I have a time-series data (Fig 1) that clearly look non-stationary on visual analysis. There is a clear trend and seasonality in the time series as shown in the Final figure below. But the ADF test and ACF/PACF show a Stationary time series.

My understanding and interpretation of the results are that over the course of the series, it maintains a mean and variance and hence statistical properties of a Stationary time series !!

Also, I read that the non-stationarity in time-series causes difficulty in prediction. But the LSTM model that I have created with this series (which look non-stationary visually, but comes out to be Stationary statistically), predicts pretty accurately. Is it because the LSTM captures the non-linearity pretty well as compared to the ARIMA models?

Time series data

The ADF test: it was clearly Stationary with p-value of 3.15e-15 and ADF stats much smaller than 1% critical value of -3.432.

ACF and PACF charts: The chart shows the stationarity as well! ACF chart touched Zero, not instantaneously, but relatively quickly! (AM I STUDYING THE ACF CORRECTLY)?

ACF and PACF charts

Trend and Seasonality decomposition


1 Answer 1


Your series does not pass all stationarity tests, it merely passes the ADF test. And even the ADF test is not a test of stationarity, it is a test of presence of a unit root. Your time series certainly does not appear to have a unit root; its level is stable. Hence, ADF rejects the null hypothesis of presence of a unit root. But that does not imply stationarity.

ACF and PACF are not tests per se, and they only reveal autocorrelations but may miss other traits of nonstationarity such as nonconstant variance. ACF and PACF indicate 24-period seasonality; no surprise given that your data seems to be hourly.

However, when adjusted for seasonality, your series might be stationary. Perhaps auto.arima (available from the forecast package in R; there is a counterpart in Python, too) or a similar algorithm will find a stationary model that is statistically adequate.

  • $\begingroup$ I read that "A time series with cyclic behaviour (but with no trend or seasonality) is stationary." Can I say, given that I do not have any mono-tonic trend over the period of time that my series is stationary? $\endgroup$
    – SJa
    Jul 25 at 17:20
  • $\begingroup$ Also, what do you mean by 'when adjusted for seasonality...' Do you mean if I normalize/remove seasonality component? $\endgroup$
    – SJa
    Jul 25 at 17:22
  • $\begingroup$ @SJa, the original series is nonstationary due to seasonality. When seasonality is removed it might become stationary. $\endgroup$ Jul 26 at 5:17
  • $\begingroup$ @RicharHardy, but when we remove the seasonality, we will entirely change our dataset as well? How do we counter that in that case? $\endgroup$
    – SJa
    Jul 26 at 13:20
  • $\begingroup$ @SJa, yes, this is always the case. If the original time series is not stationary, then that is that. A transformed series (de-seasonalized or so) can be made stationary, but it is not the original series any longer. Yet that is usually not a problem, as it is typically enough to find a transformation that is stationary, and that makes models that require stationarity work. This is e.g. how ARIMA, SARIMA and other time series models work. $\endgroup$ Jul 26 at 17:08

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