I have an hourly time series that I want to forecast. Prior to modelling it, I tested it for random walk. The ACF and PACF plots for the time series are as follows:

enter image description here

Since the PACF has a high value at lag 1 and ACF is decreasing linearly, I wanted to check its 1st differentiation for white noise to figure out if its a random walk or not. The plots for the differentiated time series and its ACF, PACF are below.

enter image description here

enter image description here

As the 1st differentiation plot has 0 mean, constant variance and the ACF values are barely above the confidence interval at lags above 0, can I conclude it the series to be a Random Walk?

  • $\begingroup$ ACF/PACF are of limited help: stats.stackexchange.com/q/595150/1352. This holds especially for long time series as yours. In addition, hourly time series often exhibit multiple seasonalities, e.g., hour of day and day of week. The tag wiki contains pointers to resources for these. What does your time series measure? $\endgroup$ Dec 30, 2023 at 17:44
  • $\begingroup$ The series depicts the hourly height of waves from 2017 to 2019. $\endgroup$
    – Om Mali
    Dec 30, 2023 at 19:48
  • $\begingroup$ the waves definitely have tidal wave cycles, so in this regard you can't call them white noise $\endgroup$
    – Aksakal
    Dec 31, 2023 at 20:45

1 Answer 1


Purely from the data / visuals shown here it's possible it's a random walk as acf/pacf don't show any clear patterns. However see Stephan Kolassa comment on better ways to quantify/test data for white noise than just looking at acf/pacf.

Having said that, based on your comment that the original data represents "he series depicts the hourly height of waves from 2017 to 2019" the answer to your question if this is potentially a random walk is - has to be - a clear 'Definitely not!'. Otherwise humanity would have been wiped out by now :)

I always recommend to do a sanity check on whether a dynamic system observed over time is stationary, trend stationary, or stochastic non stationary (ie a random walk) before looking at any data. Based on the physics / economics / ( domain knowledge) you can often rule out at least one option .

In your case, based on the variable you measure, most likely you expect to see a data generating process that consists of

  • Trend stationary $ f(t) /approx \alpha \cdot t$ for this short time period (relevant climate question/hypothesis: is $\alpha$ zero or not?)
  • Multiple seasonalities (12hr tidal patterns, 24hr day/ night - temperature changes, lunar cycle, annual seasons)
  • Stationary arma / garch like signal, where variance of garch has also similar seasonal patterns

( see also here for a good primer on time series decomposition)

The garch like patterns you can see on the difference graph with higher, clustered variance in the hurricane season. ( Was this data recorded in gulf of Mexico / carribean? Or California?)

Lastly, any chance you could share this data here ( then we can test model setup above) ? Or at least plot it ( plot a month of data + full range for comparison).

Unrelated note: we are seeing record wave heights now, Dec 30 2023, in California. Is this post inspired by it?

  • $\begingroup$ Sorry for the late reply. I am using this dataset, however I have resampled it to an hourly time series. kaggle.com/datasets/jolasa/… I wanted to test my forecasting skills and came across this dataset while exploring Kaggle. $\endgroup$
    – Om Mali
    Jan 8 at 21:03

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