# Multiple and long seasonality for a SARIMA model in R [closed]

While working on a big data set made of 10-minutes-points of information - i.e. 144 points per day, 1008 per week and 52560 per year - I encountered a few problem in R. The information concerns electricity load on a source substation during the year.

### Multiple seasonality :

The data set clearly shows multiple seasonalities, which are daily, weekly and yearly. From there I understood that R doesn't handle multiple seasonality within the ARIMA modeling functions. I would really like to work with ARIMA models though, because my previous work is based on ARIMA models and I know approximatively how to translate a model into an equation.

### Long seasonality :

Each of the seasonalities is of high value, with the shortest one being the daily seasonality at 144. Unfortunately from the SARIMA general equation which is
$\phi(B)\Phi(B^s)W_t = \theta(B)\Theta(B^s)Z_t$
I guessed that the maximum lag for a given model SARIMA(p,d,q)(P,D,Q)144 is
$max((p+P*144), (q+Q*144))$

I would really like to try and fit models with values of P and/or Q greater than 1, but R doesn't allow me since the maximum supported lag = 350. To do so I found this link which is really interesting and led to new functions in the forecast package by M. Hyndman, called fourier and fourierf which you can find here. But since I am not a specialist in forecasting nor in statistics, I have some difficulties understanding how I can make this work.

The thing is I thing this whole fourier regressors package could help me a lot. From what I understood I could use it to simulate the long-seasonality of my data set, maybe use it to simulate multiple seasonality, and even more it could allow me to introduce exogenous variables - which are the temperature and (public holiday + sundays).
I also tried doing some regression following this example but I couldn't make it work because :

Error in forecast.Arima(bestfit, xreg = fourierf(gas, K = 12, h = 1008)) :
Number of regressors does not match fitted model

I really hope somebody can help me get a better understanding of these functions. Thanks.

Edit : So I tried my best with the fourier example given here but couldn't figure out how it handles the fitting. Here is the code (I copy-pasted M. Hyndman one and adapted to my data set - unsuccessfully) :

n <- 50000
m <- 144
fourier <- function(t,terms,period)
{
n <- length(t)
X <- matrix(,nrow=n,ncol=2*terms)
for(i in 1:terms)
{
X[,2*i-1] <- sin(2*pi*i*t/period)
X[,2*i] <- cos(2*pi*i*t/period)
}
colnames(X) <- paste(c("S","C"),rep(1:terms,rep(2,terms)),sep="")
return(X)
}

library(forecast)
fit <- Arima(y[1:n,1], order=c(2,1,5), seasonal=c(1,2,8), xreg=cbind(fourier(1:n,4,m),fourier(1:n,4,1008)))
plot(forecast(fit, h=14*m, xreg=cbind(fourier(n+1:(14*m),4,m), fourier(n+1:(14*m),4,1008))))

So I wanted to "force" the model to be a SARIMA(2,1,5)(1,2,8)[144] but when I type arimodthis is the result of the Arima fitting :

> fit
Series: y[1:n, 1] ,
ARIMA(2,1,5)

sigma^2 estimated as 696895:  log likelihood=-407290.2
AIC=814628.3   AICc=814628.3   BIC=814840

It doesn't even take into consideration the seasonal part of the model, and I don't know much about the range the AIC values can take, but it seems way too high to be a good fitting model right there. I think it all comes down to my misunderstanding of the use of Fourier terms as regressors, but I can't figure out why.

Edit 2 : Also I can't seem to be able to add another exogenous variable to the Arima function. I need to use temperature - probably as a lead - to fit the SARIMAX model, but as soon as I write this :

fit <- Arima(y[1:n,1], order=c(2,1,5), seasonal=c(1,2,8), xreg=cbind(fourier(1:n,4,m),fourier(1:n,4,1008), tmp[1:n]))
plot(forecast(fit, h=14*m, xreg=cbind(fourier(n+1:(14*m),4,m),fourier(n+1:(14*m),4,1008), tmp[n+1:(14*m)])))

Nothing is plotted besides the initial data set. There is no forecast while without tmp as an xreg I still get some results.

## closed as off-topic by Nick Cox, Michael Chernick, COOLSerdash, mdewey, Peter Flom♦Jun 21 '18 at 18:37

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Structure your data as an msts (multiple seasonality time series), where you can specify

msts(your_data, start=2010, seasonal.periods=c(144,1008,52560)).

Then, when fitting fourier terms for seasonality you must specify the three seasonal periods on fourier function:

reg <- fourier(your_data, K=c(i,n,j)), this will look for fourier terms over each seasonal period.

You can loop through a linear regression where the explanatory variables are the fourier terms to obtain the best fit and then put them into the ARIMA model as regressors. These will be equivalent, because ARIMA with regressors actually is a regression with ARIMA errors.

auto.arima(your_data, xreg= as.matrix(reg), seasonal= F)

Don´t forget to specify seasonal=F to avoid maximum supported lag = 350.