Sensor Fusion and its Applications Part 2 pdf

4.3 Adaptive Determination of Information Distribution Factor By the analysis of the estimation performance of combined filter, it is known that the information distribution principle n

1,1 1, 1,

T 1,1

, ,

k k

P Φ

If taken the equal sign, that is, achieved the de-correlation of local estimates, on the one

hand, the global optimal fusion estimate can be realized by Theorem 4.1 , but on the other,

the initial covariance matrix and process noise covariance of the sub-filter themselves can

enlarged by i1 times What’s more, the filter results of every local filter will not be

optimal

4.2 Structure and Performance Analysis of the Combined Filter

The combined filter is a 2-level filter The characteristic to distinguish from the traditional

distributed filters is the use of information distribution to realize information share of every

sub-filter Information fusion structure of the combined filter is shown in Fig 4.1

Optimal fusion

Fig 4.1 Structure Indication of the Combined Filter

From the filter structure shown in the Fig 4.1, the fusion process for the combined filter can

be divided into the following four steps

Step1 Given initial value and information distribution: The initial value of the global state in

the initial moment is supposed to beX0, the covariance to beQ0, the state estimate vector

of the local filter, the system covariance matrix and the state vector covariance matrix separately, respectively to beX Q P ˆ , , ,i i i i  1, ,  N, and the corresponding master filter

to beX Q P ˆ , ,m m m.The information is distributed through the information distribution factor by the following rules in the sub-filter and the master filter

Step2 the time to update the information: The process of updating time conducted

independently, the updated time algorithm is shown as follows:

Step3 Measurement update: As the master filter does not measure, there is no measurement

update in the Master Filter The measurement update only occurs in each local sub-filter, and can work by the following formula:

Step4 the optimal information fusion: The amount of information of the state equation and

the amount of information of the process equation can be apportioned by the information distribution to eliminate the correlation among sub-filters Then the core algorithm of the combined filter can be fused to the local information of every local filter to get the state optimal estimates

, 1 1 ,

1

ˆ ( | ) ( | ) ( | ) ( | ) ˆ ( | ) ( ( | )) ( ( | ) ( | ) ( | ) ( | ))

State Optimal Estimation for Nonstandard Multi-sensor Information Fusion System 25

T 1,1

, ,

k k

P Φ

If taken the equal sign, that is, achieved the de-correlation of local estimates, on the one

hand, the global optimal fusion estimate can be realized by Theorem 4.1 , but on the other,

the initial covariance matrix and process noise covariance of the sub-filter themselves can

enlarged by i1 times What’s more, the filter results of every local filter will not be

optimal

4.2 Structure and Performance Analysis of the Combined Filter

The combined filter is a 2-level filter The characteristic to distinguish from the traditional

distributed filters is the use of information distribution to realize information share of every

sub-filter Information fusion structure of the combined filter is shown in Fig 4.1

Optimal fusion

Fig 4.1 Structure Indication of the Combined Filter

From the filter structure shown in the Fig 4.1, the fusion process for the combined filter can

be divided into the following four steps

Step1 Given initial value and information distribution: The initial value of the global state in

the initial moment is supposed to beX0, the covariance to beQ0, the state estimate vector

of the local filter, the system covariance matrix and the state vector covariance matrix separately, respectively to beX Q P ˆ , , ,i i i i  1, ,  N, and the corresponding master filter

to beX Q P ˆ , ,m m m.The information is distributed through the information distribution factor by the following rules in the sub-filter and the master filter

Step2 the time to update the information: The process of updating time conducted

independently, the updated time algorithm is shown as follows:

Step3 Measurement update: As the master filter does not measure, there is no measurement

update in the Master Filter The measurement update only occurs in each local sub-filter, and can work by the following formula:

Step4 the optimal information fusion: The amount of information of the state equation and

the amount of information of the process equation can be apportioned by the information distribution to eliminate the correlation among sub-filters Then the core algorithm of the combined filter can be fused to the local information of every local filter to get the state optimal estimates

, 1 1 ,

1

ˆ ( | ) ( | ) ( | ) ( | ) ˆ ( | ) ( ( | )) ( ( | ) ( | ) ( | ) ( | ))

correlation between sub-filters and the master filter and between the various sub-filters in

the local filter and to enlarge the initial covariance matrix and the process noise covariance

of each sub-filter byi1times, the filter results of each local filter will not be optimal But

some information lost by the variance upper-bound technique can be re-synthesized in the

final fusion process to get the global optimal solution for the equation

In the above analysis for the structure of state fusion estimation, it is known that centralized

fusion structure is the optimal fusion estimation for the system state in the minimum

variance While in the combined filter, the optimal fusion algorithm is used to deal with

local filtering estimate to synthesize global state estimate Due to the application of variance

upper-bound technique, local filtering is turned into being suboptimal, the global filter after

its synthesis becomes global optimal, i.e the fact that the equivalence issue between the

combined filtering process and the centralized fusion filtering process To sum up, as can be

seen from the above analysis, the algorithm of combined filtering process is greatly

simplified by the use of variance upper-bound technique It is worth pointing out that the

use of variance upper-bound technique made local estimates suboptimum but the global

estimate after the fusion of local estimates is optimal, i.e combined filtering model is

equivalent to centralized filtering model in the estimated accuracy

4.3 Adaptive Determination of Information Distribution Factor

By the analysis of the estimation performance of combined filter, it is known that the

information distribution principle not only eliminates the correlation between sub-filters as

brought from public baseline information to make the filtering of every sub-filter conducted

themselves independently, but also makes global estimates of information fusion optimal

This is also the key technology of the fusion algorithm of combined filter Despite it is in this

case, different information distribution principles can be guaranteed to obtain different

structures and different characteristics (fault-tolerance, precision and amount of calculation)

of combined filter Therefore, there have been many research literatures on the selection of

information distribution factor of combined filter in recent years In the traditional structure

of the combined filter, when distributed information to the subsystem, their distribution

factors are predetermined and kept unchanged to make it difficult to reflect the dynamic

nature of subsystem for information fusion Therefore, it will be the main objective and

research direction to find and design the principle of information distribution which will be

simple, effective and dynamic fitness, and practical Its aim is that the overall performance

of the combined filter will keep close to the optimal performance of the local system in the

filtering process, namely, a large information distribution factors can be existed in high

precision sub-system, while smaller factors existed in lower precision sub-system to get

smaller to reduce its overall accuracy of estimated loss Method for determining adaptive

information allocation factors can better reflect the diversification of estimation accuracy in

subsystem and reduce the impact of the subsystem failure or precision degradation but

improve the overall estimation accuracy and the adaptability and fault tolerance of the

whole system But it held contradictory views given in Literature [28] to determine

information distribution factor formula as the above held view It argued that global optimal

estimation accuracy had nothing to do with the information distribution factor values when

statistical characteristics of noise are known, so there is no need for adaptive determination

Combined with above findings in the literature, on determining rules for information distribution factor, we should consider from two aspects

1) Under circumstances of meeting conditions required in Kalman filtering such as exact statistical properties of noise, it is known from filter performance analysis in Section 4.2 that:

if the value of the information distribution factor can satisfy information on conservation principles, the combined filter will be the global optimal one In other words, the global optimal estimation accuracy is unrelated to the value of information distribution factors, which will influence estimation accuracy of a sub-filter yet As is known in the information distribution process, process information obtained from each sub-filter isi gQ1, i gP1, Kalman filter can automatically use different weights according to the merits of the quality

of information: the smaller the value ofiis, the lower process message weight will be, so the accuracy of sub-filters is dependent on the accuracy of measuring information; on the contrary, the accuracy of sub-filters is dependent on the accuracy of process information 2) Under circumstances of not knowing statistical properties of noise or the failure of a subsystem, global estimates obviously loss the optimality and degrade the accuracy, and it

is necessary to introduce the determination mode of adaptive information distribution factor Information distribution factor will be adaptive dynamically determined by the sub-filter accuracy to overcome the loss of accuracy caused by fault subsystem to remain the relatively high accuracy in global estimates In determining adaptive information distribution factor, it should be considered that less precision sub-filter will allocate factor with smaller information to make the overall output of the combined filtering model had better fusion performance, or to obtain higher estimation accuracy and fault tolerance

In Kalman filter, the trace of error covariance matrix P includes the corresponding estimate vector or its linear combination of variance The estimated accuracy can be reflected in filter answered to the estimate vector or its linear combination through the analysis for the trace

of P So there will be the following definition:

Definition 4.1: The estimation accuracy of attenuation factor of the ith local filter is:

inilocal filter, tr( ) meaning the demand for computing trace function of the matrix When introduced attenuation factor estimation accuracyEDOPi, in fact, it is said to use the measurement of norm characterizationPi inPi matrix: the bigger the matrix norm is, the corresponding estimated covariance matrix will be larger, so the filtering effect is poorer; and vice versa

According to the definition of attenuation factor estimation accuracy, take the computing formula of information distribution factor in the combined filtering process as follows:

i i

State Optimal Estimation for Nonstandard Multi-sensor Information Fusion System 27

correlation between sub-filters and the master filter and between the various sub-filters in

the local filter and to enlarge the initial covariance matrix and the process noise covariance

of each sub-filter byi1times, the filter results of each local filter will not be optimal But

some information lost by the variance upper-bound technique can be re-synthesized in the

final fusion process to get the global optimal solution for the equation

In the above analysis for the structure of state fusion estimation, it is known that centralized

fusion structure is the optimal fusion estimation for the system state in the minimum

variance While in the combined filter, the optimal fusion algorithm is used to deal with

local filtering estimate to synthesize global state estimate Due to the application of variance

upper-bound technique, local filtering is turned into being suboptimal, the global filter after

its synthesis becomes global optimal, i.e the fact that the equivalence issue between the

combined filtering process and the centralized fusion filtering process To sum up, as can be

seen from the above analysis, the algorithm of combined filtering process is greatly

simplified by the use of variance upper-bound technique It is worth pointing out that the

use of variance upper-bound technique made local estimates suboptimum but the global

estimate after the fusion of local estimates is optimal, i.e combined filtering model is

equivalent to centralized filtering model in the estimated accuracy

4.3 Adaptive Determination of Information Distribution Factor

By the analysis of the estimation performance of combined filter, it is known that the

information distribution principle not only eliminates the correlation between sub-filters as

brought from public baseline information to make the filtering of every sub-filter conducted

themselves independently, but also makes global estimates of information fusion optimal

This is also the key technology of the fusion algorithm of combined filter Despite it is in this

case, different information distribution principles can be guaranteed to obtain different

structures and different characteristics (fault-tolerance, precision and amount of calculation)

of combined filter Therefore, there have been many research literatures on the selection of

information distribution factor of combined filter in recent years In the traditional structure

of the combined filter, when distributed information to the subsystem, their distribution

factors are predetermined and kept unchanged to make it difficult to reflect the dynamic

nature of subsystem for information fusion Therefore, it will be the main objective and

research direction to find and design the principle of information distribution which will be

simple, effective and dynamic fitness, and practical Its aim is that the overall performance

of the combined filter will keep close to the optimal performance of the local system in the

filtering process, namely, a large information distribution factors can be existed in high

precision sub-system, while smaller factors existed in lower precision sub-system to get

smaller to reduce its overall accuracy of estimated loss Method for determining adaptive

information allocation factors can better reflect the diversification of estimation accuracy in

subsystem and reduce the impact of the subsystem failure or precision degradation but

improve the overall estimation accuracy and the adaptability and fault tolerance of the

whole system But it held contradictory views given in Literature [28] to determine

information distribution factor formula as the above held view It argued that global optimal

estimation accuracy had nothing to do with the information distribution factor values when

statistical characteristics of noise are known, so there is no need for adaptive determination

Combined with above findings in the literature, on determining rules for information distribution factor, we should consider from two aspects

1) Under circumstances of meeting conditions required in Kalman filtering such as exact statistical properties of noise, it is known from filter performance analysis in Section 4.2 that:

if the value of the information distribution factor can satisfy information on conservation principles, the combined filter will be the global optimal one In other words, the global optimal estimation accuracy is unrelated to the value of information distribution factors, which will influence estimation accuracy of a sub-filter yet As is known in the information distribution process, process information obtained from each sub-filter isi gQ1, i gP1, Kalman filter can automatically use different weights according to the merits of the quality

of information: the smaller the value ofiis, the lower process message weight will be, so the accuracy of sub-filters is dependent on the accuracy of measuring information; on the contrary, the accuracy of sub-filters is dependent on the accuracy of process information 2) Under circumstances of not knowing statistical properties of noise or the failure of a subsystem, global estimates obviously loss the optimality and degrade the accuracy, and it

is necessary to introduce the determination mode of adaptive information distribution factor Information distribution factor will be adaptive dynamically determined by the sub-filter accuracy to overcome the loss of accuracy caused by fault subsystem to remain the relatively high accuracy in global estimates In determining adaptive information distribution factor, it should be considered that less precision sub-filter will allocate factor with smaller information to make the overall output of the combined filtering model had better fusion performance, or to obtain higher estimation accuracy and fault tolerance

In Kalman filter, the trace of error covariance matrix P includes the corresponding estimate vector or its linear combination of variance The estimated accuracy can be reflected in filter answered to the estimate vector or its linear combination through the analysis for the trace

of P So there will be the following definition:

Definition 4.1: The estimation accuracy of attenuation factor of the ith local filter is:

inilocal filter, tr( ) meaning the demand for computing trace function of the matrix When introduced attenuation factor estimation accuracyEDOPi, in fact, it is said to use the measurement of norm characterizationPi inPi matrix: the bigger the matrix norm is, the corresponding estimated covariance matrix will be larger, so the filtering effect is poorer; and vice versa

According to the definition of attenuation factor estimation accuracy, take the computing formula of information distribution factor in the combined filtering process as follows:

i i

Obviously, i can satisfy information on conservation principles and possess a very

intuitive physical sense, namely, the line reflects the estimated performance of sub-filters to

improve the fusion performance of the global filter by adjusting the proportion of the local

estimates information in the global estimates information Especially when the performance

degradation of a subsystem makes its local estimation error covariance matrix such a

singular huge increase that its adaptive information distribution can make the combined

filter participating of strong robustness and fault tolerance

5 Summary

The chapter focuses on non-standard multi-sensor information fusion system with each kind

of nonlinear, uncertain and correlated factor, which is widely popular in actual application,

because of the difference of measuring principle and character of sensor as well as

measuring environment

Aiming at the above non-standard factors, three resolution schemes based on semi-parameter

modeling, multi model fusion and self-adaptive estimation are relatively advanced, and

moreover, the corresponding fusion estimation model and algorithm are presented

(1) By introducing semi-parameter regression analysis concept to non-standard multi-sensor

state fusion estimation theory, the relational fusion estimation model and

parameter-non-parameter solution algorithm are established; the process is to separate

model error brought by nonlinear and uncertainty factors with semi-parameter modeling

method and then weakens the influence to the state fusion estimation precision; besides, the

conclusion is proved in theory that the state estimation obtained in this algorithm is the

optimal fusion estimation

(2) Two multi-model fusion estimation methods respectively based on multi-model adaptive

estimation and interacting multiple model fusion are researched to deal with nonlinear and

time-change factors existing in multi-sensor fusion system and moreover to realize the

optimal fusion estimation for the state

(3) Self-adaptive fusion estimation strategy is introduced to solve local dependency and

system parameter uncertainty existed in multi-sensor dynamical system and moreover to

realize the optimal fusion estimation for the state The fusion model for federal filter and its

optimality are researched; the fusion algorithms respectively in relevant or irrelevant for

each sub-filter are presented; the structure and algorithm scheme for federal filter are

designed; moreover, its estimation performance was also analyzed, which was influenced

by information allocation factors greatly So the selection method of information allocation

factors was discussed, in this chapter, which was dynamically and self-adaptively

determined according to the eigenvalue square decomposition of the covariance matrix

6 Reference

Hall L D, Llinas J Handbook of Multisensor Data Fusion Bcoa Raton, FL, USA: CRC Press,

2001

Bedworth M, O’Brien J the Omnibus Model: A New Model of Data Fusion IEEE

Transactions on Aerospace and Electronic System, 2000, 15(4): 30-36

Heintz, F., Doherty, P A Knowledge Processing Middleware Framework and its Relation to

the JDL Data Fusion Model Proceedings of the 7th International Conference on Information Fusion, 2005, pp: 1592-1599

Llinas J, Waltz E Multisensor Data Fusion Norwood, MA: Artech House, 1990

X R Li, Yunmin Zhu, Chongzhao Han Unified Optimal Linear Estimation Fusion-Part I:

Unified Models and Fusion Rules Proc 2000 International Conf on Information Fusion, July 2000

Jiongqi Wang, Haiyin Zhou, Deyong Zhao, el State Optimal Estimation with Nonstandard

Multi-sensor Information Fusion System Engineering and Electronics, 2008, 30(8): 1415-1420

Kennet A, Mayback P S Multiple Model Adaptive Estimation with Filter Pawning IEEE

Transaction on Aerospace Electron System, 2002, 38(3): 755-768 Bar-shalom, Y., Campo, L The Effect of The Common Process Noise on the Two-sensor

Fused-track Covariance IEEE Transaction on Aerospace and Electronic Systems,

1986, Vol.22: 803-805 Morariu, V I, Camps, O I Modeling Correspondences for Multi Camera Tracking Using

Nonlinear Manifold Learning and Target Dynamics IEEE Computer Society Conference on Computer Vision and Pattern Recognition, June, 2006, pp: 545-552 Stephen C, Stubberud, Kathleen A, et al Data Association for Multisensor Types Using

Fuzzy Logic IEEE Transaction on Instrumentation and Measurement, 2006, 55(6): 2292-2303

Hammerand, D C ; Oden, J T ; Prudhomme, S ; Kuczma, M S Modeling Error and

Adaptivity in Nonlinear Continuum System, NTIS No: DE2001-780285/XAB Crassidis J Letal.A Real-time Error Filter and State Estimator.AIAA-943550.1994:92-102 Flammini, A, Marioli, D et al Robust Estimation of Magnetic Barkhausen Noise Based on a

Numerical Approach IEEE Transaction on Instrumentation and Measurement,

2002, 16(8): 1283-1288 Donoho D L., Elad M On the Stability of the Basis Pursuit in the Presence of Noise http:

//www-stat.stanford.edu/-donoho/reports.html Sun H Y, Wu Y Semi-parametric Regression and Model Refining Geospatial Information

Science, 2002, 4(5): 10-13 Green P.J., Silverman B.W Nonparametric Regression and Generalized Linear Models

London: CHAPMAN and HALL, 1994 Petros Maragos, FangKuo Sun Measuring the Fractal Dimension of Signals: Morphological

Covers and Iterative Optimization IEEE Trans On Signal Processing, 1998(1): 108~121

G, Sugihara, R.M.May Nonlinear Forecasting as a Way of Distinguishing Chaos From

Measurement Error in Time Series, Nature, 1990, 344: 734-741 Roy R, Paulraj A, kailath T ESPRIT Estimation of Signal Parameters Via Rotational

Invariance Technique IEEE Transaction Acoustics, Speech, Signal Processing, 1989, 37:984-98

Aufderheide B, Prasad V, Bequettre B W A Compassion of Fundamental Model-based and

Multi Model Predictive Control Proceeding of IEEE 40th Conference on Decision and Control, 2001: 4863-4868

State Optimal Estimation for Nonstandard Multi-sensor Information Fusion System 29

Obviously, i can satisfy information on conservation principles and possess a very

intuitive physical sense, namely, the line reflects the estimated performance of sub-filters to

improve the fusion performance of the global filter by adjusting the proportion of the local

estimates information in the global estimates information Especially when the performance

degradation of a subsystem makes its local estimation error covariance matrix such a

singular huge increase that its adaptive information distribution can make the combined

filter participating of strong robustness and fault tolerance

5 Summary

The chapter focuses on non-standard multi-sensor information fusion system with each kind

of nonlinear, uncertain and correlated factor, which is widely popular in actual application,

because of the difference of measuring principle and character of sensor as well as

measuring environment

Aiming at the above non-standard factors, three resolution schemes based on semi-parameter

modeling, multi model fusion and self-adaptive estimation are relatively advanced, and

moreover, the corresponding fusion estimation model and algorithm are presented

(1) By introducing semi-parameter regression analysis concept to non-standard multi-sensor

state fusion estimation theory, the relational fusion estimation model and

parameter-non-parameter solution algorithm are established; the process is to separate

model error brought by nonlinear and uncertainty factors with semi-parameter modeling

method and then weakens the influence to the state fusion estimation precision; besides, the

conclusion is proved in theory that the state estimation obtained in this algorithm is the

optimal fusion estimation

(2) Two multi-model fusion estimation methods respectively based on multi-model adaptive

estimation and interacting multiple model fusion are researched to deal with nonlinear and

time-change factors existing in multi-sensor fusion system and moreover to realize the

optimal fusion estimation for the state

(3) Self-adaptive fusion estimation strategy is introduced to solve local dependency and

system parameter uncertainty existed in multi-sensor dynamical system and moreover to

realize the optimal fusion estimation for the state The fusion model for federal filter and its

optimality are researched; the fusion algorithms respectively in relevant or irrelevant for

each sub-filter are presented; the structure and algorithm scheme for federal filter are

designed; moreover, its estimation performance was also analyzed, which was influenced

by information allocation factors greatly So the selection method of information allocation

factors was discussed, in this chapter, which was dynamically and self-adaptively

determined according to the eigenvalue square decomposition of the covariance matrix

6 Reference

Hall L D, Llinas J Handbook of Multisensor Data Fusion Bcoa Raton, FL, USA: CRC Press,

2001

Bedworth M, O’Brien J the Omnibus Model: A New Model of Data Fusion IEEE

Transactions on Aerospace and Electronic System, 2000, 15(4): 30-36

Heintz, F., Doherty, P A Knowledge Processing Middleware Framework and its Relation to

the JDL Data Fusion Model Proceedings of the 7th International Conference on Information Fusion, 2005, pp: 1592-1599

Llinas J, Waltz E Multisensor Data Fusion Norwood, MA: Artech House, 1990

X R Li, Yunmin Zhu, Chongzhao Han Unified Optimal Linear Estimation Fusion-Part I:

Unified Models and Fusion Rules Proc 2000 International Conf on Information Fusion, July 2000

Jiongqi Wang, Haiyin Zhou, Deyong Zhao, el State Optimal Estimation with Nonstandard

Multi-sensor Information Fusion System Engineering and Electronics, 2008, 30(8): 1415-1420

Kennet A, Mayback P S Multiple Model Adaptive Estimation with Filter Pawning IEEE

Transaction on Aerospace Electron System, 2002, 38(3): 755-768 Bar-shalom, Y., Campo, L The Effect of The Common Process Noise on the Two-sensor

Fused-track Covariance IEEE Transaction on Aerospace and Electronic Systems,

1986, Vol.22: 803-805 Morariu, V I, Camps, O I Modeling Correspondences for Multi Camera Tracking Using

Nonlinear Manifold Learning and Target Dynamics IEEE Computer Society Conference on Computer Vision and Pattern Recognition, June, 2006, pp: 545-552 Stephen C, Stubberud, Kathleen A, et al Data Association for Multisensor Types Using

Fuzzy Logic IEEE Transaction on Instrumentation and Measurement, 2006, 55(6): 2292-2303

Hammerand, D C ; Oden, J T ; Prudhomme, S ; Kuczma, M S Modeling Error and

Adaptivity in Nonlinear Continuum System, NTIS No: DE2001-780285/XAB Crassidis J Letal.A Real-time Error Filter and State Estimator.AIAA-943550.1994:92-102 Flammini, A, Marioli, D et al Robust Estimation of Magnetic Barkhausen Noise Based on a

Numerical Approach IEEE Transaction on Instrumentation and Measurement,

2002, 16(8): 1283-1288 Donoho D L., Elad M On the Stability of the Basis Pursuit in the Presence of Noise http:

//www-stat.stanford.edu/-donoho/reports.html Sun H Y, Wu Y Semi-parametric Regression and Model Refining Geospatial Information

Science, 2002, 4(5): 10-13 Green P.J., Silverman B.W Nonparametric Regression and Generalized Linear Models

London: CHAPMAN and HALL, 1994 Petros Maragos, FangKuo Sun Measuring the Fractal Dimension of Signals: Morphological

Covers and Iterative Optimization IEEE Trans On Signal Processing, 1998(1): 108~121

G, Sugihara, R.M.May Nonlinear Forecasting as a Way of Distinguishing Chaos From

Measurement Error in Time Series, Nature, 1990, 344: 734-741 Roy R, Paulraj A, kailath T ESPRIT Estimation of Signal Parameters Via Rotational

Invariance Technique IEEE Transaction Acoustics, Speech, Signal Processing, 1989, 37:984-98

Aufderheide B, Prasad V, Bequettre B W A Compassion of Fundamental Model-based and

Multi Model Predictive Control Proceeding of IEEE 40th Conference on Decision and Control, 2001: 4863-4868

Aufderheide B, Bequette B W A Variably Tuned Multiple Model Predictive Controller Based

on Minimal Process Knowledge Proceedings of the IEEE American Control Conference, 2001, 3490-3495

X Rong Li, Jikov, Vesselin P A Survey of Maneuvering Target Tracking-Part V:

Multiple-Model Methods Proceeding of SPIE Conference on Signal and Data Proceeding of Small Targets, San Diego, CA, USA, 2003

T.M Berg, et al General Decentralized Kalman filters Proceedings of the American Control

Conference, Mayland, June, 1994, pp.2273-2274

Nahin P J, Pokoski Jl NCTR Plus Sensor Fusion of Equals IFNN IEEE Transaction on AES,

1980, Vol AES-16, No.3, pp.320-337

Bar-Shalom Y, Blom H A The Interacting Multiple Model Algorithm for Systems with

Markovian Switching Coefficients IEEE Transaction on Aut Con, 1988, AC-33: 780-783

X.Rong Li, Vesselin P Jilkov A Survey of Maneuvering Target Tracking-Part I: Dynamic

Models IEEE Transaction on Aerospace and Electronic Systems, 2003, 39(4): 1333-1361

Huimin Chen, Thiaglingam Kirubarjan, Yaakov Bar-Shalom Track-to-track Fusion Versus

Centralized Estimation: Theory and Application IEEE Transactions on AES, 2003, 39(2): 386-411

F.M.Ham Observability, Eigenvalues and Kalman Filtering IEEE Transactions on Aerospace

and Electronic Systems, 1982, 19(2): 156-164

Xianda, Zhang Matrix Analysis and Application Tsinghua University Press, 2004, Beijing

X Rong Li Information Fusion for Estimation and Decision International Workshop on Data

Fusion in 2002, Beijing, China

Air traffic trajectories segmentation based on time-series sensor data 31

Air traffic trajectories segmentation based on time-series sensor data

José L Guerrero, Jesús García and José M Molina

X

Air traffic trajectories segmentation based on time-series sensor data

José L Guerrero, Jesús García and José M Molina

University Carlos III of Madrid

Spain

1 Introduction

ATC is a critical area related with safety, requiring strict validation in real conditions (Kennedy

& Gardner, 1998), being this a domain where the amount of data has gone under an

exponential growth due to the increase in the number of passengers and flights This has led to

the need of automation processes in order to help the work of human operators (Wickens et

al., 1998) These automation procedures can be basically divided into two different basic

processes: the required online tracking of the aircraft (along with the decisions required

according to this information) and the offline validation of that tracking process (which is

usually separated into two sub-processes, segmentation (Guerrero & Garcia, 2008), covering

the division of the initial data into a series of different segments, and reconstruction (Pérez et

al., 2006, García et al., 2007), which covers the approximation with different models of the

segments the trajectory was divided into) The reconstructed trajectories are used for the

analysis and evaluation processes over the online tracking results

This validation assessment of ATC centers is done with recorded datasets (usually named

opportunity traffic), used to reconstruct the necessary reference information The

reconstruction process transforms multi-sensor plots to a common coordinates frame and

organizes data in trajectories of an individual aircraft Then, for each trajectory, segments of

different modes of flight (MOF) must be identified, each one corresponding to time intervals

in which the aircraft is flying in a different type of motion These segments are a valuable

description of real data, providing information to analyze the behavior of target objects

(where uniform motion flight and maneuvers are performed, magnitudes, durations, etc)

The performance assessment of ATC multisensor/multitarget trackers require this

reconstruction analysis based on available air data, in a domain usually named opportunity

trajectory reconstruction (OTR), (Garcia et al., 2009)

OTR consists in a batch process where all the available real data from all available sensors is

used in order to obtain smoothed trajectories for all the individual aircrafts in the interest

area It requires accurate original-to-reconstructed trajectory’s measurements association,

bias estimation and correction to align all sensor measures, and also adaptive multisensor

smoothing to obtain the final interpolated trajectory It should be pointed out that it is an

off-line batch processing potentially quite different to the usual real time data fusion

systems used for ATC, due to the differences in the data processing order and its specific

2

processing techniques, along with different availability of information (the whole trajectory

can be used by the algorithms in order to perform the best possible reconstruction)

OTR works as a special multisensor fusion system, aiming to estimate target kinematic state,

in which we take advantage of both past and future target position reports (smoothing

problem) In ATC domain, the typical sensors providing data for reconstruction are the

following:

• Radar data, from primary (PSR), secondary (SSR), and Mode S radars (Shipley,

1971) These measurements have random errors in the order of the hundreds of

meters (with a value which increases linearly with distance to radar)

• Multilateration data from Wide Area Multilateration (WAM) sensors (Yang et al.,

2002) They have much lower errors (in the order of 5-100 m), also showing a linear

relation in its value related to the distance to the sensors positions

• Automatic dependent surveillance (ADS-B) data (Drouilhet et al., 1996) Its quality

is dependent on aircraft equipment, with the general trend to adopt GPS/GNSS,

having errors in the order of 5-20 meters

The complementary nature of these sensor techniques allows a number of benefits (high

degree of accuracy, extended coverage, systematic errors estimation and correction, etc), and

brings new challenges for the fusion process in order to guarantee an improvement with

respect to any of those sensor techniques used alone

After a preprocessing phase to express all measurements in a common reference frame (the

stereographic plane used for visualization), the studied trajectories will have measurements

with the following attributes: detection time, stereographic projections of its x and y

components, covariance matrix, and real motion model (MM), (which is an attribute only

included in simulated trajectories, used for algorithm learning and validation) With these

input attributes, we will look for a domain transformation that will allow us to classify our

samples into a particular motion model with maximum accuracy, according to the model we

are applying

The movement of an aircraft in the ATC domain can be simplified into a series of basic

MM’s The most usually considered ones are uniform, accelerated and turn MM’s The

general idea of the proposed algorithm in this chapter is to analyze these models

individually and exploit the available information in three consecutive different phases

The first phase will receive the information in the common reference frame and the analyzed

model in order to obtain, as its output data, a set of synthesized attributes which will be

handled by a learning algorithm in order to obtain the classification for the different

trajectories measurements These synthesized attributes are based on domain transformations

according to the analyzed model by means of local information analysis (their value is based

on the definition of segments of measurements from the trajectory).They are obtained for each

measurement belonging to the trajectory (in fact, this process can be seen as a data

pre-processing for the data mining techniques (Famili et al., 1997))

The second phase applies data mining techniques (Eibe, 2005) over the synthesized

attributes from the previous phase, providing as its output an individual classification for

each measurement belonging to the analyzed trajectory This classification identifies the

measurement according to the model introduced in the first phase (determining whether it

belongs to that model or not)

The third phase, obtaining the data mining classification as its input, refines this

classification according to the knowledge of the possible MM’s and their transitions,

correcting possible misclassifications, and provides the final classification for each of the trajectory’s measurement This refinement is performed by means of the application of a filter

Finally, segments are constructed over those classifications (by joining segments with the same classification value) These segments are divided into two different possibilities: those belonging to the analyzed model (which are already a final output of the algorithm) and those which do not belong to it, having to be processed by different models It must be noted that the number of measurements processed by each model is reduced with each application of this cycle (due to the segments already obtained as a final output) and thus, more detailed models with lower complexity should be applied first Using the introduced division into three MM’s, the proposed order is the following: uniform, accelerated and finally turn model Figure 1 explains the algorithm’s approach:

Fig 1 Overview of the algorithm’s approach The validation of the algorithm is carried out by the generation of a set of test trajectories as representative as possible This implies not to use exact covariance matrixes, (but estimations of their value), and carefully choosing the shapes of the simulated trajectories

We have based our results on four types of simulated trajectories, each having two different samples Uniform, turn and accelerated trajectories are a direct validation of our three basic MM’s The fourth trajectory type, racetrack, is a typical situation during landing procedures The validation is performed, for a fixed model, with the results of its true positives rate (TPR, the rate of measurements correctly classified among all belonging to the model) and false positives rate (FPR, the rate of measurements incorrectly classified among all not belonging the model) This work will show the results of the three consecutive phases using

a uniform motion model

The different sections of this work will be divided with the following organization: the second section will deal with the problem definition, both in general and particularized for the chosen approach The third section will present in detail the general algorithm, followed

Trajectory input data First phase: 

domain transformation

Second phase: data mining techniquesSynthesized attributes

Preliminaryclassifications

Third phase: 

results filteringRefined classifications

NO Apply next 

model

YESFinal segmentation results

Belongs tomodel?

Segment construction

Analyzed model

for each output  segment

Air traffic trajectories segmentation based on time-series sensor data 33

processing techniques, along with different availability of information (the whole trajectory

can be used by the algorithms in order to perform the best possible reconstruction)

OTR works as a special multisensor fusion system, aiming to estimate target kinematic state,

in which we take advantage of both past and future target position reports (smoothing

problem) In ATC domain, the typical sensors providing data for reconstruction are the

following:

• Radar data, from primary (PSR), secondary (SSR), and Mode S radars (Shipley,

1971) These measurements have random errors in the order of the hundreds of

meters (with a value which increases linearly with distance to radar)

• Multilateration data from Wide Area Multilateration (WAM) sensors (Yang et al.,

2002) They have much lower errors (in the order of 5-100 m), also showing a linear

relation in its value related to the distance to the sensors positions

• Automatic dependent surveillance (ADS-B) data (Drouilhet et al., 1996) Its quality

is dependent on aircraft equipment, with the general trend to adopt GPS/GNSS,

having errors in the order of 5-20 meters

The complementary nature of these sensor techniques allows a number of benefits (high

degree of accuracy, extended coverage, systematic errors estimation and correction, etc), and

brings new challenges for the fusion process in order to guarantee an improvement with

respect to any of those sensor techniques used alone

After a preprocessing phase to express all measurements in a common reference frame (the

stereographic plane used for visualization), the studied trajectories will have measurements

with the following attributes: detection time, stereographic projections of its x and y

components, covariance matrix, and real motion model (MM), (which is an attribute only

included in simulated trajectories, used for algorithm learning and validation) With these

input attributes, we will look for a domain transformation that will allow us to classify our

samples into a particular motion model with maximum accuracy, according to the model we

are applying

The movement of an aircraft in the ATC domain can be simplified into a series of basic

MM’s The most usually considered ones are uniform, accelerated and turn MM’s The

general idea of the proposed algorithm in this chapter is to analyze these models

individually and exploit the available information in three consecutive different phases

The first phase will receive the information in the common reference frame and the analyzed

model in order to obtain, as its output data, a set of synthesized attributes which will be

handled by a learning algorithm in order to obtain the classification for the different

trajectories measurements These synthesized attributes are based on domain transformations

according to the analyzed model by means of local information analysis (their value is based

on the definition of segments of measurements from the trajectory).They are obtained for each

measurement belonging to the trajectory (in fact, this process can be seen as a data

pre-processing for the data mining techniques (Famili et al., 1997))

The second phase applies data mining techniques (Eibe, 2005) over the synthesized

attributes from the previous phase, providing as its output an individual classification for

each measurement belonging to the analyzed trajectory This classification identifies the

measurement according to the model introduced in the first phase (determining whether it

belongs to that model or not)

The third phase, obtaining the data mining classification as its input, refines this

classification according to the knowledge of the possible MM’s and their transitions,

correcting possible misclassifications, and provides the final classification for each of the trajectory’s measurement This refinement is performed by means of the application of a filter

Finally, segments are constructed over those classifications (by joining segments with the same classification value) These segments are divided into two different possibilities: those belonging to the analyzed model (which are already a final output of the algorithm) and those which do not belong to it, having to be processed by different models It must be noted that the number of measurements processed by each model is reduced with each application of this cycle (due to the segments already obtained as a final output) and thus, more detailed models with lower complexity should be applied first Using the introduced division into three MM’s, the proposed order is the following: uniform, accelerated and finally turn model Figure 1 explains the algorithm’s approach:

Fig 1 Overview of the algorithm’s approach The validation of the algorithm is carried out by the generation of a set of test trajectories as representative as possible This implies not to use exact covariance matrixes, (but estimations of their value), and carefully choosing the shapes of the simulated trajectories

We have based our results on four types of simulated trajectories, each having two different samples Uniform, turn and accelerated trajectories are a direct validation of our three basic MM’s The fourth trajectory type, racetrack, is a typical situation during landing procedures The validation is performed, for a fixed model, with the results of its true positives rate (TPR, the rate of measurements correctly classified among all belonging to the model) and false positives rate (FPR, the rate of measurements incorrectly classified among all not belonging the model) This work will show the results of the three consecutive phases using

a uniform motion model

The different sections of this work will be divided with the following organization: the second section will deal with the problem definition, both in general and particularized for the chosen approach The third section will present in detail the general algorithm, followed

Trajectory input data First phase: 

domain transformation

Second phase: data mining techniquesSynthesized attributes

Preliminaryclassifications

Third phase: 

results filteringRefined classifications

NO Apply next 

model

YESFinal segmentation results

Belongs tomodel?

Segment construction

Analyzed model

for each output  segment

by three sections detailing the three phases for that algorithm when the uniform movement

model is applied: the fourth section will present the different alternatives for the domain

transformation and choose between them the ones included in the final algorithm, the fifth

will present some representative machine learning techniques to be applied to obtain the

classification results and the sixth the filtering refinement over the previous results will be

introduced, leading to the segment synthesis processes The seventh section will cover the

results obtained over the explained phases, determining the used machine learning

technique and providing the segmentation results, both numerically and graphically, to

provide the reader with easy validation tools over the presented algorithm Finally a

conclusions section based on the presented results is presented

2 Problem definition

2.1 General problem definition

As we presented in the introduction section, each analyzed trajectory (ܶ௜) is composed of a

collection of sensor reports (or measurements), which are defined by the following vector:

ݔԦ௝௜ൌ ൫ݔ௝௜ǡ ݕ௝௜ǡ ݐ௝௜ǡ ܴ௝௜൯, ݆ ߳ ሼͳǡ ǥ ǡ ܰ௜ሽ (1) where j is the measurement number, i the trajectory number, N is the number of

measurements in a given trajectory, ݔ௝௜ǡ ݕ௝௜ are the stereographic projections of the

measurement, ݐ௝௜ is the detection time and ܴ௝௜ is the covariance matrix (representing the error

introduced by the measuring device) From this problem definition our objective is to divide

our trajectory into a series of segments (ܤ௞௜ሻ, according to our estimated MOF This is

performed as an off-line processing (meaning that we may use past and future information

from our trajectory) The segmentation problem can be formalized using the following

notation:

ܶ௜ൌ ڂ ܤ௞௜ ܤ௞௜ ൌ ሼݔ௝௜ሽ ݆ ߳ ሼ݇௠௜௡ǡ ǥ ǡ ݇௠௔௫ሽ (2)

In the general definition of this problem these segments are obtained by the comparison

with a test model applied over different windows (aggregations) of measurements coming

from our trajectory, in order to obtain a fitness value, deciding finally the segmentation

operation as a function of that fitness value (Mann et al 2002), (Garcia et al., 2006)

We may consider the division of offline segmentation algorithms into different approaches:

a possible approach is to consider the whole data from the trajectory and the segments

obtained as the problem’s basic division unit (using a global approach), where the basic

operation of the segmentation algorithm is the division of the trajectory into those segments

(examples of this approach are the bottom-up and top-down families (Keogh et al., 2003)) In

the ATC domain, there have been approaches based on a direct adaptation of online

techniques, basically combining the results of forward application of the algorithm (the pure

online technique) with its backward application (applying the online technique reversely to

the time series according to the measurements detection time) (Garcia et al., 2006) An

alternative can be based on the consideration of obtaining a different classification value for

each of the trajectory’s measurements (along with their local information) and obtaining the

segments as a synthesized solution, built upon that classification (basically, by joining those adjacent measures sharing the same MM into a common segment) This approach allows the application of several refinements over the classification results before the final synthesis is performed, and thus is the one explored in the presented solution in this chapter

2.2 Local approach problem definition

We have presented our problem as an offline processing, meaning that we may use information both from our past and our future Introducing this fact into our local representation, we will restrict that information to a certain local segment around the measurement which we would like to classify These intervals are centered on that measurement, but the boundaries for them can be expressed either in number of measurements, (3), or according to their detection time values (4)

ܤሺݔ௠௜ ሻ ൌ ሼݔ௝௜ሽ ݆ ߳ ሾ݉ െ ݌ǡ ǥ ǡ ݉ǡ ǥ ǡ ݉ ൅ ݌ሿ (3) ܤሺݔ௠௜ ሻ ൌ ሼݔ௝௜ሽ ݐ୨߳൛ݐ୫௜ െ ǡ ǥ ǡ –୫ǡ ǥ ǡ ݐ୫௜ ൅ ൟ (4) Once we have chosen a window around our current measurement, we will have to apply a function to that segment in order to obtain its transformed value This general classification function F(ݔሬሬሬԦሻ, using measurement boundaries, may be represented with the following ఫప

formulation:

F(ݔሬሬሬሬሬԦሻ = F(ݔ୫ప ሬሬሬሬሬԦȁܶ୫ప ௜) ֜ F(ݔሬሬሬԦȁ൫š఩ ୫୧ ൯ሻ = Fp(ݔԦ୫ି௣௜ , , ݔԦ୫௜ , , ݔԦ୫ା௣௜ ) (5) From this formulation of the problem we can already see some of the choices available: how

to choose the segments (according to (3) or (4)), which classification function to apply in (5) and how to perform the final segment synthesis Figure 2 shows an example of the local approach for trajectory segmentation

Fig 2 Local approach for trajectory segmentation approach overview

2,5 3 3,5 4 4,5 5 5,5 6 6,5

Air traffic trajectories segmentation based on time-series sensor data 35

by three sections detailing the three phases for that algorithm when the uniform movement

model is applied: the fourth section will present the different alternatives for the domain

transformation and choose between them the ones included in the final algorithm, the fifth

will present some representative machine learning techniques to be applied to obtain the

classification results and the sixth the filtering refinement over the previous results will be

introduced, leading to the segment synthesis processes The seventh section will cover the

results obtained over the explained phases, determining the used machine learning

technique and providing the segmentation results, both numerically and graphically, to

provide the reader with easy validation tools over the presented algorithm Finally a

conclusions section based on the presented results is presented

2 Problem definition

2.1 General problem definition

As we presented in the introduction section, each analyzed trajectory (ܶ௜) is composed of a

collection of sensor reports (or measurements), which are defined by the following vector:

ݔԦ௝௜ൌ ൫ݔ௝௜ǡ ݕ௝௜ǡ ݐ௝௜ǡ ܴ௝௜൯, ݆ ߳ ሼͳǡ ǥ ǡ ܰ௜ሽ (1) where j is the measurement number, i the trajectory number, N is the number of

measurements in a given trajectory, ݔ௝௜ǡ ݕ௝௜ are the stereographic projections of the

measurement, ݐ௝௜ is the detection time and ܴ௝௜ is the covariance matrix (representing the error

introduced by the measuring device) From this problem definition our objective is to divide

our trajectory into a series of segments (ܤ௞௜ሻ, according to our estimated MOF This is

performed as an off-line processing (meaning that we may use past and future information

from our trajectory) The segmentation problem can be formalized using the following

notation:

ܶ௜ൌ ڂ ܤ௞௜ ܤ௞௜ ൌ ሼݔ௝௜ሽ ݆ ߳ ሼ݇௠௜௡ǡ ǥ ǡ ݇௠௔௫ሽ (2)

In the general definition of this problem these segments are obtained by the comparison

with a test model applied over different windows (aggregations) of measurements coming

from our trajectory, in order to obtain a fitness value, deciding finally the segmentation

operation as a function of that fitness value (Mann et al 2002), (Garcia et al., 2006)

We may consider the division of offline segmentation algorithms into different approaches:

a possible approach is to consider the whole data from the trajectory and the segments

obtained as the problem’s basic division unit (using a global approach), where the basic

operation of the segmentation algorithm is the division of the trajectory into those segments

(examples of this approach are the bottom-up and top-down families (Keogh et al., 2003)) In

the ATC domain, there have been approaches based on a direct adaptation of online

techniques, basically combining the results of forward application of the algorithm (the pure

online technique) with its backward application (applying the online technique reversely to

the time series according to the measurements detection time) (Garcia et al., 2006) An

alternative can be based on the consideration of obtaining a different classification value for

each of the trajectory’s measurements (along with their local information) and obtaining the

segments as a synthesized solution, built upon that classification (basically, by joining those adjacent measures sharing the same MM into a common segment) This approach allows the application of several refinements over the classification results before the final synthesis is performed, and thus is the one explored in the presented solution in this chapter

2.2 Local approach problem definition

We have presented our problem as an offline processing, meaning that we may use information both from our past and our future Introducing this fact into our local representation, we will restrict that information to a certain local segment around the measurement which we would like to classify These intervals are centered on that measurement, but the boundaries for them can be expressed either in number of measurements, (3), or according to their detection time values (4)

ܤሺݔ௠௜ሻ ൌ ሼݔ௝௜ሽ ݆ ߳ ሾ݉ െ ݌ǡ ǥ ǡ ݉ǡ ǥ ǡ ݉ ൅ ݌ሿ (3) ܤሺݔ௠௜ ሻ ൌ ሼݔ௝௜ሽ ݐ୨߳൛ݐ୫௜ െ ǡ ǥ ǡ –୫ǡ ǥ ǡ ݐ୫௜ ൅ ൟ (4) Once we have chosen a window around our current measurement, we will have to apply a function to that segment in order to obtain its transformed value This general classification function F(ݔሬሬሬԦሻ, using measurement boundaries, may be represented with the following ఫప

formulation:

F(ݔሬሬሬሬሬԦሻ = F(ݔ୫ప ሬሬሬሬሬԦȁܶ୫ప ௜) ֜ F(ݔሬሬሬԦȁ൫š఩ ୫୧ ൯ሻ = Fp(ݔԦ୫ି௣௜ , , ݔԦ୫௜ , , ݔԦ୫ା௣௜ ) (5) From this formulation of the problem we can already see some of the choices available: how

to choose the segments (according to (3) or (4)), which classification function to apply in (5) and how to perform the final segment synthesis Figure 2 shows an example of the local approach for trajectory segmentation

Fig 2 Local approach for trajectory segmentation approach overview

2,5 3 3,5 4 4,5 5 5,5 6 6,5

3 General algorithm proposal

As presented in the introduction section, we will consider three basic MM’s and classify our

measurements individually according to them (Guerrero & Garcia, 2008) If a measurement

is classified as unknown, it will be included in the input data for the next model’s analysis

This general algorithm introduces a design criterion based on the introduced concepts of

TPR and FPR, respectively equivalent to the type I and type II errors (Allchin, 2001) The

design criterion will be to keep a FPR as low as possible, understanding that those

measurements already assigned to a wrong model will not be analyzed by the following

ones (and thus will remain wrongly classified, leading to a poorer trajectory reconstruction)

The proposed order for this analysis of the MM’s is the same in which they have been

introduced, and the choice is based on how accurately we can represent each of them

In the local approach problem definition section, the segmentation problem was divided

into two different sub-problems: the definition of the ܨ௣ሺݔሬሬሬሬሬԦሻ function (to perform ௠ప

measurement classification) and a final segment synthesis over that classification

According to the different phases presented in the introduction section, we will divide the

definition of the classification function F(ݔሬሬሬԦሻinto two different tasks: a domain ఫప

transformation DtሺݔሬሬሬԦሻ (domain specific, which defines the first phase of our algorithm) and ఫప

a final classification Cl(DtሺݔሬሬሬԦሻ) (based on general classification algorithms, represented by ఫప

the data mining techniques which are introduced in the second phase) The final synthesis

over the classification results includes the refinement over that classification introduced by

the filtering process and the actual construction of the output segment (third phase of the

proposed algorithm)

The introduction of the domain transformation DtሺݔሬሬሬԦሻ from the initial data in the common ఫప

reference frame must deal with the following issues: segmentation, (which will cover the

decision of using an independent classification for each measurement or to treat segments as

an indivisible unit), definition for the boundaries of the segments, which involves segment

extension (which analyzes the definition of the segments by number of points or according

to their detection time values) and segment resolution (dealing with the choice of the length

of those segments, and how it affects our results), domain transformations (the different

possible models used in order to obtain an accurate classification in the following phases),

and threshold choosing technique (obtaining a value for a threshold in order to pre-classify

the measurements in the transformed domain)

The second phase introduces a set of machine learning techniques to try to determine

whether each of the measurements belongs to the analyzed model or not, based on the

pre-classifications obtained in the first phase In this second phase we will have to choose a

Cl(DtሺݔሬሬሬԦሻ) technique, along with its configuration parameters, to be included in the ఫప

algorithm proposal The considered techniques are decision trees (C4.5, (Quinlan, 1993))

clustering (EM, (Dellaert, 2002)) neural networks (multilayer perceptron, (Gurney, 1997))

and Bayesian nets (Jensen & Graven-Nielsen, 2007) (along with the simplified naive Bayes

approach (Rish, 2001))

Finally, the third phase (segment synthesis) will propose a filter, based on domain

knowledge, to reanalyze the trajectory classification results and correct those values which

may not follow this knowledge (essentially, based on the required smoothness in MM’s

changes) To obtain the final output for the model analysis, the isolated measurements will

be joined according to their classification in the final segments of the algorithm

The formalization of these phases and the subsequent changes performed to the data is presented in the following vectors, representing the input and output data for our three processes:

The following aspects are the key parameters for this phase, presented along with the different alternatives compared for them, (it must be noted that the possibilities compared here are not the only possible ones, but representative examples of different possible approaches):

 Transformation function: correlation coefficient / Best linear unbiased estimator

residue

 Segmentation granularity: segment study / independent study

 Segment extension, time / samples, and segment resolution, length of the segment,

using the boundary units imposed by the previous decision

 Threshold choosing technique, choice of a threshold to classify data in the

transformed domain

Each of these parameters requires an individual validation in order to build the actual final algorithm tested in the experimental section Each of them will be analyzed in an individual section in order to achieve this task

4.1 Transformation function analysis

The transformation function decision is probably the most crucial one involving this first phase of our algorithm The comparison presented tries to determine whether there is a real accuracy increase by introducing noise information (in the form of covariance matrixes) This section compares a correlation coefficient (Meyer, 1970) (a general statistic with no noise information) with a BLUE residue (Kay, 1993) (which introduces the noise in the measuring process) This analysis was originally proposed in (Guerrero & Garcia, 2008) The equations for the CC statistical are the following:

Air traffic trajectories segmentation based on time-series sensor data 37

3 General algorithm proposal

As presented in the introduction section, we will consider three basic MM’s and classify our

measurements individually according to them (Guerrero & Garcia, 2008) If a measurement

is classified as unknown, it will be included in the input data for the next model’s analysis

This general algorithm introduces a design criterion based on the introduced concepts of

TPR and FPR, respectively equivalent to the type I and type II errors (Allchin, 2001) The

design criterion will be to keep a FPR as low as possible, understanding that those

measurements already assigned to a wrong model will not be analyzed by the following

ones (and thus will remain wrongly classified, leading to a poorer trajectory reconstruction)

The proposed order for this analysis of the MM’s is the same in which they have been

introduced, and the choice is based on how accurately we can represent each of them

In the local approach problem definition section, the segmentation problem was divided

into two different sub-problems: the definition of the ܨ௣ሺݔሬሬሬሬሬԦሻ function (to perform ௠ప

measurement classification) and a final segment synthesis over that classification

According to the different phases presented in the introduction section, we will divide the

definition of the classification function F(ݔሬሬሬԦሻinto two different tasks: a domain ఫప

transformation DtሺݔሬሬሬԦሻ (domain specific, which defines the first phase of our algorithm) and ఫప

a final classification Cl(DtሺݔሬሬሬԦሻ) (based on general classification algorithms, represented by ఫప

the data mining techniques which are introduced in the second phase) The final synthesis

over the classification results includes the refinement over that classification introduced by

the filtering process and the actual construction of the output segment (third phase of the

proposed algorithm)

The introduction of the domain transformation DtሺݔሬሬሬԦሻ from the initial data in the common ఫప

reference frame must deal with the following issues: segmentation, (which will cover the

decision of using an independent classification for each measurement or to treat segments as

an indivisible unit), definition for the boundaries of the segments, which involves segment

extension (which analyzes the definition of the segments by number of points or according

to their detection time values) and segment resolution (dealing with the choice of the length

of those segments, and how it affects our results), domain transformations (the different

possible models used in order to obtain an accurate classification in the following phases),

and threshold choosing technique (obtaining a value for a threshold in order to pre-classify

the measurements in the transformed domain)

The second phase introduces a set of machine learning techniques to try to determine

whether each of the measurements belongs to the analyzed model or not, based on the

pre-classifications obtained in the first phase In this second phase we will have to choose a

Cl(DtሺݔሬሬሬԦሻ) technique, along with its configuration parameters, to be included in the ఫప

algorithm proposal The considered techniques are decision trees (C4.5, (Quinlan, 1993))

clustering (EM, (Dellaert, 2002)) neural networks (multilayer perceptron, (Gurney, 1997))

and Bayesian nets (Jensen & Graven-Nielsen, 2007) (along with the simplified naive Bayes

approach (Rish, 2001))

Finally, the third phase (segment synthesis) will propose a filter, based on domain

knowledge, to reanalyze the trajectory classification results and correct those values which

may not follow this knowledge (essentially, based on the required smoothness in MM’s

changes) To obtain the final output for the model analysis, the isolated measurements will

be joined according to their classification in the final segments of the algorithm

The formalization of these phases and the subsequent changes performed to the data is presented in the following vectors, representing the input and output data for our three processes:

The following aspects are the key parameters for this phase, presented along with the different alternatives compared for them, (it must be noted that the possibilities compared here are not the only possible ones, but representative examples of different possible approaches):

 Transformation function: correlation coefficient / Best linear unbiased estimator

residue

 Segmentation granularity: segment study / independent study

 Segment extension, time / samples, and segment resolution, length of the segment,

using the boundary units imposed by the previous decision

 Threshold choosing technique, choice of a threshold to classify data in the

transformed domain

Each of these parameters requires an individual validation in order to build the actual final algorithm tested in the experimental section Each of them will be analyzed in an individual section in order to achieve this task

4.1 Transformation function analysis

The transformation function decision is probably the most crucial one involving this first phase of our algorithm The comparison presented tries to determine whether there is a real accuracy increase by introducing noise information (in the form of covariance matrixes) This section compares a correlation coefficient (Meyer, 1970) (a general statistic with no noise information) with a BLUE residue (Kay, 1993) (which introduces the noise in the measuring process) This analysis was originally proposed in (Guerrero & Garcia, 2008) The equations for the CC statistical are the following: