# Time Series

We have a number of tools for looking at time series and searching for periodicity in the data. The Nyquist Sampling Theorem relates the frequency at which you sample the time series to the frequencies in the original data series that you can resolve.

We will consider a time series to be a sequence of readings of some parameter at equally spaced points in time, although in many cases we could also consider a series of equally spaced values of the parameter along a horizontal distance axis. Waves represent time series that we can visualize either on a time or distance axis: on a time axis we remain at a fixed point and see how the water surface varies at discrete points in time, and on a distance axis we freeze the water surface and see how the surface varies at discrete distances along it. Tides represent time series that we can visualize on a time axis: we remain at a fixed point and see how the water surface varies at discrete points in time.

We look at time series for two reasons:

• Search for periodicity in the data, both for objective confirmation that the data is periodic, and to determine the period or frequency of the regular pattern.
• Search for the relationship between two time series, and in particular to determine if there is a time “lag” between the two relationships. A change in one parameter may cause a change in another, but it may take time for the effect to be seen. For instance an El Niño will cause a change in the water levels on the east side of the Pacific, but it takes time for the water to cross the Pacific in a Kelvin wave. Thus the water level changes may lag the changes in pressure or wind speed.

If we had clean data this would be an easy process, but real world data can be very noisy and we need mathematical assistance.

Methods in MICRODEM for looking at time series available on Stats button of table display window for a database.

• FFT (Fast Fourier Transform)
• Auto Correlation
• Cross Correlation
• Fit Fourier Curve
• Plot (xy):
• Plot (1 series): if there is no explicit time variable, the program will plot the values on the y axis, with the x values being the place within the file.  This is only valid if the readings are evenly spaced.
• Linear interpolate across gaps: pick the time variable (cannot have gaps) and the series value, and any gaps will be filled by linear interpolation.

Last revised 2/21/2016