Module 6
6-1
Methods for Nonstationary Time Series
Transformations, Differencing, and ARIMA Model
Identification
Class notes for Statistics 451: Applied Time Series
Iowa State University
Copyright 2015 W. Q. Meeker.
November 11, 2015
17h 20min
Dealing with Nonconstant Variance
• A common reason: variability increases with level (e.g., σa2
is a percent of the level)
This type of variance-nonstationarity can sometimes be handled with a transformation of the response.
• Can also have variability changing independently of level.
Thousands of Passengers
1953
1955
Year
1957
1959
1961
6-3
This type of nonstationarity may be handled by models allowing for nonstationary variance (e.g. Autoregressive Conditional Heteroscedasticity (ARCH) and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) Models (not covered here in detail).
1951
Module 6
Segment 1
Nonconstant Variance and an Introduction to
Transformations
Variability Increasing with Level
6-2
• Example: If the standard deviation of sales is 10% of level,
then for sales ± σa we have
* 100 ± 10 or [90, 110] when sales are 100
* 1000 ± 100 or [900, 1100] when sales are 1000
• On the log scale,
* [log(90), log(110)] = [2.2, 2.4] when sales are 100
* [log(900), log(1100)] = [4.5, 4.7] when sales are 1000
6-4
• Common problem when the “dynamic range” of a time series [measured by the ratio max(Zt)/ min(Zt )] is large (say
more than 3 or 4).
1950
300
1952
Range-Mean Plot
1951
400
1953
10
w= Thousands of Passengers
1955
time
0
10
1956
International Airline Passengers
1954
0
1957
1958
Lag
20
PACF
Lag
20
ACF
1959
30
30
1960
6-6
1961
Plot of the airline data along with a range-mean plot
and plots of sample ACF and sample PACF.
[iden(airline.tsd)]
1949
200
Mean
1.0
0.5
600
500
Number of international airline passengers
from 1949 to 1960
1949
6-5
ACF
Partial ACF
0.0
-1.0
1.0
0.5
0.0
-1.0
w
Range
400
300
200
100
200
150
100
50
600
500
400
300
200
100
1949
5.0
1950
5.2
5.4
1952
5.6
Range-Mean Plot
1951
5.8
1953
1954
6.2
1955
time
0
0
1956
w= log(Thousands of Passengers)
International Airline Passengers
6.0
10
10
1957
1958
Lag
20
PACF
Lag
20
ACF
1959
30
30
1960
6-7
1961
Plot of the logarithms of the airline data along with a
range-mean plot and plots of sample ACF and sample
PACF. [iden(airline.tsd,gamma=0)]
4.8
Mean
Module 6
Segment 2
6-9
Box-Cox Transformations and the Range-Mean Plot
Range-Mean Plot
1. Divide realization into groups (4 to 12 in each group). If
there is a natural seasonal period (e.g., 12 observations per
year for monthly data), use it to choose the groups.
2. Transform data with given γ (and perhaps m)
3. Compute the mean and range in each group.
4. Plot the ranges versus the means
γ
−1
−.333
Zt∗ + m
?
No transformation
Count data
Slightly weaker than log
Percentage change
Slightly stronger than log
Possible Application
Very long upper tail
Examples of Power Transformations
Transformation
Zt ∼ −1/(Zt∗ + m)
q
Zt ∼ −1/ 3 Zt∗ + m
q
3 Z∗ + m
t
q
Zt ∼ log(Zt∗ + m)
Zt ∼
0
.333
Zt ∼
Zt ∼ [Zt∗ + m]2
Zt ∼ Zt∗
.5
1
2

γ 6= 0
log(Zt∗ + m) γ = 0

γ
∗
 (Zt +m) −1
γ
Box-Cox Family of Transformations
Zt =
6-8
where Zt∗ is the original, untransformed time series, γ is primary transformation parameter, and log is natural log (i.e.,
base e).
• For γ < 0, the γ in the denominator of the transformation
assures that Zt is an increasing function of Zt∗ so that a plot
of Zt has the same direction of trend as Zt∗.
• Because
(Zt∗ + m)γ − 1
lim
= log(Zt∗ + m)
γ→0
γ
the transformation is a continuous function in γ.
6 - 10
• The quantity m is typically chosen to be 0. Choose m > 0
if some Zt∗ < 0 or Zt∗ ≈ 0. Increasing m has the effect of
weakening the transformation.
1949
1950
25
1951
1952
Range-Mean Plot
30
35
1953
time
1955
0
0
10
10
1956
w= ((Thousands of Passengers)^0.5)-1)/0.5
International Airline Passengers
1954
40
1957
1958
Lag
20
PACF
Lag
20
ACF
1959
30
30
1960
6 - 12
1961
Plot of the square roots (γ = .5) of the airline data
along with a range-mean plot and plots of sample ACF
and sample PACF. [iden(airline.tsd,gamma=.5)]
20
Mean
1.0
0.5
45
40
1.0
6.5
6.0
5. Repeat for different values of γ
6 - 11
ACF
Partial ACF
0.0
-1.0
1.0
0.5
0.0
-1.0
w
Range
35
30
25
20
10
9
8
7
6
5
4
ACF
Partial ACF
0.5
0.0
-1.0
1.0
0.5
0.0
-1.0
w
Range
5.5
5.0
0.45
0.40
0.35
1949
5.0
1950
5.2
1951
Mean
1952
5.6
Range-Mean Plot
5.4
1951
1952
Range-Mean Plot
0.995
Mean
1953
5.8
1953
1954
6.2
1955
time
0
0
1956
w= log(Thousands of Passengers)
International Airline Passengers
6.0
1954
0.998
1955
time
0
0
1956
w= ((Thousands of Passengers)^(-1))-1)/(-1)
International Airline Passengers
0.997
1957
10
10
1957
10
10
1958
1958
ACF
20
Lag
PACF
20
Lag
ACF
20
Lag
PACF
20
Lag
1959
1959
1960
30
30
1960
30
30
1961
6 - 13
1961
6 - 15
1950
1951
1952
Range-Mean Plot
Mean
2.50
2.55
1953
time
1955
0
0
10
10
1956
w= ((Thousands of Passengers)^(-0.3333))-1)/(-0.3333)
International Airline Passengers
1954
2.60
1957
1958
Lag
20
PACF
Lag
20
ACF
1959
30
30
1960
6 - 14
1961
Plot of the reciprocal cube-root (γ = −.3333) of the
airline data. [iden(airline.tsd,gamma=-.3333)]
1949
2.45
Effects of Doing a Box-Cox Transformation
Nonconstant Level and Exponential Growth
Segment 3
Module 6
6 - 18
6 - 16
• Changes the shape of the distribution of the residuals (e.g.,
from lognormal to normal)
• Changes the shape of a trend line (e.g., exponential or percentage trend versus linear trend)
• Changes the relationship between amount of variability and
level (as reflected in a range-mean plot)
2.40
1.0
0.5
2.65
2.60
2.55
Plot of the logarithms (γ = 0) of the airline data along
with a range-mean plot and plots of sample ACF and
sample PACF. [iden(airline.tsd,gamma=0)]
4.8
1950
0.994
0.996
Iden plot of the reciprocal (γ = −1) of the airline data.
[iden(airline.tsd,gamma=-1)]
1949
0.993
1. Try γ = 1 (no transformation)
2. Try γ = .5, .333, 0 (moderate to strong transformations)
3. Try γ = −.333, −.5, −1 (even stronger transformations)
4. Choose one tentative value of γ.
• Use the range-mean plot as a rough guide. It is not necessary to obtain 0 correlation between means and ranges.
• Choose one value for γ and use if for initial modeling. Then
experiment with other values with your chosen model (usually the chosen model does not depend strongly on the
choice of γ).
6 - 17
• Box-Cox transformations useful in other kinds of data analysis (e.g., regression analysis).
• Some computer programs attempt to estimate γ.
ACF
Partial ACF
0.0
-1.0
1.0
0.5
0.0
w
Range
2.50
2.45
2.40
0.075
0.070
0.065
0.060
1.0
1.0
0.5
Usual Procedure for Deciding on the Use of a
Transformation
0.992
ACF
Partial ACF
0.0
-1.0
1.0
0.5
0.0
-1.0
0.998
0.996
-1.0
ACF
Partial ACF
0.5
0.0
-1.0
1.0
0.5
0.0
-1.0
6.5
6.0
5.5
5.0
0.45
0.40
0.35
w
Range
w
Range
0.994
0.992
0.0025
0.0020
0.0015
0.0010
Dealing with Nonconstant Level (Trend or Cycle)
• Fit a trend line (possibly after a transformation)
• Analyze differences (changes) instead of the actual time
series [e.g., fit a model to Wt = Zt − Zt−1]
6 - 19
• If transformation and differencing is used always do the
transformation first
Number
3
•
4
•
5
•
6
•
100% Growth per Period
•
2
7
•
1
•
2
•
3
•
Time
4
•
5
•
6
•
6 - 23
6 - 21
7
•
Log of 100% Growth per Period
Plots showing the effect of a log transformation on
exponential growth
•
1
Segment 4
Module 6
Log Number
Introduction to Time Series Differencing and
Integrated Models
Time
5
4
3
2
1
120
100
80
60
40
20
0
t
1
2
3
4
5
6
7
Difference
Amount
Zt − Zt−1
NA
2
4
8
16
32
64
Log Amount
log(Zt)
0.69
1.39
2.08
2.77
3.47
4.16
4.85
Difference
Log Amount
log(Zt) − log(Zt−1)
NA
.70
.69
.69
.70
.69
.69
Example of Exponential (Percentage) Growth
Number of Bacteria in a Dish
Doubling Every Time Period (100% Growth)
Amount
Zt
2
4
8
16
32
64
128
Differences of Zt also grow exponentially!
6 - 24
6 - 22
6 - 20
Differences of log(Zt) are constant (except for roundoff)
Exponential (Percentage) Growth
Zt∗ = β0∗ [β1∗ ]t
Zt = log(β0∗ ) + log(β1∗ )t
Zt = log(Zt∗) = log(β0∗ ) + log([β1∗ ]t )
Zt = β 0 + β 1 t
where β0 = log(β0∗ ) and β1 = log(β1∗ ).
" ∗
#
"
#
∗ ∗ t+1 − β ∗ [β ∗ ]t
Z
− Z∗
β
[β
]
t
t+1
= 100 0 1 ∗ ∗ t 0 1
Zt∗
β0[β1]
Growth is rate in percent is
100
= 100[β1∗ − 1]
Ha : µW 6= 0
H0 : µ W = 0
= 100[exp(β1) − 1] ≈ 100β1
Gambler’s Winnings
P
“Integrated”
Incremental
Zt Wt = Zt − Zt−1
100
NA
110
+10
108
−2
115
+7
119
+4
118
−1
125
+7
127
+2
125
−2
134
+9
140
+6
for small β1
t
1
2
3
4
5
6
7
8
9
10
11
H0 : µW = 0
40
Wi
=
=4
W =
10
s10
P
(Wi − 4)2
= 4.52
9
SW =
W −0
W −0
√
t=
=
= 2.79
SW
SW / 10
•
2
•
•
•
4
•
4
•
Cumulative Winnings
•
•
6
Time
•
8
•
Incremental Winnings
•
•
6
Time
•
8
•
•
•
10
•
10
•
•
6 - 25
Plots of the gambler’s cumulative winnings and
incremental winnings
•
•
2
56
10
58
60
20
62
w= Dollars
time
0
0
30
10
10
1979 Weekly Closing Price of ATT Common Stock
Range-Mean Plot
Mean
40
ACF
20
Lag
PACF
20
Lag
30
30
50
6 - 27
Forecasting the Gambler’s Cumulative Winnings
• Because Wt = (1−B)Zt = Zt −Zt−1 appears to be stationary,
we can build an ARMA model for Wt and use Zt = Zt−1 +Wt.
• If Wt is Wt = θ0 + at, then Zt = Zt−1 + θ0 + at is a “a random
walk with drift” with E(increase) = θ0 per period.
b a = SW = 4.52
• Estimated from data: θb0 = W = 4, σ
6 - 26
• A forecast (or prediction) interval for Wt can be translated
into a prediction interval for Zt (because we know Zt−1)
• One-step-ahead forecast for Zt is Zbt = Zt−1 + θb0
• One-step-ahead 100(1 − α)% prediction interval:
Zbt ± t(1−α/2,nw −1)S(W −W )
q
q
2 + S 2 = S 2 + S 2 /n
SW
w
W
W
W
• S(W −W ) =
b a for large nw
• S(W −W ) ≈ SW = σ
0
0
10
10
ACF
20
Lag
20
time
30
0
10
Lag
20
PACF
40
1979 Weekly Closing Price of ATT Common Stock
w= (1-B)^ 1 Dollars
30
Module 6
Segment 5
30
50
ARIMA Models Part 1: ARIMA(0, 1, 1)
6 - 30
6 - 28
Function iden Output for the First Differences of 1979
Weekly Closing Prices of AT&T Common Stock.
[iden(att.tsd,d=1)]
3
2
1
0
-2
1.0
Function iden Output for 1979 Weekly Closing Prices
of AT&T Common stock. [iden(att.tsd)]
0
54
Forecasting a Random Walk
• If the model for Wt = (1 − B)Zt = Zt − Zt−1 is Wt = at, then
Zt = Zt−1 + at
= at−1 + at−2 + at−3 + · · · + at
which is a “a random walk” process. This is like an AR(1)
model with φ1 = 1 (note that model is nonstationary).
• The only parameter of this model is σa
• A forecast (or prediction interval) for Wt can be translated
into a forecast (or prediction interval) for Zt (because we
know Zt−1)
Partial ACF
1.0
• One-step-ahead point forecast for Zt is Zbt = Zt−1
w
ACF
64
62
• One-step-ahead prediction interval is: Zbt ± t(1−α/2,n−1)SW
6 - 29
1.0
0.0
-1.0
ACF
Partial ACF
0.5
0.0
-1.0
1.0
0.5
0.0
-1.0
0.0
-1.0
140
120
100
2 4 6 8
-2
Dollars
Dollars
w
Range
60
58
56
54
52
4.5
4.0
3.5
3.0
2.5
20
60
100
Range-Mean Plot
120
Mean
30
140
w= Sales
time
0
0
Simulated Time Series #B
160
40
10
10
50
ACF
20
Lag
PACF
20
Lag
PACF
20
Lag
30
30
30
50
6 - 33
60
20
0
10
ACF
20
Lag
Mean
30
45
Range-Mean Plot
40
30
30
50
0
40
Simulated Time Series #B
w= (1-B)^ 1 Sales
time
w= Sales
time
40
0
0
10
10
Simulated Time Series #C
55
10
Lag
20
PACF
50
ACF
20
Lag
PACF
20
Lag
30
30
30
50
Function iden Output for Simulated Series C
35
-1.0
1.0
60
6 - 34
6 - 36
This model is also known as exponentially weighted moving
average (EWMA) and its forecast equations can be shown
to be equivalent to exponential smoothing.
0.0
Phi
which is a nonstationary “ARMA(1,1)”
Zt = Zt−1 − θ1at−1 + at
Wt = (1 − B)1Zt = (1 − θ1B)at
Special Case: ARIMA(0, 1, 1) Model [a.k.a. IMA(1,1)]
30
6 - 32
Function iden Output for the First Differences of
Simulated Series B
60
Function iden Output for Simulated Series B
80
6 - 31
AutoRegressive Integrated Moving Average Model
(ARIMA)
• An ARIMA(p, d, q) model is
10
1.0
0.5
φp(B)(1 − B)d Zt = θq (B)at
Simulated Time Series #C
w= (1-B)^ 1 Sales
40
time
0
Partial ACF
ACF
Partial ACF
can be viewed as a generalized ARMA model, with d roots
on the unit circle.
• Letting Wt = (1 − B)d Zt be the “working series” and substituting gives an ARMA model for Wt
φp(B)Wt = θq (B)at
30
1.0
0.0
-1.0
0 10
-20
1.0
0.0
-1.0
w
ACF
w
Range
50
40
30
20
30
25
0.0
-1.0
1.0
0.5
0.0
-1.0
• Can fit an ARMA model to Wt
30
ACF
20
Lag
6 - 35
1.0
Theta
0.0
-1.0
• Can unscramble the Zt for forecasting and interpretation
10
Partial ACF
20
1.0
Function iden Output for the First Differences of
Simulated Series C
20
0
1.0
0.0
-1.0
15
ACF
Partial ACF
0.5
0.0
-1.0
1.0
0.5
0.0
-1.0
180
160
140
120
100
80
60
45
40
35
30
25
20
15
w
Range
w
ACF
0 10
-20
1.0
0.0
-1.0
Module 6
Segment 6
ARIMA Models Part 2: ARIMA(1, 1, 0)
0
10
50
ACF
20
Lag
100
30
Simulated data
w= (1-B)^ 1 Data
150
time
0
200
10
250
PACF
20
Lag
30
300
6 - 37
w
0
50
Mean
10
Range-Mean Plot
5
100
15
20
w= Data
10
200
0
10
Simulated data
time
150
0
ACF
20
Lag
PACF
20
Lag
250
30
30
6 - 38
300
Function iden Output for Simulated Series D
0
-5
Special Case: ARIMA(1, 1, 0) Model
leading to
Zt = (1 + φ1)Zt−1 − φ1Zt−2 + at
• (1 − φ1B)(1 − B)1Zt = at or
(1 − φ1B − B + φ1B2)Zt = at ,
0
500
50
-2
-1
0
Phi1
1
2
′
′
Zt = φ1
Zt−1 + φ2
Zt−2 + at
1000
Range-Mean Plot
Mean
100
w= Data
time
150
0
10
10
200
0
Simulated data
1500
ACF
20
Lag
PACF
20
Lag
250
30
30
Function iden Output for Simulated Series E
′ = (1 + φ ) and φ′ = −φ or φ′ = 1 − φ′
where φ1
1
1
2
2
1
0
6 - 42
300
6 - 40
which is a nonstationary “ARIMA(2,0,0)” or “AR(2)”
1.0
Phi2
0.0
-1.0
Function iden Output for the First Differences of
Simulated Series D
0
6 - 39
How to Choose d (the amount of differencing)
1. d = 0 to d = 1 most common; d = 2 not common;
d ≥ 3 generally not useful;
2. Plot data versus time, looking for trend and cycle?
1.0
0.5
1.0
0.5
3. Consider the physical process (is the process changing?)
4. Examine the ACF for d = 0, stepping up to d = 1 or d = 2,
if needed. Look for the smallest d such that ACF “dies
down quickly.”
5. Fit AR(1) model and test H0 : φ1 = 1 using
1
φb − 1
t= 1
Sφb
6 - 41
ACF
Partial ACF
0.0
-1.0
1.0
0.5
0.0
-1.0
Need special tables [see Dickey and Fuller 1979 JASA]
6. Minimize SW
7. Compare forecasts and prediction intervals.
ACF
Partial ACF
0.0
-1.0
1.0
0.5
0.0
-1.0
30
20
10
0
-10
16
14
12
10
8
6
4
1500
Partial ACF
2
Range
w
Range
1000
500
0
200
150
100
50
0
1.0
0.0
-1.0
w
ACF
0
-2
1.0
0.0
-1.0
0
10
50
ACF
20
Lag
30
100
30
w= Sales
40
time
0
0
0
Simulated Time Series #A
180
200
10
10
10
250
Lag
20
PACF
50
ACF
20
Lag
PACF
20
Lag
30
30
30
300
6 - 43
6 - 45
60
0
10
50
ACF
20
Lag
100
30
Simulated data
w= (1-B)^ 2 Data
150
time
0
200
10
250
PACF
20
Lag
30
An Example of a Nonstationary Model
at ∼ NID(0, σa2)
300
6 - 44
Function iden Output for the Second Differences of
Simulated Series E
20
0
10
30
ACF
20
Lag
30
Simulated Time Series #A
w= (1-B)^ 1 Sales
40
time
0
10
50
PACF
20
Lag
30
6 - 48
60
Function iden Output for the First Differences of
Simulated Series A
6 - 46
Beware of differencing when differencing is not warranted.
Is there a better solution for fitting a model to Zt??
Therefore, Zt is nonstationary, but Wt is stationary. What
kind of model does Wt have??
2 = Var(W ) = 2σ 2
σW
t
a
E(Wt) = β1
= β1 − at−1 + at
= [β0 + β1t + at] − [β0 + β1(t − 1) + at−1 ]
Wt = (1 − B)Zt = Zt − Zt−1
E(Zt) = β0 + β1t
Zt = β0 + β1t + at,
Let t = 1, 2, . . . be coded time.
0
1.0
Simulated data
w= (1-B)^ 1 Data
150
time
Module 6
Segment 7
160
w
Function iden Output for the First Differences of
Simulated Series E
0
Mean
120
Range-Mean Plot
100
140
Function iden Output for Simulated Series A
80
Partial ACF
20
0
-40
1.0
Beware of Differencing When You Should Not
20
60
6 - 47
Partial ACF
4
2
0
-2
-4
1.0
0.0
-1.0
0.0
-1.0
1.0
0.0
-1.0
1.0
ACF
w
ACF
Partial ACF
ACF
Partial ACF
0.5
0.0
-1.0
1.0
0.5
0.0
-1.0
0.0
-1.0
1.0
0.0
-1.0
15
0 5
-10
1.0
0.0
-1.0
w
ACF
w
Range
150
100
50
40
35
30
25
Trivial model:
at ∼ NID(0, σa2)
Beware of Over Differencing
Zt = θ0 + at,
First difference of Zt
= [θ0 + at ] − [θ0 + at−1]
Wt = (1 − B)Zt = Zt − Zt−1
= −at−1 + at
is a noninvertible MA(1)!
Second difference of Zt
Wt = (1 − B)2Zt = Zt − 2Zt−1 + Zt−2
= −2at−1 + at−2 + at
50
ACF
20
Lag
150
time
0
200
Simulated Seried F (from Trivial Model)
w= (1-B)^ 1 Data
100
30
10
250
PACF
20
Lag
30
300
6 - 49
-0.2
50
Mean
0.2
Range-Mean Plot
0.0
0.6
w= Data
time
150
0
0
10
10
200
Simulated Seried F (from Trivial Model)
100
0.4
ACF
20
Lag
PACF
20
Lag
250
30
30
Function iden Output for Simulated Series F
0
0
0
10
50
ACF
20
Lag
150
time
0
200
Simulated Seried F (from Trivial Model)
w= (1-B)^ 2 Data
100
30
10
250
PACF
20
Lag
30
6 - 50
300
300
6 - 54
• Situation is similar, but more complicated, with higher order
differencing.
◮ The physical nature of the data-generating process
◮ H0 : µW = 0 by looking at t = (W − 0)/SW
• Use of a constant term after differencing is rare. Check:
the deterministic trend is θ0 each time period.
Zt = Zt−1 + θ0 + Wt
• If there is differencing (d = 1) then a constant term should
be included only if there is need to or evidence of deterministic trend.
With stationary Wt, E(Wt) = 0, and
• With no differencing (d = 0) include a constant term in the
model to allow estimation of the process mean.
6 - 52
Function iden Output for the Second Differences of
Simulated Series F
-0.4
1.0
0.5
When to Include θ0 in an ARIMA Model
Partial ACF
is a noninvertible MA(2)!!
0
10
1.0
0.0
-1.0
2 4 6
Function iden Output for the First Differences of
Simulated Series F
0
6 - 51
6 - 53
ACF
Partial ACF
0.0
-1.0
1.0
0.5
0.0
-1.0
2
1
0
-1
-2
4.0
3.5
3.0
2.5
2.0
w
Range
w
ACF
4
2
Using a Constant Term After Differencing and
Exponential Smoothing
Segment 8
Module 6
Partial ACF
-2
-6
1.0
0.0
-1.0
1.0
0.0
-1.0
w
ACF
0
-2
-4
1.0
0.0
-1.0
Relationship Between IMA(1,1)
and Exponential Smoothing
IMA(1,1) model, unscrambled
Zt = Zt−1 − θ1at−1 + at
IMA(1,1) forecast
= Zt−1 − θb1(Zt−1 − Zbt−1)
b t−1
Zbt = Zt−1 − θb1a
= (1 − θb1)Zt−1 + θb1Zbt−1
= αZt−1 + (1 − α)Zbt−1
= αZt−1 + α(1 − α)Zt−2 + α(1 − α)2Zt−3 + · · ·
where α = 1−θ1. Usually 0.01 < α < 0.1. This shows why the
IMA(1,1) forecast equation is sometimes called “exponential smoothing” or “exponentially weighted moving average”
(EWMA).
6 - 55
Issues in Applying Exponential Smoothing
• Choice of the smoothing constant α.
• Start-up value for the forecasts?
• Single, double, or triple exponential smoothing?
(Single exponential smoothing is equivalent to IMA(1,1),
double exponential smoothing is equivalent to IMA(2,2),
...)
• Seasonality (Winter’s method)
• Prediction intervals or bounds?
6 - 56