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The simplest statistical prediction is known as a persistence forecast: the prediction is set equal to the last available measurement. In other words, the last measured value is assumed to persist into the future without any change:

(2.49)

where y_k−1 is the measured value at step k−1 and is the prediction for the next step.

A more sophisticated prediction might be some linear combination of the last n measured values, that is,

(2.50)

This is known as an n^th order autoregressive model, or AR(n). We can now define the prediction error at step k by

(2.51)

and then use the recent prediction errors to improve the prediction:

(2.52)

This is known as an n^th order autoregressive, m^th order moving average model, or ARMA(n,m). This can be further extended to an ARMAX model, where the X stands for an ‘exogenous’ variable: another measured variable that is included in the prediction because it influences y.

The model parameters a_i, b_j can be estimated in various ways. A useful technique is the method of recursive least squares, or RLS (Ljung and Söderström 1983). Estimates of the model parameters are updated on each timestep in such a way as to minimise the expected value of the sum of squares of the prediction errors. By including a so‐called ‘forgetting factor’, the influence of older observations can be progressively reduced, leading to an adaptive estimation of the parameters, which will gradually change to accommodate variations in the statistical properties of the variable y.

Bossanyi (1985) investigated the use of ARMA models for wind speed predictions from a few seconds to a few minutes ahead, obtaining reductions in root mean square (rms) prediction errors of up to 20% when compared to a persistence forecast. The best results were obtained when predicting 10 minutes ahead from 1‐minute data.

Kariniotakis et al. (1997) compare ARMA methods against a selection of more recent techniques such as neural network, fuzzy logic, and wavelet‐based methods. The fuzzy logic method is tentatively selected as giving the best predictions over periods of 10 minutes to 2 hours, with improvements of 10–18% compared to persistence.

Nielsen and Madsen (1999) use an ARX model with RLS to predict wind farm power output based on previous values of power output, and measured wind speed as an exogenous variable, supplemented by a function describing the diurnal variations of wind speed and by meteorological forecasts of wind speed and direction. Predictions up to 48 hours ahead are considered, and the inclusion of meteorological forecasts is shown to improve the predictions significantly, especially for the longer period forecasts.