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Mobility Prediction in Cellular Network Using Hidden Markov Model Hongbo SI∗ , Yue WANG, Jian YUAN and Xiuming SHAN Department of Electronic Engineering Tsinghua University, Beijing 100084, P. R. China. ∗ Email: [email protected]

Abstract—In next generation networks, mobile communication calls for service with higher quality, which brings new challenge for mobility management. Thereinto, utilization and improvement of mobility prediction helps for preserving resource and providing better performance. So this paper aims to propose a theoretical and factual method to perform mobility prediction in cellular network. By analyzing the demand and character of this kind of personal mobility prediction in large spacial and temporal scale, it is concluded that Hidden Markov Model fits for system modeling. However, classical HMM algorithm will meet with numerical calculation problem when adopted to practical communication system. An improved algorithm is put forward to overcome possible calculating defects. Three different scenarios are set to testify HMM’s efficiency and accuracy, using factual measurement data in cellular network.

I. I NTRODUCTION Mobile communication technology has felt the progressively increasing demand of users, since they do not entirely satisfy with the quality of current service. In order to improve communication condition and provide service of better performance, researchers and technologists have paid more attention to users’ movement patterns. Mobile terminal movements are never totally random since they are constrained by local terrain and traffic condition or since they usually have a precise purpose or habitual route. This ascertainment implicated behind uncertainty is just the foundation of mobility prediction. Thus a mathematical problem could be abstracted from application scenarios as a definition of mobility prediction: learning and inference from prior knowledge. This knowledge could be road information, movement history or user preference. In consideration of different research objects and spacialtemporal scales, mobility prediction can be divided into personal prediction [1] [2]. and community prediction [3] [4], macro-prediction [5] [6] and micro-prediction [7]. This paper is proposed to provide an effective method for solving personal prediction problem from a large point of view. Current researches have given some productions on this subject in theory [6], but its application in practical scenario is still lack of study. In this paper, Hidden Markov Model is adopted for mobile prediction and its factual effect is tested in cellular network system. The rest of paper is organized as follows. Section II gives a brief introduction on basic definition and calculation method of HMM. The next section describes HMM’s application on mobility prediction in detail. Section IV exhibits the results

of simulation using different forecasting methods, where prediction accuracy and consumed time are two major factors measuring the performance of algorithm.The last section gives a conclusion of the whole paper. II. H IDDEN M ARKOV M ODEL Hidden Markov Model (HMM) [9], as a classic part of the theory of Bayesian network [8], sets two kinds of stochastic variables, state variable (hidden variable) and output variable (observable variable). State variable (q1:T in Fig.1) describes the factual situation of observed object, and the underling state-sequence forms a Markov chain. However, due to impossibility, difficulty or imprecision of observation, measured value may not reflect practical situation sometimes. So output variables (y1:T in Fig.1) are distinguished from state variables, and the one-to-one mapping between them describes their corresponding relationship. ...

Fig. 1.

...

Structure of Hidden Markov Model

Based on the structure of HMM (shown in Fig.1), there implicates two kinds of conditional independence statements. qt ⊥{q1:t−2 , y1:t−1 }|qt−1 yt ⊥{q1:T \t , y1:T \t }|qt Using these assumptions, the joint probability distribution of all variables can be simplified as follow. p(q1:T , y1:T ) =

T

p(qt |qt−1 )p(yt |qt )

(1)

t=1

Since all other forms of probability distribution can be gained by marginalizing joint probability p(q1:T , y1:T ), it is believed that the whole model can be represented by p(q1 ), p(qt |qt−1 ) and p(yt |qt ). Mark λ (π, A, B) is introduced, where π is a N × 1 vector and πq1 p(q1 ); A is a N × N matrix and Aqt ,qt+1 p(qt+1 |qt ); B is a N × N matrix and Bqt ,yt p(yt |qt ) (N is the number of states). For HMM, there are three kinds of typical problems [9], which are also basically concerned in mobility prediction, so their solution are briefly discussed in next subsections.

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A. Likelihood calculation Given a model λ = (π, A, B), how to efficiently compute the likelihood of an output-sequence p(y1:T |λ)? Traditionally, the method of brute-force is often adopted. It can be estimated that the overall complexity of calculating p(y1:T ) is O(N T ), thus this algorithm is simple and effective for small T , yet becoming invalid with the increase of T . For this, two denotations are introduced to reduce complexity by utilizing the redundancy of calculation. α(qt ) p(y1:t , qt )

β(qt ) p(yt+1:T |qt )

Then p(y1:T , qt ) = α(qt )β(qt ). The problem of likelihood calculation is changed into computing α and β. Here recursive algorithm is adopted to reduce complexity. B t=0 q1 ,y1 πq1 (2) α(qt+1 ) = Bqt+1 ,yt+1 Aqt ,qt+1 α(qt ) t > 0 qt Bqt+1 ,yt+1 Aqt ,qt+1 β(qt+1 ) t < T qt+1 β(qt ) = (3) 1 t=T It can be summarized that complexity of this recursive method is O(T N 2 ), which is remarkably reduced comparing with O(N T ) of brute-force method. B. Decoding (recognition) Given an output-sequence y1:T and a model λ, how to find the optimal state-sequence? max p(q1:T |λ, y1:T ) q1:T

Getting q1:T from y1:T is necessary and important in inference theory of probabilistic graphical model [8]. In deed, only the distribution of qT is interested in mobility prediction. Thus, the calculation method is shown as follow and another two denotations are introduced. ξ(qt , qt+1 ) p(qt , qt+1 |y1:T ) α(qt )β(qt+1 )Aqt ,qt+1 Bqt+1 ,yt+1 = α(qt )β(qt+1 )Aqt ,qt+1 Bqt+1 ,yt+1 qt qt+1

γ(qt ) p(qt |y1:T ) =

ξ(qt , qt+1 )

(4)

system structure. For HMM, using Baum-Welch algorithm [10], the answer to learning problem is shown as follow: πq∗1 = γ(q1 ) T −1 ξ(qt , qt+1 ) ∗ Aqt ,qt+1 = t=1 T −1 t=1 γ(qt ) T γ(qt )δ(yt , yt ) Bq∗t ,yt = t=1 T t=1 γ(qt )

(6) (7) (8)

Where δ() is Kronecker delta function. Baum-Welch algorithm is an iterative algorithm and every time a group of parameters is calculated, the mount of p(y1:T |λ) and p(y1:T |λ∗ ) should be compared. Only if p(y1:T |λ) ≥ p(y1:T |λ∗ ), the algorithm is completed and the optimal parameters are acquired. III. M OBILITY PREDICTION USING HMM Hidden Markov Model is widely utilized in lots of domains [9], including automatic control, artificial intelligence, finance and biology. Here we attempt to adopt it in cellular communication system. A. Overview For cellular network, its hierarchical character lends itself to adopting HMM. In most cellular network systems, mobility management is performed in BSC. So what mobility prediction concerns is just user’s path from entering the covering area of BSC to leaving. Thus the transfer matrix, distributed and managed by BSC for every user to record his movement habit, only thinks of the cells in this area, which brings about two advantages: 1) The number of states will not be too large. 2) The length of movement history will also be acceptable. B. System Modeling

Fig. 2.

System modeling of cellular network (regular arrangement)

(5)

qt+1

Thus p(qT |y1:T ) = γ(qT ). C. Learning (parameter estimation) Given an output-sequence y1:T , how to estimate the model parameters λ so as to best describe these data? max p(y1:T |λ) λ

Target of learning is utilizing existent data to adjust model parameters, so as to make new parameters better describing

Fig. 3.

System modeling of cellular network (irregular arrangement)

Structure of cellular network can be modeled as a graph. Nodes represent cells and edges represent the neighbouring relationship of cells. So the topology of cellular system, no matter regular (like Fig.2) or irregular arrangement (like Fig.3), can be abstracted into a undirected graph. From another


point of view, this graph is exactly the state-transition graph of every stochastic variable, and every one-step transition must be along the edge from one node to another. Based on this model, the path followed by a mobile is thus modeled as a string of symbols, called movement history. And the problem of mobility prediction in cellular network is converted into a problem of stochastic process. Usually, an assumption is introduced that the state sequence is stationary, based on general knowledge. C. Mobility Prediction Mobility prediction usually contains two major steps: 1) Parameter learning. The object of this process is to determine optimal parameters fitting history data. Using denotations of HMM, it means: max p(y1:T |λ) λ

2) Prediction. Actually, prediction process is inference in HMM. More concretely, in a mobility prediction process, three steps of inference are performed, including: max p(qT |y1:T ) = max γ(qT ) qT

qT

(10)

max p(yT +1 |qT +1 ) = max BqT +1 ,yT +1

(11)

qT +1

yT +1

yT +1

In cellular network, not every changing of user’s locating cell number will cost system resource. In fact, only handoff in communication process will lead system to distribute channel, so prediction is performed only for handoff. In another word, parameter learning happens as long as user steps into new cell, while prediction is only performed in communication. D. Algorithm Improvement

qt

qt

α(qt ) =

α(qt+1 ) α ¯ (qt+1 ) = α(qt+1 ) qt+1 Bqt+1 ,yt+1 Aqt ,qt+1 α ¯ (qt ) qt = Bqt+1 ,yt+1 Aqt ,qt+1 α ¯ (qt )

(13)

qt+1 qt

And especially for q1 , Bq ,y πq α ¯ (q1 ) = 1 1 1 Bq1 ,y1 πq1

(14)

q1

Similarly the recursion of β¯ is as follow: ¯ t+1 ) Bqt+1 ,yt+1 Aqt ,qt+1 β(q qt+1 ¯ t) = β(q ¯ t+1 ) Bqt+1 ,yt+1 Aqt ,qt+1 β(q

(15)

qt qt+1

And especially for qT , ¯ T) = 1 β(q T

(16)

2) The expression of likelihood p(y1:T |λ). Reconsider the denominator of calculating α ¯ (qt+1 ) and give it a denotation μt+1 , we have: Bqt+1 ,yt+1 Aqt ,qt+1 α ¯ (qt ) μt+1 qt+1 qt

=

qt+1

α(qt+1 )

p(y1:t )

=

p(y1:t+1 ) p(y1:t )

Thus

According to the method above, using classic HMM will meet with problem when adopted in mobility prediction. Normally, location update performs frequently and it leads to long movement history. It is noticed that with the growing of T , the probability p(y1:T ) is declining and finally less than the calculation precision of computer. Facing this problem, two classic numerical calculation solutions are worthy for reference. Normalized probability distribution. Using normalized distribution for α and β will keep their calculation precision in recursive processes. α(qt ) α ¯ (qt ) α(qt )

p(y1:t ) is independent of qt , we have:

(9)

max p(qT +1 |qT ) = max AqT ,qT +1 qT +1

α(qt ) = Using (2)(12), and noticing that α(qt )/¯

¯ t ) β(qt ) β(q β(qt )

(12)

qt

Logarithmic summation. Replacing product by logarithmic summation also preserves calculation precision. This action ensures data apart from calculation limit of computers. These treatments must cause changes to classic algorithm. It can be imagined that the following three aspects should be improved in new algorithm. ¯ 1) The recursion of α ¯ and β.

p(y1:T ) = μT p(y1:T −1 ) =

T

μt

(17)

t=1

When using the logarithmic form, equation (17) can be expressed as: T T 1 log = − log μt = − log μt p(y1:T ) t=1 t=1

(18)

3) The method of calculating π ∗ , A∗ and B ∗ . ¯ the form of calculating Based on the definition of α ¯ and β, ξ is not changed. So the calculating method of γ and π ∗ , A∗ , B ∗ is not radically changed accordingly. ¯ t+1 )Aq ,q Bq ,y α ¯ (qt )β(q t t+1 t+1 t+1 ξ(qt , qt+1 ) = ¯ t+1 )Aq ,q Bq ,y α ¯ (qt )β(q t t+1 t+1 t+1

(19)

qt+1 qt

E. Algorithm Pseudocode The pseudocode of improved HMM algorithm can be summarized as follow. Learning and P rediction are two kernel functions for main program.


Algorithm 2 Function Learning Require: λ, y1:T Ensure: λ∗ ¯ ξ and γ using (13) (15) (19) (5) 1: Calculate α ¯ , β, 2: Calculate log p(y1:T |λ) using (18) 3: for r = 1 to c do 4: Parameter adjust to get π ∗ , A∗ and B ∗ using (6) (7) (8) ¯ ξ and γ using (13) (15) (19) (5) 5: Recalculate α ¯ , β, 6: Recalculate log p(y1:T |λ∗ ) using (18) 7: if log p(y1:T |λ∗ ) > log p(y1:T |λ) then 8: Replace π, A and B with π ∗ , A∗ and B ∗ 9: else 10: End circulation 11: Replace π ∗ , A∗ and B ∗ with π, A and B 12: end if 13: end for 14: return λ∗ = (π ∗ , A∗ , B ∗ ) Algorithm 3 Function P rediction Require: λ∗ , y1:T Ensure: rT +1 1: Get qT from max p(qT |y1:T ) using (9) qT

2:

Get qT +1 from max p(qT +1 |qT ) using (10)

3:

Get rT +1 from max p(rT +1 |qT +1 ) using (11)

4:

return rT +1

qT +1

rT +1

records movement history and communication situation of 100 persons over one year. For personal prediction in large scale, what we interest in this database is locating cell tower information and corresponding staying time of a particular user. Using this database for mobility prediction, a particular location area with group of cells is chosen as the background of research. The system model of neighbouring relationship in this area is shown in Fig.4. Moreover, a person in this project is selected as our research object.

Fig. 4.

System model of location area in simulation

B. Simulation Result In order to exhibit the performance of HMM in different application scenarios, two different actual environments are set in our simulation, where prediction accuracy and time consumed are two major elements of algorithm concerned. It is noted that in order to realize the performance of HMM, prediction is executed as long as user’s locating cell number changes, yet in actual cellular system, only changes in communication cause mobility prediction. For comparison, Markov chain and order-2 Markov method are also contained in the simulation. Scenario One. User’s first access to this location area. Since no history data for this user can be utilized, the initial parameters of HMM are randomly generated, using a distribution close to uniform. Scenario Two. The user has accessed this area frequently and left mature data for mobility modeling, yet he suddenly changes his moving habit. Here old parameters are still adopted when initializing HMM since the changing is unpredictable, and HMM’s fitting ability is mainly concerned in this scenario. 0.8 0.7 0.6 Accuracy of prediction

Algorithm 1 Main Program Require: observation information y1:T Ensure: prediction results r2:T +1 1: Algorithm starts up when user enters appointed area. 2: if user is the first time accessing to this area then 3: Initialize λ with random parameters 4: else 5: Initialize λ with history parameters 6: end if 7: T = 1 8: Calculate α and β using (2) and (3) 9: while user is still in this area do 10: Add new cell information to y1:T 11: if user is in communication then 12: λ∗ = Learning(λ, y1:T ) 13: rT +1 = P rediciton(λ∗ , y1:T ) 14: return r2:T +1 15: else 16: λ∗ = Learning(λ, y1:T ) 17: end if 18: T =T +1 19: end while

0.5 0.4 0.3 0.2 HMM Markov order−2 Markov

0.1

IV. S IMULATION A. Data Source A database, called Reality Mining Project [11] from MIT media laboratory, is adopted in the simulation. This project

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Fig. 5.

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Time consumed in prediction

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V. C ONCLUSION In order to improve communication performance in cellular, the technology of mobility prediction is introduced. Through analysis and simulation, it is concluded that Hidden Markov Model uses more cost in calculation time to exchange for prediction accuracy. However, in actual communication system, the movement path may not be so long as our simulation scenarios, so temporal cost in prediction is not comparable to staying time in corresponding cell. Thus, Hidden Markov Model is proved to be efficient and accurate as a solution to mobility prediction, and will exhibit more merit if adopted in factual communication system.

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ACKNOWLEDGMENT This work is supported in part by the National Basic Research Program of China (973 Program) under grants 2007CB307100 and 2007CB307105.

Prediction accuracy in scenario two

0.4

Time consumed in prediction

ability to user’s change. In another word, HMM learns faster than Markov, which is determined by algorithm itself. Considering the calculation method of these algorithms, it can be analyzed that multiplication complexity of HMM is O(T N 2 ) while the one of Markov and order-k Markov is only O(1). So time consumed in prediction using Markov and order-2 Markov is far less than using HMM. This theoretical conclusion can be proved from figures, which only show the time consumed of HMM (Time curves of Markov and order-2 Markov are so close to 0 that they cannot be seen in figures). It is seen that the trend of curve is basically linear with the increase of sequence. And the fluctuation is mainly due to different attempt times in parameter learning.

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R EFERENCES

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[1] Amiya Bhattacharya and Sajal K. Das, “LeZi-update: an informationtheoretic approach to track mobile users in PCS networks”, International Conference on Mobile Computing and Networking, pp.1-12, 1999. [2] Lucian Vintan, Arpad Gellert, Jan Petzold, and Theo Ungerer, “Person Movement Prediction Using Neural Networks”, First Workshop on Modeling and Retrieval of Context, 2004. [3] Wee-Seng Soh and Hyong S. Kim, “Dynamic Bandwidth Reservation in Cellular Networks Using Road Topology Based Mobility Predictions”, INFOCOM 2004, pp.2766-2777, 2004. [4] Wee-Seng Soh and Hyong S. Kim, “Dynamic guard bandwidth scheme for wireless broadband networks”, INFOCOM 2001, pp.572-581, 2001. [5] Fei Yu and Victor Leung, “Mobility-based predictive call admission control and bandwidth reservation in wireless cellular networks”, Computer Networks, vol.38, pp.577-589, 2002. [6] Arpad Gellert and Lucian Vintan, “Person Movement Prediction Using Hidden Markov Models”, Studies in Informatics and Control, vol.15, pp.17-30, 2006. [7] Wei Cui and Xuemin Shen, “User Movement Tendency Prediction and Call Admission Control for Cellular Networks”, IEEE International Conference on Communications, pp.670-674, 2000. [8] Micheal I. Jordan, “Graphical Models”, Statistical Science, vol.19, pp.140-155, 2004. [9] Sherif Akoush and Ahmed Sameh, “Mobile User Movement Prediction Using Bayesian Learning for Neural Networks”, Proceeding of the 2007 International Conference on Wireless Communications and Mobile Computing, pp.191-196, 2007. [10] LE Baum and T Petrie, “Statistical Inference for Probabilistic Functions of Finite State Markov Chains”, Annals of Mathematical Statistics, vol.37, 1554-1563, 1996. [11] N Eagle and A Pentland, “Reality Mining: Sensing Complex Social Systems”, Personal and Ubiquitous Computing, Vol 10, pp.255-268, 2006.

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C. Result Analysis From figures above, it is concluded that HMM surely has its advantage in prediction accuracy, comparing with Markov and order-2 Markov. A precise prediction algorithm is usually measured from two aspects: insensitive to outliers and adaptive to new changing. In fact, these two characters contradict with each other, so making moderate compromise is crucial. HMM and Markov method are both based on large data of history, which makes them insensitive to outliers. On the other hand, comparing with Markov method, HMM has more adaptive