The system Main result Optimal strategies Performance Distortion-transmission trade-o ff in real-time transmission of Gauss-Markov sources Jhelum Chakravorty, Aditya Mahajan McGill University IEEE International Symposium on Information Theory, HK, June 14-19, 2015 1 / 16
The system Main result Optimal strategies Performance Motivation Sequential transmission of data Zero delay in reconstruction 2 / 16
The system Main result Optimal strategies Performance Motivation Sequential transmission of data Zero delay in reconstruction Applications Smart grids Environmental monitoring Sensor networks Sensing is cheap Transmission is expensive Size of data-packet is not critical 2 / 16
The system Main result Optimal strategies Performance The remote-state estimation setup ˆ X t Y t X t U t Gauss-Markov process Transmitter Receiver Source process X t + 1 = X t + W t , W t ∼ N ( 0 , σ 2 ) , i.i.d. Uncontrolled Gauss-Markov process. ( X t , if U t = 1 ; Transmitter U t = f t ( X 1 : t , U 1 : t � 1 ) and Y t = E , if U t = 0 , Receiver ˆ X t = g t ( Y 1 : t ) Distortion: ( X t − ˆ X t ) 2 Communication Transmission strategy f = { f t } 1 t = 0 strategies Estimation strategy g = { g t } 1 t = 0 3 / 16
The system Main result Optimal strategies Performance The optimization problem T E ( f , g ) h T � 1 1 � i X d ( X t − ˆ D ( f , g ) := lim sup X t ) � X 0 = 0 � T !1 t = 0 T E ( f , g ) h T � 1 1 � i X N ( f , g ) := lim sup U t � X 0 = 0 � T !1 t = 0 4 / 16
The system Main result Optimal strategies Performance The optimization problem T E ( f , g ) h T � 1 1 � i X d ( X t − ˆ D ( f , g ) := lim sup X t ) � X 0 = 0 � T !1 t = 0 T E ( f , g ) h T � 1 1 � i X N ( f , g ) := lim sup U t � X 0 = 0 � T !1 t = 0 The Distortion-Transmission function D ⇤ ( α ) := D ( f ⇤ , g ⇤ ) := ( f , g ): N ( f , g ) α D ( f , g ) inf Minimize expected distortion such that expected number of transmissions is less than α 4 / 16
The system Main result Optimal strategies Performance Literature overview Costly communication: analysis of optimal performance Estimation with measurement cost: estimator decides whether the sensor should transmit - Athans, 1972; Geromel, 1989; Wu et al, 2008. Sensor sleep scheduling: sensor is allowed to sleep for a pre-specified amount of time - Shuman and Liu, 2006; Sarkar and Cruz, 2004, 2005; Federgruen and So, 1991. Censoring sensors: sequential hypothesis testing setup; sensor decides whether to transmit or not - Rago et al, 1996; Appadwedula et al, 2008. 5 / 16
The system Main result Optimal strategies Performance Literature overview Remote state estimation: focus on structure of optimal strategies Gauss-Markov source with finite number of transmissions - Imer and Basar, 2005. Gauss-Markov source with costly communication (finite horizon) - Lipsa and Martins, 2011; Molin and Hirche, 2012; Xu and Hespanha, 2004. Countable Markov source with costly communication (finite horizon) - Nayyar et al, 2013. 5 / 16
The system Main result Optimal strategies Performance Literature overview Remote state estimation: focus on structure of optimal strategies Gauss-Markov source with finite number of transmissions - Imer and Basar, 2005. Gauss-Markov source with costly communication (finite horizon) - Lipsa and Martins, 2011; Molin and Hirche, 2012; Xu and Hespanha, 2004. Countable Markov source with costly communication (finite horizon) - Nayyar et al, 2013. Gauss-Markov source; infinite horizon setup; constrained optimization. 5 / 16
The system Main result Optimal strategies Performance Main result: the Distortion-Transmission function Variance: σ 2 = 1 1.5 1 D ∗ ( α ) 0.5 0 0 0.2 0.4 0.6 0.8 1 α 6 / 16
The system Main result Optimal strategies Performance Main result: the Distortion-Transmission function How to compute D ⇤ ( α ) for a given α ∈ ( 0 , 1 ) ? 6 / 16
The system Main result Optimal strategies Performance Main result: the Distortion-Transmission function How to compute D ⇤ ( α ) for a given α ∈ ( 0 , 1 ) ? Find k ⇤ ( α ) ∈ R � 0 such that M ( k ⇤ ( α )) ( 0 ) = 1 / α , where R k M ( k ) ( e ) = 1 + � k φ ( w − e ) M ( k ) ( w ) dw . Compute L ( k ⇤ ( α )) ( 0 ) where R k L ( k ) ( e ) = e 2 + � k φ ( w − e ) L ( k ) ( w ) dw . D ⇤ ( α ) = L ( k ⇤ ( α )) ( 0 ) / M ( k ⇤ ( α )) ( 0 ) . Scaling of distortion-transmission function with variance. D ⇤ σ ( α ) = σ 2 D ⇤ 1 ( α ) . 6 / 16
The system Main result Optimal strategies Performance An illustration Comparison with periodic strategy 7 / 16
The system Main result Optimal strategies Performance An illustration Source process X t t 7 / 16
The system Main result Optimal strategies Performance An illustration α = 1 / 6, Periodic strategy Source process X t t Error process E t t Distortion = 2.083 7 / 16
The system Main result Optimal strategies Performance An illustration α = 1 / 6, Threshold strategy; Threshold=2 Source process X t t Error process E t t Distortion = 1.5 7 / 16
The system Main result Optimal strategies Performance An illustration Threshold strategy Periodic strategy Distortion 0 1 α 7 / 16
The system Main result Optimal strategies Performance Proof outline We don’t proceed in the usual way to find the achievable scheme and a converse ! Instead, 8 / 16
The system Main result Optimal strategies Performance Proof outline We don’t proceed in the usual way to find the achievable scheme and a converse ! Instead, Identify structure of optimal strategies. Find the best strategy with that structure. 8 / 16
The system Main result Optimal strategies Performance Lagrange relaxation C ⇤ ( λ ) := inf ( f , g ) C ( f , g ; λ ) , where C ( f , g ; λ ) = D ( f , g ) + λ N ( f , g ) , λ ≥ 0. 9 / 16
The system Main result Optimal strategies Performance Structure of optimal strategies The structure of optimal transmitter and estimator follows from [Lipsa-Martins 2011] and [Nayyar-Basar-Teneketzis-Veeravalli 2013]. Finite horizon setup; results for Lagrange relaxation Optimal estimation Let Z t be the most recently transmitted symbol. strategy ˆ X t = g ⇤ t ( Z t ) = Z t ; Time homogeneous! Optimal transmission Let E t = X t − Z t � 1 be the error process and strategy f t be the threshold based strategy such that ( 1 , if | E t | ≥ k t f t ( X t , Y 0 : t � 1 ) = if | E t | < k t . 0 , 10 / 16
The system Main result Optimal strategies Performance Structure of optimal strategies The structure of optimal transmitter and estimator follows from [Lipsa-Martins 2011] and [Nayyar-Basar-Teneketzis-Veeravalli 2013]. Finite horizon setup; results for Lagrange relaxation Optimal estimation Let Z t be the most recently transmitted symbol. strategy ˆ X t = g ⇤ t ( Z t ) = Z t ; Time homogeneous! Optimal transmission Let E t = X t − Z t � 1 be the error process and strategy f t be the threshold based strategy such that ( 1 , if | E t | ≥ k t f t ( X t , Y 0 : t � 1 ) = if | E t | < k t . 0 , We prove that the results generalize to infinite horizon setup; the optimal thresholds are time - homogeneous. 10 / 16
The system Main result Optimal strategies Performance Performance of threshold based strategies Fix a threshold based startegy f ( k ) . Define D ( k ) : the expected distortion. N ( k ) : the expected number of transmissions. 11 / 16
The system Main result Optimal strategies Performance Performance of threshold based strategies Fix a threshold based startegy f ( k ) . Define D ( k ) : the expected distortion. N ( k ) : the expected number of transmissions. { E t } 1 t = 0 is regenerative process. 11 / 16
The system Main result Optimal strategies Performance Performance of threshold based strategies Fix a threshold based startegy f ( k ) . Define D ( k ) : the expected distortion. N ( k ) : the expected number of transmissions. { E t } 1 t = 0 is regenerative process. τ ( k ) : stopping time when the Gauss-Markov process starting at state 0 at time t = 0 enters the set { e ∈ R : | e | ≥ k } 11 / 16
The system Main result Optimal strategies Performance Performance of threshold based strategies Fix a threshold based startegy f ( k ) . Define D ( k ) : the expected distortion. N ( k ) : the expected number of transmissions. { E t } 1 t = 0 is regenerative process. L ( k ) ( e ) : the expected distortion until the first transmission, starting from state e . M ( k ) ( e ) : the expected time until the first transmission, starting from state e . 11 / 16
The system Main result Optimal strategies Performance Performance of threshold based strategies Fix a threshold based startegy f ( k ) . Define D ( k ) : the expected distortion. N ( k ) : the expected number of transmissions. { E t } 1 t = 0 is regenerative process. L ( k ) ( e ) : the expected distortion until the first transmission, starting from state e . M ( k ) ( e ) : the expected time until the first transmission, starting from state e . Renewal relationship D ( k ) = L ( k ) ( 0 ) N ( k ) = 1 M ( k ) ( 0 ) , M ( k ) ( 0 ) 11 / 16
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