Finding Seasonality automatically in Time Series

Question

I am working on predicting a demand forecast of a time series data.

dput output is saved to Data Variable

Data <- structure(list(Yr = c(2010L, 2010L, 2010L, 2010L, 2010L, 2010L, 
2010L, 2010L, 2010L, 2010L, 2010L, 2010L, 2011L, 2011L, 2011L, 
2011L, 2011L, 2011L, 2011L, 2011L, 2011L, 2011L, 2011L, 2011L, 
2012L, 2012L, 2012L, 2012L, 2012L, 2012L, 2012L, 2012L, 2012L, 
2012L, 2012L, 2012L, 2013L, 2013L, 2013L, 2013L, 2013L, 2013L, 
2013L, 2013L, 2013L), Month = c(1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 
9L, 10L, 11L, 12L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 
12L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 1L, 2L, 
3L, 4L, 5L, 6L, 7L, 8L, 9L), Demand = c(58L, 59L, 108L, 145L, 
109L, 105L, 104L, 175L, 101L, 105L, 254L, 199L, 187L, 201L, 149L, 
93L, 126L, 115L, 136L, 94L, 135L, 116L, 112L, 95L, 122L, 247L, 
188L, 121L, 237L, 190L, 187L, 206L, 206L, 156L, 198L, 154L, 231L, 
190L, 237L, 250L, 182L, 250L, 118L, 123L, 222L)), .Names = c("Yr", 
"Month", "Demand"), class = "data.frame", row.names = c(NA, -45L
 ))

str(Data)

I take a log Transformation of Demand Variable and Decompose to check Seasonality

Data$Log_Demand = log(Data$Demand)
splot <- ts(Data$Log_Demand, start=c(2010, 1),end=c(2013,9),frequency=12)              
fit <- stl(splot, s.window="period") 
monthplot(splot) 
library(forecast)
seasonplot(splot)

I get a Month plot and Seasonal plot - I am finding it tough to code the seasonal pattern observed.

Data$Seasonal_Jan = ifelse(Data$time %in% c(1,13,25,37),1,0)

My Question here is :

From the Graph i wanted to automatically find for what months seasonal patterns are observed and code a dummy variable(as above) for those seasonal to use that variable in lm model to fit a trend component and from the lm model residuals, I fit a ARIMA Model to predict the forecast.

Answer 1

This is not what you asked but as nobody has answered here are some comments that may help you.

First, I don't see the need to take logarithms on this series. This simple plot does not suggest that the variance of the series increases with the mean.

x <- ts(Data$Demand, start=c(2010, 1), end=c(2013,9), frequency=12)
lx <- split(x, gl(length(x)/6, 6))
m <- unlist(lapply(lx, mean))
r <- unlist(lapply(lx, function(x) diff(range(x))))
plot(m, r)
abline(lm(r ~ m))

A more elaborate method does not suggest taking logs either.

library("forecast")
> BoxCox.lambda(x, lower=0, upper=1)
[1] 0.9999339

As regards the seasonal pattern you are interested in, the autocorrelation functions do not show significant autocorrelation of seasonal order.

par(mfrow = c(2, 1), mar = c(3,2,3,2))
acf(x, lag.max = 60)
pacf(x, lag.max = 60)

A first-order autoregressive model, AR(1), can be appropriate for this series.

The function forecast: 'seasonaldummy' builds seasonal dummies for all seasons except one (to avoid multicollinearity) that can be included in a regression.

SD <- seasonaldummy(x)
> summary(lm(x ~ SD))
[...]
Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 149.42857   22.69524   6.584 1.52e-07 ***
SD[, -1]Feb  24.82143   37.63580   0.660    0.514    
SD[, -1]Mar  21.07143   37.63580   0.560    0.579    
SD[, -1]Apr   2.82143   37.63580   0.075    0.941    
SD[, -1]May  14.07143   37.63580   0.374    0.711    
SD[, -1]Jun  15.57143   37.63580   0.414    0.682    
SD[, -1]Jul -13.17857   37.63580  -0.350    0.728    
SD[, -1]Aug   0.07143   37.63580   0.002    0.998    
SD[, -1]Sep  16.57143   37.63580   0.440    0.662    
SD[, -1]Oct -23.76190   41.43566  -0.573    0.570    
SD[, -1]Nov  38.57143   41.43566   0.931    0.358    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
[...]

The seasonal dummies are not significant at the 5% level. Similar results are obtained with the logarithms of the original series.