Forecasting Enrollment Model Based

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Forecasting Enrollment Model Based
on First
-
Order Fuzzy Time Series

By

Melike Şah
(
*
)

Konstantin Y.

Degtiarev


İ
nternational Conference on Computational
İ
ntelligence (
İ
CC
İ
)

17
-
19 December 2004,
İ
stanbul, Turkey


Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

2

Overview


Introduction


Fuzzy Time Series


Forecasting Enrollments with a new Time
-
Invariant Fuzzy Time Series method


Forecasting Results and Discussion


Conclusion


References

Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

3

Introduction


Forecasting: weather, staff scheduling, finance


Well
-
known forecasting methods cannot solve
problems, when data are available in
linguistic

form


A new Time
-
Invariant Fuzzy Time Series method
to forecast University of Alabama enrollment


The effect of different number of fuzzy sets


Comparison with Song

&

Chissom and Chen’s time
invariant
-
methods
(
see Reference section, slide 15
)



Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

4

Fuzzy Time Series


First
-
order fuzzy time series





Fuzzy Logical Relationship



;


Forecasting





is an operator


1)
F(t
F(t)


)
t
(
F
)
1
t
(
F


)
1
t
,
t
(
R

)
1
t
,
t
(
R
)
1
t
(
F
)
t
(
F





Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

5

A New method of Time
-
Invariant
Fuzzy Time Series


Variations of University of Alabama enrollment




At fuzzification stage different number of
fuzzy sets [5
-
9] used. Intervals and linguistic
variables

of 6 fuzzy sets as



, ….



(
big decrease
),

(
decrease
),

(
no change
),

(
increase
), (
big increase
),

(
too big increase
)



]
1400

,
1000
[
U


1
A
2
A
3
A
4
A
5
A
6
A
]
600

,
1000
[
u
1



]

200

,
600
[
u
2



Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

6

Fuzzified variations of University
of Alabama enrollment

Years

Variation
s

Fuzzified

variation
s

Years

Variation
s

Fuzzified

variation
s

1971



1982


955

1
A

1972

+ 508

4
A

1983

+ 64

3
A

1973

+ 304

4
A

1984


352

2
A

1974

+ 829

5
A

1985

+ 18

3
A

1975

+ 764

5
A

1986

+ 82

5
A

1976


149

3
A

1987

+ 875

5
A

1977

+ 292

4
A

1988

+ 1291

6
A

1978

+ 258

4
A

1989

+ 820

5
A

1979

+ 946

5
A

1990

+ 358

4
A

1980

+ 112

3
A

1991

+ 9

3
A

1981


531

2
A

1992


461

2
A


Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

7

A New method of Time
-
Invariant
Fuzzy Time Series (Cont.)


First
-
order fuzzy logical relationships:


Years


Fuzzified Variations

1972 A4

1973

A4

1974

A5

1975

A5

1976

A3

… …








Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

8

A New method of Time
-
Invariant
Fuzzy Time Series (Cont.)


Group fuzzy logical relationships:










-

union of relationships in each group










3
1
A
A


3
1
2
A
,
A
A


5
4
2
3
A
,
A
,
A
A


5
4
3
4
A
,
A
,
A
A


6
5
4
3
5
A
,
A
,
A
,
A
A


5
6
A
A



1,6
i

,
R
i

Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

9

A New method of Time
-
Invariant
Fuzzy Time Series (Cont.)


Forecasting:


Deffuzification:


If MF all 0


forecasted variation is 0


If MF has one Max

midpoint of that interval


If MF has two or more consecutive Maxs


Midpoint of corresponding conjunct intervals


Otherwise


Centroid of the output


i
1
i
i
R
A
A



Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

10

Forecasted Outputs and Actual
Enrollments from 1973
-
1993

Year

Actual

Enrollments

Fuzzy Outputs

Forecasted

Enrollments

1973

13867

0 0.5 1 1 1 0.5

13963

1974

14696

0 0.5 1 1 1 0.5

14267

1975

15460

0 0.5 1 1 1 1

15296

1976

15311

0 0.5 1 1 1 1

16060

1977

15603

0.5 1 0.5 1 1 0.5

15530

1978

15861

0 0.5 1 1 1 0.5

16003

1979

16807

0 0.5 1 1 1 0.5

16261

1980

16919

0 0.5 1 1 1 1

17407

1981

16388

0.5 1 0.5 1 1 0.5

17138

1982

15433

1 0.5 1 0.5 0 0

16175

1983

15497

0 0.5 1 0.5 0 0

15433

1984

15145

0.5 1 0.
5 1 1 0.5

15716

1985

15163

1 0.5 1 0.5 0 0

14932

1986

15984

0.5 1 0.5 1 1 0.5

15382

1987

16859

0 0.5 1 1 1 1

16584

1988

18150

0 0.5 1 1 1 1

17459

1989

18970

0 0 0 0.5 1 0.5

18950

1990

19328

0 0.5 1 1 1 1

19570

1991

19
337

0 0.5 1 1 1 0.5

19728

1992

18876

0.5 1 0.5 1 1 0.5

19556

1993


1 0.5 1 0.5 0 0

18663


Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

11

Results and Discussion


The proposed method is implemented in MATLAB


Song and
Chissom

time
-
invariant
model

Chen’s

time
-
invariant
model

P
roposed
time
-
invariant
method

Average

forecasting errors

%
18
.
3

%
23
.
3

%
42
.
2


100

enrollment

act.

enrollment

forecast.

-

enrollment

act.



error

g
forecastin

Actual


Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

12

Results and Discussion (Cont.)

Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

13

Results and Discussion (Cont.)


Different number of fuzzy sets:


P
roposed

time
-
invariant
method

5 fuzzy

sets

6 fuzzy

sets

7 fuzzy

sets

8 fuzzy

sets

9 fuzzy

sets

Average

forecasting

errors

%
75
.
2

%
42
.
2

%
50
.
2

%
02
.
2

%
02
.
2


Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

14

Conclusion


Sorely available historical data used for
forecasting


Significantly improves accuracy


For all examined cases (different number of
fuzzy sets) forecasting error below 3%


Melike Şah, Konstantin
Y. Degtiarev

Forecasting Enrollment Model Based on
First
-
Order Fuzzy Time series

15

References


Q. Song and B.S. Chissom, “Fuzzy time series and its models”, Fuzzy
Sets and Systems, vol. 54, pp. 269
-
277, 1993.


Q. Song and B.S. Chissom, “Forecasting enrollments with fuzzy time
series


part 1”, Fuzzy Sets and Systems, vol. 54, pp. 1
-
9, 1993.


Q. Song and B.S. Chissom, “Forecasting enrollments with fuzzy time
series


part 2”, Fuzzy Sets and Systems, vol. 62, pp. 1
-
8, 1994.


S.
-
M. Chen, “Forecasting Enrollments Based on Fuzzy Time Series”,
Fuzzy Sets and Systems, vol. 81, pp. 311
-
319, 1996.


S.
-
M. Chen, “Temperature Prediction using Fuzzy Time Series”,
IEEE Transactions on Systems, Man, and Cybernetics


Part B:
Cybernetics, vol. 30, pp. 263
-
275, 2000.


K.Huarng, “Heuristic Models of Fuzzy Time Series for Forecasting”,
Fuzzy Sets and Systems, vol. 123, pp. 369
-
386, 2001.

Thank you for attention!



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