In this section, we apply a model-based clustering developed by [66] to the SST data described in the upcoming section. Clustering is an unsupervised learning technique that aims to identify clusters with homogeneous characteristics. Within the clustering framework, the model-based techniques assume the existence of a latent categorical random variable $Z$ defining $G$ clusters of data. This variable $Z$ leads to a probability distribution of data as a mixture of cluster distributions. Let $f$ denote the probability distribution of $X$ and by $f_g$ represent the probability distribution of $X$ given $Z=g$. Consequently, the mixture model is expressed as

$f\left(x\right)=\underset{g=1}{\overset{G}{{\displaystyle \sum }}}{\pi }_g{f}_g\left(x\right);$(1)

where ${\pi }_g=P\left(Z=g\right)$ represents the prior probability of cluster $g$.

In the context of spatial dependency, the model given in Equation (1) has been extended to incorporate the location $s$ into the prior probabilities of clusters. This modification transforms the mixture model into:

$f\left(x/s\right)=\underset{g=1}{\overset{G}{{\displaystyle \sum }}}{\pi }_g\left(s;\beta \right)f_g\left(x\right);$(2)

where $\beta$ represents a parametrization of the spatial prior. Consequently, given the cluster $Z=g$, the distribution of observations within the cluster becomes independent of location. All spatial dependencies are accounted for the priors ${\pi }_g\left(s;\beta \right)$. This concept is utilized in [67] for clustering spatio-temporal data. This paper introduces multinomial logistic regression as a model for the ${\pi }_g\left(s;\beta \right)$:

$\text{ln}\left(\frac{{\pi }_g\left(s;\beta \right)}{{\pi }_G\left(s;\beta \right)}\right)={\beta }_{0g}+{\langle {\beta }_g;s\rangle }_{{ℝ}^N}.$(3)

Within a parametric framework, the conditional distribution $f_g$ depends on parameters ${\theta }_g$. For instance, in the Gaussian model, ${\theta }_g$ represents the mean and the covariance matrix of cluster $g$. Let $\theta$ denote the set of all parameters, which also encompasses those defining the ${\pi }_g\left(s;\beta \right)$. As a result, the model is transformed into:

$f\left(x/s;\theta \right)=\underset{g=1}{\overset{G}{{\displaystyle \sum }}}{\pi }_g\left(s;\beta \right)f_g\left(x;{\theta }_g\right).$(4)

In a finite-dimensional context, the multivariate probability density function serves as the primary tool for estimating such a model using the EM algorithm. However, for functional random variables, the concept of a probability density isn’t well-defined due to the infinite dimension of the data. To address this challenge, [66] employs the expansion coefficients of $X$ with respect to a finite basis of functions. This approach enables the derivtion of a well-defined probability density function based on these coefficients. The use of functional principal component analysis helps define an approximation of the probability density for functional data.

Assuming a spatial autoregressive dynamic for the random effect, [66] introduces a functional classification criterion to identify local spatially homogeneous regions. In the subsequent section, we assume that given $Z=g$, $X$ follows a Gaussian process. Within cluster $g$, a pseudo-density is employed [68] :

$f_g^{\left(q_g\right)}\left(x;{\theta }_g\right)=\underset{j=1}{\overset{q_g}{{\displaystyle \prod }}}{f}_{gj}\left(C_{gj}\left(x\right),{\lambda }_{gj}\right)\underset{j^{\prime }=q_g+1}{\overset{d}{{\displaystyle \prod }}}{f}_{g{j}^{\prime }}\left(C_{g{j}^{\prime }}\left(x\right);{\bar{\lambda }}_g\right),$(5)

where $f_{gj}$ represents the probability density of the j-th major component $C_{gj}$ of $X$ within cluster $g$. The random variables $C_{gj}\left(j=1,\cdots ,q_g\right)$ are independent Gaussian variables with zero mean and variances equal to the eigenvalues ${\lambda }_{gj}$ of the covariance operator of $X$. Similarly, the random variables $C_{g{j}^{\prime }}\left(j^{\prime }=q_g+1,\cdots ,d\right)$ are independent Gaussian variables with zero mean and variances equal to the mean eigenvalues ${\bar{\lambda }}_g$ of eigenvalues ${\lambda }_{g{j}^{\prime }}\left(j^{\prime }=q_g+1,\cdots ,d\right)$ of the $X$ covariance operator. Consequently, the parameters ${\theta }_g=\left({\lambda }_{g1},\cdots ,{\lambda }_{g{q}_g},{\bar{\lambda }}_g\right),q_g$ and $d$ must be appropriately chosen.

Indeed, the surrogate density proposed can be regarded as an actual density when the functional data belong to a finite-dimensional space of functions spanned by a basis $\left\{{\varphi }_1,\cdots ,{\varphi }_d\right\},d\ge 1$, i.e.

$X\left(t\right)=\underset{j=1}{\overset{d}{{\displaystyle \sum }}}{\alpha }_j{\varphi }_j\left(t\right),t\in \left[0,T\right],T>0.$

Hence, we will choose $d$ as the dimension of the basis used for data smoothing. In this scenario, the principal components $C_{kj}$ of the functional PCA can be derived by conducting PCA on the expansion coefficients of $X$ in the metric $M$ defined by the inner product of the basis functions.

Let us now outline the EM algorithm for estimating $\theta$ and, consequently. similar to the finite setting, and based on Equation (5), the likelihood of the sample of curves $S=\left\{X_s,s\in ℐ\right\}$ is:

$l\left(\theta ,S\right)=\underset{s\in ℐ}{{\displaystyle \prod }}\left(\underset{g=1}{\overset{G}{{\displaystyle \sum }}}{\pi }_g\left(s,\beta \right)f_g^{\left(q_g\right)}\left(x_s,{\theta }_g\right)\right).$(6)

A common approach for maximizing the likelihood when data are missing (such as the variable $Z$) is to employ the iterative EM algorithm to maximize the likelihood (6), and modify it for update the principal components scores of each group as well as the parameters $\beta$ define ${\pi }_g\left(s\right)$ in (3).

The algorithm involves maximizing the approximate complete log-likelihood. Let $Z_g\left(s\right)$ denote the indicator random variable for the cluster $g$ at location $s$. Thus, the completed log-likelihood is as follows:

$L_c\left(\theta ;S,Z\right)=\underset{s\in ℐ}{{\displaystyle \sum }}\underset{g=1}{\overset{G}{{\displaystyle \sum }}}{Z}_g\left(s\right)\left(\log {\pi }_g\left(s,\beta \right)+\log f_g^{\left(q_g\right)}\left(x_s;{\theta }_g\right)\right);$(7)

This version is known to be easier to maximize than its incomplete counterpart. Let ${\theta }^{\left(h\right)}$ represent the estimated value of the parameter at iteration $h\ge 0$ of the algorithm [66] .

E Step:

Since the groups to which $Z_g\left(s\right)$ belong unknown, the E step involves calculating the conditional expectation of the approximated completed log-likelihood:

$\begin{array}{c}\mathcal{Q}\left(\theta ,{\theta }^{\left(h\right)}\right)={\mathbb{E}}_{{\theta }^{\left(h\right)}}\left[L_c\left(\theta ,S,Z\right)/S\right]\\ =\underset{s\in ℐ}{{\displaystyle \sum }}\underset{g=1}{\overset{G}{{\displaystyle \sum }}}{t}_g^{\left(h+1\right)}\left(s\right)\left(\log {\pi }_g\left(s;\beta \right)+\log f_g^{\left(q_g\right)}\left(x_s;{\theta }_g\right)\right),\end{array}$(8)

where $t_g^{\left(h+1\right)}\left(s\right)$ represents the probability that the curve $X_s$ belongs to the cluster $g$ given $C_{gj}=c_{gj}\left(x_s\right);j=1,\cdots ,q_g$ at iteration $h+1$:

$\begin{array}{c}{t}_g^{\left(h+1\right)}\left(s\right)={\mathbb{E}}_{{\theta }^{\left(h\right)}}\left[L\left(Z_g\left(s\right)\right)/s\right]\\ =\frac{{\pi }_g\left(s;{\beta }^{\left(h\right)}\right)f_g^{\left(q_g\right)}\left(x_s;{\theta }_g^{\left(h\right)}\right)}{{{\displaystyle \sum }}_{l=1}^G{\pi }_l\left(s,{\beta }^{\left(h\right)}\right)f_l^{\left(q_l\right)}\left(x_s;{\theta }_l^{\left(h\right)}\right)}.\end{array}$(9)

M step:

The M step involves maximizing the conditional expectation of the completed log-likelihood with respect to $\theta$

$\begin{array}{l}{\theta }_g^{\left(h+1\right)}=\arg \underset{{\theta }_g}{\max }\underset{s\in ℐ}{{\displaystyle \sum }}{t}_g^{\left(h+1\right)}\left(s\right)\log f_g^{\left(q_g\right)}\left(x_s;{\theta }_g\right);\\ {\beta }_g^{\left(h+1\right)}=\arg \underset{\beta }{\max }\underset{s\in ℐ}{{\displaystyle \sum }}{t}_g^{\left(h+1\right)}\left(s\right)\log {\pi }_g\left(s;\beta \right).\end{array}$(10)

Observe that ${\beta }_g^{\left(h+1\right)}$ is obtained as a solution of a weighted logistic regression. The EM algorithm commences with an initial random partition of the data $S$ into $G$ clusters. It’s important to note that in homoscedastic models, there’s a modification of the update ${\theta }_g^{h+1}$. Further details can be found in [66] .

To determine the number of clusters $G$ when $q_g\left(g=1,\cdots ,G\right)$ are known, we suggest maximizing the Bayesian Information Criterion (BIC) criterion defined as:

$\text{BIC}\left(G\right)=\log l\left(G\right)-\frac{{\nu }_G}{2}\log \left(n\right);$(11)

where

${\nu }_G=\left(N+1\right)\left(G-1\right)+Gd+\underset{g=1}{\overset{G}{{\displaystyle \sum }}}\left(q_g\left[d-\left(q_g-1\right)/2\right]+1\right)$

here ${\nu }_G$ is the number of parameters in the model (including spatial mixing proportions, center means, principal scores and variances) and $n=\left|ℐ\right|$ represents the number of points involved. When the values $q_g\left(g=1,\cdots ,G\right)$ are unknown, they can be determined by maximizing the BIC criterion. This can be achieved through the following modified M step, which aims to maximize the conditional expectation of the BIC criterion:

$\left(q_g,{\theta }_g^{\left(h+1\right)}\right)=\arg \underset{\left(q_g,{\theta }_g\right)}{\max }\underset{s\in ℐ}{{\displaystyle \sum }}{t}_g^{\left(h+1\right)}\left(s\right)\log f_g^{\left(q_g\right)}\left(x_s,{\theta }_g\right)-\frac{{\nu }_{q_g}}{2}\log \left(n\right),$(12)

where ${\nu }_{q_g}=q_g\left(d-\left(q_g-1\right)/2\right)$ represents the additional number of parameters needed for the model with $q_g$ main components, as discussed in [66] .

In functional data analysis, directly applying cluster analysis to observations is not often recommended. This caution arises for valid reasons; the discrete measurements intervals of the observations might be irregular, and the measurement intervals could differ among different functional observations. Consequently, conducting cluster analysis directly on such data can present challenges. To address this, a practical approach is to carry out cluster analysis based on the primary functional principal component (FPC) scores [69] .

K-centers functional clustering (KCFC) is a method grounded in the computation of principal components. In this approach, the elements of each cluster are drawn with consideration to better approximation by the first principal components. The method can be outlined in the following steps:

$X_{\left(s\right)}^{\left(c\right)}\left(t\right)=\mu \left(s\right)+\underset{k=1}{\overset{+\infty }{{\displaystyle \sum }}}{\xi }_k\left(s\right){\rho }_k\left(t\right).$(13)

The covariance function is defined as follows:

${\Gamma }^{\left(c\right)}\left(s,s^{\prime }\right)=\mathrm{cov}\left(X_{\left(s\right)}\left(t\right),X_{\left(s\right)}\left(t^{\prime }\right)/C=c\right).$(14)

From (13) and (14), we obtain

$\langle {\text{Γ}}^{\left(c\right)}\left(s,s^{\prime }\right),{\rho }_k^{\left(c\right)}\rangle ={\lambda }_k^{\left(c\right)}{\rho }_k^{\left(c\right)},$

and

${\xi }_k^{\left(c\right)}\left(s\right)=\langle X\left(t,s\right)-\mu \left(s\right),{\rho }_k^{\left(c\right)}\rangle .$

Given that the data of interest are functional in nature, dimension reduction is necessary for efficient computation. A common approach involves expanding each curve using a finite number of principal components [66] [70] .

Recalling that clustering and supervised classification are valuable tools in traditional multivariate data analysis, they present challenges in the context of functional data analysis. Clustering involves grouping a dataset into configurations where data within clusters are more similar to each other than across clusters, based on a defined metric. In contrast, supervised classification assigns an individual to a predefined group or class using labeled observations.

In machine learning terms, functional data clustering is an unsupervised learning process, while supervised classification employs a discriminant function or classifier to assign new data to predetermined groups. Functional classification typically uses training data with functional predictors and associated multi-class labels for each data point.

In the application, it’s essential to determine the percentage of variance to be explained and subsequently establish the number K_c of principal components required. Equation (13) is then modified as follows:

${\hat{X}}_{\left(s\right)}^{\left(c\right)}\left(t\right)=\mu \left(s\right)+\underset{k=1}{\overset{K_c}{{\displaystyle \sum }}}{\xi }_k\left(s\right){\rho }_k\left(t\right).$(15)

To avoid making additional distribution assumptions, the cluster membership for an observation $X_{\left(t,s\right)}$ is determined by:

$\begin{array}{c}{C}^{\star }\left(X\right)=\arg \underset{c\in \left\{1,\cdots ,C\right\}}{\min }\Vert X_{\left(s\right)}^{\left(c\right)}(.)-{\hat{X}}_{\left(s\right)}^{\left(c\right)}(.)\Vert \\ =\arg \underset{c\in \left\{1,\cdots ,C\right\}}{\min }\Vert \underset{k=K_c+1}{\overset{+\infty }{{\displaystyle \sum }}}{\left({\xi }_k\left(s\right){\rho }_k(.)\right)}^{1/2}\Vert ,\end{array}$(16)

which determines the cluster that can represent the observation with the smallest error. For the purpose of grouping, it’s essential to initially estimate the moments, eigenfunctions, eigenvalues, and functional principal component (FPC) scores. The KCFC algorithm builds upon an initial cluster assignment based on the FPC scores $\left({\hat{\xi }}_{i,1},\cdots ,{\hat{\xi }}_{i,K_c}\right)$, where $i=1,\cdots ,n$, one common approach is to use a standard classification procedure such as K-means clustering with $K_c$ representing the number of main components considered. Once the initial clustering is established, the algorithm operates as follows:

Suppose $g_i^{\left(l\right)}\in \left\{1,\cdots ,C\right\}$ is the cluster membership of the i-th observation in the th iteration $G^{\left(l\right)}=\left\{g_i^{\left(l\right)},i=1,\cdots ,n\right\}$ all the clusters, we have:

1) Choose $i\in \left\{1,\cdots ,n\right\}$ and we calculate ${\hat{\mu }}_{\left(-i\right)}^{\left(c\right)}$ and ${\hat{g}}_{\left(-i\right)}^{\left(c\right)}$ based on observations with $g_j^{\left(l\right)}$ with $j\ne i$.

2) Calculate the i-th predicted observation for cluster $c$.

${\hat{X}}_i^{\left(c\right)}\left(t\right)={\hat{\mu }}_{\left(-i\right)}^{\left(c\right)}\left(s\right)+\underset{k=1}{\overset{K_c}{{\displaystyle \sum }}}{\hat{\xi }}_k\left(s\right){\hat{\rho }}_k\left(t\right).$(17)

3) Observation number $i$ is assigned to the closest cluster.

4) Steps 1 to 3 are repeated until there is no further reclassification.

The data come from NCDC/NOAA (National Climatic Data Center/National Climatic Data Center) https://psl.noaa.gov/data/gridded/data.noaa.ersst.v4.html . They are monthly measurements of sea surface temperature (SST) off tropical African zone from January 1, 1854, to February 29, 2020. This area ( $ℐ\subset {ℝ}^2$) of interest, see Figure 1 , is covered by longitude −70˚ to 20˚ and latitude −26˚ to 24˚.

This study area includes most of the countries of West Africa, Central Africa and especially the coastal countries.

This area is divided into 4309 geographical points. At each of these points, monthly sea surface temperatures are recorded from January 1, 1854, to February 29, 2020.

Let $s_i\in ℐ$, $i=1,\cdots ,4309$ be the locations. From January 1, 1854, to February 29, 2020, we consider monthly SST at given locations $s_i$. Then the temporal index is $t\in \left[0,1994\right]$; t = month. The 4309 observations recorded in these measurement sites, are transformed into a functional object using B-splines ( Figure 2 ), see [71] - [73] for more details.

We have 4309 sites where SST measurements were taken. Figure 2 illustrates that not all SST curves are overlaid. The temporal temperature variation differs across the various sites, indicating spatial temperature heterogeneity. On average, the curves share a similar. When examining the curves shapes, there appears to be a suggestive periodicity.

This Figure 3 displays a heterogeneous spatial distribution of SST in the tropical zone, with higher temperatures observed in the central area and lower temperatures in the eastern extremes. Figure 3 corroborates the spatial heterogeneity observed in 2.

Panels (a), (b), (c) (d), (e), and (f) of Figure 4 depict distinct SST trends for the respective years 1970, 1971, 2001, 2002, 2018 and 2019. By focusing on the month of March across these six years, it becomes apparent that the SST distribution across the off-tropical African sub-zones varies in terms of spatial scale. Notably, the spatial configuration of the off-tropical African zone in March 2018 differs from that in March 2019. The clustering method outlined in Section 2 is subsequently applied to the SST functional spatial data (as shown in Figure 2 ) to discern the heterogeneity of SST.

In each step of the EM algorithm, and for each value of $q$, BIC is computed using Equation (12).

While the curves appear to share the same shape, Figure 5 depicts three distinct classes of curves. An analysis of this figure reveals that the clustering of the sea surface temperatures (SST) off tropical Atlantic consists of three groups: one distinct cluster and a combination of two clusters. To gain a clearer view of these classes, we aim to extract and represent them separately.

Figure 5 displays the outcomes of the unsupervised classification involved in three groups portraying the spatial and temporal structure of SST off tropical Atlantic.

In panel (c) of Figure 5 it is observed most curves exhibit temperature variations between the ranges: 24˚ and 30˚. Similarly, in panel (d) of Figure 5 an analysis reveals that most curves undergo temperature changes within the interval of: 22˚ and 30˚. Similarly an examination of panel (e) in Figure 5 demonstrates that most curves experience temperature fluctuations within the range of 24˚ to 30˚.

An analysis of the panel Figure 6 illustrates the spatial distribution of the measurement sites for the three temperature classes.

The average curves of the three classes ( Figure 7 ) demonstrate distinct three phases in the SST. Each phase is characterized by abrupt changes in SST. Notably, during the initial phase, the red and blue classes are intermingled, whereas in the subsequent phases, they are clearly separated. Furthermore, the red curve class dominates as the primary class, followed by the blue curve class as the intermediate class, and the green curve class as the least prominent class.

The first phase of the red curve spans from 1854 to August 1897 (at t = 500). The second phase, marked by a sharp SST decline, extends from September 1897 to April 1939. The final phase, characterized by an SST increase, covers the period from May 1939 to February 2020.

The three phases of SST variation in the green curve align with those of the red curve. A slight distinction is observed in the phases of variation of the blue curve. Notably, its first phase is longer than the first two phases of the other classes (red and green curves), extending until the year 1900. This suggests that global warming might have commenced around 1939. In summary, the descriptive analysis of Figure 6 and Figure 7 reveals the spatial distribution of measurement sites across the three distinct classes: a very hot zone (red), a moderately hot zone (blue), and a relatively less hot zone (green).

A more detailed analysis of the differences in SST curves could be beneficial through a grouping of SSTs that enables the clear differentiation of two classes ( Figure 8 and Figure 9 ). In each class, sites with similar SST curves are grouped together. Furthermore, by considering the average curves within the classes, these can be divided into two categories: the hot class and the non-hot class ( Figure 10 and Figure 11 ).

An analysis of the graph in Figure 8 reveals that the SST range fluctuates between 16˚ and 28˚. Regarding the classification into three classes, Figure 9 distinctly illustrates the heterogeneous nature of SST. To enhance visibility of the two classes, they will be presented separately in two panels ( Figure 9 ).

Figure 9 illustrates the outcomes of unsupervised classification using 2 groups to represent the spatial and temporal structure of SST of the tropical Atlantic.

Figure 10 and Figure 11 present two distinctly discernible clusters, demonstrating the heterogeneity of SST across both spatial and temporal scales. It’s noteworthy that these figures highlight the evident spatial and temporal heterogeneity of SST within the tropical zone.

The comprehensive analysis of the two curves in Figure 11 reveals three distinct phases of sea surface temperature (SST) change. The first phase spans from 1854 to August 1897 (t = 500). The second phase exhibits a sudden SST drop and covers the period from September 1897 to April 1939. The final phase extends from May 1939 to the end of February 2020. Throughout these phases, the two SST classes (represented by the red and blue curves) exhibit clear separation. The warmer class corresponds to the blue curve, while the cooler class corresponds to the red curve.