Dynamic clustering of spatial–temporal rainfall and temperature data over multi-sites in Yemen using multivariate functional approach

Abstract

Analyzing Multivariate Functional Data (MFD) presents growing challenges in the context of climate change modeling due to many issues, such as coarse resolution, model complexity, and big data processing. In this regard, we introduced a Multivariate Functional Model-Based Clustering (MFMBC) method to analyze Multivariate Functional Rainfall and Temperature (MFRT) data. The data was collected spanning four decades (Jan.1980–Apr.2022) over 37 locations in Yemen. The main objective is to identify the underlying spatial–temporal dynamic structure of MFRT data and model the association/interrelationship between data. The proposed MFMBC method consists of three key phases: projecting MFRT data variation through Multivariate Functional Principal Component Analysis (MFPCA), identifying optimal clusters with Bayesian Information Criteria (BIC), and optimizing model parameters using Expectation–Maximization (EM) algorithm. According to the findings, three ideal clusters for MFRT data profiles were identified and labeled as severe, moderate, and high temperatures, which correspond to heavy, moderate, and light rainfall patterns. Cluster 1 had a negative nexus characterized by slight changes and low-peak rainfall with high changes and large-peak temperatures. Cluster 2 exhibited a natural nexus with a mild pattern in both rainfall and temperature. Cluster 3 had positive-nexus displayed significant variations with large-volume peaks in rainfall and temperature. Overall, these results help in assessing the complex interaction between rainfall and temperature over the spatial–temporal domain and offer valuable insights for policy-makers to address climate-related challenges.

Appendices

Appendix 1 The spatial distribution of the selected sites in Yemen

Appendix 2 The cities’ names and their geographical characteristics

No	Station name	Latitude (°N)	Longitude (°E)	Elevation (m)
1	Sanaa	15.35	44.2	2257
2	Manaha	15.0742	43.7416	2233
3	Taiz	13.5789	44.0219	1369
4	Mocha	13.3167	43.250	2
5	Dhubab	12.9431	43.4102	2
6	Turbah	13.2127	44.1241	1772
7	Ḩudaydah	14.8022	42.9511	14
8	Zabid	14.200	43.3167	108
9	Mukalla	14.5333	49.1333	21
10	Tarīm	16.050	49.000	610
11	Sayun	15.943	48.7873	652
12	Addis	14.8833	49.8667	108
13	Aden	12.800	45.0333	38
14	Ibb	13.9667	44.1667	1893
15	Yarim	14.2978	44.3778	2638
16	Dhamar	14.550	44.4017	2427
17	Zinjibar	13.1283	45.3803	13
18	Aḩwar	13.5202	46.7137	34
19	Şadah	16.9358	43.7644	1875
20	Ḩajjah	15.695	43.5975	1733
21	Midi	16.321	42.813	8
22	Rada	14.4295	44.8341	2120
23	Bayḑa’	13.979	45.574	2018
24	Ataq	14.550	46.800	1158
25	Rawḑah	14.480	47.270	793
26	Bayḩan	14.8007	45.7189	1129
27	Laḩij	13.050	44.8833	132
28	Ghayz̧ah	16.2394	52.1638	14
29	Sayḩut	15.2105	51.2454	4
30	Marib	15.4228	45.3375	1089
31	Khamir	15.990	43.950	2449
32	Amrān	15.6594	43.9439	2264
33	Ḩadibu	12.6519	54.0239	7
34	Maḩwīt	15.4694	43.5453	2026
35	Ḩazm	16.1641	44.7769	1114
36	Jabīn	14.704	43.599	2034
37	Ḑali	13.6957	44.7314	1519

Appendix 3 More technical details about Sect. 3 (Theoretical framework)

• The general structure of basis functions ${\varvec{\psi}}\left( t \right)$ and its coefficients ${\mathbf{\mathcal{C}}}$ in the functional multivariate framework can be expressed as:

  $$\begin{gathered} {\mathbf{\mathcal{C}}} = \left( {\begin{array}{*{20}l} {c_{11}^{1} \ldots c_{{1R_{1} }}^{1} } \hfill & {c_{11}^{2} \ldots c_{{1R_{2} }}^{2} } \hfill & \cdots \hfill & {c_{11}^{j} \ldots c_{{1R_{j} }}^{j} } \hfill \\ {c_{21}^{1} \ldots c_{{2R_{1} }}^{1} } \hfill & {c_{21}^{2} \ldots c_{{2R_{2} }}^{2} } \hfill & \cdots \hfill & {c_{21}^{p} \ldots c_{{2R_{j} }}^{p} } \hfill \\ \vdots \hfill & \vdots \hfill & \vdots \hfill & \vdots \hfill \\ {c_{N1}^{1} \ldots c_{{NR_{1} }}^{1} } \hfill & {c_{N1}^{2} \ldots c_{{NR_{2} }}^{2} } \hfill & \cdots \hfill & {c_{N1}^{j} \ldots c_{{NR_{j} }}^{j} } \hfill \\ \end{array} } \right) \hfill \\ {\varvec{\psi}}\left( t \right) = \left( {\begin{array}{*{20}l} {\psi_{1}^{1} \left( t \right) \ldots \psi_{{R_{1} }}^{1} \left( t \right)} \hfill & {0 \ldots \ldots 0} \hfill & \cdots \hfill & {0 \ldots \ldots 0} \hfill \\ {0 \ldots \ldots 0} \hfill & {\psi_{1}^{2} \left( t \right) \ldots \psi_{{R_{2} }}^{2} \left( t \right)} \hfill & \cdots \hfill & {0 \ldots \ldots 0} \hfill \\ \vdots \hfill & \vdots \hfill & \vdots \hfill & \vdots \hfill \\ {0 \ldots \ldots 0} \hfill & {0 \ldots \ldots 0} \hfill & \cdots \hfill & {\psi_{1}^{j} \left( t \right) \ldots \psi_{{R_{j} }}^{j} \left( t \right)} \hfill \\ \end{array} } \right) \hfill \\ \end{gathered}$$

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• Consequently, using the covariance estimator $\hat{\varvec{v}}\left( {s,t} \right)$ in Eq. (5) and principal component ${\varvec{\xi}}_{l}$ in Eq. (6), the reformulating of the eigen-problem in Eq. (4) becomes:

  $$\mathop \smallint \limits_{1}^{508} \hat{\varvec{v}}\left( {s,t} \right) {\varvec{\xi}}_{l} \left( t \right) dt = \eta_{l} {\varvec{\xi}}_{l} \left( s \right)$$
  $$\begin{gathered} \mathop \smallint \limits_{1}^{508} \frac{1}{N - 1}{\varvec{\psi}}\left( s \right) {\mathbf{\mathcal{C}}}^{\prime } {\mathbf{\mathcal{C}}}{\varvec{\psi}}^{\prime } \left( t \right){\varvec{\xi}}_{l} \left( t \right)dt = \eta_{l} {\varvec{\psi}}\left( s \right){\varvec{b}}_{l}^{\prime } \Leftrightarrow \frac{1}{N - 1}{\varvec{\psi}}\left( s \right){\mathbf{\mathcal{C}}}^{\prime } {\mathbf{\mathcal{C}}}\underbrace {{\mathop \smallint \limits_{1}^{508} {\varvec{\psi}}^{\prime } \left( t \right){\varvec{\psi}}\left( t \right)dt}}_{{\text{W}}}{\varvec{b}}_{l}^{\prime } \hfill \\ \frac{1}{N - 1}{\varvec{\psi}}\left( s \right) {\mathbf{\mathcal{C}}}^{\prime } {\mathbf{\mathcal{C}}} {\varvec{W}} {\varvec{b}}_{l}^{\prime } = \eta_{l} \psi \left( s \right) {\varvec{b}}_{l}^{\prime } , \hfill \\ \end{gathered}$$

  (12)

  where ${\varvec{W}} = \mathop \smallint \limits_{1}^{508} {\varvec{\psi}}^{\prime } \left( t \right){\varvec{\psi}}\left( t \right)dt$ is a R × R Matrix (with R = $\mathop \sum \limits_{j = 1}^{p} R_{j}$) containing the inner product of our pre-defined basis functions.

• The MFPCA scores are followed a Gaussian distribution $\delta_{i}^{k} \sim {\mathbf{\mathcal{N}}}\left( {\mu_{k} ,\Delta_{k} } \right)$ with the mean function $\mu_{k} \in \mathbb{R}$ and the covariance matrix ${{\varvec{\Delta}}}_{k}$, which is structured as:

  $${{\varvec{\Delta}}}_{k} = \left( {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}c} {a_{k1} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {a_{{kd_{k} }} } \\ \end{array} } \right]} \hfill & \cdots \hfill & 0 \hfill \\ \vdots \hfill & \ddots \hfill & \vdots \hfill \\ 0 \hfill & \cdots \hfill & {\left[ {\begin{array}{*{20}c} {b_{k} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {b_{k} } \\ \end{array} } \right]} \hfill \\ \end{array} } \right) \begin{array}{*{20}c} {\left\{ {\begin{array}{*{20}c} {d_{1} } \\ \vdots \\ {d_{k} } \\ \end{array} } \right.} \\ {\left\{ {\begin{array}{*{20}c} {R - d_{1} } \\ \vdots \\ {R - d_{k} } \\ \end{array} } \right.} \\ \end{array}$$

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With the help of the covariance structure ${ }{{\varvec{\Delta}}}_{k}$, we have the ability to model the variance of the first $d_{k}$ principal component with a high degree of accuracy, while the remaining components can be viewed as noise components that can be retained and modeled via the parameter $b_{k}$.

Appendix 4 Outlier assessment of principal component (PC) scores

To assess the potential outlier values in the rainfall and temperature datasets, here we present supplementary results utilizing the graphical representation of the Bivariate Bag-Plot (BBP) method. This method involves projecting the scores of the first and second principal components and plotting them in a two-dimensional graph, as depicted in the figure below. According to this figure, the BBP results illustrate the 99% probability coverage for the rainfall and temperature data. Within the BBP framework, we can see the dark and light gray regions representing the inner and outer bag regions, respectively. The inner bag denotes the smallest depth region encompassing at least 50% of the observations, while the outer bag is the convex hull of the region obtained by inflating the inner bag using a factor α = 2.58, covering 90% of the estimated probability values. Any points located outside the fence (outer bag) regions are considered outliers. Nevertheless, no severe outliers have been identified based on our BBP findings.