Review Petteri Nurmi

Transkriptio

1 Review Petteri Nurmi

2 Overview of the Course I: Measuring and estimating location information II: Analysing and understanding location data Representing location, location systems, positioning and tracking Place identification, mobility modeling, trajectory analysis

3 Part I: Measuring and Estimating Location Information

4 Representing Location Information Absolute Coordinates with respect to a given reference system Requires an origin and a reference/unit distance Relative Location specified relative to an object Requires reference point, distance and angle Symbolic Location expressed using a semantic description (place or room)

5 Ellipsoidal Coordinate Systems Contemporary geographic reference systems rely on ellipsoidal coordinate systems Shape of Earth represented as an ellipsoid Origin specified by two orthogonal planes that intersect at the geocenter of Earth Latitude ): parallel to Equator Longitude ): perpendicular to Equator Shape of geographic area approximated using a reference ellipsoid Essential for distance calculations Specified by a combination of semi-major axis (or diameter) and inverse flattening

6 Geoid Approximation of mean sea level Equipotential surface of the gravitational field that coincides with mean sea level Standard for measuring altitudes from sea level 1) True ocean level 2) Reference ellipsoid 3) Local plumb 4) Continent 5) Geoid

7 Measuring distances between locations Euclidean distance Useful for short distances Requires an unit distance, i.e., how much a difference of 0.01 (or so) in coordinates is in meters at a given latitude Geodetic distance Distance along the surface of an ellipsoid Geodetic problem defines how differences in coordinates are mapped into distances and vice versa Direct: given an angle and distance, determine new location from a known origin Indirect: given two points, determine angle and distance between them Vincenty s formula provides an iterative algorithm for solving geodetic problems

8 Determining Location Location system Component that provides measurements that can be used for determining the position of an entity Types of measurements: Identifier: GSM cell identifier, WiFi Mac address RSS: received signal strength (-dbm) Geographic: distance or angle Positioning algorithm Technique for determining the position of an entity using the measurements provided by the location system Triangulation, trilateration, multilateration, fingerprinting

9 Geometric Positioning Algorithms Basic idea: Angle and/or distance measurements from two or more reference points used for determining position Triangulation The use of angle measurements for determining the position of an entity Trilateration The use of distance measurements for determining the position of an entity Multilateration The use of differences in distances for determining the position of an entity

10 Error of Geometric Positioning Distance and/or angle from reference points can seldom be measured exactly Obstacles cause attenuation and multi path effects Synchronization, reference point errors, interference Measurements define an error region Intersection of multiple error regions defines an area of uncertainty for position estimate Size of the area of uncertainty referred to as the Dilution of Precision (DoP) Geometry of reference points influences the size of the error region

11 Location Systems for Geometric Positioning Distance can be measured using Time-of-flight One-way: from reference point to receiver or vice versa, requires time synchronization (e.g., GPS) Round-trip time (e.g., Radar) Radio propagation models Mathematical formulation of how signals vary as a function of distance Example, log distance path loss model Alternative is to use angle measurements E.g., ultrasound, zigbee

12 Other sources of error Atmospheric effects Propagation follows speed of light only in a vacuum Refraction: change in direction of wave, occurs when a wave enters from one medium to another Multipath propagation Phenomenon where signals reach the receiver along multiple paths Reflection, diffraction or scattering, depending on wavelength and size of obstacles Clock errors Synchronization errors between two clocks can cause huge errors due to speed of signals

13 Fingerprinting Technique that exploits spatial variations in observed signal characteristics for positioning Two phases Calibration: construct a database that characterizes the signal variations at different locations Operates on measurements containing both location and signal characteristics Estimation: compare a new measurement against the radio map to determine the location of an entity Given a new signal characteristics measurement, use the radio map to estimate the most likely position where the measurement was taken

14 Deterministic Fingerprinting Basic idea: Given measurement s, calculate a distance d(s,x) between s and all measurements x in the radio map Typically Euclidean distance or correlation Use the most similar measurements x (as defined by the distance) to estimate location of client knn and WkNN Find the best k measurements, i.e., measurement x for which d(s,x) is smallest Assign a weight for each measurement (knn = uniform, WkNN = non-uniform) Position estimated as a weighted average of the locations associated with the best matching measurements

15 Probabilistic Fingerprinting Basic idea: Given measurement s, calculate the probability of seeing the measurement at different locations Estimate the location of the client based on the resulting probabilities Probability of a location given by a signal model Specifies the probability distribution of observing particular signal values at a given location Can be histogram-based or parametric (e.g., Gaussian)

16 Location Systems Proximity Sensing Position based on closeness of a reference point Examples: infrared, RFID, Bluetooth Special case: mobile call-detail records (CDR), location estimated as coordinates of current cell tower

17 Location Systems Satellite Positioning Trilateration-based positioning approach Reference points are satellites on a specific orbit Distances from satellites measured using one way timeof-flight measurements Satellites broadcast messages that contain orbital position and system time of the satellite Receivers listen for the broadcasts and estimate Distance from the satellites Clock offset Pseudorange:

18 Location Systems Mobile Networks Network divided into base stations and cells Base Transceiver Station: equipment responsible for handling communications within a geographic area Cell: antenna that serves a specific geographic area One BTS can be responsible for multiple cells Location Area Identifier (LAI): globally unique id for a cell Location Area (LA): cluster of cells, smallest unit for which the network maintains position information Position can be determined using Proximity sensing: BTS coordinates, Timing Advance Multilateration: Differences in arrival times Fingerprinting: on the handset based on observed RSS values of cells

19 Position Tracking Monitoring the location of an entity over time State space models provide a generic framework for implementing tracking Location of an entity represented using a state x k which cannot be directly observed Evolution of state over time controlled by state equation x k = Ax k-1 + v Measurements (or observations) y k provide cues about the true state of the system Measurement equation specifies how the measurements relate to the true system state y k = Ux k + w

20 Bayesian Optimal Filter Probabilistic approach to state space models Given a sequence of measurements y 1:k returns a probability distribution p(x k y 1:k ) over the current state (location of object) Evolution of state specified by a probability distribution p(x k x k-1 ), also know as (state) transition probability Measurements relate to state using another probability distribution p(y k x 1:k, y 1:k-1 ) Usually assume that Future and past independent of each other given present Measurements are conditionally independent given state

21 Kalman Filter Closed form solution of the Bayesian optimal filter Assumes noise is uncorrelated and Gaussian and that relationships between variables are linear Under these assumptions, the filtering distribution p(x k y k-1 ) is also Gaussian Implementing the filter thus requires maintaining estimates of the mean and covariance matrix Two phases Prediction: guess the most likely values of the parameters given the current values Update: correct the predictions when a new measurement comes available

22 Kalman Filter Prediction: Propagate current values using system dynamics Update m* k A k-1 m k-1 P* k A k-1 P k-1 A T k-1 + Q Calculate residuals between predicted value and estimated value and correct the predicted values

23 Particle Filters In real world applications noise often non-gaussian and the state space is non-linear x k = z(x k-1 ) + q k y k = h(x k )+ r k Particle filtering uses Monte Carlo integration recursively to approximate the filtering distribution For function g( ) and distribution f(x), we have: E(g(x)) = g(x) f(x) dx When f(x) equals the filtering density, we get: E(g(x)) = g(x) p(x k, y 1:k ) dx 1/K j g(x kj )

24 Particle Filters Sequential Importance Resampling Initialization: Draw K particles according to prior distribution, set w j = 1/ K for all particles Estimation step: Draw K samples from proposal distribution: x kj (x k x kj, y 1:k ) Update importance weight of particle j w kj = w k-1 j p(y k x kj ) p(x kj x k-1j ) / (x k j, x 1:k-1j, y 1:k ) Calculate number of effective particles N EFF = 1 / j (w kj ) 2 and resample particles if value below a threshold State of filter can be estimated using j w kj x k j

25 Part II: Analysing Location Measurements

26 Preprocessing Errors in location measurements should be rectified before data can be analyzed Measurement validity: remove extreme values or values that are otherwise known to be erroneous (e.g., less than 4 GPS satellites) Measurement uncertainty: remove measurements with high uncertainty (e.g., high HDOP value) or filter measurements over time (Kalman filter or particle filter) to smoothen them

27 Partitioning Algorithms K-Means One of the best-known clustering algorithms Iterative relocation algorithm, optimizes squared loss m i corresponds to the center of a cluster, C i is the set of points allocated to cluster i Basic structure: Initialization: generate k cluster centers according to some criterion (e.g., random selection from data) During each iteration: Allocate each point to the cluster that is closest Revise cluster centers based on the points that are assigned to the cluster Repeat until no change in values

28 Partitioning Algorithms Gaussian Mixture Models Data generated by k random variables, each variable X i characterized by probability density function f i ( i ) For each point i, a hidden and unobservable variable c i determines the cluster where i belongs to The clusters are called mixture components Each f i ( i ) is assumed to be Gaussian Mean i determines the center of the cluster Covariance matrix i determines shape of the cluster Cluster parameters can be determined using the expectation maximization (EM) algorithm

29 Density-Based Clustering Class of algorithms that represents clusters as dense regions of objects Epsilon neighborhood: collection of points that are within distance Eps from a point Dense neighborhood: Epsilon neighborhood that contains at least MinPts points Radius-based clustering Merge all points within distance Eps Prune clusters using a density criterion

30 Density-Based Clustering DBScan Algorithm that recursively merges Epsilon neighborhoods together to identify dense regions Let c be a core object, within the Epsilon neighborhood of c considered as seed points A point that has at least MinPts within its Epsilon neighborhood is called a core object Non-core objects which do not belong to the Epsilon neighborhood of any core objects are noise Cluster expanded with (previously unallocated) points that are within the Epsilon neighborhood of a seed point

31 Place Identification (from coordinate data) Place is a symbolic/semantic representation of location Physical locations linked with activities and semantics Consistent with the way people refer to location information Places can be detected from coordinate data: Spatial clustering used to identify regions where a person spends a significant amount of time Pre- or post-processing used to remove areas/points that are unlikely to be significant E.g., using temporal or spatial constraints Additionally a labeling step that assigns semantics with the identified places

32 Movement Statistics Area of influence The geographical area within which a user spends most of her time doing daily activities Diameter Maximum distance between two cell towers (BTS) Characterizes the size of the area of influence Radius of gyration Average distance that the person typically travels

33 Mobility Model Mathematical characterization of how people move Model can capture: 1. Movement between places (location) 2. Arrival times, i.e., when person arrives to places 3. Duration of stay in places Context independent vs. dependent Whether the model depends or not on factors such as time-of-day, weekend/weekday, type of location etc. Model order Specifies how much historical information needs to be considered to make predictions about future behavior

34 Location Transition Modeling Markov Predictor Stochastic state machine State of the system is assumed to evolve over time according to a probabilistic model p(x k x 1:k-1 ) Transition probabilities p(x k x 1:k-1 ) determine how likely it is that a person moves from one place to another Probability of current state assumed to depend on previous q values, where q is the order of the model In practice Use place identification to learn the places of interest for an individual Use location measurements to estimate how the person moves from one location to another

35 Lempel-Ziv (LZ) Predictor Sequence of visited locations represented as a string The string is recursively split into parts to maximize compressibility of the input Splitting of string can be represented using a so-called LZ tree Next place can be predicted by examining how often a particular location has followed the current location history 2/3 3/8 2/8 2/8 1/8 1/2 1/

36 Stationary Detection Stationary Moving Determining whether a person is moving or not, the first subtask in transportation behavior monitoring Sensor values tend to contain more variation when the user is moving compared to stationary periods Variance or intensity of accelerometer values Rate of change in the signal environment Similarity of WiFi access points seen over consecutive time windows Number of unique cell towers observed within a window

37 GPS Locomotion Detection HCR = P c / d SR = P s / d VCR = P v / d Variations in movement trajectories can be used for determining transportation mode Heading change rate (HCR): frequency that people change heading direction within unit distance Stop rate (SR): frequency that people stop / move with slow speed within unit distance Velocity change rate (VCR): frequency of observing significant changes in velocity over a unit distance Other possible features Statistical features (mean, variance, maximum etc.) characterizing velocity and acceleration, calculated over specific road segments

38 Trajectory The location of an object over time Continuous function f(t) that returns the 2D or 3D position of an object at time t Location measurements can only be collected at discrete intervals Sensed trajectory is a piece-wise continuous function f(t k ) that returns position of object at sampling interval t k s 1 s 2 s 3 s 4 s 5 s 6 s 7 s8 = s R

39 Trajectory Simplification Simplification refers to approximating the original trajectory with a simpler form Reduces storage space Provides savings in power consumption Location measurements Sensed Trajectory Simplified Trajectory Measurements retained

40 Simplification Error Error measured as the difference between the original trajectory and simplified trajectory Perpendicular distance: maximum distance between any measurement and the simplified line Angular distance: accumulated difference in angle Orthogonal distance: difference in distance e 4 e 5 e 6 e 1 e 2 e

41 Trajectory Simplification Douglas-Peucker Iterative trajectory simplification algorithm Generate segment between first and last measurement Identify point that is furthest from the segment and if it is further than a threshold, include the point Apply algorithm recursively on the two segments defined by the first, last and the added point e e e e e e

42 Trajectory Simplification Minimum Description Length Simplified trajectory considered a hypothesis Hypothesis that provides the best balance between preciseness (accuracy) and conciseness (small number of points) is the optimal simplification True s i+1 s i+2 False s i+3 False s i s i+4 = s j

43 Trajectory Similarity Measures Euclidean A B Compares measurements at the same time instance t Time offsets, dilation and noise have significant impact on resulting values A B Dynamic time warping Dynamic programming approach to time series, similar to edit distance but differences carry a dynamic penalty Longest common subsequence (LCS) Similarity measured as the total length of common subsequences

44 Trajectory Clustering KMedoids 15 Partitioning-based clustering algorithm where clusters represented using the most centrally located measurement/object medoid 0 0 medoid -5-5 medoid

45 Trajectory Clustering TRACLUS 1. Segment trajectories 2. Calculate weighted sum of distances between trajectory segments 3. Apply DBSCAN 4. Extract representative trajectory

46 Location-Based Services Computer applications that deliver information depending on the location of the device and user Numerous different categories of services Emergency services: E911 and E112 Mobile advertising Location-based games (Mobile) Augmented reality Main challenges Privacy Energy-efficiency Indoor positioning Lack of standards

47 Energy-Efficiency Minimizing power consumption of location sensing, tracking and reporting Trade-off between position accuracy and the amount of savings that can be obtained Define an error threshold E and optimize energy so that position is accurate within E most of the time Three main techniques for reducing power Duty cycling: reducing sampling frequency of sensors Sensor management: use less accurate but more energyefficient sensors whenever possible Data uploading: intelligent schemes for reducing location reporting frequency

48 Location Privacy Ability to prevent other parties from learning one s current or past location Countermeasures Anonymity: replace associated name with pseudonym or other untraceable identifier K-Anonymity Mix-Zones Obfuscation: reduce quality of information to hide sensitive details Spatial and/or temporal degradation Cloaking

49 Home Exam

50 Home Exam Exam period 22/2 16:00 7/3 16:00 Similar to the exercises, do not provide only results but also how they were derived (including code) Solutions to exam must be submitted by to the lecturer AND teaching assistants before the hard cut-off (16:00 on 7 th of March) Course grading takes into account both exercise activity and success in the exam