Ch 4. – Exponential Smoothing – Computing in JMP
Example 1: Dow Jones Index (pgs. 176 – 187)
To begin we select Modeling > Time Series which will produce a plot of the time series
and compute ACF and PACF (discussed later) for the time series. Notice the note that
appears to the right of the Modeling drop-down menu. We can see that in addition to
the ACF plots, there are options to fit models to the time series and make forecasts from
them. Smoothing models are the exponential smoothing models we are examining in
Chapter 4.
The dialog box to begin modeling the Dow Jones Index is shown below.
Notice the time series data must be evenly spaced, i.e. daily, weekly, quarterly, or
annually in general.
1
Below is the default output from the Time Series modeling option.
To fit exponential smoother models we select the model we wish to fit using the
Smoothing Model pull-out menu as shown below.
2
Simple exponential smoothing – is appropriate when there is no trend or seasonal
pattern in the data, but the mean (or level) of the time series 𝑦𝑡 is changing slowly over
time.
Double exponential smoothing – is appropriate where there is evidence of some linear
trends in the data. Simple exponential smoothing will tend to over or under estimate
the time series when there are some linear trends in the time series. Simple exponential
smoothing with a large 𝜆 will decrease the over or under estimate problem, but a
double exponential smooth will generally handle it better.
Linear exponential smoothing – is appropriate when both the mean (or level) and the
growth rate (slope of linear trends) are changing over time. This is also referred to as
Holt’s Trend Corrected exponential smoothing. In situations where a double
exponential smooth is appropriate, this method would be appropriate as well.
Seasonal exponential smoothing – is appropriate when there is a seasonal pattern in
the time series, but not much in terms of long-term linear trend. In other words, the
level is not changing a great deal but there strong seasonal patterns on top of a fairly
constant mean (or level).
Winter’s Method – is appropriate when there is a seasonal pattern on top of a long-term
trend.
3
For the Dow Jones Index time series we see there is no evidence of a seasonal effect, but
the trend does have some linear trends in, e.g. between February 2003 and February
2004. We first fit a simple exponential smooth to this time series.
Above and to the left are the results of a simple
exponential smooth fit to the DJI time series. We
can see the optimal 𝜆 chosen automatically by
JMP is 𝜆 = .842, which is quite large. The 𝑅 2 =
78.63% which is quite good. The mean
absolute percent error (MAPE) = 3.24% and the
mean absolute error/deviation (MAD) = 319.94.
The forecasts for the next 25 time periods are
shown graphically and as you can see are
constant. This is because for a simple
exponential smooth the forecasts for the next
time periods, regardless of how far into the
future we look, is 𝑦
̃𝑇 the last smoothed value
from the exponential smooth of the data. The
prediction interval gets wider the further out
we look as shown by the formula in your notes.
4
The residuals plotted versus time are shown above and the ACF plot of the residuals is
shown below. The residuals are consistent with white noise as there are no significant
autocorrelations at lags > 1.
5
You can save the actual forecasts as the prediction intervals to the spreadsheet by
selecting the Save Prediction Formula from the Model: Simple Exponential
Smoothing (Zero to One) drop-down menu.
The columns above are:
Actual DJI – original time series (𝑦𝑡 )
DJI Prediction Formula – the smoothed or fitted values (𝑦̃𝑡 )
Time – duh?
6
Predicted DJI – the same as DJI Prediction Formula
Std Error Pred DJI - s in the formulae in class unless it is a future forecast in which case
is equal to 𝑠√1 + (𝜏 − 1)𝜆2 for 𝜏 ≥ 1.
Residual DJI – are the one-step ahead forecast errors for the original time series. These
are used to compute SSE etc. which are minimized to find an optimal 𝜆.
Upper (.95) DJI – upper limit of the prediction interval.
Lower (.95) DJI – lower limit of the prediction interval.
These columns will be the same for all forecast smoothers, how the predicted values
and the standard errors for prediction are computed of course will change with the
smoothing model used.
We now consider a double exponential smooth model fit to this times series. You DO
NOT need to start from scratch, you can actually just select that option from the same
menu we used to fit the simple exponential smooth.
This will fit the double exponential smooth and show the summary statistics for the fit
right next to those from the simple exponential smooth as shown below.
7
We can see the double exponential smooth has a smaller 𝑅 2 , thus there appears to be no
advantage to this approach over the simple exponential smooth.
We see the predictions or forecasts
made from a double exponential
smooth have a linear trend. The
prediction intervals get large fast.
8
Linear Exponential Smoothing (a.k.a. Holt’s Trend Corrected)
Example: Weekly Thermostat Sales (Thermostat.JMP)
As we can see this time series as a long term increasing trend, and the growth rate
appears to fluctuate with time, thus a linear exponential smooth or possibly a double
exponential seems appropriate. Fitting both a double exponential and linear trend
exponential smooth to this time series produces very similar results. (𝑅 2 = .557)
9
The estimated smoothing parameters for double exponential and linear smooths are
shown below. There is one optimal parameter for the double exponential smooth
(simple exponential smoothing with  = .161 twice) and two optimal parameters for the
linear trend smooth (and 


The prediction intervals for future forecasts for the next 25 weeks of thermostat sales are
very similar, though it appears the linear trend exponential is a bit narrower.
The residuals from the linear trend
exponential smoother are shown to the
right. The residual series certainly
appears to be stationary.
10
The predictions for the linear exponential smooth are shown below.
…
11
Holt-Winters Method
Holt-Winter’s Method of exponential smoothing is for time series with both a long-term
trend and a seasonal trend on top of it.
Example: Monthly Liquor Sales (1980-2000) – (U.S. Monthly Liquor Sales.JMP)
The time series and ACF & PACF (soon to be discussed) are shown above. The time
series clearly exhibits a long-term trend with a strong seasonal trend on top. The
seasonal variation however seems get larger over time! This may present a problem!
12
Fitting a Holt-Winter’s exponential smooth this time series yields the results shown
below.
The 𝑅 2 = 98.8% indicating a very good fit to the time series. The trend weight is
essentially 0, suggesting the linear trend part of the smooth is not strong and level
smooth essentially takes care of the long-term trend.
13
The residuals exhibit a non-constant variation.
There is a multiplicative form of the Holt-Winter’s Method which will model the nonconstant nature of the seasonal variation, however it is not implemented in JMP. One
way to handle the non-constant seasonal variation is to take the natural log (or any
other base) of the liquor sales before smoothing. The log base 10 of the liquor sales (i.e.
log10 (𝐿𝑖𝑞𝑢𝑜𝑟 𝑆𝑎𝑙𝑒𝑠)) is shown below. Note: I just called the logged time series ln(LS).
14
The residuals now appear to have constant variation.
The forecasts of liquor sales in the log scale for next 25-months are shown below.
To back-transform the predictions to
the original scale we need
exponentiate the results, i.e.
…
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑠𝑐𝑎𝑙𝑒 = 10 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑜𝑛
15
The prediction intervals for liquor sales in the next 25 months are shown above. The
highlighted columns were obtained by using the JMP calculator to back-transform the
Upper CL and Lower CL in the two preceding columns.
The Lower PI and Upper PI formulae in the JMP calculator are shown below.
After back-transforming we have what should prove to be very good forecasts for the
liquor sales for the next 25 months; assuming the current trends continue.
16
In the forecast library from CRAN there are a number of exponential smoothers that
can be used to fit an exponential smooth model and make forecasts.
> Sales.ts = ts(Sales,start=1980,frequency=12)
> tsdisplay(Sales.ts)
Additive Holt-Winter’s Fit
> liquor.hw = hw(Sales.ts,seasonal="additive")
> summary(liquor.hw)
Forecast method: Holt-Winters' additive method
Model Information:
ETS(A,A,A)
Call:
hw(x = Sales.ts, seasonal = "additive")
Smoothing
alpha =
beta =
gamma =
parameters:
0.3335
1e-04
0.4983
Initial states:
l = 636.7332
b = 3.4108
s=513.0279 12.4489 -29.1804 -53.4223 24.0255 61.7841
1.913 -6.7716 -88.659 -96.9889 -198.7712 -139.406
sigma:
54.367
17
AIC
AICc
BIC
4671.698 4673.403 4732.772
Training set error measures:
ME
RMSE
MAE
MPE
-0.02626900 54.36696148 38.55693750 -0.05075231
Forecasts:
Point
Jan 2008
Feb 2008
Mar 2008
Apr 2008
May 2008
Jun 2008
Jul 2008
Aug 2008
Sep 2008
Oct 2008
Nov 2008
Dec 2008
Jan 2009
Feb 2009
Mar 2009
Apr 2009
May 2009
Jun 2009
Jul 2009
Aug 2009
Sep 2009
Oct 2009
Nov 2009
Dec 2009
Forecast
1551.706
1513.810
1689.579
1721.138
1805.449
1817.071
1943.510
1832.384
1770.948
1782.941
1817.635
2467.998
1592.625
1554.729
1730.498
1762.057
1846.368
1857.990
1984.429
1873.303
1811.867
1823.859
1858.554
2508.917
Lo 80
1482.032
1440.358
1612.532
1640.655
1721.669
1730.117
1853.491
1739.400
1675.089
1684.288
1716.265
2351.188
1473.509
1433.348
1606.893
1636.266
1718.427
1727.933
1852.289
1739.111
1675.652
1685.651
1718.378
2357.169
Hi 80
1621.380
1587.262
1766.625
1801.621
1889.229
1904.026
2033.530
1925.369
1866.807
1881.593
1919.005
2584.807
1711.742
1676.110
1854.103
1887.848
1974.309
1988.047
2116.570
2007.496
1948.082
1962.068
1998.729
2660.664
Lo 95
1445.149
1401.475
1571.746
1598.050
1677.318
1684.085
1805.838
1690.177
1624.344
1632.065
1662.603
2289.353
1410.452
1369.093
1541.460
1569.676
1650.699
1659.085
1782.338
1668.074
1603.544
1612.487
1644.174
2276.838
MAPE
3.30142685
MASE
0.26527642
Hi 95
1658.264
1626.145
1807.411
1844.226
1933.580
1950.057
2081.183
1974.592
1917.552
1933.816
1972.667
2646.642
1774.798
1740.365
1919.535
1954.438
2042.037
2056.895
2186.520
2078.533
2020.189
2035.231
2072.934
2740.995
18
Multiplicative Holt-Winter’s Fit
> liquor.mhw = hw(Sales.ts,seasonal=”multiplicative”)
> summary(liquor.mhw)
Forecast method: Holt-Winters' multiplicative method
Model Information:
ETS(M,A,M)
Call:
hw(x = Sales.ts, seasonal = "multiplicative")
Smoothing
alpha =
beta =
gamma =
parameters:
0.6154
0.0104
1e-04
Initial states:
l = 539.7408
b = 4.4147
s=1.389 1.0075 0.9774 0.9615 1.0217 1.0484
0.9986 0.9956 0.9295 0.9261 0.8494 0.8953
sigma:
0.0299
AIC
AICc
BIC
4396.707 4398.412 4457.781
Training set error measures:
ME
RMSE
MAE
MPE
-0.9843634 42.7434366 30.3659230 -0.1179324
Forecasts:
Point
Jan 2008
Feb 2008
Mar 2008
Apr 2008
May 2008
Jun 2008
Jul 2008
Aug 2008
Sep 2008
Oct 2008
Nov 2008
Dec 2008
Jan 2009
Feb 2009
Mar 2009
Apr 2009
May 2009
Jun 2009
Jul 2009
Aug 2009
Sep 2009
Oct 2009
Nov 2009
Dec 2009
Forecast
1572.505
1492.283
1627.464
1633.908
1750.419
1756.259
1844.296
1797.695
1692.285
1720.667
1774.231
2446.580
1577.369
1496.898
1632.496
1638.958
1755.828
1761.685
1849.993
1803.246
1697.510
1725.978
1779.706
2454.127
Lo 80
1512.278
1424.853
1543.911
1540.770
1641.347
1637.971
1711.170
1659.551
1554.585
1573.067
1614.380
2215.791
1422.001
1343.307
1458.369
1457.564
1554.519
1552.757
1623.349
1575.312
1476.372
1494.482
1534.175
2106.167
Hi 80
1632.732
1559.712
1711.018
1727.045
1859.490
1874.547
1977.423
1935.838
1829.986
1868.267
1934.082
2677.368
1732.738
1650.489
1806.623
1820.352
1957.138
1970.613
2076.637
2031.180
1918.648
1957.473
2025.236
2802.086
MAPE
2.3613634
Lo 95
1480.395
1389.158
1499.680
1491.466
1583.608
1575.354
1640.698
1586.423
1481.690
1494.933
1529.759
2093.619
1339.754
1262.001
1366.192
1361.540
1447.952
1442.157
1503.371
1454.652
1359.308
1371.936
1404.199
1921.969
MASE
0.2089212
Hi 95
1664.614
1595.407
1755.249
1776.349
1917.229
1937.165
2047.895
2008.967
1902.880
1946.402
2018.703
2799.540
1814.985
1731.795
1898.801
1916.377
2063.705
2081.213
2196.616
2151.841
2035.711
2080.020
2155.212
2986.285
19
Other exponential smoothers are also available in the forecast package.



ses – this function performs simple exponential smoothing.
holt – this function performs linear exponential smooth (Holt Trend Corrected),
which is very comparable to double exponential smoothing.
hw – does Holt-Winter’s seasonal smoothing as shown above. Both additive and
multiplicative options are available.
Even though it is in appropriate to apply simple exponential and linear exponential
smoothing to these data, I will demonstrate the use of these smoothers for the liquor
sales data.
> Sales.ses = ses(Sales.ts,h=24)
> Sales.ses
Point Forecast
Lo 80
Jan 2008
1841.114 1586.415
Feb 2008
1841.114 1584.635
Mar 2008
1841.114 1582.867
Apr 2008
1841.114 1581.111
May 2008
1841.114 1579.367
Jun 2008
1841.114 1577.634
Jul 2008
1841.114 1575.913
Aug 2008
1841.114 1574.203
Sep 2008
1841.114 1572.504
Oct 2008
1841.114 1570.815
Nov 2008
1841.114 1569.137
Dec 2008
1841.114 1567.469
Jan 2009
1841.114 1565.812
Feb 2009
1841.114 1564.164
Mar 2009
1841.114 1562.526
Apr 2009
1841.114 1560.897
Hi 80
2095.814
2097.594
2099.362
2101.118
2102.862
2104.595
2106.316
2108.026
2109.725
2111.414
2113.092
2114.760
2116.417
2118.065
2119.703
2121.332
Lo 95
1451.585
1448.863
1446.159
1443.473
1440.806
1438.156
1435.524
1432.909
1430.310
1427.727
1425.161
1422.610
1420.075
1417.555
1415.050
1412.559
Hi 95
2230.644
2233.366
2236.070
2238.755
2241.423
2244.073
2246.705
2249.320
2251.919
2254.502
2257.068
2259.619
2262.154
2264.674
2267.179
2269.670
20
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
2009
2009
2009
2009
2009
2009
2009
2009
1841.114
1841.114
1841.114
1841.114
1841.114
1841.114
1841.114
1841.114
1559.278
1557.669
1556.068
1554.476
1552.893
1551.319
1549.753
1548.195
2122.951
2124.560
2126.161
2127.753
2129.336
2130.910
2132.476
2134.033
1410.083
1407.621
1405.173
1402.739
1400.318
1397.910
1395.515
1393.134
2272.146
2274.608
2277.056
2279.490
2281.911
2284.319
2286.713
2289.095
Lo 95
1502.727
1505.844
1508.960
1512.077
1515.193
1518.308
1521.424
1524.539
1527.653
1530.768
1533.882
1536.995
1540.109
1543.222
1546.334
1549.447
1552.559
Hi 95
2265.938
2270.960
2275.982
2281.005
2286.028
2291.051
2296.075
2301.099
2306.123
2311.148
2316.173
2321.198
2326.223
2331.249
2336.276
2341.302
2346.329
> plot(Sales.ses)
> Sales.holt =
> Sales.holt
Point
Jan 2008
Feb 2008
Mar 2008
Apr 2008
May 2008
Jun 2008
Jul 2008
Aug 2008
Sep 2008
Oct 2008
Nov 2008
Dec 2008
Jan 2009
Feb 2009
Mar 2009
Apr 2009
May 2009
holt(Sales.ts,h=24)
Forecast
1884.332
1888.402
1892.471
1896.541
1900.610
1904.680
1908.749
1912.819
1916.888
1920.958
1925.027
1929.097
1933.166
1937.235
1941.305
1945.374
1949.444
Lo 80
1634.814
1638.260
1641.707
1645.153
1648.599
1652.045
1655.491
1658.936
1662.381
1665.826
1669.271
1672.715
1676.160
1679.604
1683.047
1686.491
1689.935
Hi 80
2133.851
2138.543
2143.236
2147.928
2152.621
2157.314
2162.008
2166.701
2171.395
2176.089
2180.783
2185.478
2190.172
2194.867
2199.562
2204.258
2208.953
21
Jun
Jul
Aug
Sep
Oct
Nov
Dec
2009
2009
2009
2009
2009
2009
2009
1953.513
1957.583
1961.652
1965.722
1969.791
1973.861
1977.930
1693.378
1696.821
1700.263
1703.706
1707.148
1710.590
1714.032
2213.649
2218.345
2223.041
2227.738
2232.434
2237.131
2241.828
1555.670
1558.782
1561.893
1565.003
1568.113
1571.223
1574.333
2351.357
2356.384
2361.412
2366.440
2371.469
2376.498
2381.528
> plot(Sales.holt)
22