Introduction to Sensor Data Fusion Methods and Applications

Transcription

1 Introduction to Sensor Data Fusion Methods and Applications Last lecture: Why Sensor Data Fusion? Motivation, general context Discussion of examples oral examination: 6 credit points after the end of the semester prerequisite: participation in the excercises, running programs continuation in Summer: lectures and seminar on advanced topics job opportunities as research assistant in ongoing projects, practicum subsequently: master theses at Fraunhofer FKIE, PhD theses possible slides/script: to wolfgang.koch@fkie.fraunhofer.de, download

2 Sensor & Information Fusion: Basic Task -/ information sources: defined by operational requirements

3 Sensor & Information Fusion: Basic Task information to be fused: imprecise, incomplete, ambiguous, unresolved, false, deceptive, hard to formalize, contradictory...

4 Sensor & Information Fusion: Basic Task information to be fused: imprecise, incomplete, ambiguous, unresolved, false, deceptive, hard to formalize, contradictory... information sources: defined by operational requirements

5 single sensors/network measurements kinematical parameters classification attributes data processing/fusion temporal integration / logical analysis statistical estimation / data association combination with a priori information condensed information objects represented by track structures quantitative accuracy/reliability measures

6 A Generic Tracking and Sensor Data Fusion System Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: - Sensor Performance - Object Characteristics - Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: - Track Cancellation - Object Classification / ID - Track-to-Track Fusion Man-Machine Interface: - Object Representation - Displaying Functions - Interaction Facilities Sensor System Sensor System

7 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities source of information: received waveforms (space-time)

8 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities detection: a decision process for data rate reduction

9 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities data fusion input: estimated target parameters, false plots

10 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities core function: associate sensor data to established tracks

11 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities track maintenance: updating of established target tracks

12 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities initiation: establish new tracks, re-initiate lost tracks

13 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities exploit available context information (e.g. topography)

14 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities fusion/management of pre-processed track information

15 Tracking & Fusion System Sensor System Sensing Hardware: Received Waveforms Detection Process: Data Rate Reduction Signal Processing: Parameter Estimation Sensor Data Sensor Control A Priori Knowledge: Sensor Performance Object Characteristics Object Environment Track Initiation: Multiple Frame Track Extraction Sensor Data to Track Association Track Maintenance: Prediction, Filtering Retrodiction Track File Storage Track Processing: Track Cancellation Object Classification / ID Track to Track Fusion Man Machine Interface: Object Representation Displaying Functions Interaction Facilities user interface: presentation, interaction, decision support

16 Target Tracking: Basic Idea, Demonstration Problem-inherent uncertainties and ambiguities! BAYES: processing scheme for soft, delayed decision sensor performance: resolution conflicts DOPPLER blindness environment: dense situations clutter jamming/deception target characteristics: qualitatively distinct maneuvering phases background knowledge vehicles on road networks tactics

17 pdf: t k 1 Probability densities functions (pdf) p(x k 1 Z k 1 ) represent imprecise knowledge on the state x k 1 based on imprecise measurements Z k 1.

18 pdf: t k 1 Prädiktion: t k Exploit imprecise knowledge on the dynamical behavior of the object. p(x k Z k 1 ) {z } prediction = R dx k 1 p(x k x k 1 ) {z } dynamics p(x k 1 Z k 1 ). {z } old knowledge

19 pdf: t k 1 t k : kein plot missing sensor detection: data processing = prediction (not always: exploitation of negative sensor evidence)

20 pdf: t k 1 pdf: t k Prädiktion: tk+1 missing sensor information: increasing knowledge dissipation

21 pdf: t k 1 pdf: t k t k+1 : ein plot sensor information on the kinematical object state

22 pdf: t k 1 likelihood (Sensormodell) pdf: t k Prädiktion: tk+1 BAYES formula: p(x k+1 Z k+1 ) {z } new knowledge = p(z k+1 x k+1 ) p(x k+1 Z k ) R dxk+1 p(z k+1 x k+1 ) p(x k+1 Z k ) {z} {z } plot prediction

23 pdf: t k 1 pdf: t k+1 (Bayes) pdf: t k filtering = sensor data processing

24 pdf: t k 1 pdf: t k t k+1 : drei plots ambiguities by false plots: data interpretation hypotheses ( detection probability, false alarm statistics)

25 pdf: t k 1 pdf: t k pdf: t k+1 Multimodal pdfs reflect ambiguities inherent in the data.

26 pdf: t k 1 pdf: t k pdf: t k+1 Prädiktion: t k+2 temporal propagation: dissipation of the probability densities

27 pdf: t k 1 pdf: t k pdf: t k+1 t k+2 : ein plot association tasks: sensor data $ interpretation hypotheses

28 pdf: t k 1 likelihood pdf: t k pdf: t k+1 Prädiktion: t k+2 BAYES: p(x k+2 Z k+2 )= p(z k+2 x k+2 ) p(x k+2 Z k+1 ) R dxk+2 p(z k+2 x k+2 ) p(x k+2 Z k+1 )

29 pdf: t k 1 pdf: t k pdf: t k+1pdf: t k+2 in particular: re-calculation of the hypothesis weights

30 pdf: t k 1 pdf: t k pdf: t k+1pdf: t k+2 How does new knowledge affect the knowledge in the past of a past state?

31 pdf: t k 1 Retrodiktion: t k+1 pdf: t k pdf: t k+2 retrodiction : a retrospective analysis of the past

32 t k 1 t k t k+1 t k+2 optimal information processing at present and for the past

33 Multiple Hypothesis Tracking: Basic Idea Iterative updating of conditional probability densities! kinematic target state x k at time t k, accumulated sensor data Z k a priori knowledge: target dynamics models, sensor model, road maps prediction: p(x k 1 Z k 1 ) filtering: p(x k Z k 1 ) retrodiction: p(x l 1 Z k ) dynamics model! road maps sensor data Z k! sensor model filtering output dynamics model p(x k Z k 1 ) p(x k Z k ) p(x l Z k ) finite mixture: inherent ambiguity (data, model, road network) optimal estimators: e.g. minimum mean squared error (MMSE) initiation of pdf iteration: multiple hypothesis track extraction

34 Difficult Operational Conditions object detection: small objects: detection probability P D < 1 fading: consecutive missing plots (interference) moving platforms: minimum detectable velocity

35 Difficult Operational Conditions object detection: small objects: detection probability P D < 1 fading: consecutive missing plots (interference) moving platforms: minimum detectable velocity measurements: false returns (residual clutter, birds, clouds) low data update rates (long-range radar, e.g.) measurement errors, overlapping gates

36 Difficult Operational Conditions object detection: small objects: detection probability P D < 1 fading: consecutive missing plots (interference) moving platforms: minimum detectable velocity measurements: sensor resolution: false returns (residual clutter, birds, clouds) low data update rates (long-range radar, e.g.) measurement errors, overlapping gates characteristic parameters band-/beam width group measurements: resolution probability important: qualitatively correct modeling

37 Difficult Operational Conditions object detection: small objects: detection probability P D < 1 fading: consecutive missing plots (interference) moving platforms: minimum detectable velocity measurements: sensor resolution: object behavior: false returns (residual clutter, birds, clouds) low data update rates (long-range radar, e.g.) measurement errors, overlapping gates characteristic parameters band-/beam width group measurements: resolution probability important: qualitatively correct modeling applications: high maneuvering capability qualitatively distinct maneuvering phases dynamic object parameters a priori unknown

38 Demonstration: Multiple Hypothesis Tracking

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41 Tracking Application: Ground Picture Production GMTI Radar: Ground Moving Target Indicator wide area, all-weather, day/night, real-time surveillance of a dynamically evolving ground or near-to-ground situation GMTI Tracking: Some Characteristic Aspects backbone of a ground picture: moving target tracks airborne, dislocated, mobile sensor platforms vehicles, ships, low-flyers, radars, convoys occlusions: Doppler-blindness, topography road maps, terrain information, tactical rules dense target / dense clutter situations: MHT

42 Examples of GMTI Tracks (live exercise)

43 Multiple Sensor Security Assistance Systems 1 General Task Task Covert & Automated Surveillance of a Person Stream: Identification of Anomalous Behavior Towards a Solution Exploit Heterogeneous Multiple Sensor Systems. For OFFICIAL USE ONLY

44 Multiple Sensor Security Assistance Systems 2 General Task Task Covert & Automated Surveillance of a Person Stream: Identification of Anomalous Behavior Towards a Solution Exploit Heterogeneous Multiple Sensor Systems. Sensor Surveillance Data Kinematics: Where? When? Data Fusion Attributes: What? When? Person Classification For OFFICIAL USE ONLY

45 Security Applications: Consider Well-defined Access Regions. 3 Important Task: Detect persons carrying hazardous materials in a person flow. Escalators / Stairways Tunnels / Underground Fundamental Problem: Limited Spatio-temporal Resolution of Chemical Sensors Key to Solution: Compensate poor resolution by Space-time Sensor Data Fusion For OFFICIAL USE ONLY

46 Security Applications: Consider Well-defined Access Regions. 4 Important Task: Detect persons carrying hazardous materials in a person flow. Escalators / Stairways Tunnels / Underground Fundamental Problem: Limited Spatio-temporal Resolution of Chemical Sensors Key to Solution: Compensate poor resolution by Space-time Sensor Data Fusion Track Track Extraction Extraction / Maintenance Maintenance Laser-Range-Scanner Sensors Sensors Video Video Data Data Supporting Supporting Information Information Attributes Attributes Chemical Chemical Sensors Sensors EU Project HAMLeT: Hazardous Material Localization and Person Tracking For OFFICIAL USE ONLY

47 Detection of Persons and Goods with High Threat Potential 5 Exploit the full potential of specific sensors (attributes) Associate measured attributes/signatures to individuals Tracking and classification of individuals in person streams Covert operation: avoid interference with normal public live Avoid fatigue in situations with low frequency of suspicious events Indoor Radar Video Laser IR Camera Chemical Sensors For OFFICIAL USE ONLY

48 Multiple Sensor Security Assistance Systems Experience of human security personnel remains indispensable. Technical assistance by focusing their attention to critical situations. Security assistance systems: covert operation, continuous time. Combination of strengths of automated and human data exploitation. screening: real time analysis of large data streams high decision confidence in individual situations

49 On Characterizing Tracking / Fusion Performance a well-understood paradigm: air surveillance with multiple radars Many results can be transfered to other sensors (IR, E/O, sonar, acoustics). Sensor Data Fusion: tracks represent the available information on the targets associated to them with appropriate quality measures, thus providing answers to: When? Where? How many? To which direction? How fast, accelerating? What?

50 On Characterizing Tracking / Fusion Performance a well-understood paradigm: air surveillance with multiple radars Many results can be transfered to other sensors (IR, E/O, sonar, acoustics). Sensor Data Fusion: tracks represent the available information on the targets associated to them with appropriate quality measures, thus providing answers to: When? Where? How many? To which direction? How fast, accelerating? What? By sensor data fusion we wish to establish one-to-one associations between: targets in the field of view $ identified tracks in the tracking computer Strictly speaking, this is only possible under ideal conditions regarding the sensor performance and underlying target situation. The tracking/fusion performance can thus be measured by its deficiencies when compared with this ideal goal.

51 1. Let a target be detected at first by a sensor at time t a. Usually, a time delay is involved until a confirmed track has finally been established at time t e (track extraction). A measure of deficiency is thus: extraction delay t e t a.

52 1. Let a target be detected at first by a sensor at time t a. Usually, a time delay is involved until a confirmed track has finally been established at time t e (track extraction). A measure of deficiency is thus: extraction delay t e t a. 2. Unavoidably, false tracks will be extracted in case of a high false return density (e.g. clutter, jamming/detection), i.e. tracks related to unreal or unwanted targets. Corresponding deficiencies are: mean number of falsely extracted targets per time, mean life time of a false track before its deletion.

53 1. Let a target be detected at first by a sensor at time t a. Usually, a time delay is involved until a confirmed track has finally been established at time t e (track extraction). A measure of deficiency is thus: extraction delay t e t a. 2. Unavoidably, false tracks will be extracted in case of a high false return density (e.g. clutter, jamming/detection), i.e. tracks related to unreal or unwanted targets. Corresponding deficiencies are: mean number of falsely extracted targets per time, mean life time of a false track before its deletion. 3. A target should be represented by one and the same track until leaving the field of view. Related performance measures/deficiencies: mean life time of tracks related to true targets, probability of an identity switch between targets, probability of a target not being represented by a track.

54 4. The track inaccuracy (error covariance of a state estimate) should be as small as possible. The deviations between estimated and actual target states should at least correspond with the error covariances produced (consistency). If this is not the case, we speak of a track loss.

55 4. The track inaccuracy (error covariance of a state estimate) should be as small as possible. The deviations between estimated and actual target states should at least correspond with the error covariances produced (consistency). If this is not the case, we speak of a track loss. A track must really represent a target! Challenges: low detection probability high clutter density low update rate agile targets dense target situations formations, convoys target-split events (formation, weapons) jamming, deception Basic Tasks: models: sensor, target, environment data association problems estimation problems process control, realization! physics! combinatorics! probability, statistics! computer science

56 Summary: BAYESian (Multi-) Sensor Tracking Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time.

57 Summary: BAYESian (Multi-) Sensor Tracking Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time. Objective: Learn as much as possible about the individual target states at each time by analyzing the time series which is constituted by the sensor data.

58 Summary: BAYESian (Multi-) Sensor Tracking Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time. Objective: Learn as much as possible about the individual target states at each time by analyzing the time series which is constituted by the sensor data. Problem: imperfect sensor information: inaccurate, incomplete, and eventually ambiguous. Moreover, the targets temporal evolution is usually not well-known.

59 Summary: BAYESian (Multi-) Sensor Tracking Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time. Objective: Learn as much as possible about the individual target states at each time by analyzing the time series which is constituted by the sensor data. Problem: imperfect sensor information: inaccurate, incomplete, and eventually ambiguous. Moreover, the targets temporal evolution is usually not well-known. Approach: Interpret measurements and target vectors as random variables (RVs). Describe by probability density functions (pdf) what is known about them.

60 Summary: BAYESian (Multi-) Sensor Tracking Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time. Objective: Learn as much as possible about the individual target states at each time by analyzing the time series which is constituted by the sensor data. Problem: imperfect sensor information: inaccurate, incomplete, and eventually ambiguous. Moreover, the targets temporal evolution is usually not well-known. Approach: Interpret measurements and target vectors as random variables (RVs). Describe by probability density functions (pdf) what is known about them. Solution: Derive iteration formulae for calculating the pdfs! Develop a mechanism for initiation! By doing so, exploit all background information available! Derive state estimates from the pdfs along with appropriate quality measures!

61 How to deal with probability density functions? pdf p(x): Extract probability statements about the RV x by integration!

62 How to deal with probability density functions? pdf p(x): Extract probability statements about the RV x by integration! naïvely: positive and normalized functions (p(x) 0, R dx p(x) =1)

63 How to deal with probability density functions? pdf p(x): Extract probability statements about the RV x by integration! naïvely: positive and normalized functions (p(x) 0, R dx p(x) =1) conditional pdf p(x y) = p(x,y) : Impact of information on y on RV x? p(y)

64 How to deal with probability density functions? pdf p(x): Extract probability statements about the RV x by integration! naïvely: positive and normalized functions (p(x) 0, R dx p(x) =1) conditional pdf p(x y) = p(x,y) : Impact of information on y on RV x? p(y) marginal density p(x) = R dy p(x, y) = R dy p(x y) p(y): Enter y! {z } =p(y x) p(x)

65 How to deal with probability density functions? pdf p(x): Extract probability statements about the RV x by integration! naïvely: positive and normalized functions (p(x) 0, R dx p(x) =1) conditional pdf p(x y) = p(x,y) : Impact of information on y on RV x? p(y) marginal density p(x) = R dy p(x, y) = R dy p(x y) p(y): Enter y! Bayes: p(x y)= p(y x)p(x) p(y) = R p(y x)p(x) dx p(y x)p(x) : p(x y) p(y x), p(x)!

66 How to deal with probability density functions? pdf p(x): Extract probability statements about the RV x by integration! naïvely: positive and normalized functions (p(x) 0, R dx p(x) =1) conditional pdf p(x y) = p(x,y) : Impact of information on y on RV x? p(y) marginal density p(x) = R dy p(x, y) = R dy p(x y) p(y): Enter y! Bayes: p(x y)= p(y x)p(x) p(y) = R p(y x)p(x) dx p(y x)p(x) : p(x y) p(y x), p(x)! certain knowledge on x: p(x) = (x y) = lim 1!0 p e 1 (x y)

67 How to deal with probability density functions? pdf p(x): Extract probability statements about the RV x by integration! naïvely: positive and normalized functions (p(x) 0, R dx p(x) =1) conditional pdf p(x y) = p(x,y) : Impact of information on y on RV x? p(y) marginal density p(x) = R dy p(x, y) = R dy p(x y) p(y): Enter y! Bayes: p(x y)= p(y x)p(x) p(y) = R p(y x)p(x) dx p(y x)p(x) : p(x y) p(y x), p(x)! certain knowledge on x: p(x) = (x y) = lim 1!0 p e 1 (x y) transformed RV y = t[x]: p(y) = R dx p(y, x) = R dx p(y x) p x (x)

68 How to deal with probability density functions? pdf p(x): Extract probability statements about the RV x by integration! naïvely: positive and normalized functions (p(x) 0, R dx p(x) =1) conditional pdf p(x y) = p(x,y) : Impact of information on y on RV x? p(y) marginal density p(x) = R dy p(x, y) = R dy p(x y) p(y): Enter y! Bayes: p(x y)= p(y x)p(x) p(y) = R p(y x)p(x) dx p(y x)p(x) : p(x y) p(y x), p(x)! certain knowledge on x: p(x) = (x y) = lim 1!0 p e 1 (x y) transformed RV y = t[x]: p(y) = R dxp(y, x) = R dxp(y x)p x (x) = R dx (y t[x]) px (x) =:[T p x ](y) (T : p x 7! p, transfer operator )

69 Characterize an object by quantitatively describable properties: object state object position x on a strait line: x 2 R kinematic state x =(r >, ṙ >, r > ) >, x 2 R 9 position r =(x, y, z) >, velocity ṙ, acceleration r Examples: joint state of two objects: x =(x > 1, x> 2 )> kinematic state x, object extension X z.b. ellipsoid: symmetric, positively definite matrix kinematic state x, object class class z.b. bird, sailing plane, helicopter, passenger jet,... Learn unknown object states from imperfect measurements and describe by functions p(x) imprecise knowledge mathematically precisely!

70 Interpret unknown object states as random variables, x [1D] or x, X [vector / matrix variate]), characterized by corresponding probability density functions (pdf). The concrete shape of the pdf p(x) contains the full knowledge on x!

71 Information on a random variable (RV) can be extracted by integration from the corresponding pdf.! at present: one dimensional case: How probable is it that x 2 (a, b) R holds? Answer: P {x 2 (a, b)} = Z b a dx p(x) ) p(x) 0 in particular: P {x 2 R} = Z 1 1 dx p(x) =1 (normalzation) intuitive interpretation: the object is somewhere in R loosely: p(x) dx is probabity for x having a value between x and x + dx

72 How to characterize the properties of a pdf? specifically: How to associate a single expected value to a RV? The maximum of the pdf is sometimes but not always useful!

73 How to characterize the properties of a pdf? specifically: How to associate a single expected value to a RV? The maximum of the pdf is sometimes but not always useful! (! examples) instead: Calculate the centroid of the pdf! E[x] = Z 1 1 dx x p(x) = x expectation value more generally: Consider functions g : x 7! g(x) of the RV x! E[g(x)] = Z 1 1 dx g(x) p(x), expectation value of the observable g Example: Consider the observable 1 2 mx2 (kinetic energy, x = speed)

74 An important observable: the error of an estimate Quality: How useful is an expectation value x = E[x]? Consider special obervables as distance measure: g(x) = x x oder g(x) =(x x) 2 quadratic measures: computationally more comfortable! expected error of the expectation value x: V[x] =E[(x x) 2 ], x = p V[x] variance, standard deviation Exercise 2.1 Show that V[x] =E[x 2 ] E[x] 2 holds. Expectation value of the observable x 2 also called 2nd moment of the pdf of x.

75 Exercise 2.2 Calculate expectation and variance of the uniform density of a RV x 2 R in the intervall [a, b]. p(x) =U( x {z} ZV ; a, b {z} Parameter )= 8 < : 1 b a x 2 [a, b] 0 sonst! Pdf correctly normalized? Z 1 1 dx U(x; a, b) = 1 b a Z b a dx =1 E[x] = Z 1 1 dx x U(x; a, b) =b + a 2 V[x] = 1 b a Z b dx a x2 E[x] 2 = 1 (b a)2 12

76 Important example: x normally distributed over R (Gauß) wanted: probabilities concentrated around µ quadratic distance: x µ 2 = 1 2 (x µ)2 / 2 (mathematically convenient!) Parameter is a measure of the width of the pdf: 2 = 1 2 for large distances, i.e. x µ 2 1 2, the pdf shall decay quickly. simplest approach: p(x) =e x µ 2 (> 0 8x 2 R, normalization?) Normalized for p(x) = p(x)/ R 1 1 dx p(x)! Formula collection delivers: Z 1 1 dx p(x) =p 2 An admissible pdf with the required properties is obviously given by: N (x; µ, )= 1 p 2 exp (x µ) 2 2 2!

77 Exercise 2.3 Show for the Gauß density p(x) =N (x; µ, ): E[x] =µ, V[x] = 2 E[x] = Z 1 1 dx x N (x; µ, )=µ V[x] =E[x 2 ] E[x] 2 = 2 Use substitution and partial integration! Use R 1 1 dx e 1 2 x2 = p 2!

78 How to incorporate certain knowledge into pdfs? (x; µ) = Consider the special pdf: 1ly sharp peak at µ 8 < : 1 x = µ 0 x 6= µ holding: Z 1 1 dx (x; µ) =1 Intuitively interpretable as a limit: (x; µ) = lim!0 N (x; µ, )= lim!0 1 p 2 e 1 (x µ)2 2 2 Alternative: (x; a) = lim b!a U(x; a, b) For observables g this holds: E[g(x)] = Z 1 1 dx g(x) (x; y) =g(y) in particular: V[x] =E[(x E[x]) 2 ]=E[x 2 ] E[x] 2 =0

79 Characterize an object by quantitatively describable properties: object state object position x on a strait line: x 2 R X kinematic state x =(r >, ṙ >, r > ) >, x 2 R 9 position r =(x, y, z) >, velocity ṙ, acceleration r Examples: joint state of two objects: x =(x > 1, x> 2 )> kinematic state x, object extension X z.b. ellipsoid: symmetric, positively definite matrix kinematic state x, object class class z.b. bird, sailing plane, helicopter, passenger jet,... Learn unknown object states from imperfect measurements and describe by functions p(x) imprecise knowledge mathematically precisely!