# Coupled Spatiotemporal Characterization of Monsoon Cloud Cover and Vegetation Phenology
## 3 Results
### Mapping Distinct Spatiotemporal Patterns  
The continuum of temporal patterns visually evident in the temporal feature space can be geographically represented with a temporal mixture model (TMM). Temporal mixture models represent each pixel time series as a linear combination of temporal endmember time series [43]. The temporal mixture model can be considered a linear transformation of the data from statistically derived basis functions (EOFs) to physically meaningful basis functions (endmembers). This transformation allows us to base our interpretation of the temporal patterns in each pixel time series on physically meaningful processes rather than statistically derived patterns alone [23]. Temporal mixture models are generated by inversion of a simple set of mixing equations of the following form. Using \(n\) total endmembers, for each pixel:  
\[X_{t_{i}}=F_{EM1}E_{EM1,t_{i}}+F_{EM2}E_{EM2,t_{i}}+\ldots+F_{EMn}E_{EMn,t_{i} }+\varepsilon\]where \(X_{ti}\) is the NDVI at a given pixel at time \(i\), \(F_{EM(1\dots\ n)}\) is the relative contribution of endmember 1 (2,... n) to that pixel, and \(\varepsilon\) is the model misfit. \(\varepsilon\) can be minimized using ordinary or constrained least squares. For the example dataset, we generate temporal mixture models for Jabal Al Qara and Jabal Al Qamar using the endmember ensembles that were identified from each feature space. Each pixel time series is thus characterized by a linear combination of six (or four) endmembers, plus error. For each of these models, 95% of all root-mean-square pixel misfits were below 5%. Figure 3 shows maps of these temporal mixture models. In each map, primary colors (Red/Green/Blue or Cyan/Magenta/Yellow) represent the weights of three of the endmembers. The fractions are linearly displayed in the range of 0 to 1 for each endmember. For instance, in Panel A, a pixel in the western forests of Jabal Al Qara would appear as saturated red (100% red, 0% green, 0% blue), while a more centrally located forest pixel could appear as yellow (50% red, 50% green, 0% blue). In contrast, a Jabal Al Qara grassland pixel would appear as black (0% red, 0% green, 0% blue) in Panel A since no grassland endmembers are represented in that map, but bright in Panel B. In each panel, the locations of the endmember pixels that were selected from the temporal feature space are outlined in white. The locations of endmember pixels, and the spatial distribution of endmember temporal patterns in non-endmember locations, indicate strong biogeophysical clustering with examples of both gradual and abrupt transitions across the landscape. This directly emerges from the spatiotemporal structure of the data--without any prescribed assumptions about regional geography, climate, or ecology. The consistency in geographic location of the endmember pixels selected from the apexes of the temporal feature space is an important result because it reinforces the physical meaning in temporal feature space structure, providing clear evidence that the temporal patterns are related to geographic factors like elevation, slope, and coastal proximity. The spatial coherence of the endmember pixels was not imposed _a priori_, and is not required by the analysis. The clustering of endmember locations is particularly prominent in the western forests of Jabal Al Qara (Panel A, red). The most distinct temporal patterns clearly track individual wadi subsystems and do not include the grasslands separating them. The eastern forests (green) also form a spatially coherent cluster distinct from the western forests, with little mixing between the two. Regions with minimal vegetation (blue) are most extensive in the Nejd, but also include barren portions of the coastal plain. In contrast, the phenology of the Jabal Al Qara grasslands (Panel B) shows a remarkably smooth gradation between distinct western (magenta), central (yellow), and eastern (cyan) patterns.