T-FoT SP: Trajectory function of time

1. O2  -- start from naïve, Data Only

While admitting the success of Markov-model-specific filters in many cases, I am even more interested in investigating the failure cases when they are in fact ineffective for state estimation, because of mismodeling or too much approximation. In the following paper, classic scenarios have shown that the straightforward observation-only (O2) inference that does not need state system modeling can perform better (in terms of both accuracy and computing speed) than model-specific filters. Special attention has been paid to quantitatively analyze when and why a filter will not outperform the O2 inference from the information fusion perspective.

We advocate that, the O2 inference shall serve as a benchmark to assess the effectiveness of Bayes filters:

T.Li, J.M. Corchado, J. Bajo, S. Sun and J. F. Paz, Effectiveness of Bayesian Filters: An Information Fusion Perspective, Information Sciences, 2016, 329: 670-689. @  ScienceDirect 

However, the data-driven solution does not conflict with the model-specific solution. Instead, once useful model information is feasible, we shall use it! Why not?---Surely, it does not have to be in the HMM manner -- It is shown that with the use of regression analysis /fitting to linking distant estimates obtained at discrete scans via whether the Bayesian inference or the O2/C4F inference, continuous-time trajectories (each representing a target track) can be obtained. This framework, namely fitting for smoothing (F4S), will also facilitate long term prediction, given that the trend of the track holds in the time interval of interest.  This releases the requirement of regular fixed-lag time sensoring frequency!

T. Li, J. Prieto, J. M. Corchado, Fitting for Smoothing: A Methodology for Continuous-Time Target Track Estimation, IPIN 2016, At Alcalá de Henares, Spain, Oct. 4-7, 2016.@IEEE Xplore

Furthermore, C4F may work hand-in-hand with traditional filters such as the PHD filter as shown in the following work, in which  the collection of the measurements of different sensors are converted to a set of proxy, homologous measurements -- thorough multisensor data flooding and clustering. These synthetic measurements overcome the problems of false and missing data and of unknown statistics and facilitate linear PHD updating that amounts to the standard PHD filtering with no false and missing data. 

T. Li, J. Prieto and J.M. Corchado, A Robust Multi-Sensor PHD Filter Based on Multi-Sensor Measurement Clustering, IEEE Communications Letters, vol.22, no.10, pp. 2064-2067, 2018 , Aug. 2018. @ IEEE link

2. C4F (centralized or distributed sensor network)

Thanks to the rapid development of advanced sensors and their joint application, the O2 inference using multiple sensors is not only engineering friendly and computationally fast but can also be very accurate and reliable. Given an adequate number of sensors (usually >=4), multi-sensor data clustering provides a Markov-free multi-object detection and estimation (MODE) solution in the cluttered environment with unknown and time-varying target/clutter/noise models (for which traditional filter is hard to design!) , when Markov modelling is difficult or simply does not pay off.

T.Li, J.M. Corchado, S. Sun and J. Bajo, Clustering for filtering:  Multi-target Detection and Estimation Using Multiple/Massive Sensors, Information Sciences. Vol.388–389, May 2017, Pages 172–190. @ ScienceDirect    

T. Li, F. De la Prieta Pintado, J. M. Corchado, J. Bajo, Multi-source Homogeneous Data Clustering for Multi-target Detection from Cluttered Background with Misdetection, Applied Soft Computing,Vol. 60, pp. 436-446. @ ScienceDirect.

The source code for an illustrative MODE example for a scenario with no prior information, as well as the implementation based on a distributed sensor network, is available @ C4F.  

T. Li, J.M. Corchado and H. Chen, Distributed Flooding-then-Clustering: A Lazy Networking Approach for Distributed Multiple Target Tracking, FUSION 2018, Cambridge, July 10-13 2018. @IEEE Xplore

M codes can be found @ Distributed C4F_Flooding-then-clustering

3.  HMM-free:  modeling target trajectory by a function of time (FoT) 

Moreover, we get a unified framework for joint smoothing, tracking, and forecasting (STF), particularly for a type of targets that are subject to a smooth evolving process in the space-time domain -- such as a train moves on the railway. Based on this important fact, our idea is to utilize this type of unstructured/soft/linguistic/fuzzy context information that “the trajectory is smooth” or “the target is descending”. This is the reality of a wide class of real world situations of significance such as passenger aircraft/ships and even ballistic missiles.

Fundamentally different from the conventional Markov-Bayes formulation, the state process is characterized by a function in the continuous time domain and the STF problem is formulated as an online data fitting problem with the goal of finding the trajectory FoT (function of time) that best fits the observations in a sliding time-window (in the sense of least-squares or similar), conditioned on a priori model information if any. As a result, the movement of the target at any time, whether the past (namely, smoothing), the current (filtering) or the future (forecasting), can be inferred from the trajectory function. Few assumptions are made on the statistical properties of either the target or the sensor, ensuring high flexibility and robustness of our approach. That is to say, we are now aiming to estimate the trajectory function/FoT directly rather than the discrete-time state!

Please refer to 

T. Li, H. Chen, S. Sun and J. M. Corchado,  "Joint Smoothing and Tracking Based on Continuous-Time Target Trajectory Function Fitting," in IEEE Transactions on Automation Science and Engineering, vol. 16, no. 3, pp. 1476-1483, July 2019. @ IEEE Xplore   

M codes can be found @ FoT4STF. 

4. T-FoT:  Clutter, Misdetection, and No A-Prior etc.

Furthermore, we extend the T-FoT approach to account for random finite set observations consisting of both missing and false data.  More challenging, the missing and false data are generated under unknown ratios, i.e., they can not be accurately modeled. To tackle this problem, a fully data-driven method is proposed for identifying the real measurement of the target from clutter if the target is detected and for declaring a misdetection otherwise.  Connection of the approach to the Bayesian optimal and (constrained) least square estimator. Simulation results have demonstrated that the proposed fitting method performs comparable to the Bayesian optimal filter, without using exact statistics information about the target and the sensor while favorably yielding real-time estimate of the continuous-time trajectory directly. For more, see the following papers:

• T.Li, X. Wang, Y. Liang, J. Yan and H. Fan, A Track-oriented Approach to Target Tracking with Random Finite Set Observations, ICCAIS 2019, Chengdu, China, Oct. 22-24, 2019  @ IEEE Xplore --- analyze the relationship with the benchmark KF

• T. Li,  Y. Song, et al., From Target Tracking to Targeting Track: A Data-Driven Yet Analytical Approach to Joint Target Detection and Tracking, Signal Processing, Volume 205, April 2023, 108883 @ScienceDirect  ----- with more statistical analyze on the optimality and property of the T-FoT and ability to deal with unknown-yet-low-rate clutter and misdetection!

M codes can be found @T-FoT-for-joint-detection-tracking-in-clutter 

5.  Constrained Tracking; long-term/soft constraints

Furthermore, we consider a specific type of target that moves on a deterministic, predefined trajectory which can be modeled as a curve in the planar space, such as a train/subway that moves on the railway, a car on the highway, a satellite on its orbit, etc. Apart from the a-priori “trajectory” geometry, there is no further (statistical) information about the target motion, e.g, regarding its velocity, acceleration, etc. To solve such a trajectory-constrained positioning problem,  we establish a trajectory-constrained state space model to implement the unscented filtering and a constrained MoC-trajectory function of time for weighted least squares fitting. A maneuvering target that moves on a simple deterministic trajectory is studied. The results demonstrate a significant performance improvement gained by using the trajectory constraint.

• T. Li, Single-road-constrained positioning based on deterministic trajectory geometry, IEEE Communications Letters, vol.23, no.1, pp. 80-83, 2019 @ IEEEXplore

• J. Zhou, T. Li*, X. Wang and L. Zheng, Target Tracking with Equality/Inequality Constraints Based on Trajectory Function of Time, in IEEE Signal Processing Letters, vol. 28, pp. 1330-1334, 2021 @IEEE Xplore 

M codes can be found @ Cosntrained-T-FoT 

6. T-FoT-based Sensor Selection/Control:

In this paper, we propose a computationally efficient sensor selection approach for maneuvering target tracking using a sensor network with communication bandwidth constraints, given limited prior information on the target maneuvering models. We formulate the stochastic sensor selection problem as a linear programming problem which consists of two easily implementable steps. First, the Cramér–Rao lower bound corresponding to the sensor subset is derived as the objective function of the proposed sensor selection method based on a partially observable Markov decision process. Second, the target trajectory is modeled by a function of time to enable online target tracking which is free of the conventional, a priori Markov modeling of the target dynamics.

• C. Liu, K. Di, T. Li*, V. Elvira. A sensor selection approach to maneuvering target tracking based on trajectory function of time. EURASIP J. Adv. Signal Process. 2022, 72 (2022)

Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful. 

--  Box, George E. P.; Norman R. Draper (1987). Empirical Model-Building and Response Surfaces, p. 74