One-Hour Autocorrelation Coefficient

The autocorrelation coefficient represents the strength of autocorrelation in a time series. One can calculate the autocorrelation coefficient for a lag of any number of time steps. The one-time-step autocorrelation coefficient, for example, indicates how strongly the value in any time step tends to depend on the value in the previous time step. The two-time-step autocorrelation coefficient indicates the strength of the dependence on the value two time steps earlier, the three-time-step autocorrelation coefficient refers to the dependence on the value three time steps earlier, and so on.

The one-hour autocorrelation coefficient is simply the autocorrelation coefficient at a lag of one hour. For a dataset with a 10-minute time step, that means a lag of six time steps. For a dataset with a 15-minute time step, that means a lag of four time steps. The one-hour autocorrelation coefficient is undefined if the time step is greater than one hour.

Use in Synthesizing Wind Speed Data

A sequence of values in which the current value depends only on the immediately preceding value is called a first-order autoregressive sequence. In such a sequence of numbers, the autocorrelation coefficients obey a very simple pattern:

where r₁ is the correlation coefficient for a lag of one time step.

When you synthesize data in the Synthesize Wind Speed Data window, you enter the one-hour autocorrelation coefficient and Windographer assumes the wind behaves like a first-order autoregressive sequence, so it calculates the corresponding value of r₁ using the following equation, which is just the previous equation solved for r₁:

Using that r₁ value, Windographer generates a first-order autoregressive sequence as part of the process of synthesizing wind speed data.

Example

For example, if you synthesize 10-minute data and set the one-hour autocorrelation coefficient to 0.85, then k will be 6 and the above equation gives r₁ = 0.973. That is to say, in an autoregressive sequence with r₁ = 0.973, the autocorrelation coefficient for a lag of six time steps will be 0.85.

You can use the Wind Speed Statistics for Synthesis table to calculate the monthly one-hour autocorrelation coefficients for use in the Synthesize Wind Speed Data tool.