change compute_pixel_stats to compute actual stats

src/plenoptic/simulate/models/portilla_simoncelli_masked.py

@@ -824,9 +824,21 @@ def _compute_pixel_stats(self, mask: list[Tensor], image: Tensor) -> Tensor:
             f"{self._mask_input_idx}, b c h w -> b c {self._mask_output_idx}"
         )
         mean = einops.einsum(*mask, image, weighted_avg_expr)
-        var = einops.einsum(*mask, image.pow(2), weighted_avg_expr)
-        skew = einops.einsum(*mask, image.pow(3), weighted_avg_expr)
-        kurtosis = einops.einsum(*mask, image.pow(4), weighted_avg_expr)
+        # these are all non-central moments...
+        moment_2 = einops.einsum(*mask, image.pow(2), weighted_avg_expr)
+        moment_3 = einops.einsum(*mask, image.pow(3), weighted_avg_expr)
+        moment_4 = einops.einsum(*mask, image.pow(4), weighted_avg_expr)
+        # ... which we use to compute the var, skew, and kurtosis. the formulas we use
+        # for var and skew here can be found on their respective wikipedia pages, and
+        # the one for kurtosis comes from Eero working through the algebra
+        var = moment_2 - mean.pow(2)
+        skew = (moment_3 - 3 * mean * var - mean.pow(3)) / (var.pow(1.5))
+        kurtosis = (
+            moment_4
+            - 4 * mean * moment_3
+            + 6 * mean.pow(2) * moment_2
+            - 3 * mean.pow(4)
+        ) / (var.pow(2))
         return einops.rearrange(
             [mean, var, skew, kurtosis],
             f"stats b c {self._mask_output_idx} -> b c ({self._mask_output_idx}) stats",