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:

$$\begin{split}H_0: F_{XY} &= F_X F_Y \\ H_A: F_{XY} &\neq F_X F_Y\end{split}$$

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

• Independence Tests
• Independence Power
• Independence Algorithm 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

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

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

• K-Sample Tests

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,

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

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

• Time-Series Independence Tests

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.

• Independence Simulations