forecasting: principles and practice · classical method originated in 1920s. census ii method...
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Forecasting: principles and practice 1
Forecasting: principlesand practice
Rob J Hyndman
1.3 Seasonality and trends
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Outline
1 Time series components
2 STL decomposition
3 Forecasting and decomposition
4 Lab session 5
Forecasting: principles and practice Time series components 2
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Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data.
Seasonal pattern exists when a series is influenced byseasonal factors (e.g., the quarter of the year,the month, or day of the week).
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years).
Forecasting: principles and practice Time series components 3
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Time series decomposition
Yt = f(St, Tt, Et)
where Yt = data at period tSt = seasonal component at period tTt = trend-cycle component at period tEt = remainder (or irregular or error) compo-
nent at period t
Additive decomposition: Yt = St + Tt + Et.Multiplicative decomposition: Yt = St × Tt × Et.
Forecasting: principles and practice Time series components 4
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Time series decomposition
Yt = f(St, Tt, Et)
where Yt = data at period tSt = seasonal component at period tTt = trend-cycle component at period tEt = remainder (or irregular or error) compo-
nent at period t
Additive decomposition: Yt = St + Tt + Et.Multiplicative decomposition: Yt = St × Tt × Et.
Forecasting: principles and practice Time series components 4
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Time series decomposition
Yt = f(St, Tt, Et)
where Yt = data at period tSt = seasonal component at period tTt = trend-cycle component at period tEt = remainder (or irregular or error) compo-
nent at period t
Additive decomposition: Yt = St + Tt + Et.Multiplicative decomposition: Yt = St × Tt × Et.
Forecasting: principles and practice Time series components 4
![Page 7: Forecasting: principles and practice · Classical method originated in 1920s. Census II method introduced in 1957. Basis for modern X-12-ARIMA method. STL method introduced in 1983](https://reader033.vdocuments.site/reader033/viewer/2022060605/605a2a6782ca8e550e2daec1/html5/thumbnails/7.jpg)
Time series decomposition
Additive model appropriate if magnitude of seasonalfluctuations does not vary with level.If seasonal are proportional to level of series, thenmultiplicative model appropriate.Multiplicative decomposition more prevalent witheconomic seriesAlternative: use a Box-Cox transformation, and thenuse additive decomposition.Logs turn multiplicative relationship into an additiverelationship:
Yt = St × Tt × Et ⇒ log Yt = log St + log Tt + log Et.Forecasting: principles and practice Time series components 5
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Euro electrical equipment
60
80
100
120
2000 2005 2010
New
ord
ers
inde
x
series
Data
Trend
Electrical equipment manufacturing (Euro area)
Forecasting: principles and practice Time series components 6
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Euro electrical equipmentda
tatr
end
seas
onal
rem
aind
er
2000 2005 2010
60
80
100
120
80
90
100
110
−20
−10
0
10
−8
−4
0
4
Time
Forecasting: principles and practice Time series components 7
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Euro electrical equipment
−20
−10
0
10
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
Sea
sona
l
Forecasting: principles and practice Time series components 8
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Seasonal adjustment
Useful by-product of decomposition: an easy way tocalculate seasonally adjusted data.Additive decomposition: seasonally adjusted datagiven by
Yt − St = Tt + EtMultiplicative decomposition: seasonally adjusteddata given by
Yt/St = Tt × Et
Forecasting: principles and practice Time series components 9
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Euro electrical equipment
60
80
100
120
2000 2005 2010
New
ord
ers
inde
x
series
Data
SeasAdjust
Electrical equipment manufacturing
Forecasting: principles and practice Time series components 10
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Seasonal adjustment
We use estimates of S based on past values toseasonally adjust a current value.Seasonally adjusted series reflect remainders as wellas trend. Therefore they are not “smooth” and“downturns” or “upturns” can be misleading.It is better to use the trend-cycle component to lookfor turning points.
Forecasting: principles and practice Time series components 11
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History of time series decomposition
Classical method originated in 1920s.Census II method introduced in 1957. Basis formodern X-12-ARIMA method.STL method introduced in 1983TRAMO/SEATS introduced in 1990s.
Forecasting: principles and practice Time series components 12
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Outline
1 Time series components
2 STL decomposition
3 Forecasting and decomposition
4 Lab session 5
Forecasting: principles and practice STL decomposition 13
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STL decomposition
STL: “Seasonal and Trend decomposition using Loess”,Very versatile and robust.Unlike X-12-ARIMA, STL will handle any type ofseasonality.Seasonal component allowed to change over time,and rate of change controlled by user.Smoothness of trend-cycle also controlled by user.Robust to outliersNot trading day or calendar adjustments.Only additive.
Forecasting: principles and practice STL decomposition 14
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Euro electrical equipmentelecequip %>% stl(s.window=5) %>%
autoplot
data
tren
dse
ason
alre
mai
nder
2000 2005 2010
60
80
100
120
80
90
100
110
−20
−10
0
10
−8
−4
0
4
TimeForecasting: principles and practice STL decomposition 15
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Euro electrical equipmentelecequip %>%
stl(t.window=15, s.window='periodic', robust=TRUE) %>%autoplot
data
tren
dse
ason
alre
mai
nder
2000 2005 2010
60
80
100
120
80
90
100
110
−10
0
10
−5
0
5
10
TimeForecasting: principles and practice STL decomposition 16
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STL decomposition in R
t.window controls wiggliness of trend component.s.window controls variation on seasonal component.
Forecasting: principles and practice STL decomposition 17
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Outline
1 Time series components
2 STL decomposition
3 Forecasting and decomposition
4 Lab session 5
Forecasting: principles and practice Forecasting and decomposition 18
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Forecasting and decomposition
Forecast seasonal component by repeating the lastyearForecast seasonally adjusted data using non-seasonaltime series method. E.g.,
Holt’s method — next topicRandom walk with drift model
Combine forecasts of seasonal component withforecasts of seasonally adjusted data to get forecastsof original data.Sometimes a decomposition is useful just forunderstanding the data before building a separateforecasting model.Forecasting: principles and practice Forecasting and decomposition 19
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Seas adj elec equipment
70
90
110
2000 2005 2010 2015
New orders index
eead
j level
80
95
Naive forecasts of seasonally adjusted data
Forecasting: principles and practice Forecasting and decomposition 20
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Seas adj elec equipment
40
60
80
100
120
2000 2005 2010 2015
Time
New
ord
ers
inde
x
level
80
95
Forecasts from STL + Random walk
Forecasting: principles and practice Forecasting and decomposition 21
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How to do this in R
fit <- stl(elecequip, t.window=15,s.window="periodic", robust=TRUE)
eeadj <- seasadj(fit)autoplot(naive(eeadj, h=24)) +
ylab("New orders index")
fcast <- forecast(fit, method="naive", h=24)autoplot(fcast) +
ylab="New orders index")
Forecasting: principles and practice Forecasting and decomposition 22
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Decomposition and prediction intervals
It is common to take the prediction intervals from theseasonally adjusted forecasts and modify them withthe seasonal component.This ignores the uncertainty in the seasonalcomponent estimate.It also ignores the uncertainty in the future seasonalpattern.
Forecasting: principles and practice Forecasting and decomposition 23
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Some more R functions
fcast <- stlf(elecequip, method='naive')
fcast <- stlf(elecequip, method='naive',h=36, s.window=11, robust=TRUE)
Forecasting: principles and practice Forecasting and decomposition 24
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Outline
1 Time series components
2 STL decomposition
3 Forecasting and decomposition
4 Lab session 5
Forecasting: principles and practice Lab session 5 25
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Lab Session 5
Forecasting: principles and practice Lab session 5 26