How to Detect Random Walk and White Noise in Time Series Forecasting



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Random Walks

A more challenging but equally unpredictable distribution in time series forecasting is a random walk. Unlike white noise, it has non-zero mean, non-constant std/variance, and when plotted, looks a lot like a regular distribution:

Doesn’t it resemble a plot of stocks from Yahoo Finance?

Random walk series are always cleverly disguised in this manner, but still, they are unpredictable as ever. The best guess for today’s value is yesterday’s.

A common confusion among beginners is thinking of a random walk as a simple sequence of random numbers. This is not the case because, in a random walk, each step is dependent on the previous step.

For this reason, the Autocorrelation function of random walks does return non-zero correlations.

The formula of a random walk is simple:

Whatever the previous data point is, add some random value to it and continue for as long as you like. Let’s generate this in Python with a starting value of, let’s say, 99:

Let’s also plot the ACF:

As you can see, the first ~40 lags yield statistically significant correlations.

So, how do we detect a random walk when a visualization is not an option?

Because of how they are created, differencing the time series should isolate the random addition of each step. Taking the first-order difference is done by lagging the series by 1 and subtracting it from the original. Pandas has a convenient diff function to do this:

If you plot the first-order difference of a time series and the result is white noise, then it is a random walk.

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