--- title: "State Space Reconstruction" author: "Wenbo Lyu" date: | | Last update: 2026-02-15 | Last run: 2026-04-05 output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{2. State Space Reconstruction} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ## Methodological Background Takens theory proves that for a dynamical system $\phi$, if its trajectory converges to an attractor manifold $M$—a bounded and invariant set of states—then there exists a smooth mapping between the system $\phi$ and its attractor $M$. Consequently, the time series observations of $\phi$ can be used to reconstruct the structure of $M$ through delay embedding. According to the generalized embedding theorem, for a compact $d$-dimensional manifold $M$ and a set of observation functions $\langle h_1, h_2, \ldots, h_L \rangle$, the mapping $\psi_{\phi,h}(x) = \langle h_1(x), h_2(x), \ldots, h_L(x) \rangle$ is an embedding of $M$ when $L \geq 2d + 1$. Here, *embedding* refers to a one-to-one map that resolves all singularities of the original manifold. The observation functions $h_i$ can take the form of time-lagged values from a single time series, lags from multiple time series, or even completely distinct measurements. The former two are simply special cases of the third. This embedding framework can be extended to *spatial cross-sectional data*, which lack temporal ordering but are observed over a spatial domain. In this context, the observation functions can be defined using the values of a variable at a focal spatial unit and its surrounding neighbors (known as *spatial lags* in spatial statistics). Specifically, for a spatial unit $s$, the embedding can be written as: $$ \psi(x, s) = \langle h_s(x), h_{s(1)}(x), \ldots, h_{s(L-1)}(x) \rangle, $$ where $h_{s(i)}(x)$ denotes the observation function of the $i$-th order spatial lag unit relative to $s$. These spatial lags provide the necessary diversity of observations for effective manifold reconstruction. In practice, if a given spatial lag order involves multiple units, summary statistics such as the mean or directionally-weighted averages can be used as the observation function to maintain a one-to-one embedding. ## Usage examples ### Example of spatial vector data Load the `spEDM` package and its county-level population density data: ``` r library(spEDM) popd_nb = spdep::read.gal(system.file("case/popd_nb.gal",package = "spEDM")) ## Warning in spdep::read.gal(system.file("case/popd_nb.gal", package = "spEDM")): neighbour ## object has 4 sub-graphs popd = readr::read_csv(system.file("case/popd.csv",package = "spEDM")) ## Rows: 2806 Columns: 7 ## ── Column specification ─────────────────────────────────────────────────────────────────────── ## Delimiter: "," ## dbl (7): lon, lat, popd, elev, tem, pre, slope ## ## ℹ Use `spec()` to retrieve the full column specification for this data. ## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message. popd_sf = sf::st_as_sf(popd, coords = c("lon","lat"), crs = 4326) popd_sf ## Simple feature collection with 2806 features and 5 fields ## Geometry type: POINT ## Dimension: XY ## Bounding box: xmin: 74.9055 ymin: 18.2698 xmax: 134.269 ymax: 52.9346 ## Geodetic CRS: WGS 84 ## # A tibble: 2,806 × 6 ## popd elev tem pre slope geometry ## * ## 1 780. 8 17.4 1528. 0.452 (116.912 30.4879) ## 2 395. 48 17.2 1487. 0.842 (116.755 30.5877) ## 3 261. 49 16.0 1456. 3.56 (116.541 30.7548) ## 4 258. 23 17.4 1555. 0.932 (116.241 30.104) ## 5 211. 101 16.3 1494. 3.34 (116.173 30.495) ## 6 386. 10 16.6 1382. 1.65 (116.935 30.9839) ## 7 350. 23 17.5 1569. 0.346 (116.677 30.2412) ## 8 470. 22 17.1 1493. 1.88 (117.066 30.6514) ## 9 1226. 11 17.4 1526. 0.208 (117.171 30.5558) ## 10 137. 598 13.9 1458. 5.92 (116.208 30.8983) ## # ℹ 2,796 more rows ``` Embedding the variable `popd` from county-level population density: ``` r v = spEDM::embedded(popd_sf,"popd",E = 10) v[1:5,c(4,5,10)] ## [,1] [,2] [,3] ## [1,] 962.7204 1664.756 1581.4351 ## [2,] 919.6000 2408.766 1494.8241 ## [3,] 1435.0165 1958.686 813.9077 ## [4,] 1488.2727 2066.748 1216.6986 ## [5,] 2326.8429 1290.188 1038.3864 ``` ``` r plot3D::scatter3D(v[,4], v[,5], v[,10], colvar = NULL, pch = 19, col = "red", theta = 45, phi = 10, cex = 0.35, bty = "f", clab = NA, tickmarks = FALSE) ``` ![**Figure 1**. The reconstructed attractors for the variable `popd`.](../man/figures/ssr/fig1-1.png)
### Example of spatial raster data Load the `spEDM` package and its farmland npp data: ``` r library(spEDM) npp = terra::rast(system.file("case/npp.tif", package = "spEDM")) npp ## class : SpatRaster ## size : 404, 483, 5 (nrow, ncol, nlyr) ## resolution : 10000, 10000 (x, y) ## extent : -2625763, 2204237, 1877078, 5917078 (xmin, xmax, ymin, ymax) ## coord. ref. : CGCS2000_Albers ## source : npp.tif ## names : npp, pre, tem, elev, hfp ## min values : 164.00, 384.3409, -47.8194, -122.2004, 0.03390418 ## max values : 16606.33, 23878.3555, 263.6938, 5350.4902, 44.90312195 ``` Embedding the variable `npp` from farmland npp data: ``` r r = spEDM::embedded(npp,"npp",E = 5,tau = 20) r[which(!is.na(r),arr.ind = T)[1:5],1:3] ## [,1] [,2] [,3] ## [1,] 2896.833 2933.926 2898.288 ## [2,] 3664.286 3071.789 3003.966 ## [3,] 3317.337 3090.832 2973.627 ## [4,] 3196.011 3107.388 2942.477 ## [5,] 3329.188 3080.360 2892.083 ``` ``` r plot3D::scatter3D(r[,1], r[,2], r[,3], colvar = NULL, pch = 19, col = "#e77854", theta = 45, phi = 10, cex = 0.01, bty = "f", clab = NA, tickmarks = FALSE) ``` ![**Figure 2**. The reconstructed attractors for the variable `npp`.](../man/figures/ssr/fig2-1.png)