704 words - 3 pages

2.1 Classification phase: fuzzy c –means clustering algorithm

The FCM algorithm introduced by Dunn [27] and modified by Bezdek [28] was used widely in the clustering methods. It consists of separating the data point into c clusters with respect to some given criterion for the optimization of an objective function. However, due to the presence of nonlinearity in some time series such as the hourly global solar radiation time series, it is more useful to represent the time series in higher-dimensional space to understand the underlying dynamical of the system [25]. Hence, phase space reconstruction was used to simplify the analysis of the time series. It provides a simplified, multidimensional representation of a nonlinear time series that simplifies further analysis. It consists of determining the minimum, appropriate, embedding dimension for a time series [35]. The most widely used version is the time delay embedding method [29]. A scalar time series is embedded into a m-dimensional space denoted , as expressed in Eq. (1),

(1)

Where, , is the delay time, is the embedding dimension and is the number of embedded points in the m-dimensional space given by Eq. (2). Where, N is the total number of points of the time series and is the embedded time series into an m-dimensional space.

(2)

To determine the number of delays, the mutual information method proposed by Fraser and Swinney [36] was used. The optimum delay is equal to the first minimum of the plotted mutual information expressed by the following equation,

(3)

is the mutual information and is the joint probability mass function for the marginal probability mass functions and .

In addition, the false nearest neighbour method was chosen to choose the suitable number of the embedding dimension [37]. This method determines the nearest neighbour of every point in a given dimension, and then checks if there are still close neighbours in the higher dimension.

After determining the optimal embedding dimension, the reconstructed phase space of the solar radiation data is clustered using the fuzzy c-means algorithm. In this...

Get inspired and start your paper now!