Modelling of the Markovian Behaviour of Mobile Satellite Channels with MCMC methods C. Alasseur, L. Husson
F. Perez-Fontán
Service Radioélectricité et Electronique SUPELEC Gif-sur-Yvette, France
[email protected],
[email protected]
Departamento de Teoria de la Señal y Communicaciones Universidad de Vigo Vigo, Spain
[email protected]
Abstract— Classical models for mobile satellite channels are based on Markov chains where each state is linked to special propagation conditions. In this paper, we focus on the methodology of extraction of Hidden Markov Model (HMM) from experimental data to describe the time fluctuations of received power in a mobile satellite service (MSS) context. The method developed in this paper is based on a MCMC (Monte Carlo Markov Chain) training phase associated with a k-means classification. Its complexity is reduced when compared to traditional MCMC method. Contrary to existing detection methods, nearly no prior are necessary and it enables an accurate estimation of the HMM parameters and the possibility to detect the evolution of the channel sequentially. Furthermore, a sequential method, based on estimated HMM parameters, is presented and enables to follow the on-line variation of the state sequence. Keywords-component; satellite channel modelling, MCMC, HMM
I.
INTRODUCTION
The interests of satellite channel modelling are numerous and diverse. To quote some of them, good modelling will enable for example effective link adaptation, effective fast fading compensation, or the evaluation of systems performances. It can also assess of the effects of satellite diversity, and better take into account fading effects like rain, obstacles, various types of environments etc … Existing models [1,2], that are usually based on Markov model, postulate the number of states, and their associated distribution based on physical considerations like the type of encountered environment (urban, rural …). Parameters like the transition matrix and distributions parameters of the HMM are then adjusted with a training sequence. This produces quite good models but that can not be easily applied to different contexts. Some studies present methods to extract HMM parameters directly from experimental data, e.g. F. Perez-Fontán who models rain rate time series [3]. The Markov states of the model correspond to special propagation conditions like scintillation, presence of vegetation, types of rain, e.g. rain events are various: convective, stratiform and can be split into 4 model states regarding the rain rates and event durations. Depending on the characteristics of rain, the effects on the
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attenuation between the satellite and the terrestrial receiver are different. Therefore in this paper, we are not trying to give a universal model but a method that can derive a proper HMM in any situation. It provides a better solution by making as few prior assumptions as possible and we will limit our assumptions to the number of states, assumption that will certainly be not necessary in future research. This paper develops a sequential estimation method performed in two steps: a training and an on-line (sequential) detection step. The first step consists in estimating the HMM model of the received satellite signal, i.e. the transition matrix between states and their corresponding distribution. The MCMC method, presented in this paper, is based on [4] and provides raw change-points detection because fewer iterations are performed. Hence, it is associated to a k-means classification to refine change-points location. Because the association of MCMC and k-means produces very good change-points detection, the HMM parameters can be very easily estimated. The second step of the method performs sequential detection and consists in using the estimated HMM parameters in order to follow the on-line evolution of the channel. In the following the HMM is presented in the first section. In the second section, the HMM estimation method using MCMC and k-means is detailed. We explained the sequential method in the third part and results of parameters estimation and on-line detection are given in the last section. II.
CHANNEL MODEL
We assumed that the satellite channel received power can be modelled by an HMM. An HMM model can be described by: •
its number of states n
•
a n×n transition matrix
•
n probability density functions (pdf)
At each time t, the channel is considered to be a realisation of one of the n states of the HMM. The n×n transition matrix gives the probability of transition from one state to another
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between two consecutive time samples. One probability density function (pdf) is associated to each state of the Markov chain and gives the distribution of the observations when the channel is in this particular state. Therefore, a hidden (non-observable) state sequence corresponds to a sequence of observations. A 2state HMM is presented in fig. 1: 1-p11 pdf1
p(r / y; θ) = C( y; θ)e− φSr − γK r where •
y is the observation vector
•
C( y, θ) is a normalizing constant
•
θ = (λ, V, σ 2 ) the set of hyper parameters with:
pdf2
p11
1-p22
λ the prior proportion of change-points p22
V the variance of the Gaussian distribution for the vector of means (m k )
Figure 1. example of HMM
III.
HMM ESTIMATION METHOD
The proposed method for HMM parameters estimation is composed of two stages: a raw classification using MCMC method based on [4] and then a k-means classification to refine MCMC results. The MCMC method enables to detect the change-points sequence of the received satellite signal; but only raw change-points position estimation is obtained because the algorithm is run with less iterations than recommended by [4] in order to speed the execution time. The change-points sequence is raw in the sense that in our case the MCMC method detects too many change-points. Therefore, a k-means method is used to eliminate false detections and to classify the segment defined between two consecutive change-points to its most probable state. If more iterations are performed, the MCMC may lead to perfect results. But the association of MCMC and k-means enables faster and accurate change-points detection. The number of state is a prior of the problem. A. MCMC Method 1) MCMC Principles [5]: The purpose of MCMC method is to generate an ergodic sequence (x) of associated distribution f without simulating (x) directly from f. For an arbitrary starting value x(0), an ergodic chain (x(t)) is generated using a transition kernel with stationary distribution f and for a large enough T, x(T) can be considered as distributed from f. In our case, MCMC aims at estimating the change-points sequence (rt, n≥t≥1), i.e. the transition between two states of the model. The change-points sequence (rt) is null except when a transition between two states occurs for which rt is equal to one. (rt) is assumed to be a sequence of independent and identically distributed (iid) Bernouilly random variables with parameter λ . Let (τ k ) be the sequence of change-points instants: τk is the instant t of the change-point k and y = ( y t , t ≥ 1) the observations sequence, i.e. the channel attenuation. Then it exists a sequence of means (m k ) such that for any τk −1 + 1 ≤ t ≤ τk , y t = m k , t + ε t with (ε) a sequence of zero-mean random process.
In theses conditions, the posterior distribution of r is shown to be [4]:
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(1)
σ 2 the variance of the additive noise (ε)
V σ2 + V 1 1− λ , γ = log( ) + log( ), 2 2σ (σ + V ) σ2 λ 2
•
φ=
•
K r = ∑ rt + 1
2
n −1
t =1 Kr
Sr = ∑
(the
number
of
segments)
and
τk
2 ∑ (y t − yk )
k =1 t = τ k −1 +1
2) Hastings-Metropolis Algorithm: The Hastings-Metropolis algorithm is a MCMC method based on a conditional density q( x, x ' ) . The principle of every Hastings-Metropolis algorithm is to generate an ergodic Markov chain (r i ) whose stationary distribution tends to be equal to the posterior distribution of r in virtue of the ergodic theorem.
At iteration i, the Hastings-Metropolis algorithm can be split into two steps: • a candidate ~r is proposed from the proposal kernel q(r i , ~r ) •
~r is accepted as the new value for (r ) , r i +1 = ~r , with the probability: p(~r / y; θ)q(~r , r i ) α(r i , ~r ) = min(1, ) p(r i / y; θ)q (r i , ~r )
(2)
At each iteration, candidate sequences ~r are proposed from the following four kernels: • Independent change-points sequence: ~r is completely independent from r. Then, the proposed kernel is equal ~ ~ to q(r, ~r ) = λK r −1 (1 − λ) n − K r that leads to the following acceptance ~ ~ α(r, ~r ) = min(1, e −φ( Sr −Sr ) −β( K r − K r ) ) with
β=
probability:
σ2 + V 1 log( ) σ2 2
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•
Creation of a change-point: a new change-point position is chosen randomly among ( rt ) such that rt = 0 . The corresponding kernel is 1 q(r, ~r ) = and the creation acceptance n − (K r − 1) probability ~ ~ ~ K r −1 ). is α (r, ~r ) = min(1, e −φ( Sr −Sr ) − γ ( K r − K r ) × n − (K r − 1)
•
Deletion of an existing change-points: a new changepoint position is chosen randomly among ( rt ) such that rt = 1 . The corresponding kernel is 1 q(r, ~r ) = and the creation acceptance (K r − 1) probability ~ ~ ~ n − (K r − 1) ). is α (r, ~r ) = min(1, e − φ( Sr −Sr ) − γ ( K r − K r ) × (K r − 1)
•
Update of one change-point position: two instant points (t, t’) are randomly chosen such that rt = 1 and rt ' = 0 . All remaining instant points keep the same value except these two: ~rt = 1 − rt and ~rt ' = 1 − rt ' . Then q(r, ~r ) = q(~r , r ) and the acceptance probability is now ~ equal to α (r, ~r ) = min(1, e −φ( Sr −Sr ) ) .
For any ~r acceptance, the set of hyper parameters is updated so that they maximise the complete S K − 1 ~ r ~2 = likelihood f ( y, r, θ) : λ = r σ and , n −1 n −Kr n
~ V=
2 ∑ ( y t − y) − Sr
t =1
Kr
− σ2
Hastings Metropolis is run at a fixed and low temperature [4] in order to discriminate global and local maxima of the posterior distribution p(r / y; θ) . Concretely, this induces the replacement of (φ, γ ) by the couple (φ / T, γ / T ) . To run Hastings Metropolis algorithm at low temperature enables to lower the number of necessary iterations before convergence. B. K-Means Classification with Mahalanobis Distance 1) Principle of the Method: • Projection [6, 7]: every interval between two estimated change-points is projected onto the mean and variance (in dB) subspace in order to reduce the classification complexity.
•
‘Safe’ projections: the k-means classification operates only on ‘safe’ projections, i.e. it selects only the projections that correspond to the longest time intervals. This precaution is necessary in order to avoid non-representative projections because of a too short time interval.
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•
Classification of ‘safe’ points: in the k-means method, the centers mi of the clusters i are evaluated by successive iterations and the projected points are classified to the nearest cluster in term of Euclidean distance between them and the centers of the clusters. After several iterations, this preliminary rough classification converges, and the covariance matrix Ci can be approximated for each class. The projected points x are then once again classified to the nearest cluster i but this time in term of the Mahalanobis Di defined as distance follows: D i 2 = ( x − m i ) ' C i−1 ( x − m i ) . One advantage of the Mahalanobis distance is to take into account the correlation of the projected points and makes it possible to have curved decision boundaries (unlike linear boundaries for the Euclidean distance). The purpose of k-means classification on ‘safe’ projections is to obtain correct values of centre and covariance for every distribution.
•
Classification of every projection points: the previous step has given quite accurate centre for each distribution; every projection points (not only the ‘safe’ one) are then classified to the nearest distribution’s centre according to the Mahalanobis distance. If the Euclidean distance is used, the final result is poorer: some short states are not detected.
2) HMM Parameters Estimation: With a long enough training sequence (enough state transitions) the transition matrix can be easily estimated. The probability of transition between state i and state j is equal to pij=(number of transitions (i → j) )/(total number of training samples).The distribution pdf of one state is estimated by the histogram calculated on every point allocated to this particular distribution.
IV.
SEQUENTIAL METHOD
When the HMM has been estimated within the training phase, the sequential method is able to detect the channel evolution for each new channel measures. To determine in which state the channel x is at time t, the probabilities of transition p transition and of no-transition p no − transition are evaluated. We set: p transition = p x ( t −1) =i, x ( t )= j × p x ( t )= j
(2)
p no − transition = p x ( t −1) =i ,x ( t )=i × p x ( t )=i
(3)
The transition probability between two states p x ( t −1)=i , x ( t ) = j is given by the estimated Markov transition matrix. The probability that the current measure belongs to state j: p x ( t )= j is given by the pdf estimation (chronogram). A transition event occurs when p transition > p no− transition . Hence, the channel state sequence can be easily evaluated sequentially.
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V.
RESULTS
We have carried out extensive simulations on MSS simulated signals. The experimental signals are generated according to the Lutz model [1]. The channel is composed of two different propagation conditions: NLOS state (Non Line of Sight) modelled by a Rayleigh distribution, whose parameter is σ Rayleigh = 0.8 , and LOS state (Line of Sight) modelled by a Rician distribution, which has a Rice factor equal to 10 dB [8]. These two classes define a 2-state Markov chain whose probability to remain in one particular state is 0.99 for Tsample = 1 . The classic Jakes [9] filter is used to model the correlation of the multipath components induced by the mobility of the channel. Fig. 2 illustrates the state sequence and the satellite power signal generated by the described HMM. original state chronogram
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Too many change-points are detected by MCMC and further processing is required in order to select the relevant ones. The change-points sequence defines time intervals that can be projected into a reduced 2D-subspace as shown in fig. 4. But the classification is very complicated to handle if all projections are considered: the separation between the two distributions is far from clear. Then, the k-means classification operates on ‘safe’ projections selected among all projections and represented on fig. 5. Fig. 5 also shows that the k-means classification separates the two distributions very efficiently; non-safe remaining projection points can now be classified accurately.
sample Figure 2. example of received power with its corresponding state chronogram
10
variance (dB)
The MCMC method enables to estimate the change-points sequence. Fig. 3 shows its results after 1,500 iterations following 3,500 burn-in iterations. The burn-in period is necessary to ensure the convergence. This number of iterations is quite small compared to the number of iterations recommended by [4] (15,000 iterations with 5,000 burn-in iterations). The value of the temperature T is set to 0.02 for the Hastings-Metropolis algorithm.
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Fig. 6 shows the estimated pdf of the two HMM states after the MCMC/k-means classification of the training sequence. The final results of the estimation of the states sequence
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probability density function
estimation is given in fig. 7. The 1500 first samples (corresponding to the training phase) correspond to the state and HMM parameters estimation with the MCMC/k-means method; the remaining points to the sequential detection estimation. The estimation of the state sequence during the training phase is very accurate: estimated change points locations are very close from their real time location, and even short duration states are detected. The sequential detection also gives very good transitions estimation and only detects false transitions two times. 0.08
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VI.
We propose a method to extract the HMM parameters and the state-transitions location of a received satellite signal. This method operates on a training sequence and estimates the Markov transition matrix and the distribution of every state. Once the HMM parameters are estimated, a sequential method is developed in order to follow the evolution of the channel states. This method presents lots of advantages: accurate detection of transitions and estimation of the model, reduced number of required iterations, no prior assumption except the number of possible states. It can also follow slow variations of the channel; the Markov model can vary, but its parameters can be estimated from time to time by reapplying the MCMC/kmeans method without any difficulties. Possible evolutions of the method would consist in making no prior on the number of states that would reduces the number of prior assumptions to zero. The method must also be tested on channels with different characteristics: existence of rain attenuation and larger Markov chain.
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signal level (dB) Figure 6. histograms of the two distributions
REFERENCES [1]
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[8]
estimated state sequence LOS NLOS
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sequential detection
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CONCLUSION AND FURTHER RESEARCH
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sample Figure 7. sequential detection of the states sequence
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E. Lutz , D. Cygan, M. Dippold, F. Dolainsky and W. Papke, ‘The land mobile satellite communication channel’, IEEE trans. on Vehic. Technol., vol. 40, no. 2, pp. 375-386, May 1991 C. Loo, ‘A statistical model of a land mobile satellite link’, IEEE trans. on Vehic. Technol., vol. 34, no. 3, pp. 122-127, Aug. 1985 F. Perez-Fontán, ‘Synthetic rain-rate time-series generation for radio system simulation’, ClimDiff, Fortaleza, Brazil, Nov. 2003 M. Lavielle, E. Lebarbier, “An application of MCMC methods for the multiple change-points problem”, Signal Processing 81, pp. 39-53, 2001 C. Robert, ‘Méthodes de Monte Carlo par chaînes de Markov’, Statistique Mathématique et Probabilité, ECONOMICA L. Husson, M. Pourmir, J-C. Dany and A. Beffani, “Extraction of empirical models from experimental measurements in the L-band for mobile satellite transmissions”, 19th AIAA International Communications Satellite Systems Conference, 17-20 April 2001, Toulouse, France L. Husson, J.C. Dany, S. Chambon, K. Berradi, A. Beffani, ‘Modelling of Mobile Satellite Channels by Scalable Clustering Algorithm’, VTC 2002 spring, 6-10 May 2002, Birmingham, USA E. Kubista, Fontan F.P., Castro M.A.V., Buonomo S., ArbesserRastburg B.R., Baptista J.P.V.P, ‘Ka-band propagation measurements and statistics for land Mobile Satellite Applications’, IEEE trans. on Vehic. Technol., vol. 49, no. 3, May 2000, pp. 973-983 M. Pätzold, Y. Liu and F. Laue, ‘A study of a land mobile satellite channel model with asymmetrical Doppler power spectrum and lognormally distributed line-of Sight component’, IEEE trans. on Vehic. Technol., vol. 47, no. 1, February 1998, pp. 297-309
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