# aem.time: AEM for data series In AEM: Tools to construct Asymmetric eigenvector maps (AEM) spatial variables

## Description

Function `aem.time` constructs AEM eigenfunctions for multi-scale analysis of a regular time series or spatial transect of univariate or multivariate data.

## Usage

 `1` ```aem.time(n, w=NULL, moran=FALSE,plot.moran=FALSE) ```

## Arguments

 `n` Numeric. Number of points in the series. `w` A vector of weights to be applied to the edges (columns of matrix E). Equal weights are used if no vector `w` is provided. The length of vector `w` must be (`n`-1) where `n` is the number of points in the spatial or temporal series. `moran` Logical. If `TRUE`, Moran's I are computed for all AEM. If `FALSE` (default value), Moran's I are not computed. `plot.moran` Logical. If `TRUE`, a plot of the Moran's I associated to each AEM is drawn, if `FALSE` (default value) no plot is drawn. Also, if `moran` is `FALSE`, this argument becomes invalid.

## Details

Time series represent a form of directional stochastic process. To emphasize the directional nature of the process influencing the data, AEM analysis, which was designed to take trends into account, should be applied to the non-detrended series. MEM analysis (see `scores.listw`) can be applied to data series that were detrended to remove the directional component as recommended by Blanchet et al. (2008, 2011) and Legendre & Legendre (2012, Subsection 14.1.2). Detrended palaeoecological sediment core data, for example, could be studied by MEM analysis.

No data file needs to be provided to this function. The AEM eigenvectors are constructed from a matrix E generated from the regular sequence of points along the series.

A vector of weights `w` can be provided, representing the ease of communication of matter, energy or information among the points. The most simple form would be the inverse of (d/dmax) where d is the distance between adjacent nodes and dmax is the maximum distance between adjacent nodes in the spatial or time series. More general forms of weights may represent the inverse of landscape resistance to the movement of organisms, propagules, genes, etc. If the calculation of Moran's I is requested, the point coordinates are generated from the point positions along the series.

## Value

 `E` Nodes-by-edges matrix E. `values` Eigenvalues of the principal component analysis of E. `aem` Matrix of AEM eigenfunctions normalized to unit length. `Moran` A three-column table giving: `Moran` = Moran's I statistics, `p.value` = p-values (2-tailed parametric test), `Positive` = 1 for Moran's I larger than the expected value. Computed using function `moran.I.multi` of the AEM package. `expected_Moran` The expected value of Moran's I.

## Author(s)

Pierre Legendre, Universite de Montreal, and F. Guillaume Blanchet, University of Alberta

## References

Blanchet, F. G., P. Legendre, and D. Borcard. 2008. Modelling directional spatial processes in ecological data. Ecological Modelling, 215, 325-336.

Blanchet, F. G., P. Legendre, R. Maranger, D. Monti, and P. Pepin. 2011. Modelling the effect of directional spatial ecological processes at different scales. Oecologia 166, 357-368.

Legendre, P. and L. Legendre. 2012. Numerical ecology, 3rd English edition. Developments in Environmental Modelling, Vol. 24. Elsevier Science BV, Amsterdam.

`aem`, `moran.I.multi`, `scores.listw`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```# Time series containing 20 equispaced observations out <- aem.time(20, moran=TRUE, plot.moran=TRUE) # Time series containing 20 observations with unequal spacing # Generate (n-1) random interpoint distances distances <- runif(19,1,5) # Compute weights representing the ease of communication among points w <- 1/(distances/max(distances)) # Compute the AEM eigenfunctions out <- aem.time(20, w=w, moran=TRUE, plot.moran=TRUE) ```