In mdscheuerell/muti: Calculates the Mutual Information Between Two Vectors

knitr::opts_chunk$set(echo = TRUE)
library(muti)

Background

muti computes the mutual information $(\mathrm{MI})$ contained in two vectors of discrete random variables. muti was developed with time series analysis in mind, but there is nothing tying the methods to a time index per se.

Mutual information $(\mathrm{MI})$ estimates the amount of information about one variable contained in another; it can be thought of as a nonparametric measure of the covariance between the two variables. $\mathrm{MI}$ is a function of entropy, which is the expected amount of information contained in a variable. The entropy of $X$, $\mathrm{H}(X)$, given its probability mass function, $p(X)$, is

$$ \begin{align} \mathrm{H}(X) &= \mathrm{E}[-\log(p(X))]\ &= -\sum_{i=1}^{L} p(x_i) \log_bp(x_i), \end{align} $$

where $L$ is the length of the vector and $b$ is the base of the logarithm.muti uses base-2 logarithms for calculating the entropies, so $\mathrm{MI}$ is expressed in units of "bits". In cases where $p(x_i) = 0$, then $\mathrm{H}(X) = 0$.

The joint entropy of $X$ and $Y$ is

$$ \mathrm{H}(X,Y) = -\sum_{i=1}^{L} p(x_i,y_i) \log_b p(x_i,y_i). $$

where $p(x_i,y_i)$ is the probability that $X = x_i$ and $Y = y_j$. The mutual information contained in $X$ and $Y$ is then

$$ \mathrm{MI}(X;Y) = \mathrm{H}(X) + \mathrm{H}(Y) - \mathrm{H}(X,Y). $$

One can normalize $\mathrm{MI}$ to the interval [0,1] as

$$ \mathrm{MI}^*(X;Y) = \frac{\mathrm{MI}(X;Y)}{\sqrt{\mathrm{H}(X)\mathrm{H}(Y)}}. $$

Usage

Input. At a minimum muti requires two vectors of class numeric or integer. See ?muti for all of the other function arguments.

Output. The output of muti is a data frame with the $\mathrm{MI}$ MI_xy and respective significance threshold value MI_tv at different lags. Note that a negative (positive) lag means X leads (trails) Y. For example, if length(x) == length(y) == TT, then the $\mathrm{MI}$ in x and y at a lag of -1 would be based on x[1:(TT-1)] and y[2:TT].

Additionally, muti produces a 3-panel plot of

1. the original data (top);
2. their symbolic or discretized form (middle);
3. $\mathrm{MI}$ values (solid line) and their associated threshold values (dashed line) at different lags (bottom).

The significance thresholds are based on a bootstrap of the original data. That process is relatively slow, so please be patient if asking for more than the default mc=100 samples.

Data discretization

muti computes $\mathrm{MI}$ based on 1 of 2 possible discretizations of the data in a vector x:

1. Symbolic. (Default) For 1 < i < length(x), x[i] is translated into 1 of 5 symbolic representations based on its value relative to x[i-1] and x[i+1]: "peak", "decreasing", "same", "trough", or "increasing". For example, the symbolic translation of the vector c(1.1,2.1,3.3,1.2,3.1) would be c("increasing","peak","trough"). For additional details, see Cazelles (2004).

2. Binned. Each datum is placed into 1 of n equally spaced bins as in a histogram. If the number of bins is not specified, then it is calculated according to Rice's Rule where n = ceiling(2*length(x)^(1/3)).

Installation

You can install the development version using devtools.

if(!require("devtools")) {
  install.packages("devtools")
  library("devtools")
}
devtools::install_github("mdscheuerell/muti")

Examples

Ex 1: Real values as symbolic

Here's an example with significant information between two numeric vectors. Notice that none of the symbolic values are the "same".

set.seed(123)
TT <- 30
x1 <- rnorm(TT)
y1 <- x1 + rnorm(TT)
muti(x1, y1)

Ex 2: Integer values as symbolic

Here's an example with significant information between two integer vectors. Notice that in this case some of the symbolic values are the "same".

x2 <- rpois(TT,4)
y2 <- x2 + sample(c(-1,1), TT, replace = TRUE)
muti(x2, y2)

Ex 3: Real values as symbolic with normalized MI

Here are the same data as Ex 1 but with $\mathrm{MI}$ normalized to [0,1] (normal = TRUE). In this case the units are dimensionless.

muti(x1, y1, normal = TRUE)

Ex 4: Real values with binning

Here are the same data as Ex 1 but with regular binning instead of symbolic (sym = FALSE).

muti(x1, y1, sym = FALSE)

Citation

Please cite the muti package as:

Scheuerell, M. D. (2017) muti: An R package for computing mutual information. https://doi.org/10.5281/zenodo.439391

See citation("muti") for a BibTeX entry.

mdscheuerell/muti documentation built on June 20, 2020, 12:52 p.m.