Surface Similarity Parameter

The Surface Similarity Parameter

(1)\[\mathrm{SSP} = \frac{\sqrt{\int|Y - \hat{Y}|^2 df}}{\sqrt{\int|Y|^2 df} + \sqrt{\int|\hat{Y}|^2 df}}\in[0, 1]\]

with \(Y\) the Fourier transform of \(y\), is a normalized error metric originally introduced by Perlin and Bustamante (2016). The SSP quantifies the difference between two signals in the complex Fourier space, and thus inherently penalizes deviations in magnitude and phase in a single metric.

For discrete signals, Eq. (1) collapses to

\[\mathrm{SSP} = \frac{||Y-\hat{Y}||}{||Y|| + ||\hat{Y}||}\in[0, 1].\]

Being a normalized error, the SSP is defined in the range [0, 1], where

  • \(\mathrm{SSP}=0\) indicates perfect agreement, and

  • \(\mathrm{SSP}=1\) indicates perfect disagreement among the signals.

Perfect disagreement means that either \(\hat{y}=-y\), or \(\hat{y}=0\) while \(y\neq 0\).

Using the SSP as a machine learning loss function forces the model to improve the prediction in terms of magnitude and phase in order to reduce the loss (cf. Wedler et al. (2022)). This sets the SSP apart from established Euclidean distance-based loss functions like the MSE and MAE, when training models on oscillatory spatio-temporal data. The stricter error penalization of the SSP leads to a more refined and optimizer-friendly loss surface, where local minima are sparse but meaningful. This allows the optimizer to take more confident steps, leading to faster convergence to better local minima.

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