The xmap subcommand

The explore subcommand focuses on using a the history of a time series to predict itself in the future.

On the other hand, the xmap subcommand - which is short for cross mapping - focuses on using one time series to predict the value of a different time series.

This process is central to the 'convergent cross mapping' algorithm which we use to decide if one time series has a causal effect on another.

Setup

Imagine that we use the command:

edm xmap a b, oneway

This will consider two different time series, here labelled \(a\) and \(b\).

Choose the number of observations

In tabular form, the data looks like:

Choose a value for \(E\)

Choose a value for \(\tau\)

The lagged embedding \(M_a\) is constructed:

Library and prediction sets

In xmap mode, the library set \(\mathscr{L}\) is typically the first \(L\) points of \(M_a\). The library size parameter \(L\) is set by the Stata parameter library.

Choose a value for \(L\)

However, in xmap mode the prediction set \(\mathscr{P}\) will include every point of the \(a\) embedding so:

Targets come from the other time series

The \(b\) time series will be the values which we try to predict. Here we are trying to predict \(p\) observations ahead, where the default case is actually \(p = 0\). The \(p = 0\) case means we are using the \(a\) time series to try to predict the contemporaneous value of \(b\). A negative \(p\) may be chosen, though this is a bit abnormal.

Choose a value for \(p\)

What does edm xmap a b do?

In xmap, we perform a series of predictions in very similar manner to the explore subcommand. The difference is our library and prediction sets contain values from the \(a\) time series whereas the \(\mathbf{y}_{\mathscr{L}}\) and \(\mathbf{y}_{\mathscr{P}}\) vectors contain (usually contemporaneous) values from the \(b\) time series.

\[ \begin{aligned} \text{Given } \underbrace{ \mathbf{x}_{i} }_{\text{From } a} \text{ and target }\underbrace{ y_i }_{\text{From } b} & \underset{\small \text{Find neighbours in } \mathscr{L} }{\Rightarrow} ( \underbrace{ \mathbf{x}_{[j]} }_{\text{From } a} , \underbrace{ y_{[j]} }_{\text{From } b} )_{j \in \mathcal{NN}_k(i)} \underset{\small \text{Make prediction}}{\Rightarrow} \underbrace{ \hat{y}_i }_{\text{For } b } \end{aligned} \]

This means that we are learning the mapping from (the recent history of) the \(a_t\) time series to the (contemporaneous value of the) \(b_t\) time series.

Convergent cross mapping

Typically the goal of using xmap is to specify a grid of values for the library size and look at the relationship between the library size \(L\) and the predictive performance \(\rho\).

If the xmap a b predictive performance increases significantly when more data to learn from (i.e. as \(L\) increases) then the convergent cross mapping (CCM) algorithm says that this is evidence that the time series \(b_t\) has a causal effect on \(a_t\).

This may seem backwards at first glance, but the logic is that if \(b_t\) has a causal effect on \(a_t\), then information about \(b_t\) is embedded in the time series \(a_t\).

Hence, if we get more and more data from \(a_t\), we can glean more information about the cause \(b_t\) and hence make better prediction of \(b_t\) as \(L\) increases.

The summary which the edm command prints out uses the notation b|M(a) to indicate that we use the reconstructed manifold of \(a_t\) to make predictions about \(b_t\), and so the causal effect of \(b\) to \(a\) can be read left-to-right.

See the examples/vignettes to see the CCM process in action.