Time series generators

[crangelsmith]

As described in #185 adding functionality for time series generation.

The following models will be implemented:

Independently sampled from a distribution parametrised by population level statistics

• E.g. sampling each event from N(mean_population, std_population).

Random walk with drift

A random walk with drift is one of the simplest time-series models, creating a smooth-ish stochastic trajectory.

$x_t = x_{t-1} + \delta + \varepsilon_t$

where:

• $x_t$ = value at time (t)
• $\delta$ = drift (average change per time step, the sign will tell you how the value is changing over time.)
• $\varepsilon_t$= random noise, usually ($\mathcal{N}(0,\sigma_\varepsilon^2)$)

Autoregressive Model of Order 1- AR(1)

In an AR(1) model the value at the next time step depends on the current value plus a bit of random noise.

$x_t = \mu + \phi(x_{t-1} - \mu) + \varepsilon_t$

Where:

• $x_t$ — the measurement at time (t)
• $\mu$ — the average level (mean) of the series
• $\phi$ — the autoregressive coefficient that controls how strongly the series depends on the previous value
• $\varepsilon_t$ — random noise (usually Gaussian with mean 0)

[crangelsmith]

@myyong, I've done some abstraction in this function. I didn't go too far, but it may be one step toward #184.