Exponential Smoothing Methods

Single Exponential Smoothing (SES)
Adaptive-Response-Rate Single Exponential Smoothing (ARRSES)
Double Exponential Smoothing: Brown's One-Parameter Linear Method
Double Exponential Smoothing: Holt's Two-Parameter Linear Method

A simplest case of the Single Exponential Smoothing (SES) can be written as:			(1)
In equation (1), forecast Ft +1 is based on weighting the most recent observation Xt with a weight value of (1-/n) and weighting the most recent previous forecast Ft with a weight value of (1-(1/n)). Substituting for 1/n becomes equation (2), where ei  is the  forecast error for period t. So really forecast provided by SES is simply the old forecast plus an adjustment for the error occurred in the last forecast.			(2)
As you see, it is no longer necessary to store all the historical data as in the case of MA methods. Rather we only need the most recent observation, the most recent forecast, and a value for . By substituting F with its components in equation (2), we expand it to equation (3).			(3)
Repeats this substitution by replacing Ft -1 by its components, we gets equation (4)			(4)
Equations (4) illustrates the exponential behavior. The weights, (1-) t decrease geometrically, and their sum is unity as shown in equation (5).			(5)

Suppose = 0.2, 0.3 and 0.8, we observe that the weights, (1-) t decrease exponentially (geometrically) with time. This is illustrated below and shown in the graph on the right.				Figure 1.1  For example, for =0.5, the weights tend to follow the pattern of exponential decay. Hence the name exponential smoothing, a non-linear estimation.	The speed at which the older responses are dampened (smoothed) is a function of the value of . When is close to 1, dampening is quick and when alpha is close to 0, dampening is slow. This is illustrated below.
Weight assigned to X:	= 0.2	= 0.5	= 0.8			(1-)	(1-)2	(1-)3	(1-)4
X t X t -1 X t -2 X t -3 X t -4 X t -5 X t -6 X t -7 X t -8 X t -9	.2 .16 .128 .1024 .08192 .065536 .052429 .041943 .033554 (.2)(.8)4	.5 .25 .125 .0625 .03125 .015625 .007813 .003906 .001953 (.5)(.5)4	.8 .16 .032 .0064 .00128 .000256 .0000512 .00001024 .000002048 (.8)(.2)4		0.9 0.7 0.5 0.3 0.1	.1 .3 .5 .7 .9	.01 .09 .25 .49 .81	.001 .027 .125 .343 .729	.001 .0081 .0625 .2401 .6561

From above illustration, we therefore know that when has a value close to 1, the new forecast will include a substantial adjustment for the error in the previous forecast. When is close to 0, the new forecast will include very little adjustment. SES will always trail the trend in the actual data, for the most it can do is to adjust the next forecast for some percentage of the most recent error. Table 1.1 below shows the forecasting for the sales using the Single Exponential Smoothing.

 

Table 1.1  Forecasting sales using Single Exponential Smoothing.

			Exponentially Smoothed Values, Fi				Forecast Error			Absolute Percentage Error (APE)			Squared Error ( Xi -Fi )2
Month	Period  t	Historical Sales Xt	α =0.1	α =0.5	α =0.9		α =0.1	α =0.5	α =0.9	α =0.1	α =0.5	α =0.9	α =0.1	α =0.5	α =0.9
Jan-06	1	224
Feb-06	2	188	200	200	200		-12	-12	-12	6.38	6.38	6.38	144.00	144.00	144.00
Mar-06	3	198	199	194	189		-1	4	9	0.40	2.02	4.44	0.64	16.00	77.44
Apr-06	4	206	199	196	197		7	10	9	3.53	4.85	4.31	53.00	100.00	78.85
May-06	5	203	199	201	205		4	2	-2	1.75	0.99	1.04	12.62	4.00	4.46
Jun-06	6	238	200	202	203		38	36	35	16.05	15.13	14.62	1459.00	1296.00	1210.26
Jul-06	7	228	204	220	235		24	8	-7	10.69	3.51	2.86	594.24	64.00	42.53
Aug-06	8	231	206	224	229		25	7	2	10.80	3.03	1.02	621.97	49.00	5.51
Sep-06	9	221	209	228	231		12	-7	-10	5.63	2.94	4.42	154.89	42.25	95.36
Oct-06	10	259	210	224	222		49	35	37	19.00	13.42	14.29	2420.73	1207.56	1370.74
Nov-06	11	273	215	242	255		59	32	18	21.46	11.62	6.62	3443.44	1009.65	327.69
Dec-06	12		221	258	272
										95.70	63.89	60.01	8904.53	3932.46	3356.85
Download SES worksheet							Figure 1.2  Actual Sales and Forecasts  Using Exponential Smoothing
			Test Periods
			2-11	2-11	2-11
Statistics Summay:			α =0.1	α =0.5	α =0.9
Mean Error			20.59	11.50	7.95
Mean Absolute Error			23.15	15.20	14.03
Mean Abs. Percentage Error			9.57	6.39	6.00
Standard Deviation of Error			31.45	20.90	19.31
Mean Squared Error			890.45	393.25	335.68

One point of concern for SES is the initial estimate values. Since F1 is not known, I use a conservative estimate of 200 (or you can assume F1 = X1 = 224, or use the average the last 4 or 5 values of  X t -n). You can see the smoothing effect of in Figure 1.2 - a large (.9) gives very little smoothing in the forecast, whereas a small (.1) gives con considerable smoothing. To choose the best value for , MSE is computed over a test set, and then another value is tried. The MSEs are then compared to find the value that gives the smallest MSE. From the summary statistics, you see that MSE and MAPE decreases as approaches 1. This is because the data exhibit a trend. If only the data are at random, the smaller the value of , the smaller would be MSE.

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Adaptive-Response-Rate Single Exponential Smoothing (ARRSES)
ARRSES has an advantage over SES in that it allows the value of to change, in a controlled manner or programmed automatically, as changes in the pattern of data occur. This is especially useful when there are tens of thousands of items require forecasting.

The equation for forecasting with ARRSES method is the same as equation (4) except that is replaced by t -1			(6)
Instead of using t + 1 we could have used t in equation (7). Reason is ARRSES is often too responsive to change in the data pattern, thus using  t + 1 allows a lag of one period, and forecast in a more conservative manner.		where     and are parameters between 0 and 1.  Et is a smoother error term, Mt is a smoothed absolute error term	(7)    (8)    (9)    (10)

Table 1.2  Forecasting sales using Adaptive-Response-Rate Single Exponential Smoothing.

						=0.8						=0.55						=0.15
Period  t	Historical Sales	Forecast Ft	Error ( ei )	Smoothed Error ( Et )	Absolute Smoothed Error ( Mt )	Value		Forecast Ft	Error ( ei )	Smoothed Error ( Et )	Absolute Smoothed Error ( Mt )	Value		Forecast Ft	Error ( ei )	Smoothed Error ( Et )	Absolute Smoothed Error ( Mt )	Value
1	253															-5.0	20.0
2	171	253.0	-82.0	-65.6	65.6	0.200		253.0	-82.0	-45.1	45.1	0.400		230.0	-59.0	-13.1	25.9	0.750
3	327	236.6	90.4	59.2	85.4	1.000		220.2	106.8	38.4	79.0	1.000		185.8	141.3	10.1	43.2	0.507
4	249	327.0	-78.0	-50.6	79.5	0.693		327.0	-78.0	-25.6	78.5	0.486		257.3	-8.3	7.3	37.9	0.233
5	392	273.0	119.0	85.1	111.1	0.636		289.1	102.9	45.1	91.9	0.326		255.4	136.6	26.7	52.7	0.192
6	221	348.7	-127.7	-85.1	124.4	0.766		322.6	-101.6	-35.6	97.3	0.491		281.7	-60.7	13.6	53.9	0.506
7	196	250.9	-54.9	-60.9	68.8	0.684		272.8	-76.8	-58.3	86.0	0.366		251.0	-55.0	3.3	54.1	0.252
8	165	213.3	-48.3	-50.8	52.4	0.886		244.7	-79.7	-70.0	82.5	0.677		237.1	-72.1	-8.0	56.8	0.061
9	278	170.5	107.5	75.8	96.5	0.970		190.7	87.3	16.5	85.1	0.849		232.7	45.3	0.0	55.1	0.141
10	351	274.8	76.2	76.1	80.3	0.786		264.8	86.2	54.8	85.7	0.194		239.1	111.9	16.8	63.6	0.000
11	212	334.7	-122.7	-82.9	114.2	0.949		281.5	-69.5	-13.5	76.8	0.640		239.1	-27.1	10.2	58.1	0.264
			Test Periods					With =0.8, α =0.2, F1=253, and E1=M1=0, below are the example workouts:
			2-11	2-11	2-11
			β =0.8	β =0.55	β =0.15			F3  =(0.2*171)+(1-0.2)*253   =236
			F2 =X1	F2 =X1	F2 =230			e3  =(327-236)   =90.4
			α2 =0.2	α2 =0.4	α2 =0.75			E3  =(0.8*90.4)+((1-0.8)*(-65.6))   =59.2
			E1= 0	E1= 0	E1= - 5			M3  =(0.8*ABS(90.4)+((1-0.8)*65.6))   =85.4
			M1= 0	M1= 0	M1= 20			M2  =(0.8*ABS(-82)+((1-0.8)*0))  =65.6
Mean Absolute Error:			90.67	87.08	44.54			11 =(-76.1 / 80.3)  =0.786
Mean Squared Error:			9912.63	8585.21	7708.39			F11 =((0.786*351)+((1-0.786)*274.8))  =334.7

Download ARRSES woeksheet.
 

The three graphs plotted in Figure 1.2 are using altogether the different values of and . Note that the values of fluctuate very differently for all the three scenarios. As you can see it is no longer true that a smaller value of would give larger effect of smoothing. If you were to use different initializing values, including the estimate values of F2, E1 and M1, entirely different sets of values would have been generated. One way to control the changes in the values of is to change the value of . Note that when I used a smaller value of (0.15) combined with a higher (0.75), I was able to obtain relatively smaller series of values; and simultaneously, a greater smoothing effect had also been achieved for the forecasts. This is in fact true when looking at the MAE (44.54) and MSE (7708.4) which have the smallest values among the three.

Summing up, ARRSES is an SES method with values systematically changed from period to period to allow for the changes in the data pattern. Extra care need to be taken when evaluating the fluctuation in . An alternative is to put an upper bound value on how much is allowed to change (using Excel Conditional Formatting to flag you an alert - with red fonts, blink, etc) from one period to the next.

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Linear Exponential Smoothing method has great advantages over Linear Moving Averages, mainly because of the limitations of single moving averages that need to save the last n values - more data storage. With Linear Exponential Smoothing, you need to compute only 3 data values and a single value for . Also, this method gives decreasing weights to the more distant past observations.

The computational rationale of this linear exponential smoothing is quite similar to linear moving averages - both the single and double-smoothed values lag the actual data when a trend exists. The single and double-smoothed values are to be added to the single smoothed values and then adjust for trend. See the equations below for the One-Parameter Linear Exponential Smoothing.
 

Single exponential smoothed value is denoted as St , and Double exponential smoothed value is denoted by Dt which will only be computed after values for St are determined. at refers to the adjustment of the St by the difference of St - Dt (meaning the adjusted smoothed estimation value for period t). The trend component bt is the estimate of the trend from one time period to another. Ft + m refers to the derivation of forecast for m periods ahead of t . The forecast for m periods ahead of t is equal to at + bt (m). Notice that at has a factor of / (1-n), whereas at in Linear Moving Average has a factor of 2 / (n-1). The reason for the difference in the two factors is because the average age of data in an n period moving average is (n-1) / 2, and the average age of data in a simple exponential smoothing is (1-) / .

Table 1.3 Application of Brown's One-Parameter Linear Exponential Smoothing to forecast Sales
	Period  t	Historical Sales	Single Exponential Smoothing St	Double Exponential Smoothing Dt	Value of a	Value of b	Value of a+b(m)		Error Difference ei	| ei |	ei 2	APE
	1	125	125.00	125.00
	2	149	129.80	125.96	133.64	0.960
	3	136	131.04	126.98	135.10	1.016	134.6
	4	157	136.23	128.83	143.64	1.851	136.1
	5	173	143.59	131.78	155.39	2.952	145.5
	6	131	141.07	133.64	148.50	1.858	158.3
	7	177	148.25	136.56	159.95	2.924	150.4
	8	188	156.20	140.49	171.92	3.929	162.9
	9	154	155.76	143.54	167.98	3.055	175.8
	10	179	160.41	146.92	173.90	3.373	171.0		8	7.96	63.41	4.449
T	11	180	164.33	150.40	178.26	3.482	177.3		3	2.72	7.41	1.513
E	12	150	161.46	152.61	170.31	2.213	181.7		-32	31.74	1007.40	21.160
S	13	182	165.57	155.20	175.94	2.592	172.5		9	9.47	89.76	5.206
T	14	192	170.86	158.33	183.38	3.130	178.5		13	13.47	181.49	7.017
	15	224	181.48	162.96	200.01	4.630	186.5		37	37.49	1405.61	16.737
	16	178	180.79	166.53	195.05	3.565	204.6		-27	26.64	709.46	14.964
P	17	198	184.23	170.07	198.39	3.540	198.6		-1	0.61	0.37	0.309
E	18	206	188.58	173.77	203.40	3.703	201.9		4	4.07	16.55	1.975
R	19	156	182.07	175.43	188.70	1.659	207.1		-51	51.10	2611.13	32.756
I	20	248	195.25	179.40	211.11	3.965	190.4		58	57.64	3322.08	23.241
O	21	228	201.80	183.88	219.73	4.481	215.1		13	12.92	167.02	5.668
D	22	231	207.64	188.63	226.65	4.753	224.2		7	6.79	46.10	2.939
S	23	175	201.11	191.13	211.10	2.497	231.4		-56	56.41	3181.83	32.233
	24	224	205.69	194.04	217.34	2.913	213.6		10	10	108.21	4.644
	25						220.3	m=1	-4	329	12918	175
	26						223.2	m=2
	27						226.1	m=3
	28						229.0	m=4
	29						231.9	m=5
	30						234.8	m=6
	Descriptive Statistics Analysis:			Values:
	Mean Error			-0.24
	Mean Absolute Error			21.96
	MAPE			11.65
	MSE			861.19
	Std Deviation of Error			30.38

In Table 1.3 the initializing value for is 0.2 and to forecast one period ahead. At t =1, values of St-1 and Dt-1 are not known, but they must be initialized at the outset. In this example, I simply take value of Xt =St =Dt. Alternatively, there are other ways you can initialize the values, for example, by taking some average of the first few data points. A point to note of initialization procedure - if the smoothing parameter is close to 0, the influence of the initialization process would allow trailing and not lagging by too much smoothing effect for many time periods ahead. Below are examples of the equation workout. S2  =X2 + (1-)S1  =(0.2*149)+((1-0.2)*125)     =129.8 D2  =S2 + (1-)D1  =(0.2*129.8)+((1-0.2)*125)  =125.96 a2  =2S2 - D2  =(2*129.8)-126   =133.64 b2  = /(1-) (S2 - D2)  =(0.2/0.8)*(129.8-126)   =0.960 F25 =a24 + b24(1)    =217.3+(2.913*1)   =220.3 F30 =a24 + b24(6)    =217.3+(2.913*6)   =234.8

Figure 1.4   Applying Forecasts  Using Brown's One-Parameter Exponential Smoothing.

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Double Exponential Smoothing: Holt's Two-Parameter Linear Method
This two-parameter linear exponential smoothing method is similar in principle to the earlier one-parameter method. The difference is that it does not apply the double smoothing formula. It smooths the trend separately by using different parameters than that used on the original series. There are two smoothing constants to be used in the equations ( and with values lie between 0 and 1).

St in equation (16) smooths the data by helping to eliminate the lag and brings St to the approximate base of the current data value. Next bt updates the trend, which is expressed as the difference between the last two smoothed values. This is appropriate because if there is a trend in the data, new values should be higher or lower than the previous ones. The constant then smooth some of the randomness remaining in the trend in the last period (St - St - 1), adding that to the previous estimate of the trend, multiplied by (1 - ).
Table 1.3 shows the application of this method to a series with trend, using = 0.2 and = 0.3. To initialize the values, just remember Holt's linear exponential smoothing requires two estimates - to get the smoothed value for St and to get bt to smooth the trend. In this case, I let S1=X1=145, and the trend b1=X2 - X1 =5.
Download worksheet.

Table 1.3   Application of Holt's Two-Parameter Linear Exponential Smoothing ( =0.2, =0.3)
	Period  t	Historical Sales Xt	Smoothing Data St	Smoothing Trend bt	Forecast F t +m when m=1		ei	| ei |	ei 2	APE
	1	145	145.00	5.000
	2	150	150.00	5.000
	3	161	156.20	5.360	155.00
	4	138	156.85	3.946	161.56
	5	142	157.04	2.819	160.79
	6	192	166.28	4.747	159.85
	7	142	165.22	3.006	171.03
	8	141	162.78	1.372	168.23
	9	162	163.72	1.242	164.16
	10	180	167.97	2.144	164.97		15	15.03	225.98	8.351
T	11	152	166.49	1.057	170.12		-18	18.12	328.27	11.920
E	12	158	165.64	0.484	167.55		-10	9.55	91.24	6.046
S	13	191	171.10	1.977	166.13		25	24.87	618.73	13.023
T	14	178	174.06	2.272	173.08		5	4.92	24.23	2.766
	15	156	172.27	1.052	176.33		-20	20.33	413.46	13.034
	16	203	179.26	2.833	173.32		30	29.68	880.96	14.621
	17	224	190.47	5.348	182.09		42	41.91	1756.61	18.711
P	18	210	198.65	6.198	195.82		14	14.18	201.13	6.753
E	19	189	201.68	5.247	204.85		-16	15.85	251.31	8.388
R	20	212	207.94	5.552	206.93		5	5.07	25.71	2.392
I	21	190	208.80	4.142	213.50		-23	23.50	552.03	12.366
O	22	198	209.95	3.246	212.94		-15	14.94	223.14	7.544
D	23	228	216.16	4.134	213.20		15	14.80	219.16	6.493
	24	239	224.03	5.256	220.29		19	19	350.04	7.828
	25				229.29	m=1	67	271	6162	140
	26				234.55	m=2
	27				239.80	m=3
	28				245.06	m=4
	29				250.31	m=5
	30				255.57	m=6
	S23 =0.2(X23)+((1 - 0.2)(S22+b22)						Descriptive Statistics Analysis:			Values:
	S23 =0.2*(228)+(0.8*(209.95+3.25))  =216.16						Mean Error			4.46
	b23 =0.3(S23 - S22)+((1 - 0.3)b22						Mean Absolute Error			18.10
	b23 =0.3(216.16 - 209.95)+(0.7*(3.25)) =4.138						Mean Abs. Percentage Error (MAPE)			9.35
	F24 =S23+b23(1) =216.16+4.138  =220.29						Mean Squared Error (MSE)			410.80
	F27 =S24+b24(3) =224.03+(5.256*3)  =239.80						Std Deviation of Error			20.98
Figure 1.5   Applying Forecasts to Sales Using Holt's Two-Parameter Linear Exponential Smoothing.

 

 

 

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This site was created in February 2007.
contact Tan, William     email: vbautomation@yahoo.com