Habilitation à diriger des recherches +1cm Resampling methods for

Transcription

Habilitation à diriger des recherches
Resampling methods for periodic and almost
periodic processes
Anna Dudek
Université Rennes 2
[email protected]
Anna Dudek (
Université Rennes
Habilitation
2 à[email protected]
diriger des recherches Resampling )methods for periodic and almost periodic
1 / 45 p
Main research areas:
1
stochastic processes with some periodic/almost periodic structure;
2
resampling methods.
Today:
why periodic processes
why bootstrap
overview of bootstrap methods for periodic processes
almost periodic case
some open problems and current projects
Anna Dudek (
Université Rennes
Habilitation
2 / 45 p
Examples
Number of incoming packets in one hour non-overlapping bins. The
measurement was conducted at the border between the network of
University of Waikato and the internet provider. Time of observation is
20 workdays (480 hours).
6
4·10
6
3·10
6
2·10
6
1·10
100
200
300
400
Data from R. Nelson and B. Jones from Univ. of Waikato.
Université Rennes
Habilitation
3 / 45 p
Anna Dudek (
Examples
Volumes of energy traded hourly on the Nord Pool Spot Exchange (6
July - 31 August 2010, N = 984 records) - without weekends
Source: http://www.npspot.com
Anna Dudek (
Université Rennes
Habilitation
4 / 45 p
Examples
Mean monthly flow of Fraser River (1913-1990)
Source: http://www.umass.edu/statdata/statdata/stat-time.html
Anna Dudek (
Université Rennes
Habilitation
5 / 45 p
Examples
Garden blower signal
Source: http://www.reliableplant.com
Anna Dudek (
Université Rennes
Habilitation
6 / 45 p
Characteristics of interest
Periodic time series
Definition
Time series Xt , t ∈ Z is strictly periodic with period d , if for each
t, τ, . . . , τr ∈ Z
L
Xt , Xt+τ1 , . . . , Xt+τr = Xt+d , Xt+τ1 +d , . . . , Xt+τr +d
for each r ∈ Z.
Definition
Time series Xt , t ∈ Z is weakly periodic of order r with period d , if
E|Xt |r < ∞ for each t, τ, . . . , τr −1 ∈ Z
E Xt Xt+τ1 . . . Xt+τr −1 = E Xt+d Xt+τ1 +d . . . Xt+τr −1 +d .
Source: Synowiecki (2007)
Anna Dudek (
Université Rennes
Habilitation
7 / 45 p
First and second order characteristics:
overall mean
seasonal means
seasonal variances
autocovariance function
Fourier coefficients of the mean and the autocovariance functions
Problems:
usually parameters of interest are multidimensional −→
asymptotic simultaneous confidence intervals;
they are difficult to obtain: issues with asymptotic variance and
calculation of quantiles.
Anna Dudek (
Université Rennes
Habilitation
8 / 45 p
Why bootstrap
to construct pointwise confidence intervals
to construct simultaneous confidence intervals
for testing (e.g., detection of significant frequencies)
Which bootstrap method should be used?
Anna Dudek (
Université Rennes
Habilitation
9 / 45 p
Bootstrap methods
MBB
Moving Block Bootstrap (MBB) algorithm - Künsch
(1989), Liu and Sight (1992)
(X1 , . . . , Xn ) - observed sample
b = bn - block length
n = lb, l ∈ N
Block Bi is defined as
Bi = (Xi , . . . , Xi+b−1 ),
where i = 1, . . . , n − b + 1.
1 2 3
12
15
24
36
n-b+1
n
B1
B2
B3
Anna Dudek (
Bn-b+1
Université Rennes
Habilitation
10 / 45 p
Bootstrap methods
MBB
Moving Block Bootstrap algorithm
From the set {B1 , . . . , Bn−b+1 } select randomly with replacement l
blocks.
1
, i, j = 1, . . . , n − b + 1.
P Bi∗ = Bj =
n−b+1
Joining the blocks we get
X ∗ = (B1∗ , . . . , Bl∗ ).
Anna Dudek (
Université Rennes
Habilitation
11 / 45 p
Bootstrap methods
MBB
MBB - disadvantages
method designed for stationary time series
destroys the periodic structure of the considered time series
its application for periodic time series turned out to be limited to
the overall mean case (Synowiecki (2007))
WE NEED METHODS DESIGNED FOR PERIODIC DATA!
(X1 , . . . , Xn ) - sample from the periodic (in distribution or moments)
time series with period d
Anna Dudek (
Université Rennes
Habilitation
12 / 45 p
Bootstrap methods
SBB
Seasonal Block Bootstrap algorithm - Politis (2001)
1
d
2d
3d
4d
5d
B1
Bd+1
B2 d+1
1.5
B1*
B2*
B3*
1.0
0.5
50
100
150
200
250
-0.5
-1.0
-1.5
Figure: B1∗ = Bd+1 , B2∗ = B1 , B3∗ = Bd+1
Anna Dudek (
Université Rennes
Habilitation
13 / 45 p
Bootstrap methods
SBB
SBB - disadvantages
minimal block length is equal to period length
block length is always an integer multiple of the period length
Anna Dudek (
Université Rennes
Habilitation
14 / 45 p
Bootstrap methods
PBB
Periodic Block Bootstrap algorithm - Chan et al. (2004)
1.5
A1
B1
C1
A2
B2
C2
A3
B3
C3
1.0
0.5
50
100
150
-0.5
-1.0
-1.5
1.5
A*1
B*1
C1*
A*2
B*2
C2*
A*3
B*3
C3*
1.0
0.5
50
100
150
-0.5
-1.0
-1.5
Figure: A∗1 = A2 , B1∗ = B1 , C1∗ = C2 , A∗2 = A3 , B2∗ = B2 , C2∗ = C2 , A∗3 = A1 , B3∗ =
B3 , C3∗ = C3
Anna Dudek (
Université Rennes
Habilitation
15 / 45 p
Bootstrap methods
PBB
PBB - disadvantages
is designed for periodic time series that have long periodicities,
since it is assumed that the block length b is much smaller
compared to the period d
Leśkow and Synowiecki (2010) showed the consistency of PBB
procedure assuming that the period length d → ∞ as the sample
size n → ∞ increases
Anna Dudek (
Université Rennes
Habilitation
16 / 45 p
Bootstrap methods
What did we need?
a new bootstrap procedure that is suitable for periodic time series
with fixed length periodicities of arbitrary size as related to block
size and sample size
Since 2011 we introduced:
Generalized Seasonal Block Bootstrap (Dudek et al. (2014a))
Generalized Seasonal Tapered Block Bootstrap (Dudek et al.
(2016))
Extension of Moving Block Bootstrap (Dudek (2015))
Anna Dudek (
Université Rennes
Habilitation
17 / 45 p
Bootstrap methods
Any second order random process that is generated by the mixing
(in the workings of a system) of randomness and periodicity will
likely have the structure of PERIODIC CORRELATION.
Definition
A random sequence {Xt , t ∈ Z} with finite second moments is called
periodically correlated (PC) or cyclostationary with period d if it has
periodic mean and covariance functions e.g.
E (Xt ) = E (Xt+d )
and
Cov (Xt , Xs ) = Cov (Xt+d , Xs+d )
for each t, s ∈ Z. To avoid ambiguity, the period d is taken as the
smallest positive integer such that above holds.
Source: Gladyshev (1961), Hurd and Miamee (2007); Applications: Gardner,
Napolitano, Paura (2006), Antoni (2009), Napolitano (2016).
Anna Dudek (
Université Rennes
Habilitation
18 / 45 p
Bootstrap methods
GSBB
GSBB - Generalized Seasonal Block Bootstrap
(Dudek et al. (2014a))
(X1 , . . . , Xn ) - a sample from the periodic (in distribution or
moments) time series with period d
n = lb, l ∈ N
Bi = (Xi , . . . , Xi+b−1 ),
where i = 1, . . . , n − b + 1.
Anna Dudek (
Université Rennes
Habilitation
19 / 45 p
Bootstrap methods
GSBB
1.5
1.0
0.5
10
20
30
40
50
60
-0.5
-1.0
d = 12, b = 15, n = 60
Anna Dudek (
Université Rennes
Habilitation
20 / 45 p
Bootstrap methods
1
12 13
15
24 25
GSBB
27
36 37
39
48
51
60
B1
B13
B25
B37
1
4
12
16
18
24
28
30
36
40
42
48
54
60
B4
B16
B28
B40
First step choice:
Anna Dudek (
B1∗
= B13 . Second step choice: B2∗ = B4 .
Université Rennes
Habilitation
21 / 45 p
Bootstrap methods
1
7
12
19
21
24
GSBB
31
33
36
43
45
48
57
60
B7
B19
B31
B43
1
10
12
22
24
34
36
46
48
60
B10
B22
B34
B46
Third step choice:
Anna Dudek (
B3∗
= B31 . Fourth step choice: B4∗ = B46
Université Rennes
Habilitation
22 / 45 p
Bootstrap methods
1.5
B*1
GSBB
B*2
B*3
B*4
1.0
0.5
10
20
30
40
50
60
-0.5
-1.0
-1.5
Anna Dudek (
Université Rennes
Habilitation
23 / 45 p
Bootstrap methods
GSBB
Comments:
GSBB is a generalization of existing before bootstrap methods:
Seasonal Block Bootstrap (block length is an integer multiple of
the period length) - Politis (2001)
Periodic Block Bootstrap (block length is smaller than the period
length) - Chan et al. (2004)
GSBB: there is no relation between block and period lengths.
Anna Dudek (
Université Rennes
Habilitation
24 / 45 p
Bootstrap methods
GSBB
GSBB consistency results
PC time series:
overall mean, seasonal means (Dudek et al. (2014a))
seasonal variances, autocovariance function (Dudek (2016) submitted)
(Dudek et al. (2014b))
Period d → ∞ as sample size n → ∞:
overall mean (Dudek et al. (2014a))
(Dudek (2016))
Anna Dudek (
Université Rennes
Habilitation
25 / 45 p
Bootstrap methods
TBB
TBB - Tapered Block Bootstrap (Paparoditis and Politis
(2001))
Method designed for stationary time series; modification of
MBB-Moving Block Bootstrap (Künsch (1989))
(X1 , . . . , Xn ) - a sample from the stationary time series
n = lb ∈ N
Anna Dudek (
Université Rennes
Habilitation
26 / 45 p
Bootstrap methods
TBB
Tapered Block Bootstrap algorithm
e = X − X , where X = 1/n
center the data, i.e. let X
t
t
Pn
i=1 Xt ;
e ,...,X
e
from the set {B1 , . . . , Bn−b+1 }, where Bi = (X
i
i+b−1 ) select
randomly with replacement l blocks. Probability of choosing any
block is 1/(n − b + 1);
join selected blocks to get X ∗ = (B1∗ , . . . , Bl∗ );
for m = 0, . . . , l − 1
√
∗
Ymb+j
:= wb (j)
b
X∗ ,
||wb ||2 mb+j
where wn (t) = w t−0.5
, w : R → [0, 1], symmetric about t = 0.5,
n
nondecreasing for t ∈ [0, 0.5], w(t) > 0 for t in a neighbourhood
of 0.5.
Anna Dudek (
Université Rennes
Habilitation
27 / 45 p
Bootstrap methods
TBB
TBB advantages
In the setting of linear or approximately linear statistics, usually block
resampling methods (MBB, CBB, Stationary Block Bootstrap) have a
MSE of variance estimator of order O(n−2/3 ), while TBB has a MSE of
order O(n−4/5 ), outperforming all methods.
HOW TO APPLY TBB FOR PERIODIC DATA?
Anna Dudek (
Université Rennes
Habilitation
28 / 45 p
Bootstrap methods
GSTBB
Generalized Seasonal Tapered Block Bootstrap
algorithm - seasonal means case
Let (X1 , . . . , Xn ) be a sample from the periodic (in distribution or
moments) time series with period d; b block length, n = lb.
e =X −µ
b<t> , where < t >= (t mod d) denotes the
define X
t
t
season associated with t;
the bootstrap sample X1∗ , . . . , Xn∗ is generated using GSBB to the
e ,...,X
e ;
sample X
n
1
for m = 0, . . . , l − 1
√
∗
Ymb+j
Anna Dudek (
:= wb (j)
b
X∗ .
||wb ||2 mb+j
Université Rennes
Habilitation
29 / 45 p
Bootstrap methods
GSTBB
Comments
In practical applications, the GSTBB should not be used with
b ≤ d such that d = kb for k ∈ N especially for simultaneous
confidence intervals. In such a case, the GSTBB provides too high
or too low coverage probabilities. If b = d observations from the
first and the last season are used with lower weights. By contrast,
if b = 2d, then lower weights are no longer assigned to all
observations from aforementioned seasons, and this negative
effect disappears.
Tapering idea is applied to residuals obtained after removing
seasonal means from the data. The usual TBB was designed for
stationary time series and hence to have zero-mean observations
it was enough to subtract the overall mean from each observation.
In a case of other characteristics (e.g., Fourier coefficients of
the autocovariance function) further modifications of
algorithm are essential.
Anna Dudek (
Université Rennes
Habilitation
30 / 45 p
Bootstrap methods
GSTBB
GSTBB consistency results
PC time series:
overall mean, seasonal means (Dudek et al. (2016))
(Dudek et al. (2016)) - modified algorithm!
GSTBB advantages:
it provides often the actual coverage probability curves
(comparing to the GSBB ones) flatter and closer to the nominal
coverage level; especially for short samples;
Anna Dudek (
Université Rennes
Habilitation
31 / 45 p
Bootstrap methods
EMBB
Problems
GSBB requires knowledge of period length, but sometimes it
may happen that period length is not known or considered signal
is a composition of two components with incommensurable
periods. To model such data almost periodically correlated (APC)
processes are used. These are processes which mean and
autocovariance functions are almost periodic.
The MBB method in contrary to the GSBB does not keep the
periodic structure of the original sample. Applied directly to PC
data it provides often inconsistent estimators. The main problem is
that having the MBB sample one cannot identify which seasons
the bootstrap observations come from. The only case for which
the MBB can be used for PC/APC time series is the overall mean
case.
Anna Dudek (
Université Rennes
Habilitation
32 / 45 p
Bootstrap methods
EMBB
Extension of MBB (Dudek(2015))
Idea: modify the MBB to make it suitable for PC and APC data.
EMBB algorithm
Let {Xt , t ∈ Z} be PC or APC time series.
Define a bivariate series Yi = (Xi , i) and then
do the MBB on the
∗
∗
sample (Y1 , . . . , Yn ) to obtain Y1 , . . . , Yn .
Comments:
In the second coordinate of the series Y1∗ , . . . , Yn∗ we preserve
the information on the original time indices of chosen
observations. Using it one may define consistent estimators of
time and frequency domain characteristics of PC/APC time series.
Anna Dudek (
Université Rennes
Habilitation
33 / 45 p
Bootstrap methods
EMBB
EMBB consistency results
PC time series:
seasonal means, seasonal variances, autocovariance function
(Dudek et al. (2016) - submitted)
PC/APC time series:
(Dudek (2015))
Anna Dudek (
Université Rennes
Habilitation
34 / 45 p
Bootstrap methods
Circular versions of GSBB, GSTBB, EMBB
Usual MBB:
Bi = (Xi , . . . , Xi+b−1 ),
1 2 3
12
15
where
24
i = 1, . . . , n − b + 1.
36
n-b+1
n
B1
B2
B3
Bn-b+1
Problem:
Observations that are in the center of the considered sample are
present in b different blocks, while the observations from the beginning
and the end of the sample appear more rarely e.g., X1 belongs only
to B1 . This often leads to increase of the bias of the considered
estimator.
Anna Dudek (
Université Rennes
Habilitation
35 / 45 p
Bootstrap methods
Circular Block Bootstrap (CBB) - Politis and Romano
(1992)
the CBB is a modification of the MBB;
data are treated as wrapped on the circle;
each observation is present in the same number of blocks;
we have exactly n blocks of the length b.
1 2 3
b
n-b+1
n
B1
Bn-b+1
Bn-b+5
Anna Dudek (
Université Rennes
Habilitation
36 / 45 p
Bootstrap methods
Circular versions of GSBB, GSTBB, EMBB:
the only change in algorithms is the set of possible block length
choices. Now it is of the form {B1 , . . . , Bn }. All rules how the
blocks can be select remain unchanged.
Anna Dudek (
Université Rennes
Habilitation
37 / 45 p
Bootstrap methods
For stationary time series:
b
the CBB is providing usually the unbiased estimator E∗ (θb∗ ) = θ.
PC/APC time series:
If n = lb the circular EMBB estimators of the overall mean, the
seasonal means, the seasonal variances and Fourier coefficients
of the mean function (PC and APC case) are unbiased;
If additionally n = wd, then the corresponding cGSBB estimators
are also unbiased;
b∗ (λ, τ ) for τ 6= 0 is
Independently on the bootstrap approach a
always biased.
In general:
Circular block bootstrap algorithms allow to reduce computational
cost comparing with the standard approaches;
Circular bootstrap algorithms are easier and more clear.
Anna Dudek (
Université Rennes
Habilitation
38 / 45 p
Bootstrap methods
Some open problems
modification of Stationary Bootstrap for PC/APC time series (joint
project with D. Politis and E. Paparoditis);
block length choice;
bootstrap applicability for data with jitter effect (joint project with
D. Dehay and M. Elbadaoui);
testing (joint project with Ł. Lenart)
choice of bootstrap approach for particular problem;
heavy-tailed data;
...
Anna Dudek (
Université Rennes
Habilitation
39 / 45 p
Bootstrap methods
Other projects
bootstrap in testing smoothness parameter of a density function
(joint project with B. Ćmiel)
application of harmonizable processes for EEG data analysis
(joint project with J. Aston, D. Dehay and J.M. Freyermuth);
bootstrap confidence sets in matrix completion problem (joint
project with A. Carpentier, J.M. Freyermuth and M. Zhilova);
Anna Dudek (
Université Rennes
Habilitation
40 / 45 p
Bootstrap methods
References
ANTONI, J. (2009). Cyclostationarity by examples. Mech. Syst Sig
Process., 23(4), 987-1036.
CHAN, V., LAHIRI, S.N., and MEEKER, W.Q (2004). Block
bootstrap estimation of the distribution of cumulative outdoor
degradation. Technometrics, 46, 215-224.
DUDEK, A.E. (2015). Circular block bootstrap for coefficients of
autocovariance function of almost periodically correlated time
series. Metrika, 78(3), 313-335.
DUDEK, A.E. (2016). First and second order analysis for periodic
random arrays using block bootstrap methods. Electron. J. Statist.,
10(2), 2561-2583.
DUDEK, A.E. (2016). GSBB and MBB for periodic characteristics
of PC time series - submitted.
DUDEK, A.E., LEŚKOW, J., PAPARODITIS, E. and POLITIS, D.
(2014a). A generalized block bootstrap for seasonal time series. J.
Time Ser. Anal., 35, 89-114.
Anna Dudek (
Université Rennes
Habilitation
41 / 45 p
Bootstrap methods
References
DUDEK, A.E., MAIZ, S. and ELBADOUI, M. (2014b). Generalized
Seasonal Block Bootstrap in frequency analysis of cyclostationary
signals. Signal Process., 104C, 358-368.
DUDEK, A.E., PAPARODITIS, E. and POLITIS, D. (2016).
Generalized Seasonal Tapered Block Bootstrap, Statistics and
Probability Letters, 115, 27-35.
GARDNER, W.A., NAPOLITANO, A., PAURA, L. (2006).
Cyclostationarity: half a century of research. Signal Processing,
86, 639–697.
GLADYSHEV, E.G. (1961). Periodically Correlated Random
Sequences. Soviet mathematics, 2, 385–388.
HURD, H.L., MIAMEE, A.G. (2007).Periodically Correlated
Random Sequences: Spectral. Theory and Practice. Wiley.
KUNSCH, H. (1989) The jackknife and the bootstrap for general
stationary observations, Ann. Statist., 17, 1217-1241.
Anna Dudek (
Université Rennes
Habilitation
42 / 45 p
Bootstrap methods
References
LENART, Ł., LEŚKOW, J. and SYNOWIECKI, R. (2008).
Subsampling in testing autocovariance for periodically correlated
time series. J. Time Ser. Anal., 29 995-1018.
NAPOLITANO, A. (2016). Cyclostationarity: New trends and
applications. Signal Process., 120, 385-408.
PAPARODITIS, E. and POLITIS, D. (2001). Tapered block
bootstrap Biometrika, 88, 1105-1119.
POLITIS, D.N. (2001). Resampling time series with seasonal
components, in Frontiers in Data Mining and Bioinformatics:
Proceedings of the 33rd Symposium on the Interface of
Computing Science and Statistics, Orange County, California,
June 13-17, pp. 619-621.
Anna Dudek (
Université Rennes
Habilitation
43 / 45 p
Bootstrap methods
References
Politis, D.N. and Romano, J.P. (1992). A circular block-resampling
procedure for stationary data. Exploring the Limits of Bootstrap,
Wiley Ser. Probab. Math. Statist. Probab. Math. Statist. Wiley, New
York, pp 263-270.
SYNOWIECKI, R. (2007). Consistency and application of moving
block bootstrap for nonstationary time series with periodic and
almost periodic structure. Bernoulli, 13(4), 1151-1178.
Anna Dudek (
Université Rennes
Habilitation
44 / 45 p
Bootstrap methods
This project has received
funding from the European
Union’s Horizon 2020 research
and innovation programme under
the Marie Skłodowska-Curie
grant agreement No 655394.
Anna Dudek (
Université Rennes
Habilitation
45 / 45 p

Habilitation à diriger des recherches +1cm Resampling methods for

Transcription

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