# Numerical Smoothing of Time Series Data

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.

We can use numerical and/or graphical smoothing of time series plots to see overall trends, where we may not be able to see trends otherwise. We implement averaging or a process of moving means to smooth a time plot.

## Smoothing a Time Plot -Movement of a Nanoparticle Through an Electric Field

Say that a nanoparticle, for instance, an isotope of silver is exposed to an ever-fluctuating electric field over the course of 15 seconds, so that the nanoparticle moves up and down vertically through 1 micrometer. If we are given a table of the nanoparticle's vertical height given in increments of one 10th of a micrometer through 15 seconds, how might we smooth the table's time plot using a 3-mean smooth or a 5-mean smooth?

Before we can perform this smoothing of the time plot representing vertical movement of the nanoparticle, let's look at the table of data giving its position through time.

If we were to plot this data, we have a graph that looks like the following, where our x-axis represents time in seconds and our y-axis is representative of the nanoparticle's position through 1 micrometer -numbered in 1/10ths of a micrometer.

### 3-Mean and 5-Mean Time Plot Smoothing

Notice how the time plot above, representing the nanoparticle's vertical movement through an electric field, yields sharp peak and valleys. It may be hard to spot any overall trends in this time plot or, perhaps, we might want to extrapolate more information from this time plot of this given data series. In either case, the process of smoothing this time plot can expose overall trends or trends through smaller intervals.

#### 3-Mean Smoothing

Let's look at the table below, showing the operations for numerical smoothing of position versus time. In the case of the 3-mean smoothing operation in column 3, we merely take the average (or mean) of every group of 3 measurements for height or vertical distance from column 2.

As seen in the 3 -mean column below, our 1st mean value is (0 + 2 + 1) / 3 = 1 and our 2nd is (2 + 1 + 4) / 3 = 2.33, etc. We merely continue averaging groups of 3 until we have no more groups of 3 to average. Our original time plot with the 3-mean smoothed time plot in orange (labeled series 1, superimposed) is shown as:

#### 5-Mean Smoothing

With 5-mean smoothing, we do essentially the same thing as we do with the 3-mean smoothing; however, as opposed to groups of 3, we mean groups of 5. For example, our first mean is (0 + 2 + 1 + 4 + 2) / 5 = 1.8 and our 2nd is (2 + 1 + 4 + 2 + 5) / 5 = 2.8, etc. Superimposing the 5-mean plot (labeled series 2) onto our last plot, we see

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