The geometry of the statistical model for The estimation problem - PowerPoint PPT Presentation

Geometry of range–based localization Marco Compagnoni The Range Model The Kummer’s The geometry of the statistical model for The estimation problem range-based localization Conclusions and Perspectives Extra Marco Compagnoni Algebraic Statistics 2015 Genova, Italy June 9, 2015

Geometry of range–based localization Marco Compagnoni The Range Model The Kummer’s The Joint work with Roberto Notari, Andrea Ruggiu, Fabio estimation problem Antonacci and Augusto Sarti. Conclusions and Perspectives Extra

Geometry of range–based Range–based Localization localization Marco Compagnoni The Range Problem: find the position of a point x from the range Model measurements between x and a set of given points The Kummer’s m i , i = 1 , . . . , n . The estimation problem Conclusions and Perspectives Extra Examples of applications: • radar and active sonar • molecular conformation • wireless sensor networks

Geometry of range–based The Range Model localization Marco Compagnoni d i ( x ) = x − m i d i ( x ) = � d i ( x ) � m 3 The Range d ji = m j − m i d ji = � d ji � Model The R r R n T r , n : − → Kummer’s d 3 ( x ) d 23 The d 13 x �− → ( d 1 ( x ) , . . . , d n ( x )) estimation problem ˆ d i ( x ) = measured range x Conclusions ⇒ d 2 ( x ) and ǫ i = measurement error Perspectives d 1 ( x ) Extra ˆ d i ( x ) = d i ( x ) + ǫ i m 1 d 12 m 2 T r , n ( x ) = ( ˆ � d 1 ( x ) , . . . , ˆ The model: d n ( x )) ∼ N ( T ( x ) , Σ ) • Deterministic problem: if ǫ i = 0, find the conditions for existence and uniqueness of x (the identifiability problem). • Statistical problem: if ǫ i � = 0, efficiently estimate x .

Geometry of range–based Euclidean Distance Geometry localization Marco Compagnoni The deterministic problem is a The Range main topic of Euclidean Dis- Model m 3 tance Geometry (DG) [Liberti The Kummer’s and others, 2014] . The estimation Given a weighted graph G = problem d 3 ( x ) d 23 ( V , E , W ) , with Conclusions d 13 and Perspectives • V the points m i and x x Extra d 2 ( x ) • E the available distances d 1 ( x ) • W the measured ranges d 12 m 1 m 2 is G embeddable into some k - dimensional Euclidean space? In DG the answer is usually given in terms of Cayley–Menger determinant .

Geometry of range–based The set of feasible ranges localization Marco Hypothesis: Compagnoni • a point x ∈ R 2 , thus r = 2; The Range • three known points m 1 , m 2 , m 3 ∈ R 2 , thus n = 3. Model The Kummer’s T = ( T 1 , T 2 , T 3 ) ∈ Im( T 2 , 3 ) if and only if The � � estimation � � 0 1 1 1 1 problem � � � d 2 d 2 T 2 � 1 0 Conclusions � � 12 13 1 and � � d 2 d 2 T 2 1 0 = 0 , T 1 , T 2 , T 3 ≥ 0 . Perspectives � � 12 23 2 � � d 2 d 2 T 2 1 0 Extra � � 13 23 3 � � T 2 T 2 T 2 1 0 1 2 3 Proposition: the set of feasible ranges is the semialgebraic surface X ⊂ R 3 defined by  d 2 32 T 4 1 + d 2 31 T 4 2 + d 2 21 T 4 3 − 2 d 32 · d 31 T 2 1 T 2 2 +2 d 32 · d 21 T 2 1 T 2 3 − 2 d 31 · d 21 T 2 2 T 2  3 − − 2 d 21 · d 31 d 2 32 T 2 1 +2 d 32 · d 21 d 2 31 T 2 2 − 2 d 32 · d 31 d 2 21 T 2 3 + d 2 21 d 2 31 d 2 32 =0 ,  T 1 , T 2 , T 3 ≥ 0 .

Geometry of range–based The Kummer’s surface I localization Marco Compagnoni r − r + 3 1 The Range Model The Kummer’s m 2 The T 2 , 3 Γ 1 Γ 3 − − → estimation r 0 r 0 1 3 problem r − r 0 r + 2 2 2 Conclusions m 3 m 1 and Γ 2 Perspectives Extra r − r + 1 3 • ¯ X is a quartic surface with 16 nodes, thus ¯ X is a Kummer’s surface . The nodes on X are the images of m 1 , m 2 , m 3 . • There exist 16 conics on ¯ X . The conics on X are the images of r ± and Γ i , i = 1 , 2 , 3 . They are asymptotic curves of X i and divide the positive and negative curvature regions of X .

Geometry of range–based The Kummer’s surface II localization Marco Compagnoni The Range Model The Kummer’s The estimation problem Conclusions and Perspectives Extra • There exist 16 planes (the tropes), each one tangent to ¯ X along one conic. The 12 tropes tangent to X come from the triangular inequalities plus some other geometrical arguments and they define a convex polyhedron Q 3 containing X . • The boundary of the convex hull of X is the union of the positive curvature regions of X and slices of each facet of Q 3 .

Geometry of range–based Pseudorange–based localization localization Marco Compagnoni T + R 0 1 The Range Model U 0 T + 0 T + 2 The Kummer’s The estimation R 1 E − problem U 1 Conclusions and Perspectives T − 2 T − Extra U 2 0 T − R 2 1 • In some applications only the range differences or pseudoranges are available: τ 1 ( x ) = d 1 ( x ) − d 3 ( x ) , τ 2 ( x ) = d 2 ( x ) − d 3 ( x ) [Compagnoni and others, 2013] . • The set of feasible pseudoranges is the projection π ( X ) of X from its ideal singular point, where π ( T 1 , T 2 , T 3 ) = ( T 1 − T 3 , T 2 − T 3 ).

Geometry of range–based Near and Far Field localization Marco Compagnoni The Range In several applications one distinguishes between near and far Model The field scenarios (e.g. distributed sensors versus compact arrays). Kummer’s The • Near Field: the point x is closed to (at least one) estimation problem m i , i = 1 , 2 , 3 . The range model is singular . Conclusions and • Far Field: the point x is far away from m i , i = 1 , 2 , 3 . A Perspectives good approximation of the Kummer’s surface is given by the Extra tangent cone to the ideal singular point of X , i.e. the elliptic cylinder C having equation d 2 32 T 2 1 + d 2 31 T 2 2 + d 2 21 T 2 3 − − 2 d 31 · d 32 T 1 T 2 +2 d 21 · d 32 T 1 T 3 − 2 d 21 · d 31 T 2 T 3 −� d 31 ∧ d 32 � 2 = 0 .

Geometry of range–based Far Field estimation localization Marco Compagnoni Maximum Likelihood Estimation (MLE): The Range � � T − T � 2 T = argmin Model T ∈ X The Kummer’s • asymptotically efficient estimator; The estimation • nonconvex optimization; problem • X has Euclidean Distance degree 20. Conclusions and Perspectives Squared–Range–based Least Square (SR–LS): Extra [Beck,Stoica,Li 2008] 2 − T 2 � 2 � � T = argmin T T ∈ X • it is not first order efficient; • although nonconvex, there exist efficient solution methods; • it is equivalent to MLE with respect to Cayley–Menger variety, an elliptic paraboloid with Euclidean Distance degree 5.

Geometry of range–based SR-LS performance localization Marco Compagnoni Scenario : The Range √ √ 2 , − 1 3 2 , − 1 3 m 1 =( − 2 ) , m 2 =( 2 ) , m 3 =(0 , 1) Model T ( x ) ∼ N ( T ( x ) ,σ 2 I ) , σ =0 . 1 � The Kummer’s Asymptotic Inference : The estimation problem • the inverse G ( x ) of the Conclusions Fisher matrix gives the and Perspectives asymptotic mean square Extra error of the MLE; • by Cram´ er-Rao inequality, the asymptotic mean square error ¯ G ( x ) of any consistent and unbiased estimator satisfies ¯ G ( x ) − G ( x ) � 0 . Proposition : ¯ G ( x ) − G ( x ) has only a non–zero eigenvalue λ ( x ) .

Geometry of range–based Orthogonal projection on C localization Marco Compagnoni Algorithm (OPC): � T • find the nearest point The Range T Model T ∗ ∈ C to � T ; The T ∗ Kummer’s • find the line L � T containing The � T , T ∗ ; estimation problem • the estimate T is the Conclusions and intersection of L � T and X Perspectives closest to � T . Extra • OPC is a consistent estimator ; • the orthogonal projection on C is a two dimensional problem with Euclidean Distance degree 4, then to find L � T ∩ X we have to solve a degree 4 polynomial equation; • in far field regime we expect to have existence and uniqueness of the solution of OPC (at least in a local setting).

Geometry of range–based OPC performance localization Marco Compagnoni Scenario : √ √ 2 , − 1 3 2 , − 1 3 m 1 =( − 2 ) , m 2 =( 2 ) , m 3 =(0 , 1) The Range Model T ( x ) ∼ N ( T ( x ) ,σ 2 I ) , σ =0 . 1 � The Kummer’s Results : The estimation • OPC performs better than problem SR-LS in far field regime , Conclusions and while it is not suitable for Perspectives near field localization; Extra • OPC has a lower algebraic computational complexity with respect to MLE; • similar results have been obtained for more general sensor configurations and in the analysis of the bias.

Geometry of range–based Conclusions and Perspectives localization Marco Compagnoni The Range In our work: Model • we studied the range-based localization problem with two and The Kummer’s three sensors in terms of real algebraic geometry; The • we have characterized the measurements space using classical estimation problem results on Kummer’s surfaces; Conclusions and • we began the study of the estimation problem. Perspectives Extra In future works we will: • complete the analysis of near and far field estimation (singular model, second order efficient estimators [Kobayashi,Wynn 2013] ); • extend our analysis to the cases with n > 3 sensors.