chapter 5 exponential smoothing methods l 2015
TRANSCRIPT
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 1
5.1
I
.
I ,
;, .
B ,
.
D .
: M 2000.
D
.
: N 1995 2000.
A :
1. ( ),
I , , , ,
. A ,
.
. .
. H, ,
, .
2. ,
.
.
.
3. ,
. I . .
,
.
4. ,
, , . . B
, ,
.
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 2
.
. H, : () ()
.
A M
( ), ( ), ( )
( ).
+++=
.
I ,
( ), ( ) ( ).
++= ( ) .
.
.
M M
( ), ( ), ( ) ( ).
×××=
.
. I ,
. ××=
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 3
5.1
5.1 .
.
↓
• .
1+− ... 2− 1−
• • • •
. F
1+ 2+ +
• • • • • • •
. F
1+− ... 2− 1−
• • • • . F
( 11 +−+− − ),,( 11 −− − ), ( − )
( ),..
. F E ( 1+ , 2+ , , )
( 11 ++ − ),( 22 ++ − ),
( ),.
O ,
. O ,
. F
.
5.2: C
N E
1
A
2
M
3
N E
A
A
B
M
C
5.2
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 4
I ,
.
5.2.1
.
∑=
+ =
1
1
1
, 1+ , 2+
:
∑+
=
++
=1
1
2
1
1
=
1
11
+
+ ++
( )
I , ( , 1+ , , 1+ )
. ,
.
: ( A1 )
() ;() .
A ,
( ), .
5.2.2
A , )( ,
∑+−=
+ =
1
1
1
,
. .
C , :
1. I ( ).
2. .
B :
1. I ,
.
2. I , .
M , O MA(3) MA(5)
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 5
J 1 200
F 2 135
M 3 195
A 4 197.5(200+135+195/3)
=176.67
M 5 310(135+195+197.5/3)=
175.83
J 6 175 234.17(200+135+195+197.5+310/5)
= 207.5
J 7 155 227.50(135+195+197.5+310+175)/5)
=202.5
A 8 130 213.33 206.5
9 220 153.33 193.5
O 10 277.5 168.33 198
N 11 235 209.17 191.5
D 12 244.17 203.5
)(3 4 . F ,
A ( ) J, F, M. D )(3
244.17 , O, N.
1. ,
, .
2. , .
5.3 ( )
()
()
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 6
, .
( /1 ) .
, .
. O .
.
...,,, 21=
.
)1(1
)(1 α α α −+=−+=+
α 0 1,
. )( − .
1. I
.
2. α 1,
. C, α 0,
.
N 1+
)( α α −+=+ 11
111
44
33
22
1
12
1
11
111
111
11
11
)()()(
)()()(
)()(
)()(
α α α α α
α α α α α α α
α α α α
α α α α
−+−++−+
−+−+−+=
−+−+=
−+−+=
−−
−−−
−−
−−
L
1. ( 1+ ) ( )
(α ) ( )
)( α −1 .
2. .
3. A , ,
.
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 7
I . F ,
, ,
1++ = ( ...,, 32= )
I 1= , %)( 1001 α − 1+
21 /α ±+
I 2= , %)( 1001 α − 2+
)(/2
21 1 α α +±+
I , , %)( 1001 α − +
))((/ 221 11 α α −+±+
1−
=
=
( )
1
1
2
−
∑ −=
.
5.1
1 2 3 4 5
1000 900 990 909 982
0.9.
) F 5 10001 = .
) F 5
.
() 10011 ==
3.916)982(1.0)909(9.0)1(
982)910(1.0)990(9.0)1(
910)1000(1.0))900(9.0)1(1000)1000(1.0)1000(9.0)1(
445
334
223
112
=+=−+=
=+=−+=
=+=−+=
=+=−+=
α α
α α
α α α α
() 75.9494
90999090010001 =
+++=
295.916)95.981(1.0)982(9.0)1(
95.981)4975.909(1.0)990(9.0)1(
4975.909)975.994(1.0)900(9.0)1(
975.994)75.949(1.0)1000(9.0)1(
445
334
223
112
=+=−+=
=+=−+=
=+=−+=
=+=−+=
α α
α α
α α
α α
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 8
5.2
11= , 10.=α , 3234383.=
M
,
A
,
F E
F
E
J 1 1= 200 1 = 200 0 0
F 2 135
112 )1( α α −+= =
0.1(200) + 0.9(200) =
200 2 2 = 65 652 = 4225
M 3 195
223 )1( α α −+= =
0.1(135) + 0.9 (200)
= 193.5 3 3 = 1.5 1.52 = 2.25
A 4 197.5
334 )1( α α −+= =0.1(195) + 0.9(193.5)
= 193.65
197.5
193.65 = 3.85
3.852 =
14.82
M 5 310 194.04 115.97 13447.88
J 6 175 205.63 30.63 938.29
J 7 155 202.57 47.57 2262.75
A 8 130 197.81 67.81 4598.40
9 220 191.03 28.97 839.24
O 10 277.5 193.93 83.57 6984.39
N 11 235 202.28 32.72 1070.30
D 12
12 = 0.1(235) +
0.9(202.28)
= 205.55 ∑ 32.34383
F
) 12 (D)12 .
) A 95% 11(N) 12 .
) A 95% 11(N) 13
.
) A 95% 11(N) 14 .
() 111112 )1( α α −+=
12 = 0.1 (235) + 0.9(202.28)
= 205.55
() 95% I 11 12
)48.320,62.90(
93.11455.205
332.343896.155.205
025.012
=
±=±=
±
332.3438
10
32.34383
1
==
−=
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 9
() 95% I 11 13, 13 = 12
( )
)05.321,05.90(50.11555.205
)1.01(38332.3496.155.205
1
2
2
025.012
=
±=
+±=
+± α
() 95% I 11 14, 14 = 12
( )
)62.321,48.89(
07.11655.205
))1.0(21(38332.3496.155.205
21
2
2
025.012
=
±=
+±=
+± α
C:#
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 10
N:
1. I ,
)( −+=+ α 1 )( α α −+=+ 11 .
1− , , 2 1 ,
1112211 1111 )()()()( α α α α α α α α −+−++−+−+= −−−+ L , 1− , , 1 α , )( α α −1 ,
, 11 −− )( α α , , . F
.
2. O . F ,
1 112 1 )( α α −+= .
1 , ( 1 )
(11 = ) .
E 5.1.
A .
3. α ( )
.
4. N 11
)( α − . 1
. B 1
)( α −1 .
α , α .
A, 1 .
5. I α ,
. H, α ,
.
5.4
H
. H
, α β ( 0 1), :
))(( 111 −− +−+= α α
11 1 −− −+−= )()( β β +=+
...,,, 21=
/ () .
( ) , 1− 1− 1−
, .
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 11
N:
1. , 1− ,
, 1− .
.
2. , . ,
.
, β )( 1−−
)( β −1 .
A
+=+ , ( ...,,, 321= )
I 1= , %)( 1001 α − 1+
2/)( α ±+
I 2= , %)( 1001 α − 2+
))(()( /22
2 112 β α α ++±+
I 2> , %)( 1001 α − +
))(()( / ∑ ++±+ −
=
1
1
222 11
β α α
1−
=
=
( )
1
)(1
2
−
+−∑=
.
1. H
1 1 . O
11 = 121 −= 3141 /)( −= .
2. A 1 1 .
3. α β .
4. H
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 12
5.3
H
1 2 3 4 5
1000 1200 1500 1600 2000
90.=α 40.= β . F 6
, 11 = 121 −= .
( )
( )
( )
64.190
)236(6.0)6.122(4.0
)1(
6.1612
)1726(1.0)1600(9.0
))(1(
236
)200(6.0)290(4.0
)1(
1490
)1400(1.0)1500(9.0
))(1(
200
)200(6.0)10001200(4.0
)1(
1200
)1200(1.0)1200(9.0
))(1(
200
10001200
1000
3344
3344
2233
2233
1122
1122
121
11
=
+=
−+−=
=
+=
+−+=
=
+=
−+−=
=
+=
+−+=
=
+−=
−+−=
=
+=
+−+=
=
−=
−=
==
β β
α α
β β
α α
β β
α α
( )
8536.2241
)1(4736.261324.1980
4736.261
)64.190(6.0)724.367(4.0
)1(
324.1980
)24.1803(1.0)2000(9.0
))(1(
556
4455
4455
=
+=
+=
=
+=
−+−=
=
+=
+−+=
β β
α α
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 13
5.4
1 2 3 4 5 6 7 8 9
143 152 161 139 137 174 142 141 162
10 11 12 13 14 15 16 17 18
180 164 171 206 193 207 218 229 225
19 20 21 22 23 24
204 227 223 242 239 266
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 14
() 95% I 24 F26:
( ) ( )
( )
)397.306,9578.230(7196.376774.268
)07230122.01(5010719.013911.28796.1))2076.6(22622.256(
)1(12
22
22
025.02424
=
±=
++±+=
++±+ β α
5.5
...,,, 21=
() , ,
. , ,
L: ))(()( 111 −−− +−+−= α α
: 11 1 −− −+−= )()( β β
: −−+−= )()( γ γ 1
+−+ ++=
α , β γ 0 1, 1− 1−
1− , , − −
. H ( = 12 , =4
).
N:
(
), , , .
A
+−+ ++= ( ...,,, 321= )
A %)( 1001 α − + 2/α ±+
I 1= , 11 = .
I ≤≤2 , ( )
++= ∑
−
=
1
1
22 11
β α
I > , ( )( )∑−
=
−+++=1
1
2
, )1(11
γ α β α
1= , 0
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 15
= =
( )
∑=
+−−− ++−1
2
111 )(
(.. ). .
1. ( )
+++= L211
.
2.
−++
−+
−= +++
L
22111
3. −= 11 , −= 22 , , −=
.
5.5 K50
:
K50 M B
1 2 3 4
1 10 11 13 15
2 31 33 34 37
3 43 45 48 51
4 16 17 19 21
() . H
.
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 16
α = 0.2, γ = 0.1, β = 0.1
,
L, G
,
F,
F
++=
1 1 10 1= 15
2 31 2= 6
3 43 3= 18
4 16 4 = 25 4 = 0.375 4 = 9
2 5 11 5 = 25.5 5= 0.3875 5= 14.95 5= 10.9375
6 33 6= 26.11 6= 0.4098 6= 6.089 6= 32.6088
7 45 26.6158 0.4194 18.0384 7 = 45.0736
8 17 26.8281 0.3987 9.0828 18.1440
3 9 13 27.3714 0.4131 14.8921 12.8924
10 34 27.8098 0.4156 6.0991 34.3246
11 48 28.5727 0.4504 18.1773 47.2004
12 19 28.8350 0.4316 9.1580 20.1085
4 13 15 29.3917 0.4441 14.8421 14.9937
14 37 30.0488 0.4654 6.1843 36.6985
15 51 30.9759 0.5115 18.3620 49.8494
16 21 31.2215 0.4850 9.2644 22.4421
92516
18254362531
152510
375.0
4
1617
4
4345
4
3133
4
1011
4
1
44444
1
25
4
16433110
444
433
422
411
483726154
4
−=−=−=
=−=−==−=−=
−=−=−=
=
−+
−+
−+
−=
−+
−+
−+
−=
=
+++
=
L: ))(()( 111 −−− +−+−= α α
: 11 1 −− −+−= )()( β β
: −−+−= )()( γ γ 1
+−+ ++= ++=
F
.
F + ,
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 17
3875.0
)375.0)(9.0()255.25(1.0
)1()(
5.25
)375.025)(2.01())15(11(2.0
))(1()(
4455
444555
=
+−=
−+−=
=
+−+−−=
+−+−= −
β β
α α
089.6
)6(9.0)11.2633(1.0
)1()(
4098.0
)3875.0(9.0)5.2511.26(1.0
)1()(11.26
)3875.05.25(8.0)633(2.0
))(1()(
95.14
)15(9.0)5.2511(1.0
)1()(
2666
5566
55266
1555
=
+−=
−+−=
=
+−=
−+−==
−+−=
−−+−=
−=
−+−=
−+−=
γ γ
β β
α α
γ γ
0736.45
0384.184194.06158.26
6088.32
089.64098.011.26
9375.10
95.143875.05.25
7777
6666
5555
=
++=
++=
=
++=
++=
=
−+=
++=
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
ECM2263 A M C 5 18
C:B
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CHAPTER 5: EXPONENTIAL SMOOTHING METHODS
() 95% I 17 16:
)3631.18,3657.15(
4987.18644.16
5847.0)(1(96.18644.16
025.0116
=
±=
±=
±+
() 18= 16+16+ 164+2 = 16+16+ 14
= 31.2215 + 0.485(2) + 6.1843
= 38.3758
( ) 0484.1))1.0)12(1(2.01()1(1 22222 =−++=++= β α 95% I F18 16:
)9104.39,8412.36(
5346.13758.38
)5847.0(0484.196.13758.38
2025.018
=
±=
±=
±
= 1= 21