# Tutorials¶

## Independence Tests¶

The independence testing problem is generalized as follows: consider random variables \(X\) and \(Y\) that have joint density \(F_{XY} = F_{X|Y} F_Y\). We are testing:

These tutorials overview how to use these tests as well as benchmarks comparing the algorithms included against each other.

*K*-sample Tests¶

The *k*-sample testing problem is generalized as follows: consider random variables
\(X_1, X_2, \ldots, X_k\) that have densities
\(F_1, F_2, \ldots, F_k\). Then, we are testing

This tutorial overview how to use *k*-sample tests in `hyppo`

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## Time-Series Tests¶

Time-series tests of independence consider the following problem: consider random variables \(X\) and \(Y\) with joint density \(F_{XY}\) and marginal densities \(F_X\) and \(F_Y\). Let \(F_{X_t}\), \(F_{Y_s}\), and \(F_{X_t Y_s}\) represent the marginal and joint distributions of time-indexed random varlables \(X_t\) and \(Y_s\) at timesteps \(t\) and \(s\). Let \(\{ (X_t, Y_t) \}_{t = -\infty}^\infty\) be a full jointly-sampled strictly stationary time series with the observed sample \(\{ (X_1, Y_1), \ldots (X_n, Y_n) \}\). Choose some nonnegative integer \(M\) as the maximium lag hyperparamater. Then we are testing,

This tutorial overview how to use time_series based tests in `hyppo`

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## Sims¶

To evaluate existing implmentations and benchmark against other packages, we have developed a suite of 20 dependency structures. The simulation settings include polynomial (linear, quadratic, cubic), trigonometric (sinusoidal, circular, ellipsoidal, spiral), geometric (square, diamond, w-shaped), and other functions. We also include 3 sample Gaussian simulations as well, which are sampled from multivariate normal distribusions.