What model to fit given ACF and PACF (seasonal data)

Question

I have highly seasonal data, (it's energy consumption) with mostly 24 hour and 168 hour (=1 week) periods and I have applied differencing by 168 hours (diff(time_series,lag=168)) to obtain something more stationary

because I don't see how I can fit any model on a series which has this many seasonalities (there is also a less obvious 12 hour period). Now I have plotted the acf and pacf and I was wondering about how to interpret these:

Lag=0 is not included because I used forecast::Acf. So on the pacf the obvious outliers are for lag=1,3 and 168.

I don't know if we can say that there is geometric or exponential decay in any of the correlograms, so any comments are welcome.

Answer 1

Selecting ARIMA p,d,q paramerters for hourly data with 24 hour cycle lays out a strategy for building 24 daily models and then using daily predictions from each of these models to predict hourly values. Pay particularly close attention to this reference Forecasting data with multiple seasonality and https://autobox.com/cms/index.php/blog/entry/forecasting-at-an-hourly-level.

In summary there is more to forecasting than simply weighting the past (SARIMA) when latent factors ( holidays, level shifts , local time trends ) along with possible predictor series like temperature are critically important to detect and incorporate. Methods that extract latent deterministic structure are in effect pseudo-causal.

Relying on the past of the output series alone is equivalent to "drivers that navigate using only the rear-window " rather than both the rear-window and the front window.

Answer 2

ARIMA, ACF and PACF are not useful for data with multiple-seasonalities. I strongly recommend you look at models that were designed for this use case, like bats and tbats.

Don't try to specify these models by hand (as you could specify ARIMA orders by looking at the ACF/PACF plots in the obsolete Box-Jenkins method). Instead, use a good implementation like the one in the forecast package for R.