OnlineMoments.jl

Calculating conditional moments (and variance) from data streams OnlineMoments.jl is a Julia package enabling the esimation of conditional moments and conditional variance from empirical streamed time-series data. This package enables the estimation of drift and diffusion functions using both histogram- and kernel-based regression.

This package accompanies the paper:

Reconstruction of Stochastic Dynamics from Large Datasets, PRE 108, 054110, Davis, William DOI:10.1103/PhysRevE.108.054110

Background

Consider the scalar stochastic differential equation

$\frac{d}{dt}X(t) = f(X) + g(X)\Gamma(t).$

The evolution of this process can be described by a Fokker-Planck equation

$\frac{\partial}{\partial t} p(x,t|x^\prime,t^\prime) = \bigg[-\frac{\partial}{\partial x} D^{(1)}(x) + \frac{\partial^2}{\partial x^2} D^{(2)}(x) \bigg] p(x,t|x^\prime,t^\prime)$

which contain the Kramers-Moyal (KM) coefficients

$D^{(k)}(x) = \lim_{\tau\rightarrow0} \frac{1}{n!\tau} \int_{-\infty}^{\infty} \big[x^\prime - x\big]^k p(x^\prime,t+\tau|x,t)\ dx^\prime.$

Here $k=1$ is the drift coefficient and $k=2$ is the diffusion coefficient. These expressions contain moments of the conditional probability density, or "conditional moments"

$M^{(k)}(\tau,x) = \int_{-\infty}^\infty [x^\prime - x\big]^k p(x^\prime, t + \tau| x,t)\ dx^\prime.$

These conditional moments are typically estimated in an offline fashion

$\hat{M}^{(k)}_{ij} = \frac{\sum\limits_{n=1}^{N-i} K_h(\mathcal{X}_j - X_n)\big[X_{n+i} - X_n\big]^k}{\sum\limits_{n=1}^{N-i} K_h(\mathcal{X}_j - X_n)}.$

In this work, I present online formulae for sequential updating

$\hat{M}^{(k)}_{ij}\big|_{N} = \hat{M}^{(k)}_{ij}\big|_{N-1} + K_h(\mathcal{X}_j - X_{N-i})\times\left(\left[X_N - X_{N-i}\right]^k - \hat{M}^{(k)}_{ij}\big|_{N-1}\right)\Big/W_{ij}\big|_{N},$

$W_{ij}\big|_{N} = W_{ij}\big|_{N-1} + K_h\left(\mathcal{X}_j - X_{N-i}\right).$

The online method, which I call "Online Kernel-Based Regression (OKBR)" scales $\mathcal{O}(1)$ in space, compared with Kernel-Based Regression (KBR) which scales as $\mathcal{O}(N)$.

See the paper for further details (DOI:10.48550/arXiv.2307.00445).

Version

• Version 0.1.0 - First release