Croston

Croston’s method for forecasting intermittent time series.

Implements the method proposed by Croston in [1] and described in [2].

Croston’s method is a modification of (vanilla) exponential smoothing to handle intermittent time series. A time series is considered intermittent if many of its values are zero and the gaps between non-zero entries are not periodic.

Croston’s method will predict a constant value for all future times, so Croston’s method essentially provides another notion for the average value of a time series.

The method is (equivalent to) the following:

• Let \(v_0,\ldots,v_n\) be the non-zero values of the time series

• Let \(v\) be the exponentially smoothed average of \(v_0,\ldots,v_n\)

• Let \(z_0,\ldots,z_n\) be the number of consecutive zeros plus 1 between the \(v_i\) in the original time series.

• Let \(z\) be the exponentially smoothed average of \(z_0,\ldots,z_n\)

• Then the forecast is \(\frac{v}{z}\)

The intuition is that \(v\) is a weighted average of the non-zero time series values and \(\frac{1}{z}\) estimates the probability of getting a non-zero value.

Example to illustrate the \(v\) and \(z\) notation.

• If the original time series is \(0,0,2,7,0,0,0,-5\) then:

  • The \(v\)’s are \(2,7,-5\)

  • The \(z\)’s are \(3,1,4\)