ENTREPRENEURSHIP AND SUSTAINABILITY ISSUES

. In this paper, a fuzzy volatility labeling algorithm is offered to detect the periods with abnormal activities on daily share returns. Considering the vagueness in the switches of the time periods, the membership functions of high and normal volatility classes are introduced. In the assignments, both the density structure and membership degree are used. It is believed that this algorithm may be helpful to construct different estimation models for the time periods with normal and abnormal activities. Authors offer algorithm, which can be used as a tool for sustainable risk management.


Introduction
Investment into securities requires sustainable risk management technique, which is crucial for sustainable development of business and for overall process of sustainable entrepreneurship (Baikovs, Zariņš 2013;Šimelytė, Antanavičienė 2013;Wahl, Prause 2013;Laužikas, Krasauskas 2013;Laužikas, Mokšeckienė 2013;Litvaj, Poniščiaková 2014;Figurska 2014;Šabasevičienė, Grybaitė 2014;Dzemyda, Raudeliūnienė 2014;Tvaronavičienė 2014;Garškaitė-Milvydienė 2014).Time series clustering is a popular technique which is used in monitoring on brain activities, commercial consumption, retail pattern, gene expression, financial data, robot sensor data and so on (Zhang et al. 2011).It can be applied in varies fields with different purposes such as, the grouping of the time series of star brightness in the astronomy; the grouping of countries by considering their time series of gross national product (GNP) in the economy; the grouping of companies by taking into account their time series of returns in the finance; the segmenting of the markets by handling the consumption behavior of consumers during a time period in the marketing (D'urso and Maharaj 2009).
In the literature, there are many researches about the generation of time series clustering methods.For example, Zhang et al. (2011) suggested a novel algorithm for shape based time series clustering by using dynamic time International Centre for Entrepreneurship Research warping distance function and hierarchical clustering algorithm.D'urso and Maharaj (2009) introduced a fuzzy clustering approach which is used the autocorrelation functions of time series.Maharaj and D'Urso (2011) proposed a fuzzy clustering approach for time series based on the estimated cepstrum which is the spectrum of the logarithm of the spectral density function.Rodpongpun et al. (2012) offered Selective Subsequence Time Series (SSTS) clustering which use data encoding to cluster only essential subsequences.Their technique automatically detects an appropriate number of clusters without user's intervention.Lai et al. (2010) put forward a novel twolevel clustering method which takes into account both the whole time series and the subsequence information of time series.Chen and Tanuwijaya (2011) developed a multivariate fuzzy forecasting method which includes automatic clustering technique to forecast the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX).Cheng et al. (2008) produced a novel multiple-attribute fuzzy time series method which includes a fuzzy clustering to partition datasets objectively and allows processing of multiple attributes.Bang and Lee (2011) suggested a hierarchical classification technique in which the cross-correlation clustering algorithm is merged with the k-means clustering algorithm (HCKA) to overcome the problem dealing with non-linear data.Liao (2007) proposed the exploratory mining of multivariate time series using partition-based clustering methods.D'urso et al. (2013) handled the clustering of financial time series, they used an autoregressive representation of GARCH models in the partitioning around medoids algorithm.Alaeddini et al. (2009) focused on control charts and change point estimation problem.They suggested clustering as a potential tool for change point estimation and proposed a novel hybrid approach based on the concepts of fuzzy clustering and statistical methods.Nasibov and Peker (2011) proposed a time series labeling algorithm based on fuzzy c-means and k-nearest neighborhood rule.Among these researches, D'urso and Maharaj (2009) underlined the fact that time series has dynamic behavior over time and the switch from one time state to another is generally vague.So a fuzzy approach may be more appropriate in the evaluation of it.Considering this fact, in this paper, a fuzzy volatility labeling algorithm is offered.In the algorithm, the membership functions of high and normal volatility classes are constructed by taking into account the time series.Each trading day is assigned to one of the classes, considering maximum membership degree.The density of trading days with abnormal activities, p , is calculated.The s trading days is handled and if the density of high activities in s trading days is greater than p , this period is labeled as "the period with high volatility"; otherwise it is labeled as "the period with normal volatility".The last procedure is applied on each s trading days.

Preliminaries
The Fuzzy Set Theory, which was firstly introduced by Zadeh (1965), is a very useful theory to handle vague and ambiguous problems, expressions of linguistic values and human subjective judgments of natural language (Cheng et al. 2008).To consider both vague and imprecise data, the characteristics of fuzzy sets allow a degree of membership to be associated with a value in a set (Bang and Lee 2011).
A fuzzy number A is a fuzzy subset of the real line  with the membership function A  which is normal, fuzzy convex, upper semi-continuous, supp A is bounded, where and cl is closure operator (Ban 2008) where (Grzegorzewski 2008) . (2) The most common membership functions are triangular, trapezoidal and Gaussian membership functions.They can be described by Eq.( 3)-( 5), respectively (Ang and Quek 2012;Lee and Pan 2009).(3) (5) Their visual demonstrations are given in Figure 1.After constructions of membership functions, each daily share return is reassigned to a class in which it has maximum membership degree.After all assignments, the density of high activities is calculated from whole time series.Then, the sub-periods of time series is handled, and the density of high activities of each sub-period is calculated.If the density of sub-period is greater than the density of whole time series, the sub-period is labeled as "the period with high volatility"; otherwise it is labeled as "the period with normal volatility".

Algorithm
Step 1: Calculate the daily share returns by Eq.( 6), Step 2: Obtain the standardized values of t Y by Eq.( 7) Step 4: Obtain the membership function of the class of high volatility, HV C , via Eq.( 8) where Step 7: Take into account s trading days,  if the density of high activities in s trading days, s p , is greater than p , label these s trading days as "the period with high volatility";  if the density of high activities in s trading days, s p , is smaller than or equal p , label these s trading days as "the period with normal volatility" Repeat step for next s trading days.
For new observations, just only step 5 and step 7 may be applied for assignments and the other calculations and evaluations in other steps can be taken as constants.Or step 5-7 or whole algorithm can be processed, interactively.

Application
In this section, the offered fuzzy volatility labeling algorithm is applied on FTSE 100 index over the period of 20.08.2007-20.08.2012.In step 1, the daily share returns are obtained.The related descriptive statistics are calculated (Table 1).In step 3, the daily share returns which have standardized values in the interval of [-2, 2] are stored in the class of normal volatility, NV C ; the other ones are stored in the class of high volatility, HV C .In step 4, the mean and the standard deviation of daily share returns in the class of normal volatility, NV C , are calculated as 0.000199 and 0.011495, respectively.The maximum value and the standard deviation of daily share returns in the class of high volatility, HV C , are found as 0.093842 and 0.049178, respectively.Via these values, the membership functions of classes are constructed as following, In step 5, each daily share return is reassigned to a class in which it has a greater membership degree, considering membership functions in step 4.
In step 6, it is determined that 138 daily share returns fall high volatility class.By using this value, the density of high activities is calculated as 0.10935024 1262 / 138   p .
In Step 7, one month periods are taken as sub-periods and the density of high activities of each month is calculated.The periods which have a greater density value than 0.10935024 are labeled as "the period with high volatility".The remaining ones are labeled as "the period with normal volatility".The visual demonstration of this algorithm is given in Figure 2. The black lines at the 0 value represent "the periods with normal volatility"; the black lines at the 0.1 value represent "the periods with high volatility".In the black lines, 0 and 0.1 values have not the mathematical meanings; they are just used arbitrarily for the visual representations of the periods of two different groups.

Fig. 1 .
Fig. 1.The triangular (i), trapezoidal (ii), Gaussian membership (iii) functions Source: authors to "the class of high volatility, HV C ". Assign the other remaining t Y values to "the class of normal volatility, NV C ".
Find the density of high activities by Eq.(10),

Fig. 2 .
Fig. 2. The periods of FTSE 100 with normal and high volatilities Source: authors In Figure 2, the daily share returns are shown with grey lines.The labels of periods are shown with black lines.The black lines at the 0 value represent "the periods with normal volatility"; the black lines at the 0.1 value represent "the periods with high volatility".In the black lines, 0 and 0.1 values have not the mathematical meanings; they are just used arbitrarily for the visual representations of the periods of two different groups.

Table 1 .
Descriptive statistics of daily share returns