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# 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:

H_0: F_{XY} &= F_X F_Y \\
H_A: F_{XY} &\neq F_X F_Y


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

.. toctree::
:maxdepth: 1

tutorials/independence/independence
tutorials/independence/indep_power
tutorials/independence/indep_alg_speed



## 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

H_0:\ &F_1 = F_2 = \ldots F_k \\
H_A:\ &\exists \ j \neq j' \text{ s.t. } F_j \neq F_{j'}


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

.. toctree::
:maxdepth: 1

tutorials/ksample/ksample



## 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,

H_0: F_{X_t Y_{t - j}} &= F_{X_t} F_{Y_{t - j}} \text{ for each } j \in \{ 0, 1, \ldots, M \} \\
H_A: F_{X_t Y_{t - j}} &\neq F_{X_t} F_{Y_{t - j}} \text{ for some } j \in \{ 0, 1, \ldots, M \}


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

.. toctree::
:maxdepth: 1

tutorials/time_series/time_series



## 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.

.. toctree::
:maxdepth: 1

tutorials/sims/indep_simulations

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