Week-4 Exponential Smoothing Methods
Some of the slides are adapted from the lecture notes provided by Prof. Antoine Saure and Prof. Rob Hyndman
Business Forecasting Analytics
ADM 4307 – Fall 2021
Exponential Smoothing Methods
Ahmet Kandakoglu, PhD
04 October, 2021
Outline
• Review of last lecture
• Exponential smoothing
• Simple exponential smoothing
• Models with trend
• Models with seasonality
• Taxonomy of exponential smoothing methods
• Innovations state space models
• Forecasting with exponential smoothing
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Review of Last Lecture
• Transformations and adjustments
• Time series components
• Seasonal adjustment
• Decomposition methods
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Trend and Seasonality Patterns
• Patterns based on Pegel’s (1969) classification
• An inappropriate forecasting model, even when optimized, will be inferior to a
more appropriate model
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The Forecasting Scenario
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The Mean as a Forecasting Tool
• The mean as a forecasting tool:
• Appropriate when the data is stationary
• Unable to capture the data pattern if the time series involves other
components
• There is a need to improve upon the mean as the forecast for the next
period(s)
• Classification of forecasting methods discussed in this lecture:
• Averaging methods (equally weighted observations)
• Exponential smoothing methods (unequal weights)
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Averaging Methods
• The mean:
• Good results if the underlying process has no noticeable trend and no
noticeable seasonality
• Cell A-1 in Pegel’s table
• The larger the number of observations, the more stable it becomes
• Only two values need to be stored as times moves on
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Averaging Methods
• Moving averages:
• MA(𝑘) denotes a moving average forecast of order 𝑘 and 𝑘 MA a moving
average smoother of order 𝑘
• It deals with the latest 𝑘 periods of known data
• It requires more storage because all of the 𝑘 latest observations must be
stored
• It cannot handle trend or seasonality very well, although it can do better
than the mean
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Averaging Methods
• Moving average:
• A forecaster must choose the number of periods in the moving average
• The more observations included in the moving average, the greater the
smoothing effect
• Extreme cases
• 𝑘 = 1 (naïve forecast)
• 𝑘 = 𝑛 (mean forecast)
• Lags behind the pattern by one or more periods
• Exponential smoothing methods are generally superior
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Exponential Smoothing
• An obvious extension to the moving average method
• Exponential smoothing methods are weighted averages of past observations, with
weights decaying exponentially as the observations get older
• The most recent observations usually provide the best guide as to the future
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Exponential Smoothing
• Combine a “level”, “trend” and “seasonal” component to describe a time series.
• The rate of change of the components are controlled by “smoothing parameters”: α, β
and γ respectively.
• Need to choose best values for the smoothing parameters (and initial states).
• α controls the flexibility of the level
• If α = 0, the level never updates (mean)
• If α = 1, the level updates completely (naive)
• β controls the flexibility of the trend
• If β = 0, the trend is linear
• If β = 1, the trend changes suddenly every observation
• γ controls the flexibility of the seasonality
• If γ = 0, the seasonality is fixed (seasonal means)
• If γ = 1, the seasonality updates completely
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ExponenTial Smoothing (ETS) Models
General notation
• Error: Additive (“A”) or multiplicative (“M”)
• Trend: None (“N”), additive (“A”), multiplicative (“M”), or damped (“Ad” or “Md”).
• Seasonality: None (“N”), additive (“A”) or multiplicative (“M”)
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E T S
Error Trend Seasonality
Outline
• Review of last lecture
• Exponential smoothing
• Simple exponential smoothing
• Models with trend
• Models with seasonality
• Taxonomy of exponential smoothing methods
• Innovations state space models
• Forecasting with exponential smoothing
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Simple Exponential Smoothing
• Takes the forecast for the previous period and adjusts it using the forecast error:
𝐹𝑡+1 = 𝐹𝑡 + α 𝑌𝑡 − 𝐹𝑡
• The effect of a large or small α is analogous to that of including a small or a large
number of observations when computing a moving average
• Trails any trend in the actual data, since the most the method can do is adjust the
next forecast for some percentage of the most recent error
• Suitable for forecasting data with no trend or seasonal pattern, although the mean
of the data may be changing slowly over time
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Single Exponential Smoothing
• Weights the most recent observation with a weight α and the most recent
forecast with a weight 1 − α
𝐹𝑡+1 = α𝑌𝑡 + 1 − α 𝐹𝑡
• Reduces any storage problem and requires few computations
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Single Exponential Smoothing
• 𝐹𝑡+1 is a weighted moving average of all the past observations
𝐹𝑡+1 =
𝑗=0
𝑡−1
α 1 − α 𝑗 𝑌𝑡−𝑗 + 1 − α
𝑡𝐹1
• The rate at which the weights decrease is controlled by the parameter α (the
sum is approximately equal to one)
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Single Exponential Smoothing
• Simple exponential smoothing has a flat forecast function:
𝐹𝑡+ℎ = 𝐹𝑡+1 = 𝑙𝑡 , ℎ = 2, 3,…
• 𝐹1 plays a role in all forecasts generated by the process. If α is small and/or
the time series is relatively short, its weight will have a noticeable effect on the
resulting forecasts
• Initialization (Since the value of 𝐹1 is not known):
• Use the first observed value (𝑌1) as the first forecast or
• Use the average the first four or five values in the data set or
• Use optimization
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Single Exponential Smoothing
• The unknown parameter α and the initial value 𝑙0 for any exponential
smoothing method can be estimated from the observed data by minimizing the
SSE
• This involves a non-linear minimization problem and the use of an optimization
tool
• An alternative approach is trial and error (and considering other measures of
forecast error)
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ETS(A,N,N): Specifying the Model
A model for SES
ETS(y ~ error(“A”) + trend(“N”) + season(“N”))
• By default, an optimal value for α and 𝑙0 is used.
• α can be chosen manually in trend()
trend(“N”, alpha = 0.5)
trend(“N”, alpha_range = c(0.2, 0.8))
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Example: Algerian Exports
algeria_economy <- global_economy %>% filter(Country == “Algeria”)
fit <- algeria_economy %>% model(ETS(Exports ~ error(“A”) + trend(“N”) + season(“N”)))
report(fit)
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Series: Exports
Model: ETS(A,N,N)
Smoothing parameters:
alpha = 0.84
Initial states:
l[0]
39.5
sigma^2: 35.6
AIC AICc BIC
447 447 453
Example: Algerian Exports
fc <- fit %>% forecast(h = 5)
fc
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# A fable: 5 x 5 [1Y]
# Key: Country, .model [1]
Country .model Year Exports .mean
1 Algeria “ETS(Exports ~ error(\”A\”) + trend(\”N\”) + ~ 2018 N(22, 36) 22.4
2 Algeria “ETS(Exports ~ error(\”A\”) + trend(\”N\”) + ~ 2019 N(22, 61) 22.4
3 Algeria “ETS(Exports ~ error(\”A\”) + trend(\”N\”) + ~ 2020 N(22, 86) 22.4
4 Algeria “ETS(Exports ~ error(\”A\”) + trend(\”N\”) + ~ 2021 N(22, 111) 22.4
5 Algeria “ETS(Exports ~ error(\”A\”) + trend(\”N\”) + ~ 2022 N(22, 136) 22.4
Example: Algerian Exports
fc %>% autoplot(algeria_economy) +
geom_line(aes(y = .fitted), col=“red”, data = augment(fit)) +
labs(y=”% of GDP”, title=”Exports: Algeria”) + guides(colour = “none”)
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Outline
• Review of last lecture
• Exponential smoothing
• Simple exponential smoothing
• Models with trend
• Models with seasonality
• Taxonomy of exponential smoothing methods
• Innovations state space models
• Forecasting with exponential smoothing
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Holt’s Linear Trend Method
• Extends single exponential smoothing to allow forecasting of data with trends
• Forecast
• Level
• Trend
• Choose values for 𝛼, 𝛽, 𝑙0 and 𝑏0 to minimize SSE.
• Non-linear optimization
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ETS(A,A,N): Specifying the Model
ETS(y ~ error(“A”) + trend(“A”) + season(“N”))
• By default, optimal values for 𝛽 and 𝑏0 are used.
• 𝛽 can be chosen manually in trend()
trend(“A”, beta = 0.004)
trend(“A”, beta_range = c(0.0, 0.1))
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Example: Australian Population
aus_economy <- global_economy %>% filter(Code == “AUS”) %>% mutate(Pop = Population/1e6)
fit <- aus_economy %>% model(ETS(Pop ~ error(“A”) + trend(“A”) + season(“N”)))
report(fit)
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Series: Pop
Model: ETS(A,A,N)
Smoothing parameters:
alpha = 1
beta = 0.327
Initial states:
l[0] b[0]
10.1 0.222
sigma^2: 0.0041
AIC AICc BIC
-77.0 -75.8 -66.7
Example: Australian Population
fc <- fit %>% forecast(h = 10)
fc
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# A fable: 10 x 5 [1Y]
# Key: Country, .model [1]
Country .model Year Pop .mean
1 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2018 N(25, 0.0041) 25.0
2 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2019 N(25, 0.011) 25.3
3 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2020 N(26, 0.023) 25.7
4 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2021 N(26, 0.039) 26.1
5 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2022 N(26, 0.061) 26.4
6 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2023 N(27, 0.09) 26.8
7 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2024 N(27, 0.13) 27.2
8 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2025 N(28, 0.17) 27.6
9 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2026 N(28, 0.22) 27.9
10 Australia “ETS(Pop ~ error(\”A\”) + trend(\”A\”) ~ 2027 N(28, 0.29) 28.3
Example: Australian Population
fc %>% autoplot(aus_economy) + labs(y = “Millions”, title= “Population: Australia”)
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Outline
• Review of last lecture
• Exponential smoothing
• Simple exponential smoothing
• Models with trend
• Models with seasonality
• Taxonomy of exponential smoothing methods
• Innovations state space models
• Forecasting with exponential smoothing
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Holt-Winters’ Method
• Extends Holt’s method to capture seasonality
• There are two variations to this method:
• Multiplicative seasonality (preferred when seasonal variations are changing
proportional to the level)
• Additive seasonality (preferred when the seasonal variations are roughly
constant)
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Holt-Winters Additive Method
• Additive seasonality:
• Forecast
• Level
• Trend
• Seasonal
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Holt-Winters Multiplicative Method
• Multiplicative seasonality:
• Forecast
• Level
• Trend
• Seasonal
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Specifying the Model
• Holt-Winters additive method with additive errors: ETS(A,A,A)
ETS(y ~ error(“A”) + trend(“A”) + season(“A”))
• Holt-Winters multiplicative method with multiplicative errors: ETS(M,A,M)
ETS(y ~ error(“M”) + trend(“A”) + season(“M”))
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Example: Domestic Overnight Trips in Australia
# Prepare the data
aus_holidays <- tourism %>% filter(Purpose == “Holiday”) %>% summarise(Trips =
sum(Trips)/1000)
# Fit the model
fit <- aus_holidays %>%
model(
additive = ETS(Trips ~ error(“A”) + trend(“A”) + season(“A”)),
multiplicative = ETS(Trips ~ error(“M”) + trend(“A”) + season(“M”))
)
# Forecast the next 3 years
fc <- fit %>% forecast(h = “3 years”)
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Example: Domestic Overnight Trips in Australia
fc %>% autoplot(aus_holidays, level = NULL) + labs(title=”Australian domestic tourism”,
y=”Overnight trips (millions)”)
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Example: Domestic Overnight Trips in Australia
components(fit)
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# A dable: 168 x 7 [1Q]
# Key: .model [2]
# : Trips = lag(level, 1) + lag(slope, 1) + lag(season, 4) + remainder
.model Quarter Trips level slope season remainder
1 additive 1997 Q1 NA NA NA 1.50 NA
2 additive 1997 Q2 NA NA NA -0.294 NA
3 additive 1997 Q3 NA NA NA -0.670 NA
4 additive 1997 Q4 NA 9.79 0.0211 -0.534 NA
5 additive 1998 Q1 11.8 9.94 0.0425 1.50 0.496
6 additive 1998 Q2 9.28 9.88 0.0245 -0.294 -0.415
7 additive 1998 Q3 8.64 9.75 -0.000841 -0.670 -0.588
8 additive 1998 Q4 9.30 9.77 0.00298 -0.534 0.0884
9 additive 1999 Q1 11.2 9.75 -0.00124 1.50 -0.0976
10 additive 1999 Q2 9.61 9.79 0.00552 -0.294 0.157
# … with 158 more rows
Example: Domestic Overnight Trips in Australia
components(fit) %>% autoplot()
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Outline
• Review of last lecture
• Exponential smoothing
• Simple exponential smoothing
• Models with trend
• Models with seasonality
• Taxonomy of exponential smoothing methods
• Innovations state space models
• Forecasting with exponential smoothing
Fall 2021 ADM 4307 Business Forecasting Analytics 39
Exponential Smoothing Methods
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Taxonomy of Exponential Smoothing
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Outline
• Review of last lecture
• Exponential smoothing
• Simple exponential smoothing
• Models with trend
• Models with seasonality
• Taxonomy of exponential smoothing methods
• Innovations state space models
• Forecasting with exponential smoothing
Fall 2021 ADM 4307 Business Forecasting Analytics 42
State Space Models
• Exponential smoothing methods presented so far generate point forecasts
• There are statistical models that generate the same point forecasts, but can
also generate forecast intervals
• A statistical model is a stochastic (or random) data generating process that
can produce an entire forecast distribution
• Each model consists of a measurement equation that describes the observed
data and some transition equations that describe how the unobserved
components or states (level, trend, seasonal) change over time. Hence these
are referred to as state space models
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State Space Models
• Each model has an observation equation and transition equations, one for each state
(level, trend, seasonal), i.e., state space models.
• Two models for each method: one with additive and one with multiplicative errors, i.e.,
in total 18 models.
• ETS(Error, Trend, Seasonal):
• Error = {A, M}
• Trend = {N, A, Ad}
• Seasonal = {N, A, M}
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State Space Models
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Estimating ETS models
• Smoothing parameters 𝛼, 𝛽, 𝛾 and ∅, and the initial states 𝑙0, 𝑏0, 𝑠0, 𝑠−1, . . , 𝑠−𝑚+1
are estimated by maximizing the “likelihood”.
• The likelihood is the probability of the data arising from the specified model.
• For models with additive errors equivalent to minimizing SSE.
• For models with multiplicative errors, not equivalent to minimizing SSE.
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Model Selection
• Which ETS model is to use?
• The AIC, AICc and BIC can be used here to determine which of the ETS
models is most appropriate for a given time series.
Akaike’s Information Criterion (AIC)
• AIC = −2 log(L) + 2k (where L is the likelihood of the model and k is the total number of
parameters and initial states that have been estimated)
Corrected AIC
• AICc = AIC + (2k(k + 1) / (T − k − 1))
Bayesian Information Criterion (BIC)
• BIC = AIC + k[log(T) − 2]
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The ETS() Function
• Automatically chooses a model by minimizing the AICc
• Can handle any combination of trend, seasonality and damping
• Ensures the parameters are admissible (equivalent to invertible)
• Produces an object of class “ETS”
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Example: Australian Holiday Tourism
aus_holidays <- tourism %>% filter(Purpose == “Holiday”) %>% summarise(Trips = sum(Trips)/1000)
fit <- aus_holidays %>% model(ETS(Trips))
report(fit)
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Series: Trips
Model: ETS(M,N,A)
Smoothing parameters:
alpha = 0.3484054
gamma = 0.0001000018
Initial states:
l[0] s[0] s[-1] s[-2] s[-3]
9.727072 -0.5376106 -0.6884343 -0.2933663 1.519411
sigma^2: 0.0022
AIC AICc BIC
226.2289 227.7845 242.9031
Example: Australian Holiday Tourism
fit %>% augment()
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# A tsibble: 80 x 6 [1Q]
# Key: .model [1]
.model Quarter Trips .fitted .resid .innov
1 ETS(Trips) 1998 Q1 11.8 11.2 0.560 0.0498
2 ETS(Trips) 1998 Q2 9.28 9.63 -0.353 -0.0367
3 ETS(Trips) 1998 Q3 8.64 9.11 -0.468 -0.0514
4 ETS(Trips) 1998 Q4 9.30 9.10 0.201 0.0221
5 ETS(Trips) 1999 Q1 11.2 11.2 -0.0535 -0.00476
6 ETS(Trips) 1999 Q2 9.61 9.39 0.214 0.0227
7 ETS(Trips) 1999 Q3 8.91 9.07 -0.159 -0.0176
8 ETS(Trips) 1999 Q4 9.03 9.17 -0.143 -0.0156
9 ETS(Trips) 2000 Q1 11.1 11.2 -0.105 -0.00940
10 ETS(Trips) 2000 Q2 9.20 9.33 -0.130 -0.0140
# … with 70 more rows
Example: Australian Holiday Tourism
fit %>% forecast(h = 8)
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# A fable: 8 x 4 [1Q]
# Key: .model [1]
.model Quarter Trips .mean
1 ETS(Trips) 2018 Q1 N(13, 0.35) 12.7
2 ETS(Trips) 2018 Q2 N(11, 0.3) 10.9
3 ETS(Trips) 2018 Q3 N(10, 0.31) 10.5
4 ETS(Trips) 2018 Q4 N(11, 0.35) 10.6
5 ETS(Trips) 2019 Q1 N(13, 0.48) 12.7
6 ETS(Trips) 2019 Q2 N(11, 0.43) 10.9
7 ETS(Trips) 2019 Q3 N(10, 0.44) 10.5
8 ETS(Trips) 2019 Q4 N(11, 0.48) 10.6
Example: Australian Holiday Tourism
fit %>% forecast(h = 8) %>% autoplot(aus_holidays)
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Example: Australian Holiday Tourism
components(fit) %>% autoplot()
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Example: Australian Holiday Tourism
gg_tsresiduals(fit)
accuracy(fit)
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# A tibble: 1 x 10
.model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
1 ETS(Trips) Training 0.0520 0.428 0.331 0.334 3.45 0.798 0.793 -0.0642
Example: Corticosteroid Drug Sales
h02 <- PBS %>% filter(ATC2 == “H02”) %>% summarise(Cost = sum(Cost))
h02 %>% autoplot(Cost)
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Example: Corticosteroid Drug Sales
h02 %>% model(ETS(Cost ~ error(“A”) + trend(“A”) + season(“A”))) %>% report()
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Series: Cost
Model: ETS(A,A,A)
Smoothing parameters:
alpha = 0.17
beta = 0.00631
gamma = 0.455
Initial states:
l[0] b[0] s[0] s[-1] s[-2] s[-3] s[-4] s[-5] s[-6] s[-7] s[-8] s[-9] s[-10] s[-11]
409706 9097 -99075 -136602 -191496 -174531 -241437 210644 244644 145368 130570 84458 39132 -11674
sigma^2: 3.5e+09
AIC AICc BIC
5585 5589 5642
Example: Corticosteroid Drug Sales
h02 %>% model(ETS(Cost)) %>% report()
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Series: Cost
Model: ETS(M,Ad,M)
Smoothing parameters:
alpha = 0.307
beta = 0.000101
gamma = 0.000101
phi = 0.978
Initial states:
l[0] b[0] s[0] s[-1] s[-2] s[-3] s[-4] s[-5] s[-6] s[-7] s[-8] s[-9] s[-10] s[-11]
417269 8206 0.872 0.826 0.756 0.773 0.687 1.28 1.32 1.18 1.16 1.1 1.05 0.981
sigma^2: 0.0046
AIC AICc BIC
5515 5519 5575
Example: Corticosteroid Drug Sales
h02 %>% model(ETS(Cost)) %>% forecast() %>% autoplot(h02)
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Example: Corticosteroid Drug Sales
fit <- h02 %>%
model(
auto = ETS(Cost),
AAA = ETS(Cost ~ error(“A”) + trend(“A”) + season(“A”))
)
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> fit %>% accuracy()
# A tibble: 4 x 10
.model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
1 auto Training 2461. 51102. 38649. -0.0127 4.99 0.638 0.689 -0.0958
2 AAA Training -5780. 56784. 43378. -1.30 6.05 0.716 0.766 0.0258
> fit %>% glance()
# A tibble: 4 x 9
.model sigma2 log_lik AIC AICc BIC MSE AMSE MAE
1 auto 4.59e-3 -2740. 5515. 5519. 5575. 2611438596. 2787191547. 5.01e-2
2 AAA 3.50e+9 -2776. 5585. 5589. 5642. 3224448256. 3441212841. 4.34e+4
Business Forecasting Analytics
ADM 4307 – Fall 2021
Exponential Smoothing Methods
Fall 2021 ADM 4307 Business Forecasting Analytics 60