Introduction

The goal of co-prediction is to quantify dynamic similarity between two time series. Given two time series, \(x\) and \(y\), we assume that the dynamics can be represented as: \[ x_{t+tp} = F\left(\vec{x}_t\right) = F\left(\langle x_t, x_{t-\tau}, \dots, x_{t-(E-1)\tau} \rangle \right)\] and \[ y_{t+tp} = G\left(\vec{y}_t\right) = G\left(\langle y_t, y_{t-\tau}, \dots, y_{t-(E-1)\tau} \rangle \right)\] . Then co-prediction is a way to quantify how closely \(F\) and \(G\) resemble each other.

Method

We can accomplish this task using rEDM by constructing concatenated time series and applying simplex or s-map to make predictions with appropriately defined libs and preds.

First, we grab some demo time series from the block_3sp data.frame:

library(rEDM)
data(block_3sp)

x <- block_3sp$x_t
y <- block_3sp$y_t

Concatenate the time series and record which portions correspond to x and y:

concatenated_xy <- c(x, y)
lib_x <- c(1, length(x))
lib_y <- length(x) + c(1, length(y))

Use simplex to identify the optimal embedding dimension for x and use it to co-predict from x to y:

simplex_out_x <- simplex(concatenated_xy, lib = lib_x, pred = lib_x)

## Warning in model$run(): Found overlap between lib and pred. Enabling cross-
## validation with exclusion radius = 0.

best_E_x <- simplex_out_x$E[which.max(simplex_out_x$rho)]

copred_x_to_y <- simplex(concatenated_xy, lib = lib_x, pred = lib_y, E = best_E_x)

and in the reverse direction:

simplex_out_y <- simplex(concatenated_xy, lib = lib_y, pred = lib_y)

## Warning in model$run(): Found overlap between lib and pred. Enabling cross-
## validation with exclusion radius = 0.

best_E_y <- simplex_out_y$E[which.max(simplex_out_y$rho)]

copred_y_to_x <- simplex(concatenated_xy, lib = lib_y, pred = lib_x, E = best_E_y)

Results

We can interpret the strength of dynamic similarity in comparison to the univariate predictability of x and y.

First, merge the output into a single data.frame:

to_plot <- data.frame(label = c("prediction of x (from x)", 
                                "coprediction of x (from y)", 
                                "prediction of y (from y)", 
                                "coprediction of y (from x)"), 
                      rbind(simplex_out_x[which.max(simplex_out_x$rho), ], 
                            copred_y_to_x, 
                            simplex_out_y[which.max(simplex_out_y$rho), ], 
                            copred_x_to_y)
                      )

Plot the output

library(ggplot2)
ggplot(to_plot, aes(x = label, y = rho)) + 
    geom_col() + theme_bw() + 
    theme(axis.text.x = element_text(angle = 90, hjust = 1))