Decomposition of time series data

Question

Can anybody help be decipher the output of ucm. My main objective is to check if the ts data is seasonal or not. But i cannot plot and look and every time. I need to automate the entire process and provide an indicator for the seasonality.

I want to understand the following output

ucmxmodel$s.season

# Time Series:
#     Start = c(1, 1) 
#     End = c(4, 20) 
#     Frequency = 52 
#       [1] -2.391635076 -2.127871717 -0.864021134  0.149851212 -0.586660213 -0.697838635 -0.933982269  0.954491859 -1.531715424 -1.267769820 -0.504165631
#      [12] -1.990792301  1.273673437  1.786860414  0.050859315 -0.685677002 -0.921831488 -1.283081922 -1.144376739 -0.964042949 -1.510837956  1.391991657
#      [23] -0.261175626  5.419494363  0.543898305  0.002548125  1.126895943  1.474427901  2.154721023  2.501352782  0.515453691 -0.470886132  1.209419689

ucmxmodel$vs.season

# [1] 1.375832 1.373459 1.371358 1.369520 1.367945 1.366632 1.365582 1.364795 1.364270 1.364007 1.364007 1.364270 1.364795 1.365582 1.366632 1.367945
# [17] 1.369520 1.371358 1.373459 1.375816 1.784574 1.784910 1.785223 1.785514 1.785784 1.786032 1.786258 1.786461 1.786643 1.786802 1.786938 1.787052
# [33] 1.787143 1.787212 1.787257 1.787280 1.787280 1.787257 1.787212 1.787143 1.787052 1.786938 1.786802 1.786643 1.786461 1.786258 1.786032 1.785784
# [49] 1.785514 1.785223 1.784910 1.784578 1.375641 1.373276 1.371175 1.369337 1.367762 1.366449 1.365399 1.364612 1.364087 1.363824 1.363824 1.364087
# [65] 1.364612 1.365399 1.366449 1.367762 1.369337 1.371175 1.373276 1.375636 1.784453 1.784788 1.785101 1.785392 1.785662 1.785910 1.786136 1.786339

ucmxmodel$est.var.season

# Season_Variance 
#     0.0001831373 

How can i use the above info without looking at the plots to determine the seasonality and at what level ( weekly, monthly, quarterly or yearly)?

In addition, i am getting NULL in est

ucmxmodel$est

# NULL

Data

The data for a test is:

structure(c(44, 81, 99, 25, 69, 42, 6, 25, 75, 90, 73, 65, 55, 
9, 53, 43, 19, 28, 48, 71, 36, 1, 66, 46, 55, 56, 100, 89, 29, 
93, 55, 56, 35, 87, 77, 88, 18, 32, 6, 2, 15, 36, 48, 80, 48, 
2, 22, 2, 97, 14, 31, 54, 98, 43, 62, 94, 53, 17, 45, 92, 98, 
7, 19, 84, 74, 28, 11, 65, 26, 97, 67, 4, 25, 62, 9, 5, 76, 96, 
2, 55, 46, 84, 11, 62, 54, 99, 84, 7, 13, 26, 18, 42, 72, 1, 
83, 10, 6, 32, 3, 21, 100, 100, 98, 91, 89, 18, 88, 90, 54, 49, 
5, 95, 22), .Tsp = c(1, 3.15384615384615, 52), class = "ts")

and

structure(c(40, 68, 50, 64, 26, 44, 108, 90, 62, 60, 90, 64, 120, 82, 68, 60,
26, 32, 60, 74, 34, 16, 22, 44, 50, 16, 34, 26, 42, 14, 36, 24, 14, 16, 6, 6,
12, 20, 10, 34, 12, 24, 46, 30, 30, 46, 54, 42, 44, 42, 12, 52, 42, 66, 40,
60, 42, 44, 64, 96, 70, 52, 66, 44, 64, 62, 42, 86, 40, 56, 50, 50, 62, 22,
24, 14, 14, 18, 18, 10, 20, 10, 4, 18, 10, 10, 14, 20, 10, 32, 12, 22, 20, 20,
26, 30, 36, 28, 56, 34, 14, 54, 40, 30, 42, 36, 52, 30, 32, 52, 42, 62, 46,
64, 70, 48, 40, 64, 40, 120, 58, 36, 40, 34, 36, 26, 18, 28, 16, 32, 18, 12,
20), .Tsp = c(1, 4.36, 52), class = "ts")

Answer 1

I think the most straightforward approach would be to follow Rob Hyndman's approach (he is the author of many time series packages in R). For your data it would work as follows,

require(fma)
# Create a model with multiplicative errors (see https://www.otexts.org/fpp/7/7).
fit1 <- stlf(test2)
# Create a model with additive errors.
fit2 <- stlf(data, etsmodel = "ANN")

deviance <- 2 * c(logLik(fit1$model) - logLik(fit2$model))
df <- attributes(logLik(fit1$model))$df - attributes(logLik(fit2$model))$df

# P-value
1 - pchisq(deviance, df)

# [1] 1

Based on this analysis we find the p-value of 1 which would lead us to conclude there is no seasonality.

Answer 2

I quite like the stl() function provided in R. Try this minimal example:

# some random data
x <- rnorm(200)
# as a time series object
xt <- ts(x, frequency = 10)
# do the decomposition
xts <- stl(xt, s.window = "periodic")
# plot the results
plot(xts)

Now you can get an estimate of the 'seasonality' by comparing the variances.

vars <- apply(xts$time.series, 2, var)
vars['seasonal'] / sum(vars)

You now have the seasonal variance as a proportion of sum of variances after decomposition.

I highly recommend reading the original paper so that you understand whats happening under the hood here. Its very accessible and I like this method as it is quite intuitive.