ARIMA-models for non-stationary time series
Consider again the data material from Exercise 8.8 in the textbook (weekly sales
figures of thermostats)
Time Series Plot of y
350
y
300
250
200
150
1
5
10
15
20
25
30
Index
35
40
45
50
This series is obviously non-stationary as it possesses a trend.
SAC and SPAC
Autocorrelation Function for y
Partial Autocorrelation Function for y
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
(with 5% significance limits for the autocorrelations)
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.0
1
2
3
4
5
6
7
Lag
8
9
10
11
12
13
1
2
3
4
5
6
7
Lag
The first impression is that this points towards an AR(2)-model.
What will happen if we try such a model?
8
9
10
11
12
13
We may ask for forecast for weeks (53, 54, 55,) 56 and 57 like was the task in
exercise 8.8.
Note that we have to manually enter the columns where we wish the forecasts and the
prediction limits to be stored (columns are not generated automatically like for other
modules).
ARIMA Model: y
Estimates at each iteration
Iteration
SSE
Parameters
0
85100.7
0.100
0.100
182.480
1
61945.7
0.250
0.187
129.078
2
48376.0
0.400
0.272
75.777
3
44295.6
0.534
0.346
28.278
4
44267.8
0.542
0.348
26.509
5
44267.5
0.542
0.347
26.800
6
44267.5
0.542
0.347
26.837
Relative change in each estimate less than 0.0010
* WARNING * Back forecasts not dying out rapidly
Back forecasts (after differencing)
Lag
-97 - -92
241.106
241.105
241.105
241.104
241.103
241.103
Lag
-91 - -86
241.102
241.101
241.100
241.099
241.098
241.096
Lag
-85 - -80
241.095
241.094
241.092
241.090
241.088
241.086
Lag
-79 - -74
241.084
241.081
241.079
241.076
241.073
241.069
Lag
-73 - -68
241.065
241.061
241.057
241.052
241.047
241.041
Lag
-67 - -62
241.035
241.028
241.020
241.012
241.004
240.994
Lag
-61 - -56
240.984
240.972
240.960
240.947
240.932
240.916
Lag
-55 - -50
240.899
240.880
240.860
240.838
240.814
240.788
Lag
-49 - -44
240.759
240.728
240.694
240.658
240.618
240.574
Lag
-43 - -38
240.527
240.475
240.419
240.359
240.292
240.220
Lag
-37 - -32
240.142
240.057
239.964
239.863
239.753
239.633
Lag
-31 - -26
239.503
239.362
239.208
239.041
238.859
238.660
Lag
-25 - -20
238.445
238.210
237.955
237.678
237.376
237.047
Lag
-19 - -14
236.690
236.301
235.878
235.418
234.917
234.373
Lag
-13 -
-8
233.780
233.136
232.434
231.671
230.841
229.940
Lag
-7 -
-2
228.951
227.899
226.692
225.545
223.855
223.190
Lag
-1 -
0
219.355
223.431
Back forecast residuals
Lag
-97 - -92
-0.001
-0.001
-0.002
-0.002
-0.002
-0.002
Lag
-91 - -86
-0.002
-0.002
-0.003
-0.003
-0.003
-0.003
Lag
-85 - -80
-0.004
-0.004
-0.004
-0.005
-0.005
-0.005
Lag
-79 - -74
-0.006
-0.006
-0.007
-0.008
-0.008
-0.009
Lag
-73 - -68
-0.010
-0.011
-0.012
-0.013
-0.014
-0.015
Lag
-67 - -62
-0.016
-0.018
-0.019
-0.021
-0.023
-0.025
Lag
-61 - -56
-0.027
-0.029
-0.032
-0.035
-0.038
-0.041
Lag
-55 - -50
-0.044
-0.048
-0.053
-0.057
-0.062
-0.068
Lag
-49 - -44
-0.074
-0.080
-0.087
-0.095
-0.103
-0.112
Lag
-43 - -38
-0.122
-0.133
-0.145
-0.157
-0.171
-0.186
Lag
-37 - -32
-0.203
-0.220
-0.240
-0.261
-0.284
-0.309
Lag
-31 - -26
-0.336
-0.366
-0.398
-0.433
-0.471
-0.512
Lag
-25 - -20
-0.557
-0.606
-0.659
-0.717
-0.780
-0.849
Lag
-19 - -14
-0.924
-1.005
-1.093
-1.189
-1.294
-1.408
Lag
-13 -
-8
-1.532
-1.666
-1.813
-1.972
-2.146
-2.332
Lag
-7 -
-2
-2.545
-2.748
-3.043
-3.170
-3.820
-3.172
Lag
-1 -
0
-6.060
0.325
Final Estimates of Parameters
Type
Coef
SE Coef
T
P
AR
1
0.5420
0.1437
3.77
0.000
AR
2
0.3467
0.1460
2.38
0.022
Constant
26.837
4.485
5.98
0.000
Mean
241.11
40.30
Number of observations:
Residuals:
52
SS =
44137.6 (backforecasts excluded)
MS =
900.8
DF = 49
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
12
24
36
48
8.6
19.8
27.1
34.5
9
21
33
45
0.473
0.532
0.753
0.873
DF
P-Value
Forecasts from period 52
95% Limits
Period
Forecast
Lower
Upper
53
310.899
252.062
369.736
54
314.956
248.033
381.878
55
305.330
228.528
382.132
56
301.520
218.517
384.523
57
296.117
207.816
384.418
Actual
Time Series Plot for y
(with forecasts and their 95% confidence limits)
400
350
y
300
250
200
150
1
5
10
15
20
25
30
Time
35
40
45
50
ACF of Residuals for y
PACF of Residuals for y
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
(with 5% significance limits for the autocorrelations)
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
2
3
4
5
6
7
Lag
8
9
10
55
11
12
13
1
2
3
4
5
6
7
Lag
8
9
10
11
12
13
Residuals after fitting looks nice, Ljung-Box’ statistics are in order
but..
the forecasts do not seem to be consistent with the development of the sales figures
and…
we have indications of problems in the fitting (back-forecasts are not dying out
rapidly which they should)
We do not go any deeper into the subject of back-forecasting, but a signal from the
software should be taken seriously.
As we have clearly seen a trend, we can force a model which takes this into account.
 Calculate first-order differences
Calculate SAC and SPAC
for the differences series!
Autocorrelation Function for differences
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
One significant spike in SAC,
one significant spike in SPAC.
0.4
0.2
0.0
-0.2
Both are negative
consistence!
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
7
Lag
8
9
10
11
12
13
Most presumable models for
the differenced data:
AR(1) , MA(1) or ARMA(1,1)
Partial Autocorrelation Function for differences
(with 5% significance limits for the partial autocorrelations)
1.0
Partial Autocorrelation
0.8
0.6
When fitting such models to
differenced data, constant term
should be excluded as the
differences are expected to
vary around 0.
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
7
Lag
8
9
10
11
12
13
AR(1):
Type
AR
1
Coef
SE Coef
T
P
-0.4042
0.1356
-2.98
0.004
MS =
905.0
DF = 50
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
P-Value
MA(1):
Type
MA
1
12
24
36
48
12.6
23.8
30.3
38.1
0.318
0.413
0.695
0.820
Coef
SE Coef
T
P
0.6331
0.1133
5.59
0.000
MS =
813.1
DF = 50
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
P-Value
ARMA(1,1):
Type
12
24
36
48
10.7
20.4
28.2
36.2
0.471
0.617
0.785
0.873
Coef
SE Coef
T
P
AR
1
0.0948
0.2376
0.40
0.692
MA
1
0.6751
0.1763
3.83
0.000
MS =
825.7
DF = 49
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
P-Value
12
24
36
48
9.1
19.1
27.5
35.6
0.525
0.641
0.775
0.866
Seems best!
Fitting the model directly on the original observations.
This time series seems to after first-order differencing apply to a MA(1)-model.
The time-series is then said to apply to an ARIMA(0,1,1)-model
For non-seasonal time series the notation is
ARIMA(p,d,q)
Order (p ) of the ARpart in the
differenced series
Order (q ) of the MApart in the
differenced series
Order (d ) of the differencing
ARIMA(0,1,1)
Relevant again, as the original time
series may have an “intercept”
ARIMA Model: y
Estimates at each iteration
Iteration
SSE
Parameters
0
49361.5
0.100
2.825
1
45310.4
0.250
2.496
2
42249.3
0.400
2.245
3
39884.7
0.550
2.106
4
38533.0
0.687
2.124
5
38448.9
0.717
2.220
6
38447.7
0.719
2.248
7
38447.7
0.720
2.251
8
38447.7
0.720
2.252
Relative change in each estimate less than 0.0010
No longer any problems with back-forecasts!
Final Estimates of Parameters
Type
MA
SE Coef
T
P
0.7198
0.1010
7.13
0.000
2.252
1.127
2.00
0.051
1
Coef
Constant
Differencing: 1 regular difference
Number of observations:
Residuals:
Original series 52, after differencing 51
SS =
38356.2 (backforecasts excluded)
MS =
782.8
DF = 49
Note that information is given about the order of the differencing.
MS is the smallest so far (due to the inclusion of the constant term)
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
DF
P-Value
12
24
36
48
10.9
21.1
29.5
37.5
10
22
34
46
0.366
0.513
0.689
0.809
Forecasts from period 52
95% Limits
Period
Forecast
Lower
Upper
53
313.544
258.696
368.392
54
315.796
258.836
372.756
55
318.048
259.052
377.045
56
320.300
259.335
381.265
57
322.552
259.681
385.424
Actual
L-B’s are in order
Time Series Plot for y
(with forecasts and their 95% confidence limits)
400
Forecasts are now more
consistent with the
development of the sales
figures.
350
y
300
250
200
SAC and SPAC of residuals
are still satisfactory.
150
1
5
10
15
20
25
30
Time
35
40
45
50
55
ACF of Residuals for y
PACF of Residuals for y
(with 5% significance limits for the autocorrelations)
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
Partial Autocorrelation
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.0
1
2
3
4
5
6
7
Lag
8
9
10
11
12
13
1
2
3
4
5
6
7
Lag
8
9
10
11
12
13
Sometimes the non-stationary can be identified directly from the SAC and SPAC
plots.
Monthly consumer price index Sweden (1980-2005)
300
CPI_Swe
250
Note! Monthly data, but of the kind
that usually do not contain seasonal
variation within a year.
200
150
100
1
31
62
93
124
155
Index
186
217
248
279
310
SAC and SPAC usually indicate an
AR(1)-model with slowly decreasing
autocorrelations and with first value
very close to 1
Autocorrelation Function for CPI_Swe
Partial Autocorrelation Function for CPI_Swe
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
(with 5% significance limits for the autocorrelations)
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Seasonal ARIMA-models
(Weak) stationarity is often (wrongly) connected with a series that seems to vary
non-systematically around a constant mean
6
10
4
8
2
6
4
0
2
-2
0
-4
1
30
60
90
120
150
Index
180
Stationary?
210
240
270
300
1
30
60
90
120
150
Index
180
210
240
Non-Stationary?
270
300
6
4
2
0
-2
-4
1
30
60
90
120
150
Index
180
210
240
270
300
Autocorrelation Function
Partial Autocorrelation Function
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
(with 5% significance limits for the autocorrelations)
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
10
8
6
4
2
0
1
30
60
90
120
150
Index
180
210
240
270
Partial Autocorrelation Function
Autocorrelation Function
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
(with 5% significance limits for the autocorrelations)
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
5
10
15
20
25
30
35
Lag
40
300
45
50
55
60
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Autocorrelation Function
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Are the spikes outside the red border evidence of non-stationarity?
We can always try to differentiate the series:
zt=yt – yt-1

10
5
0
-5
Autocorrelation Function
(with 5% significance limits for the autocorrelations)
-10
1
30
60
90
120
150
Index
180
210
240
270
1.0
300
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
No improvement!!
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Autocorrelation Function
(with 5% significance limits for the autocorrelations)
1.0
0.8
A utocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Note that the spikes (besides the first ones) lie around the lags 12, 24, 36, 48 and
60.
Could it have something to do with seasonal variation?
Seasonal AR-models:
yt    1  yt 1     p  yt  p  1, L  yt  L    P , L  yt  PL  at
where L is the number of seasons (during a year)
Such a model takes care of both short-memory and long-memory relations within
the series yt .
More correct terms are nonseasonal and seasonal variation.
The series can still be stationary.
We differ between stationarity at the nonseasonal level and stationarity at the
seasonal level.
We do not consider the model as an AR(P L)-model!
In a stationary Seasonal AR-process (SAR(p,P) )
• ACF spikes at nonseasonal level (scale), i.e. between 1 and L die down in an
exponential fashion (possibly oscillating).
• PACF spikes at non-seasonal level (scale) cuts off after lag p.
• ACF spikes at seasonal level (scale), i.e. at lags L, 2L, 3L, 4L, … die down
in an exponential fashion (possibly oscillating).
• PACF spikes at seasonal level (scale) cuts off after lag PL.
• Moderate ACF and PACF spikes usually exist around L, 2L, 3L, 4L, …
A more correct formulation of the model is
1   B  
1
2


 B 2     p  B p  1  1, L  B L  2, L  B 2L    P, L  B PL yt    at
where Byt = yt – 1 , B2yt = yt – 2 , …, BLyt = yt – L , … (the backshift operator)
In the special case of p=1 and P=1 we get
1  1  B   1  1,L  B L yt    at

 1    B  

 1  1  B  1, L  B L  1  1, L  B  B L yt    at
1

L
L 1

B





B
yt    at
1, L
1
1, L
 yt  1  yt 1  1, L  yt  L  1  1, L  yt  L 1    at
 yt    1  yt 1  1, L  yt  L  1  1, L  yt  L 1  at
i.e. we should model a dependency at lags 1, 12 and 13 to take into account the
”double” autoregressive structure
Seasonal MA-models (SMA(q,Q))
yt    at  1  at 1     q  at  q  1, L  at  L     Q , L  at Q
• ACF spikes at nonseasonal level cuts off after lag q.
• PACF spikes at nonseasonal level, i.e. between 1 and L die down in an
exponential fashion (possibly oscillating).
• ACF spikes at seasonal level cuts off after lag QL.
• PACF spikes at seasonal level, i.e. at lags L, 2L, 3L, 4L, … die down in
an exponential fashion (possibly oscillating).
•
• Moderate ACF and PACF spikes usually exist around L, 2L, 3L, 4L, …
The model can be written with backshift operator B analogously with
SAR-models.
Seasonal ARMA-models (SARMA(p,P,q,Q))
Expression becomes more condensed with backshift operator:
1    B    B      B  1    B  
   1    B    B      B  1    B
2
1
p
2
L
p
1, L
2
1
2
q
q
1, L
2 L
P L
yt 

B





B
2, L
P,L
L
  2, L  B 2L     Q , L  B QL at
Note that the expressions within parentheses are polynomials either in B or in BL.
A more common formulation is therefore to denote these polynomials
 p B,  P B L ,q B and Q B L 
 
 
  p B  P B L yt    q B Q B L at
SARMA-models have similar patterns at non-seasonal scale and at seasonal scale
as those of ARMA-models, i.e. a mix of sinusoidal and exponentially decreasing
spikes.
Non-stationary series?
yt ~ ARIMA(p,d,q,P,D,Q)L
means taking dth order differences at nonseasonal level  zt = (1 – B)d yt
(so-called regular differences) and Dth order differences at seasonal level
wt = (1 – BL)D zt
wt = (1 – BL)D (1 – B)d yt
Then, model the differenced series with SARMA(p,P,q,Q)

Have another look at the SAC and SPAC of the series with obvious seasonal
variation:
SAC spikes at exact seasonal
lags die down
Autocorrelation Function
(with 5% significance limits for the autocorrelations)
1.0
0.8
SAC and SPAC spikes close to exact
seasonal lags are pronounced
0.4
0.2
0.0
-0.2
-0.4
SPAC spikes at exact
seasonal lags guts off at
lag 1
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Partial Autocorrelation Function
(with 5% significance limits for the partial autocorrelations)
1.0
SAC nonseasonal spikes die down
SPAC nonseasonal spikes
might cut off at lag 1
ARIMA(1,0,0,1,0,0)12 ??
0.8
Partial Autocorrelation
Autocorrelation
0.6
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Minitab: StatTime Series…ARIMA…
ARIMA( 1 , 0 , 0 , 1 , 0 , 0 ) 12
Final Estimates of Parameters
Type
AR
1
SAR
12
SE Coef
T
P
-0.3089
0.0554
-5.57
0.000
0.8475
0.0340
24.91
0.000
1.17077
0.05320
22.01
0.000
5.8672
0.2666
Constant
Coef
Mean
Number of observations:
Residuals:
OK!
300
SS =
251.964 (backforecasts excluded)
MS =
0.848
DF = 297
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
DF
P-Value
12
24
36
48
20.8
51.3
62.6
81.2
9
21
33
45
0.014
0.000
0.001
0.001
Not OK !
The time series in question has actually been generated with the model
1  0.3  B   1  0.8  B12 yt  1.5  at
with at i.i.d N(0.1)
This model is stationary, as conditions for stationarity in AR(1)-models are fulfilled
at both nonseasonal and seasonal level.
Type
10
8
6
SE Coef
T
P
AR
1
-0.3089
0.0554
-5.57
0.000
SAR
12
0.8475
0.0340
24.91
0.000
1.17077
0.05320
22.01
0.000
5.8672
0.2666
Constant
4
Coef
Mean
2
0
1
30
60
90
120
150
Index
180
210
240
270
300
Still there might be problems with the Ljung-Box statistics!
An example with real data:
Monthly registered men at work (labour statistics) in pulp and paper related industry
from January 1987 to March 2005
Time Series Plot of Employed (AKU), times 100
500
Employed (AKU), times 100
450
400
350
300
250
200
150
Month jan
Year 1987
jan
1990
jan
1993
jan
1996
jan
1999
jan
2002
The series possesses a downward trend and seasonal pattern.
jan
2005
Autocorrelation Function for Employed (AKU), times 100
(with 5% significance limits for the autocorrelations)
1.0
Obvious signs of nonstationarity.
0.8
Autocorrelation
0.6
0.4
0.2
Try 1 regular difference:
0.0
(1 – B)yt
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
55
and additionally 1 seasonal
difference
(1 – B12)(1 – B)yt
Partial Autocorrelation Function for Employed (AKU), times 100
(with 5% significance limits for the partial autocorrelations)
1.0
Partial Autocorrelation
0.8
MTB > diff c5 c6
0.6
0.4
MTB > diff 12 c6 c7
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
55
Time Series Plot of C7
Autocorrelation Function for C7
(with 5% significance limits for the autocorrelations)
100
1.0
0.8
0.6
-50
-100
-150
Month jan
Year 1987
jan
1990
jan
1993
jan
1996
jan
1999
jan
2002
jan
2005
Autocorrelation
0
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
45
50
Partial Autocorrelation Function for C7
AR(2) at nonseasonal level?
(with 5% significance limits for the partial autocorrelations)
1.0
MA(1) at seasonal level?
0.8
Partial Autocorrelation
C7
50
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
Final Estimates of Parameters
Type
Coef
SE Coef
T
P
AR
1
-0.8199
0.0505
-16.24
0.000
AR
2
-0.7120
0.0499
-14.28
0.000
SMA
12
0.6275
0.0558
11.24
0.000
-0.0484
0.7754
-0.06
0.950
Constant
Differencing: 1 regular, 1 seasonal of order 12
Number of observations:
Residuals:
Original series 219, after differencing 206
SS =
176265 (backforecasts excluded)
MS =
873
DF = 202
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
DF
P-Value
12
24
36
48
20.0
32.0
52.6
73.4
8
20
32
44
0.010
0.044
0.012
0.004
ACF of Residuals for Employed (AKU), times 100
(with 5% significance limits for the autocorrelations)
1.0
0.8
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
6
12
18
24
Lag
30
36
42
48
PACF of Residuals for Employed (AKU), times 100
(with 5% significance limits for the partial autocorrelations)
1.0
0.8
Partial Autocorrelation
Autocorrelation
0.6
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
6
12
18
24
Lag
30
36
42
48