ASCMOAdvances in Statistical Climatology, Meteorology and OceanographyASCMOAdv. Stat. Clim. Meteorol. Oceanogr.2364-3587Copernicus PublicationsGöttingen, Germany10.5194/ascmo-2-1-2016Comparison of hidden and observed regime-switching autoregressive models for (u,v)-components of wind fields in the northeastern AtlanticBessacJuliejbessac@anl.govAilliotPierreCattiauxJulienMonbetValerieInstitut de Recherche Mathématiques de Rennes, UMR 6625, Université de Rennes 1, Rennes, FranceMathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, USALaboratoire de Mathématiques de Bretagne Atlantique, UMR 6205, Université de Brest, Brest, FranceCNRM-GAME, UMR 3589, CNRS/Météo France, Toulouse, FranceINRIA Rennes, ASPI, Rennes, FranceJulie Bessac (jbessac@anl.gov)29February20162111618August201523December201510February2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://ascmo.copernicus.org/articles/2/1/2016/ascmo-2-1-2016.htmlThe full text article is available as a PDF file from https://ascmo.copernicus.org/articles/2/1/2016/ascmo-2-1-2016.pdf
Several multi-site stochastic generators of zonal and meridional components
of wind are proposed in this paper. A regime-switching framework is
introduced to account for the alternation of intensity and variability that
is observed in wind conditions due to the existence of different weather
types. This modeling blocks time series into periods in which the series is
described by a single model. The regime-switching is modeled by a discrete
variable that can be introduced as a latent (or hidden) variable or as an
observed variable. In the latter case a clustering algorithm is used before
fitting the model to extract the regime. Conditional on the regimes, the
observed wind conditions are assumed to evolve as a linear Gaussian vector
autoregressive (VAR) model. Various questions are explored, such as the
modeling of the regime in a multi-site context, the extraction of relevant
clusterings from extra variables or from the local wind data, and the link
between weather types extracted from wind data and large-scale weather
regimes derived from a descriptor of the atmospheric circulation. We also
discuss the relative advantages of hidden and observed regime-switching
models. For artificial stochastic generation of wind sequences, we show that
the proposed models reproduce the average space–time motions of wind
conditions, and we highlight the advantage of regime-switching models in
reproducing the alternation of intensity and variability in wind conditions.
Introduction
In this section, we present the context of our work and then the data used to
compare the proposed Markov-switching autoregressive models.
Stochastic weather generators have been used to generate artificial sequences
of small-scale meteorological data with statistical properties similar to the
data set used for calibration. Various wind condition generators at a single
site have been proposed in the literature; see ,
, and . However, few models have been
introduced in a multi-site context . Artificial
sequences of wind conditions provided by stochastic weather generators enable
assessment risks in impact studies; see, for instance, .
Here we propose a multi-site generator for Cartesian components of surface
wind. As far as we know, only a few models have been proposed to simulate
time series of Cartesian coordinates of wind {ut,vt}. Except in
, these models are designed for short-term wind prediction
and not for the generation of artificial conditions of {ut,vt}.
Consequently they are not focused on reproducing the same statistics we are
interested in, namely, the marginal distribution of {ut,vt} and
its spatiotemporal dynamics. In , a stochastic generator for
multiple temporal and spatial scales is proposed. The proposed
Markov-switching vector autoregressive model enables reproduction of many
spatial and temporal features; however, complex dependencies between
intensity and direction remain hard to model.
In the northeastern Atlantic, the spatiotemporal dynamics of the wind field
is complex. This area is under the influence of an unstable atmospheric jet
stream whose large-scale fluctuations induce local alternations between
periods with high wind intensity and strong temporal variability, and less
intense and variable periods. Scientists have proposed describing the North
Atlantic atmospheric dynamics through a finite number of preferred states,
namely, weather regimes or weather types . However,
introducing regime-switching in the modeling of local wind, as we propose in
this paper, enables us to better reproduce the spatiotemporal characteristics
observed in the wind data. In practice, describing a time series by regimes
involves a partitioning into time periods in which the series is homogeneous
and can be described by a single model. In this paper, we propose various
vector autoregressive (VAR) models with regime-switching. One of the
challenges is to achieve a regime-switching that is physically consistent and
that enables appropriate describing of the local observation by a VAR model.
To this end, we introduce several frameworks of regime-switching and compare
them in terms of simulation of wind data.
Depending on the availability of good descriptors of the current weather
state, regime-switching can be introduced with either observed or latent
regimes. Regimes are said to be observed when they are identified a priori,
before the modeling of the local dynamics. In this case, clustering methods
are run on adequate variables to obtain relevant regimes: either the local
variables or extra variables characterizing the large-scale weather
situation, such as descriptors of the large-scale atmospheric circulation
or variables enabling the separation into dry and
wet states . For wind models, the wind
direction can be considered since it is a good descriptor of synoptic
conditions. In , the wind direction is used both to extract
regimes and to parameterize of the predictive distribution. In this paper, we
propose a priori clusterings based on both large-scale and local variables.
When the regimes are said to be latent, they are introduced as a hidden
variable in the model. This framework is more complex from a statistical
point of view and the conditional distribution of wind given that the regime
has to be simple and tractable. Hidden Markov models (HMMs) have been widely
used for meteorological data . Hidden
Markov-switching autoregressive (MS-AR) models are a generalization of HMMs
allowing temporal dynamics within the regimes . Models with
regime-switching improve the modeling of wind intensity time series with
classical autoregressive–moving-average (ARMA) models; see
, where the wind speed is modeled at one site. Here we
propose a hidden MS-AR model and compare it with several models with observed
regime-switching.
To the best of our knowledge, no comparison between observed and latent
regime-switching has been proposed in the field of stochastic generators of
wind conditions. In , a comparison is presented in terms of
wind prediction between models with hidden regimes and models driven by
observed regimes. In this work, we compare both kinds of models in a
simulation framework.
In the multi-site context, the regime can either be common to all sites
(i.e., scalar; see ) or introduced as a site-specific
regime , which enables one to
account for a wide range of space–time dependencies. However, a
site-specific regime appears to be computationally challenging
. We will show that the choice of a regional regime is
reasonable when a homogeneous area is selected.
The paper is organized as follows. MS-AR models are introduced in
Sect. , and their inference is described in cases of both
observed and latent regime-switching. The question of a regional regime is
addressed in Sect. . In Sect. , we introduce and
discuss different sets of a priori regimes obtained by clustering. In
Sects. and , respectively, we discuss the
advantages of the proposed models and highlight the differences between
observed and latent regime-switching models.
Wind data
The data under study are zonal (west–east) and meridional (north–south)
surface wind components {ut,vt} at 10 m a.s.l. (above sea
level) extracted from the ERA-Interim data set produced by the European
Centre for Medium-Range Weather Forecasts (ECMWF). It can be freely
downloaded from the url http://apps.ecmwf.int/datasets/data/interim-full-daily/ and used for scientific purposes.
Left: spatial hierarchical clustering of the moving variance
associated with wind speed with four clusters (symbols). Right: joint and
marginal distribution of {ut,vt} at the central location 10;
contour lines of the estimated joint density.
We focus on gridded locations between latitudes 46.5 and 48∘ N
and longitudes 6.75 and 10.5∘ W (15×7 grid points; see
Fig. ). The data set we have extracted consists of 32
December–January blocks of wind data from December 1979 to January 2011
picked every 6 h. Furthermore, the statistical inference is based on the
assumption that the 32 December–January blocks of wind components are
32 independent realizations of the same stationary process, a reasonable
assumption given the strong interannual variability of the wintertime
atmospheric dynamics at such a local scale. The training data set is then
composed of 32 independent blocks and each block has 4×62 observations. In order to study the relevance of using common regimes for
all the locations, a spatial hierarchical clustering has been used to choose
a homogeneous area (see Fig. ). The clustering is run on the
process of moving standard deviation of wind speed, which is described more
precisely in Sect. . This process is a good descriptor of the
temporal characteristics of wind time series (see Fig. ), and it is
computed as the standard deviation of wind speed over nine consecutive time
steps (i.e., 2 days). The dendogram associated with the clustering suggests
the use of four clusters that are depicted in Fig. . These four
clusters are likely to be divided into an inland cluster (+), an
intermediate cluster between ocean and land (△), a cluster
corresponding to flows that propagate into the Bay of Biscay (∘),
and a cluster for flows that propagate toward northern Europe (×).
Components {ut} and {vt} show a complex relationship, as
partially reflected by the joint distribution of {ut,vt}
(Fig. ). The margin of {ut} reveals two separate modes,
whereas that of {vt} does not exhibit a clear bimodality. The contour
lines show that the density is low around the point (0,0). It indicates that
the transitions between the two modes of each component are not realized
through a vanishing of the field, but rather through a rotation of the field.
The following transformation is used on both components {ut} and
{vt}. This transformation with α>1 facilitates the modeling
of the bimodality:
ũt=Utαcos(Φt),ṽt=Utαsin(Φt),
where {Ut} and {Φt}, respectively, denote wind speed and
wind direction. In practice, α is chosen empirically equal to 1.5.
This transformation has proven helpful in modeling the distribution of
{ut,vt} in .
Markov-switching vector autoregressive models
In this section, we introduce the proposed models and discuss their parameter
estimation in cases of both observed and latent regimes.
The models
In this paper, we consider the following class of models. Let St be a
discrete Markov chain with values in {1,…,M} describing the current
weather type as a function of time t. Conditional on the weather type, the
observed wind conditions are modeled as a vector autoregressive model. Given the current value of St,
the observation Yt is written as
Yt=A0(St)+A1(St)Yt-1+A2(St)Yt-2+…+Ap(St)Yt-p+(Σ(St))-1/2ϵt.Y∈R2K represents the observed power-transformed wind
components {ut,vt} at the K locations, given by the system
(Eq. ). For i∈{1,…,M}, A0(i) is a
2K-dimensional vector,
A1(i),…,Ap(i),Σ(i) are 2K×2K
matrices, and ϵ is a Gaussian white noise of dimension 2K.
Conditional independencies between S and Y are displayed on the
following directed acyclic graph (DAG) for p=1 (see , for
additional information about DAGs).
In this model, the regime S can be latent or observed; both
cases are discussed, respectively, in Sects. and .
The parameter estimation of the model can be performed by maximum likelihood
but in a different way in each framework.
For both kinds of models, covariates can be included. The easiest way is to
include them in the intercept parameter A0 or in transitions between
regimes. Transitions between regimes can be parameterized with a covariate
(when regimes are latent, a parameterization with an extra covariate is given
in and with the studied variable in and in
when regimes are defined a priori). In the context of
multi-site models, the choice of the covariate of non-homogeneous transitions
is delicate. We do not discuss this topic here and consider only homogeneous
transition models.
To avoid over-parameterization of the conditional models, we first work with
a reduced data set. In the following, all the proposed models will be fitted
on the subset of sites (1, 6, 10, 13, 18), the extension to a wider region
being left for future studies.
Estimation by maximum likelihood
First, let us suppose that the complete set of observations
(y1,…yT,s1,…sT) is available, which is
the case in Sect. . Assume that s0, y-1, and y0
are observed. Then the complete log-likelihood, associated with an
autoregressive order p=2 (we choose p=2 according to a previous work –
), is written as
log(L(θ;y1,…yT,s1,…sT|y-1,y0,s0))=log(L(θ(Y);y1T|y-1,y0,s0T))+log(L(θ(S);s1T|y-1,y0,s0)),
where θ=(θ(S),θ(Y)). θ(Y)
corresponds to the parameters of the VAR models,
θ(S)=Π=(πi,j)i,j=1,⋯,M the transition matrix
Π of the Markov chain S, and
y1T=(y1,…,yT). Let us denote ni,j the
number of occurrences of the event {(St,St+1)=(i,j)} for t∈{1,…,T-1}, ni,.=∑j=1Mni,j and ni=ni,.+δ{sT=i}, where δ is the Kronecker symbol, the
total number of occurrences of the regime i:
log(L(θ(Y);y1,…,yT|y-1,y0,s0T))=∑t=1Tlog(p(yt|yt-1,yt-2,st))=∑i=1M∑t∈{t|st=i}log(p(yt|yt-1,yt-2,st))=∑i=1Mni(-d2log(2π)-12log(det(Σ(i)))-∑t∈{t|st=i}12et′(Σ(i))-1et,
where et=(yt-A0(i)-A1(i)yt-1-A2(i)yt-2).
For each i∈{1,…,M}, each function
θ(Y,i)→ni(-d2log(2π)-12log(det(Σ(i)))-∑t∈{t|st=i}12et′(Σ(i))-1et
can be maximized separately, where
θ(Y,i)=(A0(i),A1(i),A2(i),Σ(i)).
The optimal estimates of A1(i) and A2(i) are computed by
writing the VAR(2) model as VAR(1): for all t∈{t|st=i},
YtYt-1=A1(i)A2(i)IdK0Yt-1Yt-2+ϵt0,
where IdK is the K×K-identity matrix.
Let us write
A(i)=A1(i)A2(i)IdK0
and
Zt=YtYt-1;
expressions of A^1(i) and A^2(i)
are extracted from the estimate
A^(i)=(∑t∈{t|st=i}ZtZt-1′)(∑t∈{t|st=i}Zt-1Zt-1′)-1.
The other optimal estimates are
A^0(i)=(IdK-A^1(i)-A^2(i))μ^(i),
where μ^(i)=1ni∑t∈{t|st=i}yt is the empirical mean of Y in regime i
and
Σ^(i)=1ni∑t∈{t|st=i}e^te^t′.Σ^(i) is the empirical variance of the empirical
residuals defined as e^t=(yt-A^0(i)-A^1(i)yt-1-A^2(i)yt-2).
Concerning the Markov chain S,
log(L(θ(S);s1,…,sT|y-1,y0,s0))=∑i,j=1Mni,jlog(πi,j),
the associated maximum likelihood estimator is
π^i,j=ni,jni,..
Time series of wind speed in January 2012 and a posteriori regimes
from the fitting of a H-MS-VAR. The lighter is the grey; the smaller
is the determinant of Σ(i). From top to bottom: sites 1, 10, and
18 when the model is fitted at a single location; fourth panel from the top:
extracted regimes when the model is fitted at the five locations (1, 6, 10,
13, 18). Bottom panel: wind direction and regimes at site 10.
When observations only of process Y are available and the realizations of
S are not given a priori, as in Sect. , one inference method
is to use the expectation–maximization (EM) algorithm, which is commonly run
to estimate the parameters of models with latent variables by maximum
likelihood. Since S is not observed, the EM algorithm aims at maximizing
the incomplete log-likelihood function based on the observations Y:
θ→Eθ(log(L(θ;Y1,…,YT,S1,…,ST))|Y-1T=y-1T,S0=s0).
It is proven that through the iterations of the algorithm, a convergent
sequence of approximation of the maximum likelihood estimator of θ is
computed.
The EM algorithm cycles through two steps: the expectation step and the
maximization step . The E step is performed through
forward–backward recursions (see for hidden MS-AR
models) that enable one to compute the smoothing probabilities
P(St|Y-1T=y-1T,S0=s0). At the M step, optimal
expressions of parameters of θ(Y), given in Eqs. (), (), and (),
are used. In each regime i, however, each observation yt is weighted
by the probability P(St=i|Y-1T=y-1T,S0=s0), for
instance,
μ^(i)=1∑t=1TP(St=i|Y-1T=y-1T,S0=s0)∑t=1TP(St=i|Y-1T=y-1T,S0=s0)yt.
The transition matrix is estimated from quantities
P({St=i,St+1=j}|Y-1T=y-1T,S0=s0) that are
derived at the E step.
Leftmost panel: matrix with the number of the station is printed;
then, from left to right, conditional probabilities of occurrence of regime
i=1,2,3 at all sites conditional on the simultaneous occurrence of the same
regime at site 10; in each pixel, the value of the conditional probability is
plotted.
In this paper, we use AP-MS-VARC to denote the a
priori regime-switching model associated with the clustering C,
and we use H-MS-VAR to denote the hidden regime-switching model.
Regime definition in a multi-site context
When the current weather state is not estimated a priori, it is introduced as
a latent variable. Hidden regime-switching models have been used in various
fields; see for a wide range of applications of hidden
Markov models. In a previous work a single-site model for
{ut,vt} was proposed; the proposed hidden Markov-switching
autoregressive model reveals good qualities to describe both marginal and
joint distributions of {ut,vt} as well as the temporal dynamics of
the wind at one location. In this paper we propose an extension of this
model, when the process {ut,vt} is multi-site. In a multi-site
context, the regime can be site-specific or common to all stations.
Here, the assumption of a common regional regime is investigated, and we show
that this assumption is acceptable when the considered area is homogeneous.
The homogeneous single-site MS-AR model introduced in for
{ut,vt} with M=3 regimes and an autoregressive order p=2 has
been fitted at each site. The most likely regimes associated with the data
are extracted from the estimation procedure of H-MS-VAR models
described in the previous section. At each time, the regime corresponds to
argmaxj∈{1,⋯,M}P(St=j|Y-1T=y-1T,S0=s0); see . In
order to properly compare the regimes, they are ordered according to the
increasing value of the determinant of the matrix Σ(i). The
intuition for sorting regimes according the determinant of Σ(i)
is that we expect the innovations to be more volatile, and consequently
Σ(i) to have greater eigenvalues, in cyclonic weather regimes.
Conversely, we expect to observe innovations more persistent in time in calm
weather regimes; this is associated with smaller eigenvalues of
Σ(i). The spatiotemporal coherence of the regimes of each of the
18 sites is checked and reveals a strong homogeneity that motivates the use
of a regional regime in this area.
The sequences of regimes are compared in Fig. ; time series of a
posteriori regimes and wind speed are depicted. The last two regimes are less
coherent from one site to another. This effect is partly explained by the
fact that these regimes are less persistent in time, especially the third one
(see Table ).
Moreover, we can notice an eastward propagation in wind events, the darkest
regimes often being observed at western stations (station 1) prior to eastern
sites (10 and 18). The bottom panel of Fig. , which depicts the
sequences of regimes associated with the model fitted on the set of all
locations with a regime common to all locations, reveals that this regional
regime is coherent with the local ones, although it is less persistent.
Indeed, when fitting the model to several stations, the regime has to embed
some spatial heterogeneity that is likely to decrease the temporal
persistence.
In Fig. , probabilities of occurrence of a given regime conditional
on the simultaneous occurrence of the same regime at site 10 are depicted for
all sites. In each picture, conditional probabilities should be compared with
the reference value given at location 10, which is 1 by construction. The
first regime has the best spatial coherence, and the third regime, which is
the least persistent regime, is less coherent spatially. The ranges of values
of these probabilities indicate a satisfying consistency between the regimes
across sites.
At each site, the physical interpretation of each regime is similar. Indeed,
the first regime corresponds mainly to anticyclonic conditions with easterly
winds and a slowly varying intensity (the variance of the innovation of the
AR model is lower than in the two other regimes, and the first AR coefficient
is larger; see Table ).
Parameter values obtained when fitting a H-MS-VAR at the
different sites: diagonal of the transition matrix Π, coefficients
of the autoregressive model in each regime, and logarithm of the determinant
of Σ(i).
The two other regimes correspond to cyclonic conditions with westerly winds
and a higher temporal variability in the intensity (see Fig. ).
These two regimes are discriminated mainly by the temporal variability, which
is higher in the third regime. Moreover, the wind direction, not depicted
here, slightly differs: from southwesterlies in the second regime to
northwesterlies in the third regime.
Top panel: moving mean of wind speed computed on 2-day intervals
(nine time steps) for each regime of the H-MS-VAR model fitted at
site 10. Bottom panel: same for moving standard deviation.
In Fig. , we can notice that wind conditions with weak temporal
variability observed in the first regime are associated with weak values of
the moving mean and variance processes, whereas more volatile periods in the
second and third regimes are characterized by higher values of moving mean
and variance. To the best of our knowledge, few statistics enable us to
characterize the alternation associated with regime-switching. These two
processes of moving mean and standard deviation enable us to characterize the
alternation of variability associated with the observed regime-switching, and
will be used in the following sections.
Coefficients of the autoregressive process Y in each regime and the
transition matrix at each site are comparable and spatially coherent (see
Table ). Other criteria such as the average field of
{ut,vt} in each regime and the distribution of {Φt}
in each regime were also explored and suggest similarities between regimes at
all locations.
The assumption of a regional regime seems appropriate in the considered area
and is thus kept for the modeling of the multi-site wind in the following.
Observed regime-switching autoregressive models
Conversely to the previous section, one may derive the regimes separately
from the fitting of the conditional model. For such a priori regime-switching
models, the derivation of observed regimes can be done with appropriate
clustering methods. We seek weather states that are distinct from one other
and in which the data are homogeneous. Clustering can be run either on the
local variables under study or on extra variables: the former leads to
weather states that are more appropriate to the local data, while the latter
can provide more meteorologically consistent regimes, for example, with more
information about the large-scale situation. In this subsection, we propose
three clusterings, which differ by the clustering method and/or by the
variables used to derive the a priori regimes.
Derivation of observed regimes from extra variables: CZ500
As a first clustering, we use a classification into four large-scale weather
regimes that is commonly used in climate studies to characterize the
wintertime atmospheric dynamics over the North Atlantic/European sector
. These regimes can be described as
follows.
The positive phase of the North Atlantic Oscillation (hereafter NAO+),
characterized by a strengthening of both the Azores High and the Islandic Low, which reinforces the
westerlies.
The negative phase of the NAO (NAO-), its symmetrical counterpart
The Scandinavian blocking (BL), characterized by a strong anticyclone over
northern Europe able to totally block the westerly flow over western Europe
The Atlantic Ridge (AR), characterized by a strong west–east pressure dipole bringing polar air masses over western Europe
At the local scale of our area of study, these regimes are, respectively,
associated with strong southwesterly flows (NAO+), weak westerly flows
(NAO-), stable southerly or easterly flows (BL), and northerly flows (AR).
To derive these regimes, we use the same methodology as in
. We perform a k-means clustering on the
3607 daily mean maps of 500 mb geopotential height (Z500)
anomalies (i.e., mean-corrected fields) over the North Atlantic/European
sector (90∘ W–30∘ E, 20–80∘ N) corresponding to
days of December, January, and February 1981–2010. Daily Z500 data are
downloaded from the ERA-Interim archive. In order to reduce the computational
time, the k-means algorithm is performed on the first 10 principal
components (PCs) of the Z500 anomalies time series. These PCs are time series
corresponding to the projections of the Z500 anomalies onto the empirical
orthogonal functions (EOFs), which are eigenvectors of the spatial covariance
matrix of the Z500 field. Such a decomposition enables extraction of the main
modes of variability of the spatiotemporal process; here, the first 10 EOFs
explain 90 % of the total variance. Eventually, the obtained daily
classification is converted to a four-times-daily classification by repeating
the same regime for the four time steps of each day, a reasonable approach
given the smoothness of the Z500 both in time and space. In the following, we
denote this clustering as CZ500.
Derivation of observed regimes from the local variables: CEOF(u,v) and CDiff(u,v)
To derive observed regimes from local wind variables, one can first use a
k-means clustering procedure similar to the one used for CZ500.
However, while CZ500 provides persistent regimes in which the
conditional model satisfyingly describes {ut,vt}, local
regimes resulting from such a k-means clustering are not persistent enough to
reliably estimate the conditional VAR model. Consequently, in this
subsection, we perform the local clustering via a hidden Markov model with
Gaussian probability of emission.
The hidden structure of the Markov chain provides more stable regimes than
with a k-means clustering. It corresponds to an H-MS-VAR model
with VAR models of order p=0. The EM algorithm is used to process the
clustering, and the number of regimes is chosen to be 3. This number provides
the most physically relevant local regimes; a greater number of regimes
indeed leads to less discriminative regimes in terms of local wind conditions
(not shown).
Then two sets of descriptors of the data (i.e., local variables) are
proposed. The first partition, denoted CEOF(u,v), is
obtained by clustering the time series associated with the first two EOFs of
the anomalies of {ut,vt}, which explain 94% of the total
variance. The second partition involves descriptors of the conditional
distribution of p(Yt|Yt-1), in order to find a clustering
that may be better adapted to the description of the conditional distribution
by an autoregressive model. A simplified way to describe the dynamics is to
consider the bivariate process {ut-ut-1,vt-vt-1}.
This set of variables enables construction of regimes that discriminate well
the temporal variability of the process {ut,vt}. Let us denote
this second local clustering as CDiff(u,v).
Time series of wind speed in January 2012 and a priori regimes
extracted from the proposed methods above. The darker is the grey; the
smaller is the determinant of Σ(i). From top to bottom:
CZ500, CEOF(u,v), CDiff(u,v), and
regimes from the fitting of the H-MS-VAR model.
Analysis of the proposed clusterings
The proposed clusterings are compared through various analyses. We seek a
clustering that is physically meaningful and appropriate in terms of
conditional autoregressive models. For a proper comparison, for all
clusterings, we decide to order regimes from the more persistent to the less
persistent. This is done according to the determinant of the matrix
Σ(i).
First visual comparison
Sequences of regimes from the proposed clusterings are shown in
Fig. . The top panel shows that CZ500 has very
persistent regimes. This result is expected because it describes the
alternation between the preferred states of the large-scale atmospheric
dynamics, whose typical timescale is a few days. One can see that the less
volatile wind conditions are associated with the BL and AR phases, whereas
the most variable wind conditions occur during the two NAO phases; see
Fig. . The three bottom panels correspond to local clusterings.
For all of them, the first regime is associated with the less volatile
conditions with weakest intensity, whereas the second and third regimes are
generally associated with moderate and high intensities of wind. However, the
behavior of the regime-switching differs from one clustering to another,
probably because of the different choice of descriptors
({ut,vt} vs. {ut-ut-1,vt-vt-1}) and/or
methods (observed vs. latent) used in the clustering. The bottom panel of
Fig. shows that the second regime is a precursor to the third one
(which is confirmed by the transition probabilities between regimes), and
that this second regime is associated most of the time with rises in wind
speed intensity.
In Fig. , the average fields corresponding to each regime of the
four clusterings are plotted. The top row highlights the difficulty of
discriminating local wind features when using regimes defined from a
large-scale circulation variable. While the AR and NAO+ regimes of
CZ500 are associated with strong local wind signatures (as
described in Sect. ), the BL and NAO- regimes have a weaker
discriminatory power in the local wind data. This issue was also observed in
.
Since different descriptors are used, CDiff(u,v) and
CEOF(u,v) lead to very different results.
CEOF(u,v) leads to the most physically consistent regimes: a
northeasterly regime, a northwesterly one, and a southwesterly one, which are
flows corresponding to several of the large-scale weather regimes. The last
two regimes are associated with stronger intensities. From the derivation of
this clustering, one naturally finds regimes that correspond to the main mean
patterns of variability of the fields.
The regimes of CDiff(u,v) have less persistence, which
complicates their meteorological interpretation. The first regime corresponds
to periods of weak wind intensities. The last two regimes are southwesterly
regimes with a different intensity from one to the other. The averaged fields
of the regimes extracted from H-MS-VAR are similar to the ones of
CDiff(u,v) despite some punctual discrepancies in their
time series (Fig. ). The first regime of these two clusterings
seems associated with blocking situations.
Average fields of
{ut,vt} in each regime of the clusterings, from top to bottom:
CZ500, CEOF(u,v),
CDiff(u,v), and from the fitting of H-MS-VAR on
the set of five locations.
Np is the number of parameters. Values are computed from
models fitted on {ut,vt} at the five locations (1, 6, 10, 13,
18).
BIClog-Llog-LNplog(det(Σ(i)))% of time spent Modelof Sof YR1R2R3R4R1R2R3R4Unconditional VAR542 640–-269 82526536.4–––––––AP-MS-VARCZ500542 730-1510-263 808107229.830.33938.10.270.180.20.34AP-MS-VARCEOF(u,v)545 730-2331-266 01580128.933.338.9–0.310.420.27–AP-MS-VARCDiff(u,v)520 759-4762-251 09980120.234.148.1–0.440.410.15–H-MS-VAR459 458–-229 61680118.432.148.4–0.430.410.16–
To compare the associations between the different classifications, a multiple
correspondence analysis is made between the four categorical variables that
represent each classification. This analysis can be viewed as an analog of a
principal component analysis for categorical variables where the associations
between the variables are measured with the Chi-squared distance. The regimes
of each classification are projected on the first two components and
displayed in Fig. . These two axes enable one to account for
44 % of the variance, which is not low for such an analysis. The other
axes are not considered because they do not bring enough useful information.
Note that this analysis does not account for the temporal dependence in each
classification. The overall structure tends to associate the three
classifications CEOF(u,v),
CDiff(u,v), and H-MS-VAR, except for the third
regime of CEOF(u,v). The classifications
CDiff(u,v) and H-MS-VAR are very close in this
projection, which means that their regimes mainly occur at the same time, and
this coincides with Fig. . The first axis contrasts time-persistent
regimes with less persistent ones. Regime BL is close to regimes R1 of
CEOF(u,v), CDiff(u,v), and
H-MS-VAR; this is also seen in Table and is in
agreement with the average fields of these regimes displayed in
Fig. . The second axis contrasts regimes R2 of H-MS-VAR
and CDiff(u,v) with regimes R3, which is also a contrast
between persistent and less persistent regimes. Most of these similarities
between the regimes are also seen in Table through the logarithm
of the covariance of the innovations and the percentage of time spent in each
regime. Regime AR from CZ500 seems more difficult to associate
with other regimes. Regime R3 from CEOF(u,v) is
associated with weather regime NAO+, which coincides with Table
and Fig. .
First plan of the multiple correspondence analysis made of the four
classifications. Each regime of the four classifications is
depicted.
Quantitative analyzing
Quantitative criteria are considered in order to complete this analysis. The
optimal value of the complete log-likelihood of the model is generally a good
measure of the statistical relevance of a model. The complete log-likelihood,
given in Eq. (), evaluated at the maximum likelihood estimator of
θ^, is written in the case of observed regime-switching as the
sum of the following two terms:
log(L(θ^(Y);y1T|s1T))=-Tdlog(2π)2-Td2-∑i=1Mnilog(det(Σ^(i)))
and
log(L(θ^(S);s1,…,sT))=∑i,j=1Mni,jlog(ni,jni,.).
Note that the first term is a function of the total time spent in each regime
and the associated determinant of covariance matrix of innovation (note that
the one-step-ahead error of the forecast is linked to this quantity). The
longer the time spent in a regime with a weak determinant of covariance of
innovation, the greater the log-likelihood (see Table ). The
maximal log-likelihood of θ(S) is equal to the opposite of the
conditional entropy of St given St-1. The conditional entropy is
classically used as a quality measure of clustering. In prediction, the
weaker the entropy, the stronger the predictability of St given
St-1. More generally one tends to minimize this measure. Because of the
range of values of the log-likelihood of θ(Y), the value of that
of θ(S) has a low contribution to the complete log-likelihood. If
the complete log-likelihood is used to select models, the persistence of the
Markov chain has a low impact. BIC indexes are also given in
Table , where BIC=-2logL+Nplog(Nobs), with L the likelihood
of the model, Np the number of parameters and
Nobs the number of observations. The BIC index enables
one to consider a compromise between a model with a high likelihood and its
parsimony. Note that one should not compare BIC indexes of a priori and
latent regime-switching models. However, the BIC indexes of these two classes
of models can be compared with that of the unconditional VAR model, since it
is a particular case.
Joint probability of occurrence of the three local regimes
identified by the proposed models in rows and the four large-scale regimes in
columns.
Left: joint and marginal distributions of simulated data at site
10 from the model H-MS-VAR. Central and right panels:
autocorrelation functions of {ut} and {vt} at site 10 for the
reference data, and simulated data from the VAR(2),
AP-MS-VARCDiff(u,v), and H-MS-VAR
models.
The clustering CDiff(u,v) provides the greatest value of
complete log-likelihood. The lower value of log-likelihood of S, with
shorter persistence in the different regimes compared with the other models,
is compensated for by a larger value of log-likelihood of Y and thus a
longer time spent in regimes with low variances of innovation. The three
proposed AP-MS-VAR models lead to a satisfying description of the
marginal and joint distributions and space–time covariances (not shown). The
model AP-MS-VARCDiff(u,v), which exhibits the
best likelihood, performs the most accurately among the AP-MS-VAR
models to reproduce the moving average and moving variance processes; see
Sect. . Besides, in terms of BIC indexes, the smallest value among
the AP-MS-VAR models is that of
AP-MS-VARCDiff(u,v), and it is also greater
than that of the VAR model. In the following, the VAR model with shifts
defined by CDiff(u,v) is kept for further comparisons
with the H-MS-VAR model in simulation; see Sect. . We
choose this model although it is not the most physically meaningful because
it leads to better results according to our criterion.
Link between large-scale weather regimes and local ones
In this section we quantitatively compare the large-scale regimes described
by CZ500 with the local ones derived from the hidden MS-VAR. To
this end, we compute the joint probability of occurrence of large-scale
regimes (CZ500) and local regimes (successively
CEOF(u,v), CDiff(u,v), and
H-MS-VAR; Table ).
For the three clusterings, the local regimes seem to appear in preferential
large-scale weather regimes. The strongest link with CZ500 is
found for CEOF(u,v): the first regime coincides mainly
with BL, the second one with AR, and the third one with NAO+. These results
are not surprising because regimes of CEOF(u,v) are also
easier to interpret physically. However, the association is not systematic:
for instance, the second regime is observed not only during AR conditions,
but also during NAO+ conditions. Note that NAO- conditions split rather
equiprobably among the three local regimes.
The regimes of H-MS-VAR and CDiff(u,v) are more
difficult to link with large-scale regimes. The fact that they are less
persistent than the CEOF(u,v) ones may explain why their
joint occurrences with CZ500 are weaker. As previously said,
H-MS-VAR regimes are driven mainly by the conditional autoregressive
model in the sense of the likelihood, which results in a more difficult
physical interpretation. Some links can nevertheless be made: for both
H-MS-VAR and CDiff(u,v), the second regime
coincides mainly with NAO+, and to a lesser extent the first regime is
connected to BL.
Top: correlation of between {ut} at site 1 and {ut}
at the other locations (sorted according to increasing distance) at various
time lags. Bottom: similar quantities for {vt}. From the top panel to
the bottom one: data and simulation from VAR(2), from
AP-MS-VARCDiff(u,v), and from
H-MS-VAR.
Moving standard deviation of the value {Ut} against its
moving mean at location 10. From left to right: data and simulation from
the VAR(2), AP-MS-VARCDiff(u,v), and
H-MS-VAR.
Comparison in simulation of the multi-site wind models
In this section, we compare models VAR(2),
AP-MS-VARCDiff(u,v), and H-MS-VAR in
terms of reproducing the various scales of the spatiotemporal wind
variability. We focus on the alternation between periods with different
temporal variability of wind conditions, and we highlight the benefit of
using appropriate regime-switching in reproducing such an alternation.
N=100 sequences of the length of the data are generated with the fitted
models, and several statistics are computed on these data.
First, marginal statistics at the central site 10 are investigated (see
Fig. ). Comparing Fig. and Fig. , one can
notice that the distribution of {ut} is well reproduced by model
H-MS-VAR, while the {vt} one is less accurately described.
Results in are slightly more satisfying because of
non-homogeneous transitions between regimes. The description of this
distribution by AP-MS-VARCDiff(u,v) is also
satisfying and is not shown here. Concerning the temporal dependence, the
regime-switching models are most able to accurately reproduce the
autocorrelation functions of both {ut} and {vt}. All the models
tend to behave similarly in reproducing the correlation of {ut}.
However, the VAR model tends to underestimate the dependence of {vt}
between 2 and 5 days, and the regime-switching models improve the
description of this dependence.
The space–time correlation function of the multivariate process
{ut,vt} and its simulated replicates reveals that both models
reproduce satisfyingly the general shape of this function and especially the
non-separable and anisotropic patterns; see Fig. . The
non-separability reflected in the asymmetry around the vertical axis at lag
0 is captured by the proposed models.
To study patterns at an instantaneous timescale, we focus on the ability of
the models to reproduce the alternation of temporal variability. Indeed, the
alternation of different weather states induces an alternation in the
intensity and temporal variability of wind. In Fig. , the moving
standard deviation of wind speed around its moving mean at the central site
10 is depicted as a function of its moving mean. Observations reveal a
higher variability when the intensity is high, although a high variability
may also be associated with weaker values when the moving window overlaps the
transition time. Models with regime-switching enable the reproduction of more
temporal variability associated with moderate and high intensity of wind,
which is not captured by an unconditional VAR model. For instance, the
regime-switching models reproduce high variability around 5 and
10 m s-1, which corresponds to transitions between weather states.
This is ensured by the alternation, driven by a Markov chain, of periods
associated with different parameters of the conditional model.
Similar diagnostics to Fig. indicate that the distributions of the
moving standard deviation and the moving mean within each simulated regime of
the CDiff(u,v) and of H-MS-VAR are clearly
distinct from one regime to the other, which indicates characteristic
behaviors of these two simulated processes within each regime (not shown).
Moreover, the behavior in each simulated regime is close to the observed one.
Discussions and perspectives
In Sect. , we compare site-specific regimes to common
regional regimes. We conclude according to mainly qualitative criteria that
for this data set the use of a regime common to all locations is reasonable.
To go one step further, one would settle some likelihood-ratio test, to
quantify more precisely to what extent the assumption of a regional regime
against a site-specific regime is acceptable.
In this paper we have introduced an observed and latent regime-switching
framework, and we have shown that both types of regime-switching models have
various advantages. Models with observed switchings may account for relevant
regimes that correspond to characteristic meteorological conditions in
Europe. The choice of the clustering method and of the descriptors of the
data is crucial, as discussed in Sect. , where a
k-means clustering led to irrelevant regimes in terms of estimation of the
associated conditional model.
The hidden regime-switching framework seems to overcome this insufficiency by
providing regimes that are driven by the conditional distribution and
therefore adapted to the estimation. When considering hidden regime-switching
models, however, the estimation procedure may become challenging when
sophisticated marginal models are considered. The extracted regimes are
driven mainly by the local data and the proposed conditional distribution,
and consequently they might have less physical interpretation than do regimes
derived from other clusterings. Nevertheless, in this study we saw that for
the proposed model and studied data set, the associated regimes were not
physically inconsistent. Moreover, the use of hidden regime-switching models
saves effort in choosing an appropriate observed a priori clustering.
Concerning the proposed observed regime-switching models, there seems to be a
compromise between physically interpretable regimes and a good description of
the conditional model by a VAR, as highlighted in Sect. when
comparing the AP-MS-VARCDiff(u,v) and
AP-MS-VARCEOF(u,v) models. Indeed, we have
chosen AP-MS-VARCDiff(u,v) because it provides
the best BIC index despite the fact that CDiff(u,v)
has less physical interpretation. This highlights the difficulty in finding
relevant regimes that are adapted to the description of the data by
conditional vector autoregressive models. The proposed hidden
regime-switching model seems to respond to this compromise by providing more
interpretable regimes than the ones of CDiff(u,v) and a
similar description of temporal patterns. The improvement of BIC from the
AP-MS-VARCDiff(u,v) with respect to the
unconditional VAR is 4 %, whereas the improvement from the
H-MS-VAR is 15.3 %.
Future work may involve investigating reduced parameterizations of the
autoregressive coefficients and of the matrices of covariance of innovations,
thus helping to adapt the model to a larger data set. Indeed, the number of
parameters is already high with the small data set under consideration, and
attempts to use parametric shapes for parameters reveal that a huge effort
will be needed to extract consistent results. Furthermore, when looking at
the autoregressive matrices, one sees generally privileged predictors
according to the regimes, a situation that motivates the use of constraint
matrices in each regime.
Acknowledgements
The submitted paper has been created by UChicago Argonne, LLC, Operator of
Argonne National Laboratory (“Argonne”). Argonne, a US Department of Energy
Office of Science laboratory, is operated under contract
no. DE-AC02-06CH11357.
Edited by: W. Kleiber
Reviewed by: one anonymous referee
References
Ailliot, P. and Monbet, V.: Markov-switching autoregressive models for wind
time series, Environ. Modell. Softw., 30, 92–101, 2012.
Ailliot, P., Monbet, V., and Prevosto, M.: An autoregressive model with
time-varying coefficients for wind fields, Environmetrics, 17, 107–117,
2006.
Ailliot, P., Thompson, C., and Thomson, P.: Space time modeling of
precipitation using a hidden Markov model and censored Gaussian
distributions, J. Roy. Stat. Soc. C-App., 58, 405–426, 2009.
Ailliot, P., Bessac, J., Monbet, V., and Pene, F.: Non-homogeneous hidden
Markov-switching models for wind time series, J. Stat.
Plan. Infer., 160, 75–88, 2015.
Bardossy, A. and Plate, E. J.: Space-time model for daily rainfall using
atmospheric circulation patterns, Water Resour. Res., 28, 1247–1259,
1992.
Bessac, J., Ailliot, P., and Monbet, V.: Gaussian linear state-space model for
wind fields in the North-East Atlantic, Environmetrics, 26, 29–38,
2015.
Brown, B. G., Katz, R. W., and Murphy, A. H.: Time series models to simulate
and forecast wind speed and wind power, J. Clim. Appl. Meteorol., 23, 1184–1195, 1984.
Cassou, C.: Intraseasonal interaction between the Madden–Julian
oscillation and the North Atlantic oscillation, Nature, 455, 523–527,
2008.Cattiaux, J., Douville, H., and Peings, Y.: European temperatures in CMIP5:
origins of present-day biases and future uncertainties, Clim. Dynam., 41,
2889–2907, 10.1007/s00382-013-1731-y, 2013.
Dempster, A. P., Laird, N. M., and Rubin, D. B.:
Maximum likelihood from
incomplete data via the EM algorithm, J. Roy. Stat.
Soc. B, 39, 1–38, 1977.
Durand, J.-B.: Modèles à structure cachée: inférence,
estimation, sélection de modèles et applications, PhD thesis,
Université Joseph-Fourier-Grenoble I, 2003.Flecher, C., Naveau, P., Allard, D., and Brisson, N.: A stochastic daily
weather generator for skewed data, Water Resour. Res., 46, W07519, 10.1029/2009WR008098,
2010.
Fuentes, M., Chen, L., Davis, J. M., and Lackmann, G. M.: Modeling and
predicting complex space–time structures and patterns of coastal wind
fields, Environmetrics, 16, 449–464, 2005.
Gneiting, T., Larson, K., Westrick, K., Genton, M. G., and Aldrich, E.:
Calibrated probabilistic forecasting at the stateline wind energy center:
The regime-switching space–time method, J. Am. Stat. Assoc., 101, 968–979, 2006.
Hamilton, J. D.: A new approach to the economic analysis of nonstationary time
series and the business cycle, Econometrica, 57, 357–384, 1989.
Hamilton, J. D.: Analysis of time series subject to changes in regime, J. Econometrics, 45, 39–70, 1990.
Haslett, J. and Raftery, A. E.: Space-time modelling with long-memory
dependence: Assessing Ireland's wind power resource, Applied
Statistics, 38, 1–50, 1989.
Hering, A. S. and Genton, M. G.: Powering up with space-time wind forecasting,
J. Am. Stat. Assoc., 105, 92–104, 2010.
Hering, A. S., Kazor, K., and Kleiber, W.: A Markov-switching vector
autoregressive stochastic wind generator for multiple spatial and temporal
scales, Resources, 4, 70–92, 2015.
Hofmann, M. and Sperstad, I. B.: NOWIcob–A tool for reducing the
maintenance costs of offshore wind farms, Energy Procedia, 35, 177–186,
2013.
Hughes, J. P. and Guttorp, P.: A class of stochastic models for relating
synoptic atmospheric patterns to local hydrologic phenomenon, Water Resour. Res., 30, 1535–1546, 1994.
Hughes, J. P., Guttorp, P., and Charles, S. P.: A non-homogeneous hidden
Markov model for precipitation occurrence, J. Roy. Stat.
Soc. C-App., 48, 15–30, 1999.
Khalili, M., Leconte, R., and Brissette, F.: Stochastic multisite generation of
daily precipitation data using spatial autocorrelation, J. Hydrometeorol., 8, 396–412, 2007.Kleiber, W., Katz, R. W., and Rajagopalan, B.: Daily spatiotemporal
precipitation simulation using latent and transformed Gaussian processes,
Water Resour. Res., 48, W01523, 10.1029/2011WR011105,
2012.Michelangeli, P. A., Vautard, R., and Legras, B.: Weather regimes: recurrence
and quasi stationarity, J. Atmos. Sci., 52, 1237–1256,
1995.
Najac, J.: Impacts du changement climatique sur le potentiel éolien en
France: une étude de régionalisation, PhD thesis, Université
Paul Sabatier-Toulouse III, 2008.
Pinson, P., Christensen, L. E. A., Madsen, H., Sörensen, P. E., Donovan,
M. H., and Jensen, L. E.: Regime-switching modelling of the fluctuations of
offshore wind generation, J. Wind Eng. Ind. Aerod., 96, 2327–2347, 2008.
Richardson, C. W.: Stochastic simulation of daily precipitation, temperature,
and solar radiation, Water Resour. Res., 17, 182–190, 1981.
Thompson, C. S., Thomson, P. J., and Zheng, X.: Fitting a multisite daily
rainfall model to New Zealand data, J. Hydrol., 340, 25–39,
2007.
Vautard, R.: Multiple weather regimes over the North Atlantic: Analysis
of precursors and successors, Mon. Weather Rev., 118, 2056–2081, 1990.
Vrac, M., Stein, M., and Hayhoe, K.: Statistical downscaling of precipitation
through nonhomogeneous stochastic weather typing, Clim. Res., 34, 169–184,
2007.
Wikle, C. K., Milliff, R. F., Nychka, D., and Berliner, L. M.: Spatiotemporal
hierarchical Bayesian modeling tropical ocean surface winds, J. Am. Stat. Assoc., 96, 382–397, 2001.
Wilks, D. S.: Multisite generalization of a daily stochastic precipitation
generation model, J. Hydrol., 210, 178–191, 1998.
Wilson, L. L., Lettenmaier, D. P., and Skyllingstad, E.: A hierarchical
stochastic model of large-scale atmospheric circulation patterns and multiple
station daily precipitation, J. Geophys. Res.-Atmos., 97, 2791–2809, 1992.
Wu, C. F. J.: On the convergence properties of the EM algorithm, Ann. Stat., 11, 95–103, 1983.
Zucchini, W. and Guttorp, P.: A hidden Markov model for space-time
precipitation, Water Resour. Res., 27, 1917–1923, 1991.
Zucchini, W. and MacDonald, I.: Hidden Markov models for time series: An
introduction using R, no. 110 in Monographs on statistics and applied
probability, CRC Press, 2009.