Computational Sensing = Modeling + Optimization CENS seminar Jan 28, 2005 Miodrag Potkonjak Key Contributors: Bradley Bennet, Alberto Cerpa, Jessica Feng, FarinazKoushanfar, Sasa Slijepcevic, Jennifer L. Wong

Goals Why Modeling? Why Non-parametric Statistical Modeling? Beyond Non-parametric Statistical Modeling? Do We Really Need Models? Tricks For Fame and Fun Applications: Calibration

Why Modeling: No OF, No Results L 1 : 1.272m L 2 : 5.737m L  : 8.365m Gaussian: 0.928m Stat. Error Model: 1.662x10 -3 m Location discovery

Why Modeling: What to Optimize? Packet size

Why Modeling: What to Optimize? Receiver/Transmitter quality

Why Modeling:Localized vs. Centralized Reception rate predictability

Why Modeling: Optimization Mechanism One unknown node Two unknown nodes Atomic localization

Why Modeling: What paradigm to use? Maximum Likelihood: Distance measurements correlation

Why Modeling: Protocol Design Lagged autocorrelation

Why Modeling: Executive Summary Objective Function and Constraints: What to Optimize? - Consistency Problem Identification Formulation - Variability Localized vs. Centralized - Time variability Optimization Mechanism - Topology of Solution space: Monotonicity, Convexity Optimization Paradigm - Correlations Design of Protocols - High Impact Features First

How to Model? Most likely value: regression Probability of given value of target variable for predictor variable Validation Evaluation Parametric and Non-parametric Exploratory and Confirmatory

Model Construction: Samples of Techniques Independent of Distance (ID) Normalized Distance (ND) Kernel Smoothing (KS) Recursive Linear Regression (LR) Data Partitioning (DP)

Independent of Distance (ID)

Normalized Distance (ND)

Kernel Smoothing (KS)

Recursive Linear Regression (LR)

Data Partitioning (DP)

Statistical Evaluation of Models

Statistical Evaluation of OFs

Location Discovery: Experimental Results

Location Discovery: Performance Comparison ROBUST – D. Niculescu and B. Nath. Ad Hoc Positioning System (APS). GLOBECOM N-HOP – A. Savvides, C. Han, M.B. Strivastava. Dynamic Fine-Grained Localization in Ad-Hoc Networks of Sensors. MOBICOM. pages APS – C. Savarese, K. Langendoen and J. Rabaey. Robust Positioning Algorithms for Distributed Ad-Hoc Wireless Sensor Networks. WSNA. pages K. Langendoen and N. Reijers. Distributed Localization in Wireless Sensor Networks: A Quantitative Comparison. Tech. Rep. PDS Technical University, Delft

Combinatorial Isotonic Regression: CIR Statistical models using combinatorics Hidden covariate problem Univariate CIR – Problem Formulation: –Given data (x i, y i,  i ), i=1,…,K –Given an error measure  p and x 1 <x 2 <x 3 <…<x K –  p isotonic regression is set (x i, ŷ i ), i=1,…,K, s.t. –Objective Function: Min  p (x i, ŷ i,  i ) –Constraint: ŷ 1 <ŷ 2 <ŷ 3 <…<ŷ K

 p (x i, ŷ j ) =  k | ŷ j – ŷ k |*h ik Univariate CIR Approach Histogram  build error matrix E, e ij =  p (x i, ŷ j ) Histogram Error Matrix XX YY 20

43 CE(x i, y j ) = E(x i,y j ) + min k  j E(x i-1,y k ) Univariate CIR Approach Histogram  build error matrix E, e ij =  p (x i, ŷ j ) Build the cumulative error matrix CE Error Matrix Cumulative Error X YY X

Univariate CIR Approach Histogram  build error matrix E, e ij =  p (x i, ŷ j ) Build the cumulative error matrix CE Map the problem to a graph combinatorial! Cumulative Error X Y Error Graph

Multivariate CIR Approach - ILP Given a response variable Y, and two explanatory X1, X2  3D error matrix E Objective Function: If ŷ k is the predicted value for X 1 =x 1 and X 2 =x 2 Otherwise Constraints: –C1:  one Ŷ for X1=x1, X2=x2 –C2: Ordering constraint:

CIR Prediction Error on Temperature Sensors at Intel Berkeley Prediction error over all nodes: Limiting number of parameters - AIC criteria

Combinatorial Regression: Flavors Minimal and Maximal Slope Number of Segments Convexity Unimodularity Locally Monotonic Symmetry y = f(x), x = g(y)  x = g(f(x)) Transitivity y = f(x), z = g(y), z = h(x)  h(x) = g(f(x))

Combinatorial Regression: Symmetry

Time Dependant Models

Do We Really Need Models?

Modeling Without Modeling: Consistency x 1 x 2  f(x 1 ) > f(x 2 )

On-line Model Construction

Statistics for Sensor Networks: Executive Summary Large Scale Time Dependent Modelling Hidden Covariates: Monotonicity, Convexity,... Go to Discrete and Graph Domains Interaction: Data Collection - Modeling Properties of Networks Simulators

Tricks – Modeling and Sensor Fusion Hide Nodes Split Nodes Weight Nodes Additional Dimensions Additional Sources

Hiding Beacons

Splitting Nodes

Modeling Networks for Fame & Fun

Perfect Neighbors

Applications Calibration Location Discovery Data Integrity Sensor Network Compression Sensor Network Management Low Power Wireless Ad-hoc Network: Lossy Links

Calibration Minimal Maximal Error Minimal Average Error: median Minimal L 2 Error: average Most Likely Value Error PDF LL ML

Calibration Model for Light

Interval of Confidence

Summary: Recipe for SN Research Collect Data Model Data Understand Data...