mcsi 2025.0.0

Calculates skewness of a distribution based on the mean, Mean-Centered Skewness Index (MCSI)

Mean-Centered Skewness Index (MCSI)

The Mean-Centered Skewness Index (MCSI) is a statistical operator for analyzing the skewness of tensor data. Built with PyTorch and C++, the mcsi function efficiently quantifies the directional imbalance of tensor values relative to their mean across one or more dimensions.

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Key Features

• Compatible with PyTorch tensors.
• Configurable reduction dimensions (dim) and shape preservation (keepdim).
• Efficient implementation as a PyTorch extension using C++ and Pybind11.
• Provides meaningful insights into the asymmetry of data distributions.

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Mathematical Representation

Let $\mathbf{X} \in \mathbb{R}^{n_1 \times n_2 \times \dots \times n_k}$ represent an input tensor. The Mean-Centered Skewness Index for dimension(s) $d$ is defined as:

$$ \text{MCSI}(\mathbf{X}, d) = \frac{N_-(\mathbf{X}, d) - N_+(\mathbf{X}, d)}{N(\mathbf{X}, d)} $$

Where:

• $N_-(\mathbf{X}, d)$: Number of elements in $\mathbf{X}$ less than the mean along the specified dimensions $d$.
• $N_+(\mathbf{X}, d)$: Number of elements in $\mathbf{X}$ greater than the mean along $d$.
• $N(\mathbf{X}, d)$: Total number of elements in $\mathbf{X}$ along $d$.

The mean along dimension(s) $d$ is given by: $$ \mu(\mathbf{X}, d) = \frac{1}{N(\mathbf{X}, d)} \sum_{i \in d} X_i $$

MCSI effectively measures the imbalance in the count of values above and below the mean, normalized by the total number of elements.

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Function Signature

mcsi(input: torch.Tensor, dim: Optional[Union[int, List[int]]] = None, keepdim: bool = False) -> torch.Tensor