
## Ii Basic Concepts Of Fuzzy Time Series:

There are number of definitions for fuzzy time series. 

Definition 1: Imprecise data at equally spaced discrete time points are modeled as fuzzy variables. The set of this discrete fuzzy data forms a fuzzy time series. Definition 2: Chronological sequences of imprecise data are considered as time series with fuzzy data. A time series with fuzzy data is refereed to as fuzzy time series. 

Definition 3: Let Y(t), (t=...0,1,2,…) be the universe of discourse and Y(t) R . Assume that fi(t), i=1,2,… is defined in the universe of discourse Y(t) and F(t) is a collection of f(ti), 
( i=...0,1,2,…), then F(t) is called a fuzzy time series of Y(t), 
i=1,2,..Using fuzzy relation, we define F(t)=F(t-1) o R(t,t-1) 
where R(t,t-1) is a fuzzy relation and "o" is the max min composition operator, then F(t) is caused by F(t-1) where F(t) and F(t-1) are fuzzy sets. Let F(t) be a fuzzy time series and let R(t, t-1) be a first order model of F(t). If R(t, t-1)=R(t1,t-2) for any time t then F(t) is called a time-invariant fuzzy time series. If R(t,t-1) is dependent on time t, that is, R(t, t-1) 
may be different from R(t-1,t-2) for any t, then F(t) is called a time-variant fuzzy time series. 

## Iii. Proposed Method Using Fuzzy Time Series:

 
 In this section, we present our method to forecast the data of car road accidents based on means. The historical data of car road accidents are given in Table I. First, based on [1], we defined the partition the universe of discourse into equal length intervals. Then based on means of original data and means of frequency of interval data, we redefine the intervals. After this, define some membership function for each interval of the historical data to obtain fuzzy data to form a fuzzy time series. Then, it establishes fuzzy logical relationships (FLRs) based on the fuzzified data in Table IV. Finally, it uses our proposed method. 

Step 1: Define the universe of discourse U and partition it into four equal intervals u1,u2,u3,……un. For example, assume that the universe of discourse U=[900, 17000] is partitioned into four even and equal length intervals. 

Step 2: Get a mean of the original data. Get the means of frequency of each interval. Compare the means of original and frequency of each interval and then split the four interval into number of sub-interval respectively. 

Step 3: Define each fuzzy set Ai based on the re-divided intervals and fuzzify the historical data shown in Table I, 
where fuzzy set Ai denotes a linguistic value of the accident data represented by a fuzzy set. We have used triangular membership function to define the fuzzy sets Ai [10]. The reason for fuzzifying the accident data into fuzzified data is to translate crisp values into fuzzy sets to get a fuzzy time series. 

Step 4: Establish fuzzy logical relationships based on the fuzzified data where the fuzzy logical relationship "Ap, Aq, Ar 

-> As " denotes that "if the fuzzified data of year p, q and r are Ap, Aq and Ar respectively, then the fuzzified data of the year 
( r) is Ar". 
 1+0.5 
1+0.5  .................................  (1/ a${}_{1}$ ) +(0.5/ a${}_{2}$ )  0.5 + 1 + 0.5  ..........................................  (0.5/ a${}_{j-1}$ ) +( 1/ a${}_{j}$ ) + (0.5/ a${}_{j+1}$ )
 0.5 + 1 + 0.5 
 0.5 + 1 ----------------------- if j= n 
(10.5/a_n-1)+(1/a_n)
 
where aj-1,aj, aj+1 are the mid points of the fuzzy intervals Aj-1, Aj, Aj+1 respectively. Based on the fuzzify historical enrollments obtained in step 3, we can get the fuzzy logical relationship group (FLGR) as shown in Table IV. 

 Divide each interval derived in step 2 into subintervals of equal length with respect to the corresponding means of the interval. We have assumed twenty nine partitions of the universe of discourse of the main factor fuzzy time series. 

Assuming that 0 ≠ Ai, for every Ai, i=1,2,…29.

## Table - I

| ---------------------------------------------------------------------   Year Killed Year Killed   2004 953 1984 1,369  2003 1,035 1983 1,479  2002 1,145 1982 1,464  2001 1,288 1981 1,564  2000 1,253 1980 1,616  1999 1,173 1979 1,572  1998 1,224 1978 1,644  1997 1,150 1977 1,597  1996 1,122 1976 1,536  1995 1,228 1975 1,460  1994 1,415 1974 1,574  1993 1,346  1992 1,380  1991 1,471  1990 1,574  1989 1,488  1988 1,432  1987 1,390  1986 1,456  1985 1,308   |
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