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S-measure

# CausalityTools.s_measure — Function.

s_measure(x, y, m::Int, τ, K::Int; distance_metric::Metric = SqEuclidean())

S-measure test [1] for the directional dependence between data series.

Algorithm

1. Create an m-dimensional embeddings of both x and y, resulting in N different m-dimensional embedding points $X = \{x_1, x_2, \ldots, x_N \}$ and $X = \{y_1, y_2, \ldots, y_N \}$. τ controls the embedding lag.
2. Let $r_{i,j}$ and $s_{i,j}$ be the indices of the K-th nearest neighbors of $x_i$ and $y_i$, respectively.
3. Compute the the mean squared Euclidean distance to the $K$ nearest neighbors for each $x_i$, using the indices $r_{i, j}$.

$$R_i^{(k)}(x) = \dfrac{1}{k} \sum_{i=1}^{k}(x_i, x_{r_{i,j}})^2$$

• Compute the y-conditioned mean squared Euclidean distance to the $K$ nearest neighbors for each $x_i$, now using the indices $s_{i,j}$.

$$R_i^{(k)}(x|y) = \dfrac{1}{k} \sum_{i=1}^{k}(x_i, x_{s_{i,j}})^2$$

• Define the following measure of independence, where $0 \leq S \leq 1$, and low values indicate independence and values close to one occur for synchronized signals.

$$S^{(k)}(x|y) = \dfrac{1}{N} \sum_{i=1}^{N} \dfrac{R_i^{(k)}(x)}{R_i^{(k)}(x|y)}$$

References

Quian Quiroga, R., Arnhold, J. & Grassberger, P. [2000] “Learning driver-response relationships from synchronization patterns,” Phys. Rev. E61(5), 5142–5148.

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