used to model a variety of phenomena inspired by real-world situations. What happens, as if often the case in real-world situations, when we are given data sets that are not precisely linear, but still have a definite linear trend? An example of this is Skippy’s time and temperature data from Example timetempex1 in Section FunctionsandtheirRepresentations.

In that example, \(t\) represented the time (number of hours after 6 a.m.) and \(T\) represented the outdoor temperature in degrees Fahrenheit. The data Skippy collected along with a plot of the function \(T = f(t)\) are given below. Even though the data points as \(t\) varies from \(t = 0\) to \(t = 8\) do not all lie on the same line - a fact we could prove analytically by checking slopes - there does appear to be a linear trend evident. The same can be said for the data as \(t\) varies from \(t =8 \) to \(t = 12\). As we’ll see, there are statistical methods which can produce linear functions that are in some sense ‘closest’ to all of the data, and they are represented below by the dashed lines below on the right.

\(\begin{array}{|c||c|} \hline \text {$t$: hours after 6 a.m.} & \text {$T$: temperature $^{\circ }$F} \\ \hline 0 & 64 \\ 2 & 67 \\ 4 & 75 \\ 6 & 80 \\ 8 & 83 \\ 10 & 83 \\ 12 & 82 \\ \end{array}\)

How do we measure how ‘close’ a set of points is to a given line? Let’s leave Skippy’s data for the moment and focus on a smaller data set. Suppose we collected three data points: \(\{(1, 0.5), (3, 2), (4, 3)\}\). At the top of the next page (on the left) we plot these points along with the line \(y = 0.5 x + 0.5\). The way we measure how close the line is to these points is by computing the total squared (vertical) error between the data points and the line as follows. For each of our data points, we find the vertical distance between the point and the line. To accomplish this, we need to find a point on the line directly above or below each data point. In other words, we need a point on the line with the same \(x\)-coordinate as our data point.

For example, to find the point on the line directly above \((1,0.5)\), we plug \(x=1\) into \(y=0.5x + 0.5\) and we get the point \((1,1)\). Similarly, we find \((3,2)\) is on the line already and \((4,2.5)\) is the point on the line directly beneath \((4,3)\). We find the total squared error \(E\) by taking the sum of the squares of the differences of the \(y\)-coordinates of each data point and its corresponding point on the line. For the data and line in this discussion \(E = (0.5-1)^2+(2-2)^2+\left (3-2.5\right )^2 = 0.5\).

Using advanced mathematical machinery, it is possible to find the line which results in the lowest value of \(E\). This line is called the least squares regression line, or sometimes the ‘line of best fit’. The formula for the line of best fit requires notation we won’t present until Chapter SequencesandtheBinomialTheorem, so we will revisit it then. Most graphing utilities have a built-in regression feature, so at this point we turn the computations over to the technology. A screenshot from desmos is given on the right at the top of the next page.

[Picture]

Our graphing utility produces the model \(y=mx+b\) where the slope is \(m \approx 0.821\) and the \(y\)-coordinate of the \(y\)-intercept is \(b \approx -0.357\). The value \(r\) is the correlation coefficient and is a measure of how close the data is to being on the same line. The closer \(|r|\) is to \(1\), the better the linear fit. Having \(r \approx 0.997\) tells us that the points have a strong, positive correlation - that is, they are very close to being on a line with a positive slope, namely \(y = 0.821x - 0.357\). Indeed, the total squared error between our data set and this line is \(E \approx 0.018\). The mathematics tells us that this is the smallest we can get \(E\) by modifying the parameters \(m\) and \(b\), even though none of the data points actually lie on the line.

Now that we have this new mathematical machinery, let’s revisit Skippy’s time and temperature data.

Examples PortaBoyCost, PortaBoyDemand and timetempregressionex (among others) represent three different levels of mathematical modeling. In Example PortaBoyCost, the mathematical model (the cost function) was provided and our task was to use the model to interpret the mathematics in that context. In Example PortaBoyDemand, we were given a minimal amount of information, namely, two data points, and then asked to construct a model which fit those data exactly. Lastly, in Example timetempregressionex, we were given several data points and we used statistical methods to construct a best fit model to the data.

The validity of the models rests on the validity of the underlying assumptions used to create the models. For instance, is there any reason to assume a price-demand function would be linear? Is it reasonable to assume that the temperature changes at a constant rate? These are questions for economists and scientists. Mathematicians often take on a role of equal parts translator and prophet: they codify ideas into formulas and then use them to make predictions about yet-to-be observed phenomena.