- Perform exponential regression
- Convert between exponential and continuous growth
- Compare exponential and linear regressions for best fit
Exponential Regression and Continous Growth
While linear regression is a commonly and widely used tool in modeling and data analysis, some data sets are better modeled by non-linear equations. There are several non-linear equations that can be used to model growth. We’ll take a look in this section at one of them: exponential growth.
Imagine the growth in a population of bacteria. This kind of single-cell life propagates via binary fission. One becomes two, two become four, four become eight, and so on. The population expands rapidly. This is an example of exponential growth. That is, a quantity that grows (or decays) in proportion to itself per unit of input.
We see exponential growth (and decay) in the real world in other situations as well. It shows up in short-run population growth among people and animals, interest earned in banking, radioactive decay, and even in the temperature of a cake cooling from the oven. Anything that grows or decays at a rate proportional to itself experiences exponential growth or decay.
exponential regression
Exponential regression is the process of fitting an exponential function to a set of data. It is performed using the software in a spreadsheet or graphing calculator.
The exponential regression includes a base of [latex]e[/latex]. This base is an irrational number that appears commonly in nature and in exponential growth and decay. But you learned in the previous section that the explicit form of the exponential growth formula is [latex]P_{n}=P_{0}(1+r)^{n}[/latex]. Now we are presented with [latex]y=ae^{x}[/latex], which is called continuous growth. How do these two formulas compare? Let’s look at them side by side.
[latex]P_{n}=P_{0}(1+r)^{n} \qquad \text{ vs. } \qquad y = ae^{rx}[/latex]
We’ll need a fact about exponents before we can discuss them, though.
Product Rule [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]
Quotient Rule [latex]\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]
Power Rule [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]
There are several properties of exponents that allow us to rewrite them in useful ways. Recall that an exponent is a power above a number called a base in the form [latex]a^{m}[/latex]. This tells us to multiply [latex]a[/latex] by itself [latex]m[/latex] times. But, what happens if we raise a base to a power to another power? What does it mean to write [latex]\left(a^{m}\right)^{n}[/latex]?
Consider [latex]\left(a^{3}\right)^{4}[/latex]. Certainly this notation indicates that we should multiply [latex]\left(a^{3}\right)[/latex] by itself [latex]4[/latex] times.
[latex]\left(a^{3}\right) \ast \left(a^{3}\right) \ast \left(a^{3}\right) \ast \left(a^{3}\right)[/latex]
That yields [latex]\qquad a \cdot a \cdot a \ast a \cdot a \cdot a \ast a \cdot a \cdot a \ast a \cdot a \cdot a = a^{12}[/latex]
So it seems that [latex]\left(a^{3}\right)^{4}=a^{3 \cdot 4}=a^{12}[/latex].
This is a fact. [latex]\left(a^{m}\right)^{n} = a^{m \cdot n}[/latex].
We can leverage this fact to rewrite the form of the exponential equation we obtained above.
[latex]P_{n}=P_{0}(1+r)^{n} \qquad \text{ vs. } \qquad y = ae^{rx}[/latex]
Let [latex]P_{n} = y \text{ and } P_{0} = a \text{ and } n = x[/latex]. Then we have
[latex]y=a(1+r)^{x} \qquad \text{ vs. } \qquad y=ae^{rx}[/latex]
But by the fact that enables us to rewrite exponents, we have that [latex]e^{rx} = \left(e^{r}\right)^{x}[/latex]. This yields
[latex]y=a(1+r)^{x} \qquad \text{ vs. } \qquad y=a\left(e^{r}\right)^{x}[/latex].
Equating the bases in the equations, we can see that certainly, given the same models,
[latex]\left(1+r\right)=e^{r}[/latex].