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HMMStochastic wind generators
Forecast correction
Hidden Markov Models for wind time series
Pierre Ailliot1 Valérie Monbet2
1Université de Brest2Université de Bretagne Sud
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
ModelStatistical inference in HMMHMM with finite hidden stateState-space models
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
ModelStatistical inference in HMMHMM with finite hidden stateState-space models
Hidden Markov Models
Definition
{Xt} = {St , Yt} ∈ {S × Y} , with {St} not observable ("hidden") and
P(St |S0 = s0, · · · , St−1 = st−1, Y1 = y1, · · · , Yt−1 = yt−1) = P(St |St−1 = st−1)
P(Yt |S0 = s0, · · · , St = st , Y1 = y1, · · · , Yt−1 = yt−1) = P(Yt |St = st)
State · · · → St−1 → St → St+1 → · · ·↓ ↓ ↓
Observation · · · Yt−1 Yt Yt+1 · · ·
Parametrization of a HMM
pθ(st |st−1) : transition probability
pθ(yt |st ) : emission probability
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
ModelStatistical inference in HMMHMM with finite hidden stateState-space models
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
ModelStatistical inference in HMMHMM with finite hidden stateState-space models
Statistical inference in HMM
Likelihood function
pθ(Y1 = y1, · · · , Yt = yT ) =
∫ST+1
pθ(s0)ΠTt=1pθ(st |st−1)pθ(yt |st)ds0ds1...dsT
= ΠTt=1
∫S
pθ(yt |st )pθ(st |y1, · · · , yt−1)dst
Prediction : evaluate pθ(st |y1, · · · , yt−1)
pθ(st |y1, · · · , yt−1) =
∫S
pθ(st |st−1)pθ(st−1|y1, · · · , yt−1)dst−1
Filtering : evaluate pθ(st |y1, · · · , yt )
pθ(st |y1, · · · , yt ) ∝ pθ(yt |st)pθ(st |y1, · · · , yt−1)
Smoothing : evaluate pθ(st |y1, · · · , yt , · · · , yT )
pθ(st |y1, · · · , yT )
= pθ(st |y1, · · · , yt)
∫S
pθ(st+1|st)
pθ(st+1|y1, · · · , yt )pθ(st+1|y1, · · · , yT )dst+1
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
ModelStatistical inference in HMMHMM with finite hidden stateState-space models
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
ModelStatistical inference in HMMHMM with finite hidden stateState-space models
HMM with finite state
Example :Yt = m(St ) + σ(St )Wt
Wt ∼ iidN (0, 1)
m(1) = 1 , m(2) = 2 , σ(1) = 0.2 , σ(2) = 0.5 , Q =
(0.95 0.050.1 0.9
)
1
3
0
1
P(S
t=1|
y 1,...,y
T)
Likelihood function, filtering and smoothing : forward-backward algorithm
Maximization of the likelihood function : EM and/or gradient-based algorithm
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
ModelStatistical inference in HMMHMM with finite hidden stateState-space models
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
ModelStatistical inference in HMMHMM with finite hidden stateState-space models
State-space models
Gaussian linear state-space models{
St = αSt−1 + β + σV VtYt = aSt + b + σW Wt
Vt ∼ iidN (0, 1), Wt ∼ iidN (0, 1)Likelihood function, filtering and smoothing : Kalman filterMaximization of the likelihood function : EM and/or gradient-based algorithm
Non-linear state-space modelsLikelihood function, filtering and smoothing : Monte-Carlo approximations
Particle filters
Maximization of the likelihood function : MCEM and/or stochastic gradient algorithm
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Motivations
Many natural phenomena and human activities depend on wind conditions
Production of electricity by wind turbines
Evolution of a coast line
Maritime transport
Drift of objects in the ocean
...
Wind data generally available on short periods of time
50 years of data maximum
Not enough to compute reliable estimates of the probability of complex events
Stochastic model used to simulate artificial wind conditions
Monte-Carlo methods
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Example : drift of an object in the ocean
ref : Ailliot, Frenod, Monbet, Multiscale Modeling and Simulation (2006)
What is the probability that a lost container ends up on the coast ?
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Example : drift of an object in the ocean
Object’s trajectory depends on current and wind conditions
Example
−3
0
3
0 0.2 0.4 0.6 0.8 1−3
0
3
Wind time series
→ODE→
0.9
1
1.1
1.2
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
1.1
Object’s trajectory
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Object’s drift can last several weeks
No wind forecast available
Artificial wind conditions simulated with a stochastic model
0.7 0.8 0.9 1 1.1 1.2 1.3
0.7
0.8
0.9
1
1.1
1.2
1.3
5%10%15%20%
E
N
W
S
5% 10%15%
E
N
W
S
< 01< 02< 03< 04
50 trajectories Wind rose Running aground locations(1000 trajectories)
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
A HMM for the wind direction
22 years of data, ∆t = 6hfocus on January : stationarity ?
Existence of different weather types ?Wind direction (Jan. 2000)
0 5 10 15 20 25 30W
S
E
N
W
Marginal distribution : 2 modes ?
S
W
N
E
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
A HMM for the wind direction
AssumptionsSt ∈ {1, · · · , M} : weather typeYt = Φt : wind directionVon-Mises distribution for P(Φt |St = st ) :
P(Φt = φ|St = st ) =1
2πI0(κ(st ))exp
(κ(st ) cos(φ − µ(st ))
)
Model selection
M 1 2 3 4 5 6BIC 9630 7454 6587 5690 5253 5324
Maximum likelihood estimates (M=2)
Regime 1 : µ(1) = 60o (ENE) , κ(2) = 1.79 - Easterlies, anticyclonic conditions
Regime 2 : µ(2) = 242o deg (WSW), κ(1) = 2.17 - Westerlies, cyclonic conditions
Transition matrix :[
0.953 0.0470.034 0.966
], stationary distribution :
[0.420.58
]
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
A HMM for the wind direction
Marginal distribution
S
W
N
E
S
W
N
E
S
W
N
E
Regime 1 (42%) Regime 2 (58%) Mixed ditributionSmoothing probabilities (Jan. 2000)
0
0.5
1
P(S
t=1|
y 1,...,y
T)
0 5 10 15 20 25 30WSENW
y t
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
A HMM for the wind direction
Simulated time series (slightly better results with M = 5)
0 5 10 15 20 25 30W
S
E
N
W
Observed time series (Jan. 2000)
0 5 10 15 20 25 30W
S
E
N
W
The model can not reproduce "small scale" dynamicsConditional independence assumptions too strong ?
P(Yt |S0 = s0, S1 = s1, Y1 = y1, · · · , St−1 = st−1, Yt−1 = yt−1) = P(Yt |St = st , Yt = yt−1)
Realistic simple parametric model for P(Yt |St = st , Yt = yt−1) ?
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
A HMM for the wind speed
Existence of different weather types ?Wind speed (Jan. 2000)
0 5 10 15 20 25 300
5
10
15
20
Wind direction (Jan. 2000)
0 5 10 15 20 25 30W
S
E
N
W
Higher volatility in cyclonic conditions ?
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
A HMM for the wind speed
AssumptionsSt ∈ {1, · · · , M} : weather typeYt = Ut : wind speedMarkov-switching autoregressive model :
P(Yt |S0 = s0, · · · , St = st , Y0 = y0, · · · , Yt−1 = yt−1) = P(Yt |St = st , Yt = yt−1)
· · · → St−1 → St → St+1 → · · ·↓ ↓ ↓
· · · → Yt−1 → Yt → Yt+1 → · · ·Gamma distribution for P(Yt |St = st , Yt = yt−1), with
mean : a(st )yt−1 + b(st )
standard deviation : σ (st )
Model selection
M 1 2 3 4 5BIC 10485 10301 10307 10343 10387
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Maximum likelihood estimatesRegime 1 : low volatility, anticyclonic conditions
E [Yt |Yt−1 = yt−1, St = 1] = .79yt−1 + 1.46
var [Yt |Yt−1 = yt−1, St = 1] = 1.372
Regime 2 : higher volatility, cyclonic conditionsE [Yt |Yt−1 = yt−1, St = 1] = .77yt−1 + 2.24
var [Yt |Yt−1 = yt−1, St = 1] = 2.42
Transition matrix :[
0.98 0.020.03 0.97
], stationary distribution
[0.400.60
]
Smoothing probabilities (Jan. 2000)
0 5 10 15 20 25 300
10
20
0
1
0 5 10 15 20 25 300
1
Model improves results obtained with ARMA modelsMarginal distribution, autocorrelation function, storm and inter-storm durations...
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
A joint HMM for the wind speed and the wind direction ?
Correspondence between regimes identified on the wind speed and the winddirection ?
HMM for the wind direction
00.5
1
5 10 15 20 25 300
0.51
0 5 10 15 20 25 30WSENW
Wind direction in the 2 regimesS
W
N
E
S
W
N
E
HMM for the wind speed
0 5 10 15 20 25 300
10
20
0
1
0 5 10 15 20 25 300
1
Wind direction in the 2 regimes
3 regimes ?
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
A joint HMM for the wind speed and the wind direction ?
A realistic model structure ?
· · · → St−1 → St → St+1 → · · ·↓ ↓ ↓
· · · → (Ut−1, Φt−1) → (Ut , Φt) → (Ut+1, Φt+1) → · · ·
Realistic simple parametric model for P(Ut , Φt |Ut−1, Φt−1, St = s) ?
Another model
· · · Φt−1 Φt Φt+1 · · ·↓ ↓ ↓
· · · → St−1 → St → St+1 → · · ·↓ ↓ ↓
· · · → Ut−1 → Ut → Ut+1 → · · ·
Ref Ailliot, Monbet , preprint (2007)
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
MotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
Conclusion (first part)
Major virtues of HMM ?
Distributional versatility
Ability to model diverse time scales
Open structure which allows for more physical models
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Example
Correct numerical weather predictions given in situ observations
Data from satellites, buoys, ships....
−40 −30 −20 −10 0
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50
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1
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5
6
8
10
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
A first model
Linear Gaussian spate-space model
bt = αbt−1 + β + σV Vt Error (hidden)
Y truet = Y for
t + bt "True" Yt (hidden)Yobs
t = Y truet + σW Wt Observed Yt
Vt ∼ iidN (0, 1), Wt ∼ iidN (0, 1)
Inference
Kalman Filter (1D)
Initializations : ma, PaFor t = 1,...,T,% predictionmpred (t) = αma + β
Ppred (t) = αPaα′ + σ2V
% analysisK = Ppred (t)/(Ppred (t) + σ2
W )
ma = (1 − K )mpred (t) + KO(t) =σ2
WPpred (t)+σ2
Wmpred (t) +
Ppred (t)
Ppred (t)+σ2W
O(t)
Pa = (1 − K )Ppred (t)End
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Need for a more advanced model ?
Two types of errors ?Intensity errorPosition error
Example : wind fields
−12 −10 −8 −6 −4 −2 043
44
45
46
47
48
49
50
51
52Champs analysé 09/15 18 H
"Observed" field (nowcast)
−12 −10 −8 −6 −4 −2 043
44
45
46
47
48
49
50
51
52Champs prédit 09/15 18 H
Predicted field (18 hforecast)
Position error not included in the linear model !
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
First step : single location
Position error ↔ phase error
0 6 12 18 240
2
4
6
8
10
12
14
16
observed, forecast, * : nowcast
−6 −4 −2 042
43
44
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50
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52
Ouessant
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Introduction of a phase correction
Model
∆t = α∆∆t−1 + β∆ + σ∆V∆(t) Phase error (hidden)bt = αbbt−1 + βb + σbVb(t) Intensity error (hidden)
Y truet = Y for
t+∆t+ bt "True" Yt (hidden)
Yobst = Y true
t + σW W (t) Observed Yt
V∆(t) ∼ iidN (0, 1), Vb(t) ∼ iidN (0, 1), W (t) ∼ iidN (0, 1)
InferenceParameter estimation : maximum likelihood by a stochastic gradient algorithm
θk = θk−1 + γk ∂θLT (θ)
Monte Carlo approximation of LT (θ) and ∂θLT (θ)
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Approximation of logLT (θ)
One needs to approximate
pθ(st |y1, · · · , yt ) ∝ pθ(yt |st )
∫S
pθ(st |st−1)pθ(st−1|yt−11 )dst−1
Bootstrap particle filter
Initialization :{si
0}i∈{1,...,N} ∼ pθ(S0)For t = 1 to T% predictionsi
t,pred ∼ pθ(St |sit−1)
% resamplingν i ∝ pθ(yt |si
t,pred ) with∑N
i=1 ν i = 1
pθ(st |yt1) ≈
∑Nj=1 ν jδ
sjt,pred
{sit}i∈{1,...,N} ∼ pθ(St |yt
1) : sit = sφ(i)
t,predEnd
−15 −10 −5 0 5 10−2
−1
0
1
2
−15 −10 −5 0 5 10−2
−1
0
1
2Prediction
−15 −10 −5 0 5 10−2
−1
0
1
2Redistribution
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Approximation of gradient ∂θ logLT (θ)
logLT (θ) =T∑
t=1
log∫
Spθ(yt |st)pθ(st |yt−1
1 )dst
∂θ logLT (θ) =T∑
t=1
∫S p′
θ(yt |st)pθ(st |yt−11 ) + pθ(yt |st)p′
θ(st |yt−11 )dst∫
S pθ(yt |st)pθ(st |yt−11 )dst
At time t , {sit−1}{i∈{i,··· ,N} ∼ pθ(st−1|yt−1
1 ) hence
∫S
h(st−1)p′θ(st−1|yt−1
1 )dst−1
=
∫S
h(st−1)p′
θ(st−1|yt−11 )
pθ(st−1|yt−11 )
pθ(st−1|yt−11 )dst−1
≈ 1
N
N∑i=1
ωit h(si
t−1)
with ωit =
dpit
pit
et pit ≈ pθ(si
t |yt1), dpi
t ≈ p′θ(si
t |yt1)
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Algorithm to compute the gradient
% predictionsi
t,pred ∼ p(.|sit−1)
% Weigths
pθ(st |yt1 − 1) =
∫S pθ(st |st−1)pθ(st−1|yt−1
1 )dst−1 → pit,pred = 1
N
∑Nj=1 pθ(si
t,pred |sjt−1)
dpit,pred = 1
N
∑Nj=1 p′
θ(sit,pred |sj
t−1) + 1N
∑Nj=1 pθ(si
t,pred |sjt−1)ω
jt−1
ωit,pred = dpi
t,pred /pit,pred
lt = lt−1 + log(
1N
∑Ni=1 pθ(yt |si
t,pred ))
dlt = ...
% Correction - resampling
ν i ∝ pθ(yt |sit,pred ) et si
t = sφ(i)t,pred
pθ ∝ pθ(yt |sit,pred )pθ(st |yt−1
1 ) → pit =
pθ(yt |sit )p
φ(i)t,pred∑
νi /N
ωit = ...
dpit = ωi
t pit
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Outline
1 Hidden Markov Models (HMM)ModelStatistical inference in HMMHMM with finite hidden stateHMM with continuous hidden state : state-space models
2 Stochastic wind generatorsMotivationsA HMM for the wind directionA HMM for the wind speedA joint HMM for the wind speed and the wind direction ?Conclusion (first part)
3 Numerical weather forecast correctionIntroductionA non linear state-space modelSome numerical results
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Another dataset : wave propagation
Two Hs time series
40 60 80 100 120 140 1600
2
4
6 largecote
Inference : ∂θLT (a∆)
−8 −6 −4 −2 0 2 4 6 8−72
−70
−68
−66
−64
−62
Values of the parameters :α∆ = αb = 1, β∆ = βb = 0, σ∆ = 1,σb = 1, σW = 0.5
−10 −8 −6 −4 −2 043
44
45
46
47
48
49
50
51
52
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Some results on wave propagation
One time step prediction
40 60 80 100 120 140 1600
2
4
6LargeLarge+biaisCote
40 60 80 100 120 140 1600
2
4
6LargeLarge recalle en tpsLarge recalle en tps+biaisHs cote
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Some results on wave propagation
Root Mean Square Errors
0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Prediction horizon
RM
SE
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Back to Wind data (on going work !)
Values of the parameters : α∆ = αb = 1, β∆ = βb = 0, σ∆ = 1, σb = 1.5, σW = 0.5
Good results in some situations
Linear Model
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
Non linear Model
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
0 6 12 18 240
5
10
15
observation (plain line), assimilated observations (points), forecast (plain line)
phase+bias correction (plain line), phase correction (dashed line)
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Back to Wind data (on going work !)
Values of the parameters : α∆ = αb = 1, β∆ = βb = 0, σ∆ = 1, σb = 1.5, σW = 0.5
Good results in some situations
Root Mean Square Errors
0 2 4 6 8 100
0.5
1
1.5
2
2.5
Ailliot, Monbet Hidden Markov Models for wind time series
HMMStochastic wind generators
Forecast correction
IntroductionA non linear state-space modelSome numerical results
Concluding remarks
Lack of information to identify bias and phase from the dataBreaks in the data (assimilation at 00 :00 each day) : burning periodsQuality of the data : discretization step, measurement error
Next step : use spatial dataSpace-time model for the true wind fields including the motion of the fieldsAilliot et al., Baxevani et al.Conditional model for in-situ observations
Ailliot, Monbet Hidden Markov Models for wind time series