# Analyzing Data With Nonlinear Regression Models

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

There are many nonlinear regression models that can be used to solve a variety of real-world problems. Data in finance, business and physical science can be analyzed using nonlinear regression functions.

## Nonlinear Regression Models

We see a whole host of problems in many areas of study, where nonlinear regression models, or regression analyses, are used to analyze data and solve problems. For example, we see nonlinear regression in studies involving:

• Finance - specifically exponential functions in continually compounded interest
• Physical science - specifically power and quadratic functions in motion

In this lesson, we'll look at two such examples to illustrate just how data may be interpreted using nonlinear regression models.

### Exponential Functions in Finance

In a finance problem, like one involving a compounding interest calculator (specifically in continually compounding interest), we see the function of the form:

A = Pert

Here, A is the total amount of money in dollars (\$) over a period of time (t) in years.

Note that the amount (A) can be likened to the principal investment plus the amount of return on the principal by way of a continually compounded interest rate over a given period.

In this equation:

• P is the starting principal in dollars.
• r is the compounding interest rate.
• e is a constant called Euler's number.

### Sample Problem

If given a principal or an initial investment of \$1,000 with a continually compounding interest rate of 10% (r = .10) to be compounded over a period of 7 years, what values for A might we expect to see over the course of 7 years, where in the first-year, t = 1?

A = 1000e(.10)(1)

A = \$1105.17, or \$1105.20

Therefore, the amount after one year is \$1105.20. For the 6 subsequent years thereafter, please see the table below. Please note that time (t) is represented by x and A is represented by y1.

### Graphing the Function

If we were to graph the function A = 1000e.10t, we would come up with a line of an exponential function, a type of nonlinear regression, for amount vs. time. Here time (t) is represented by x as well.

### Parabolic Functions in Physical Science

In problems regarding motion in physical science, such as the displacement of an object vs. time, we sometimes see parabolic functions or regression models used. In this case, a power function of the form

y = t2

might be used to trace the vertical position of an object through y meters during a given time (t) period in seconds. Note that if we have an object moving vertically through a distance with respect to time represented by the parabolic function or nonlinear regression

y = -t2 + 7

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