## 3.1 The Nyquist Frequency

For any discrete time series the number of frequencies at which the data can vary will be limited by the Nyquist freqency, which is the highest frequency of variation that the observed data can provide any variation. For example, with the Baltimore temperature data, we have daily measurements, which represent 24-hour averages. As a result, the data cannot provide any information about variation in the data at a time scale less than 24 hours or daily. If there is a diurnal pattern in the data, so may be the it’s always hotter at 4pm than at 4am, we cannot learn this from the data.

If we have \(n\) observations, then the most number of cycles that we can observe in the time series is \(n/2,\) or one cycle every other data point. Another way of saying that is that the Nyquist frequency is \(f=\frac{1}{2}\), i.e. a half a cycle per day or one cycle per 2 days. If we wanted information about higher frequency variation in temperature, we would need to make more observations. For example, if we had hourly measurements of temperature, then we would have information about the within-day variation in temperature.

If the Nyquist frequency is the highest frequency about which the data can inform us, then on the other side of the spectrum is the lowest frequency, which is simply 1 cycle per \(n\) observations. In the full Baltimore dataset, we have daily observations for 19 years. Therefore, the lowest frequency we can observe is 1 cycle per 19 years. Sadly, there is no special name for the lowest frequency.