Improved Gath–Geva clustering for fuzzy segmentation of hydrometeorological time series

Abstract

In this paper, an improved Gath–Geva clustering algorithm is proposed for automatic fuzzy segmentation of univariate and multivariate hydrometeorological time series. The algorithm considers time series segmentation problem as Gath–Geva clustering with the minimum message length criterion as segmentation order selection criterion. One characteristic of the improved Gath–Geva clustering algorithm is its unsupervised nature which can automatically determine the optimal segmentation order. Another characteristic is the application of the modified component-wise expectation maximization algorithm in Gath–Geva clustering which can avoid the drawbacks of the classical expectation maximization algorithm: the sensitivity to initialization and the need to avoid the boundary of the parameter space. The other characteristic is the improvement of numerical stability by integrating segmentation order selection into model parameter estimation procedure. The proposed algorithm has been experimentally tested on artificial and hydrometeorological time series. The obtained experimental results show the effectiveness of our proposed algorithm.

Appendices

Appendix 1: Gath–Geva clustering algorithm

Inputs: data set ${\cal{X}}=\{{\bf x}_{k}| 1\leq k\leq n\},$ number of clusters 1 < c < n, weighting exponent m > 1, termination tolerance $\varepsilon>0,$ initial parameters. $\widehat{\varvec\theta}(0)=\{\widehat{\varvec\theta}_{1},\ldots,\widehat{\varvec\theta}_{c},\widehat{P}_{1},\ldots,\widehat{P}_{c}\}$

Output: optimal parameters $\widehat{\varvec\theta}_{opt}.$

Initialize: partition matrix U(0) = [μ ⁽⁰⁾ _i,k ] _c × n such that Eq. 3 holds.

Repeat for $l=1,2,\ldots$

Calculate parameters $\widehat{\varvec\theta}(l).$

$$ \begin{aligned} {\mathbf v}_{i}(l)&=\frac{\sum_{k=1}^{n}\left(\mu_{i,k}^{(l-1)}\right)^{m}{\mathbf x}_{k}}{\sum_{k=1}^{n}\left(\mu_{i,k}^{(l-1)}\right)^{m}},\\ {\mathbf F}_{i}(l)&=\frac{\sum_{k=1}^{n}\left(\mu_{i,k}^{(l-1)}\right)^{m} \left({\mathbf x}_{k}-{\mathbf v}_{i}(l)\right)\left({\mathbf x}_{k}-{\mathbf v}_{i}(l)\right)^{T}}{\sum_{k=1}^{n}\left(\mu_{i,k}^{(l-1)}\right)^{m}},\\ P_{i}(l)&=\frac{1}{n}\sum_{k=1}^{n}\mu_{i,k}^{(l-1)}, \quad 1\leq i \leq c. \end{aligned} $$

(47)

Compute the distance measurement D(x _k ,v _i )².

$$ \begin{aligned} D({\mathbf x}_{k},{\mathbf v}_{i})^{2} &=\frac{1}{P_{i}(l)G\left({\mathbf x}_{k}; {\mathbf v}_{i}(l), {\mathbf F}_{i}(l)\right)} =\frac{(2\pi)^{q/2}\sqrt{det({\mathbf F}_{i}(l))}}{P_{i}(l)}\\ &\cdot\exp\left(\frac{1}{2}({\mathbf x}_{k}-{\mathbf v}_{i}(l))^{T}({\mathbf F}_{i}(l))^{-1}({\mathbf x}_{k}-{\mathbf v}_{i}(l))\right). \end{aligned} $$

(48)

Update the partition matrix ${\bf U}(l)=\big[\mu_{i,k}^{(l)}\big]_{c\times n}.$

$$ \mu_{i,k}^{(l)}=\frac{1}{\sum_{j=1}^{c}(D({\mathbf x}_{k},{\mathbf v}_{i})/D({\mathbf x}_{k},{\mathbf v}_{j}))^{2/(m-1)}},\quad 1 \leq i\leq c, 1 \leq k\leq n. $$

(49)

until $\|{\bf U}(l)-{\bf U}(l-1)\|< \varepsilon.$

Appendix 2: Bottom-up segmentation method

Create initial fine approximation by segment boundaries $0=t_{n_{0}}<t_{n_{1}}<\cdots<t_{n_{c}}=t_{n}.$

Find the cost of merging for each pair of segments: ${\it {mergecost}}(i) = cost(t_{{n}_{i}}+1, t_{{n}_{i+2}})$

while min(mergecost) < maxerror

Find the cheapest pair to merge: $i = \arg\min_{i} (mergecost(i)).$

Merge the two segments, update the $(t_{{n}_{i}}, t_{{n}_{i+1}})$ boundary indices, and recalculate the merge costs.

${\it {mergecost}}(i) = cost(t_{{n}_{i}}+1, t_{{n}_{i+2}})$,

${\it {mergecost}}(i-1) = cost(t_{{n}_{i-1}}+1, t_{{n}_{i+1}})$.

end

Let covariance matrix F ^x _i decompose to the matrix $\varvec\Uplambda_{i}$ that includes the eigenvalues of F ^x _i in its diagonal in decreasing order, and to the matrix U _i that includes the eigenvectors corresponding to the eigenvalues in its columns, i.e., ${\bf F}_{i}^{x}={\bf U}_{i}\varvec\Uplambda_{i}{\bf U}_{i}^{T}.$ The segmentation cost can be equal to the reconstruction error of this segment

$$ cost(t_{n_{i}}+1, t_{n_{i+1}})=\frac{1}{t_{n_{i+1}}-t_{n_{i}}+1}\sum_{k=t_{n_{i}}+1}^{t_{n_{i+1}}}Q_{i,k}, $$

where Q _i,k  = x ^T _k (I − U _i,p U ^T _i,p )x _k , and U _i,p is the eigenvectors corresponding to the first few p nonzero eigenvalues. The segmentation cost can also be equal to the Hotelling T ² measure of this segment

$$ cost(t_{n_{i}}+1, t_{n_{i}+1})=\frac{1}{t_{n_{i+1}}-t_{n_{i}}+1}\sum_{i=t_{n_{i}}+1}^{t_{n_{i+1}}}T_{i,k}^{2}, $$

where $T_{i,k}^{2}={\bf y}_{i,k}^{T}{\bf y}_{i,k}, {\bf y}_{i,k}=\varvec\Uplambda_{i,p}^{-\frac{1}{2}}{\bf U}_{i,p}^{T}{\bf x}_{k}.$ The interested reader can find more details about the bottom-up method in Abonyi et al. (2005).