GMSE – Geometric Mean Squared Error

Calculates the geometric mean squared error (GMSE) between the forecast and the eventual outcomes.



is the eventual outcome time series sample data (a one-dimensional array of cells e.g., row or column).
is the forecast time series data (a one-dimensional array of cells (e.g., row or column).


  1. The time series is homogeneous or equally spaced.
  2. The time series X and F must be of identical size.
  3. The time series X or F may include observations with missing values (e.g., #N/A or blank).
  4. Observations with missing values in Y or F are excluded from the GMSE calculation.
  5. The geometric mean squared error (MSE) is defined as follows: $${\displaystyle{\mathrm{GMSE}= \sqrt[N]{\prod_{t=1}^{N}e_t^2}=\sqrt[N]{\prod_{t=1}^{N}(y_t-f_t)^2}}}$$
  6. The GMSE is more robust to outliers than MSE.
  7. In general, the biggest advantage of the geometric mean is that the mean absolute of two methods or models can be compared by computing their geometric means. If one geometric mean is 10 and the other is 12, it can be inferred that the mean absolute error of the second method is 20% higher than those of the first.
  8. The main drawback is the scale dependency. If the forecast task includes objects with different scales or magnitudes then the GMSE measure could not be applied.
  9. The GMSE function is available starting with version 1.65 HAMMOCK.

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