My goal is to cluster time-series which may present daily/weekly/yearly seasonalities. To do so, I plan to use different variables and among them, the following ones: $F_{s_{daily}}$, $F_{s_{weekly}}$ and $F_{s_{yearly}}$ should measure strength of daily, weekly and yearly seasonalities, and $F_{t}$ should measure the strength of the trend.

To quantify strength, I am using the definition provided by Hyndman in his book Forecasting: Principles and Practice (ref: https://otexts.com/fpp2/seasonal-strength.html):

Strength of trend: $F_t=max(0,1-\frac{Var(R_t)}{Var(T_t+R_t)})$ where $R_t$ is the time-series' remainder component and $T_t$ its trend component (the closer to 1, the higher the strength of trend).

Strength of seasonality: $F_t=max(0,1-\frac{Var(R_t)}{Var(S_t+R_t)})$ where $R_t$ is the time-series' remainder component and $S_t$ its seasonal component (the closer to 1, the higher the strength of seasonality).

To test these coefficients in the case of a multi-seasonal time-series, I made an artificial time-series with 3 seasonal components and no trend:



$y_{daily}(t)=cos(\frac{2\pi*t}{24})$, $y_{weekly}(t)=cos(\frac{2\pi*t}{24*7})$, $y_{yearly}(t)=cos(\frac{2\pi*t}{24*365})$

See this artificial data set as a two-year time-series sampled every hour (thus 24 points / hour).

To decompose the time-series into its different seasonal components using an additive model, I used the Python's seasonal_decompose function from statsmodel library.

After doing daily decomposition (ie. period=24) of y(t), I found $F_{t_{daily}} = 0.999$ and $F_{s_{daily}} = 0.999$

After doing weekly decomposition (ie. period=24*7) of y(t), I found $F_{t_{weekly}} = 1$ and $F_{s_{weekly}} = 1$

After doing yearly decomposition (ie. period=24*365) of y(t), I found $F_{t_{yearly}} = 0$ and $F_{s_{yearly}} = 0.763$

But if I firstly remove two of the three seasonal components, and compute $F_{s}$ and $F_{t}$ for the remaining one, I got different results:

For the daily case ($y(t) - y(t).seasonal_{weekly} - y(t).seasonal_{yearly}$), $F_{t_{daily}} = 0$ and $F_{s_{daily}} = 0.59$

For the weekly case ($y(t) - y(t).seasonal_{daily} - y(t).seasonal_{yearly}$), $F_{t_{weekly}} = 0$ and $F_{s_{weekly}} = 1$

For the yearly case ($y(t) - y(t).seasonal_{daily} - y(t).seasonal_{weekly}$), $F_{t_{yearly}} = 0$ and $F_{s_{yearly}} = 1$

I am confused as the results I got are not the ones I would have expected (expected: $F_{s_{yearly}}$, $F_{s_{yearly}}$ and $F_{s_{yearly}}$ close to 1, and $F_{t}$ close to 0) Can tell me what is wrong and need to be changed in my approach, and any recommendation to measure $F_{t}$. Thank you


I think the problem lies with the choice to use seasonal_decompose from statsmodels. It is a very naive implementation (it even says so in their documentation!) The trend component is a moving average which typically eats a lot of extra signal up and it doesn't handle multiple seasonal signals. I would look into something that handles multiple seasonalities naturally like fbProphet or some other GAM setup.

For general purpose time series clustering I probably wouldn't reinvent the wheel, there are time series feature extraction libraries out there (like tsfresh for python) and a lot come with clustering as an additional feature.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.