Time series seasonality test What are the most simple seasonality tests for time series?   
Being more specific, I want to test if in specific time series the seasonal component is meaningful.
What are the recommended packages in Python/ R?
 A: Before you test for seasonality you should reflect which type of seasonality you have. Note that there are many different types of seasonality:


*

*Additive vs. Multiplicative seasonality

*Single vs. Multiple seasonalities

*Seasonality with even vs. uneven number of periods. Each year has twelve months, but 52,1429 weeks. 

*Trend vs. Seasonality: A seasonality pattern always appears in the same period, but a trend may appear a little bit later or earlier and not exactly each 5 years. One example for a trend are business cycles.



One of the most common methods to detect seasonality is to decompose the time series into several components.
In R you can do this with the decompose() command from the preinstalled stats package or with the stl() command from the forecast package.
The following code is taken from A little book of R for time series
births <- scan("http://robjhyndman.com/tsdldata/data/nybirths.dat")
birthstimeseries <- ts(births, frequency = 12, start = c(1946,1))
birthstimeseriescomponents <- decompose(birthstimeseries)
plot(birthstimeseriescomponents)


You can check the single components with 


*

*birthstimeseriescomponents$seasonal

*birthstimeseriescomponents$random

*birthstimeseriescomponents$trend

An other method is to include seasonal dummies and to check whether they have significant p-values when you compute the regression. If the single months have siginificant coefficients your monthly time series is seasonal.

An other method to detect seasonality is either to plot the data itself or to plot the ACF (autocorrelation function). In our case you can easily notice, that there is seasonality.



And last, but not least there are some "formal" hypothesis tests in order to detect seasonality such as the Student T-Test and the Wilcoxon Signed Rank Test. 
A: My thoughts are to check the amplitude of the:


*

*ACF autocorrelation function 

*PACF partial autocorrelation function

*Fourier Coefficients


(Fourier Coefficients are related to ACF via Wiener-Khinchin theorem.)
