# Sliding Window Neural Generated Tracking Based on Measurement Model

Haya Ejjawi
Sorbonne University-Abu Dhabi
Department of Science and Engineering
Abu Dhabi  UAE
<EMAIL_ADDRESS>Amal El Fallah Seghrouchni
Mohammed VI Polytechnic University
Ai movement
Rabat  Morocco
Sorbonne University
Paris  France
<EMAIL_ADDRESS>Frederic Barbaresco
Thales Group
Paris  France
<EMAIL_ADDRESS>Raed Abu Zitar
Thales/SCAI Senior Scientist
Sorbonne University-Abu Dhabi
Abu Dhabi  UAE
<EMAIL_ADDRESS>978-1-6654-9032-0/23/$\$31.00$ ©2023 IEEE

###### Abstract

In the pursuit of further advancement in the field of target tracking, this
paper explores the efficacy of a feedforward neural network in predicting
drones’ tracks, aiming to eventually, compare the tracks created by the well-
known Kalman filter and the ones created by our proposed neural network. The
unique feature of our proposed neural network tracker is that it is using only
a measurement model to estimate the next states of the track. Object model
selection and linearization is one of the challenges that always face in the
tracking process. The neural network uses a sliding window to incorporate the
history of measurements when applying estimations of the track values. The
testing results are comparable to the ones generated by the Kalman filter,
especially for the cases where there is low measurement covariance. The
complexity of linearization is avoided when using this proposed model.

###### Contents

  1. 1 Introduction
  2. 2  Methodology of Training and Testing
  3. 3  Simulations and Results: KF and NN
    1. 3.1 Training & Testing
  4. 4 Experimenting With Different Noise Covariances
    1. 4.1 Setting the Covariance to 0.001
  5. 5 Summary

##  1 Introduction

Many variations of the Kalman filter have been derived with each being the
most convenient according to the problem at hand. The most well-known types of
Kalman filters are the linear Kalman filter, the Extended Kalman filter, and
the Unscented Kalman filter[1]. Linear Kalman filter is a recursive algorithm
that fuses measurements from several sensors to estimate in real time the
state of a robotic system when the estimation system is linear and Gaussian.
An estimation system is considered linear when both the motion model and the
measurements model are linear [2]. The state equation can be generalized to:

$x_{k+1}=A_{k}x_{k}+B_{k}u_{k}+G_{k}v_{k}$

where $A_{k}$ is the state transition matrix and $B_{k}$ is the control
matrix, which accounts for any unknown forces acting on the object. $v_{k}$
represents process noise following a Gaussian distribution of mean 0 and
covariance $Q_{k}$. $G_{k}$ is the process noise gain matrix.
Measurements are what is observed or measured in a system. A linear
measurement can be represented as:

$z_{k}=H_{k}x_{k}+w_{k}$

Where $H_{k}$ is the measurement matrix and $w_{k}$ represents measurement
noise at the current time step. The measurement noise is Gaussian and of zero
means. The covariance matrix of the measurement noise is assumed to be
$R_{k}$. The filter works in two steps, prediction, and correction. In the
prediction part, the system model is used to calculate the a priori state
$x_{k+1\mid k}$ and its covariance matrix $P_{k+1\mid k}$, given initial
estimates $x_{0\mid 0}$ and $P_{0\mid 0}$. The subscript $k+1\mid k$ indicates
that the corresponding quantity is the estimate of the k+1 step propagated
from the $k^{th}$ step. In the second step the filter uses the a priori
estimates calculated in the first part to compute the posteriori estimates
$x_{k+1\mid k+1}$ and $p_{k+1\mid k+1}$. The process can be summarized in the
table below:

Table 1: Kalman Filter Algorithm

Prediction | Correction
---|---
$x_{k+1\mid k}=A_{k}x_{k\mid k}+B_{k}u_{k}$ | $S_{k+1}=H_{k+1}P_{k+1\mid k}H_{k+1}^{T}+R_{k+1}$
$P_{k+1\mid k}=F_{k}P_{k\mid k}F_{k}^{T}+G_{k}Q_{k}G_{k}^{T}$ | $K_{k+1}=P_{k+1\mid k}H_{k+1}^{T}S_{k+1}^{-1}$
$z_{k+1\mid k}=H_{k+1}x_{k+1\mid k}$ | $x_{k+1\mid k+1}=x_{k+1\mid k}+K_{k+1}(z_{k+1}-z_{k+1\mid k})$
| $P_{k+1\mid k+1}=P_{k+1\mid k}-K_{k+1}S_{k+1}K_{k+1}^{T}$

Where $K_{k+1}$ is the Kalman gain, $S_{k+1}$ is the measurement predicted
covariance matrix, and $x_{k+1\mid k+1}$ is the corrected state at step k+1,
and $P_{k+1\mid k+1}$ is its corrected covariance matrix at step k+1.
Where $K_{k+1}$ is the Kalman gain, $S_{k+1}$ is the measurement predicted
covariance matrix, and $x_{k+1\mid k+1}$ is the corrected state at step k+1,
and $P_{k+1\mid k+1}$ is its corrected covariance matrix at step k+1.
The Extended Kalman filter (EKF) is the nonlinear version of the Kalman filter
[3]. EKF is used in this work and it works on two major assumptions, the
system is linear and Gaussian [4]. In order to use the Kalman filter with
nonlinear functions, linearization is required. The problem that arises when
dealing with nonlinear functions is that feeding a Gaussian distribution with
a nonlinear function results in a non-Gaussian distribution and therefore
Kalman filter cannot be used. To overcome this issue, a linear approximation
of nonlinear functions is needed and can be achieved with the help of the
Taylor series
for linearizing, only first-order approximation is required. The approximation
of the nonlinear curve is done at the mean of the Gaussian distribution. In
this problem, the extended Kalman filter is used since measurements are
collected from a simulated 2-D bearing range sensor as polar coordinates.
On the other hand, a feed-forward neural network will be used to be trained
with the measurements data [5], [6], [7]. Feedforward neural networks are
artificial neural networks where the connections between units do not form a
cycle. In a feed-forward neural network, information only travels forward in
the network, first through the input nodes, then through the hidden layers,
and finally through the output nodes. Backpropagation algorithm is used to
find the optimal parameters [8]. The paper is organized as follows; section 3
will cover the methodology, section 4 will cover simulations and results, and
section 5 will summarize the conclusion and results.

##  2  Methodology of Training and Testing

Stone soup, a software that provides a framework for target tracking and state
estimation and enables the development and testing of state estimation
algorithms, is first used to create a target that moves with a near-constant
velocity from the (0,0) location point. [9]. Measurements are then obtained
through a 2-D bearing and range sensors. Measurements are given as polar
coordinates so conversion to Cartesian coordinates is required:

$\begin{bmatrix}x\\\ y\end{bmatrix}$ = $\begin{bmatrix}rcos(\theta)\\\
rsin(\theta)\end{bmatrix}$

However, the sensor offset must be taken into account when converting. In this
case, the sensor is placed at (50,0). For obtaining the correct x-coordinate,
50 must be added to the X-location value. The EKF is then used to obtain a
track. Several data sets are generated by setting different seeds and based on
different given ground truth tracks.

For initial training and testing [10], data is divided into training and
testing sets for better initial evaluation of the model. The model we are
proposing is called the sliding window method. This method is implemented such
that $n$ measurements are fed into the neural network (NN) at a time $t$. For
example, if $n=3$ (number of measurement values), then measurement values at
time $t$, $t-1$, and $t-2$ are fed simultaneously to the neural network. The
neural network calculates the predicted output $\hat{Y}_{t}$. The given ground
truth value $Y_{t}$ is the target value at $t$. Please, see Figure 1 for more
clarifications. The error that is used in adjusting the weights of the neural
net is evaluated by the Mean Squared Error (MSE). This measure is based on the
sum of the euclidean distances between the ground truth values and the neural
network output values for a sequence of $m$ length, (see equations 1 and 2):

$MSE=\frac{1}{m}\sum_{i=1}^{m}(Y_{i}-\hat{Y}_{i})^{2}\\\ $ (1)

MSE = mean squared error
n = number of data points
$Y_{i}$ = observed values (neural network targets)
$\hat{Y}_{i}$ = predicted values (neural network outputs

$d(p,q)=\sqrt{\sum_{i=1}^{n}(q_{i}-p_{i})^{2}}$ (2)

p,q = two points in Euclidean n-space
$q_{i},p_{i}$ = $i^{th}$ coordinate of the points p & q
n = dimension of the space

First, weights are initialized randomly in order to start the forward
propagation process. In forward propagation, an input feature vector is
initially multiplied with a weight matrix and a bias vector is added. Then, in
order to achieve non-linearity, the weighted sum is fed into an activation
function [11], [12]. The nonlinear Relu Activation function $f(Z)$ is used for
all units of layers [l]; l ¡ L:

$f(Z)=max(0,Z)$ (3)

and its first order derivative:

$f^{\prime}(Z)=\begin{cases}0&\text{if }Z<0\\\ 1&\text{if }Z>0\\\ \end{cases}$
(4)

This provides the output of the $j^{th}$ unit of the layer. Multiplications
are simultaneously performed by multiple units of layer [l] providing an
output vector of length equal to the number of units. This output vector is
then multiplied by the weight matrix of the subsequent layer [l+1] and fed
into the activation function of layer [l+1]. After repeating this process L-1
times (for an L-Layer Neural Network), the output vector (at layer L-1) is fed
into a final activation function. In this case, the final activation function
used is the identity in order to obtain a real-numbered continuous outcome.
The cost function measures how well the parameters perform on the training set
[13]. In this case, the aim is to minimize the sum of least squares, also
known as the Ordinary Least Squares method (OLS):

$L(a,y)=\frac{1}{m}\sum_{i=1}^{m}\ (a_{i}-y_{i})^{2}\,$ (5)

The back-propagation computes the gradient of the loss function with respect
to the weights [14]. The gradients are computed iteratively starting at the
final Layer L [15]. The algorithm used in this problem for getting the best
model parameters is called Gradient Descent. The general work procedure can be
summarized in the following flow chart:

Figure 1: Flow of Tracks Generation.

Figure 2: Sliding Window Neural Network

The sliding window neural network, as shown in Figure 2, implicitly includes a
short history of the measurement values,in addition to the current input (for
example $t$, $t-1$, $t-2$ for $n=3$), and use them as the input vector. Then
it slides up one time step and uses the measurements at $t+1$, $t$, and $t-1$
as a new vector of measurement inputs and the new target will be the ground
truth at $t+1$. It keeps doing this for a predefined sequence of times $m$
during the training phase. During the testing phase, the process is repeated
but with no adjustment for weights and using data that never was used in
training. Keep in mind that every input and the target value is a 2-tuple
vector of $X_{Cartesian}$ and $Y_{Cartesian}$ location values. Originally
Stone Soup software provides the range and the bearing values, as mentioned
earlier.

##  3  Simulations and Results: KF and NN

Below is depicted the flow of simulations and testing processes shown in the
flowchart of Figure .2. The process uses both the neural network and the
Extended Kalman filter (EKF) model that is provided by stone soup open source
software https://stonesoup.readthedocs.io/en/v0.1b5/ in order to generate
input measurements and EKF tracking results. The tracked variables are only
the locations (converted to $X_{cartesian}$ and $Y_{cartesian}$) for drone
(flying object) and in 2-D.

###  3.1 Training & Testing

Through the sliding window method, the number of sequences used in training
and testing is set to $m=5$ so data is read such that every 3 measurement
points ($n=3)$ (a point has two tuples of X and Y) is mapped to one ground
truth point (another X and Y tuple). You can say that memory of length $3$ is
implicitly used here, see Figure 1, please. Initially, the first 4 rows are
used for training and the $5th$ is kept for testing and not used in training.
Each row contains 3 tuples for training and the last tuple as a target (ground
truth). The number of layers, number of nodes, and the learning rate are
chosen empirically. Weights are initially set randomly from a standard normal
distribution and the biases are set to zero. Cost is traced every 100
iterations.

Table 2:  Five rows of the training data (each row is 3 Measurements + 1
Ground truth)

X(M).1 | Y(M).1 | X(M).2 | Y(M).2 | X(M).3 | Y(M).3 | X(G) | Y(G)
---|---|---|---|---|---|---|---
2.2218 | -0.95 | 1.549 | -2.188 | 4.1186 | 11.127 | 2.2755 | 2.0307
1.549 | -2.188 | 4.1186 | 11.127 | 4.5342 | 4.7703 | 3.156 | 3.2077
4.1186 | 11.127 | 4.5342 | 4.7703 | 4.473 | 4.422 | 4.1115 | 4.6815
4.5342 | 4.7703 | 4.473 | 4.422 | 5.8321 | 9.1323 | 5.0689 | 6.4241
4.473 | 4.422 | 5.8321 | 9.1323 | 4.6718 | 9.0161 | 6.1341 | 8.2916

The inputs of the training model are:

  * •

Number of hidden layers: 3

  * •

Number of nodes in each hidden layer: 7 & 5 & 4

  * •

Number of iterations: 20000

  * •

Learning rate: 0.0001

After 20000 iterations, the cost drops to 0.017( See Figure 3, please):

Figure 3: Cost Plot

The predicted output of the training set is shown in Table III below:

Table 3: Training Set NN Output (predicted track) x | 2.18 | 3.036 | 4.09 | 5.19
---|---|---|---|---
y | 2.11 | 3.30 | 4.69 | 6.33

Plotting the neural network track obtained from the training set output along
with the ground truth and EKF track yields tracks in Figure 4. The testing is
implemented on measurement values

$(x,y)$

that was never used in training. Usually, around 70% of the measurement values
are used in training, and the rest are used in testing :

Figure 4: NN training output, (Ground truth) GT, measurements,& EKF

The mean squared error for the training set is: 0.01
The Euclidean distance between the neural network output for the training set
and the ground truth is: 0.465
The Euclidean distance between the Extended Kalman Filter track corresponding
to the training set (measurements) and the ground truth is: 7.465

The output of the testing set is shown in TABLE IV:

Table 4: Testing Set NN Output after using the fifth row in Table II as input test x | 4.96
---|---
y | 5.84

The mean squared error for the testing set is: 1.38
The Euclidean distance between the neural network output of the testing set
and the ground truth is 2.722
The Euclidean distance between the Extended Kalman Filter track corresponding
to the testing set and the ground truth is 1.364

The neural network is then used to train 15 rows of data, and 4 rows are kept
for testing(see Tables V and VI). The ground truth is provided below for
comparison:

Table 5: Training Set GT

X(G) | 2.28 | 3.16 | 4.11 | 5.07 | 6.13 | 7.34 | 8.71 | 9.66 | 10.23 | 10.91 | 11.44 | 12.087 | 12.866 | 13.66 | 14.66
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Y(G) | 2.03 | 3.21 | 4.68 | 6.42 | 8.29 | 10.21 | 12.64 | 15.08 | 17.46 | 19.92 | 22.44 | 25.23 | 28.32 | 31.65 | 34.96

Table 6: Testing Set GT

X(G) | 15.76 | 16.99 | 18.30 | 19.72
---|---|---|---|---
Y(G) | 38.33 | 41.65 | 44.95 | 48.07

The inputs of the training model are:

  * •

Number of hidden layers: 3

  * •

Number of nodes in each hidden layer: 7 & 5 & 4

  * •

Number of iterations: 2000000

  * •

Learning rate: 0.0001

After 2000000 iterations, the cost drops to 0.097. The predicted output of the
training set is (Table VII):

Table 7: Training Set NN Output

X | 2.33 | 3.25 | 4.29 | 5.19 | 5.99 | 7.21 | 8.49 | 9.38 | 10.79 | 10.55 | 11.21 | 12.39 | 13.09 | 13.88 | 14.63
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Y | 2.04 | 3.2 | 4.69 | 6.51 | 8.19 | 10.21 | 12.49 | 14.82 | 17.47 | 19.84 | 23.03 | 25.21 | 28.88 | 31.68 | 34.29

Plotting the neural network track obtained from the training set output along
with the ground truth and EKF track yields (Figure. 5):

Figure 5: NN Training Output, GT, & EKF

The mean squared error for the training set is: 0.015
The neural network predictions of the training set are visually and by
inspection close to the ground truth values.
It is worth informing that in all previous simulations, the default covariance
values for the bearing angle and the range have the following values(
$\sigma_{11}$ is for the bearing, and $\sigma_{22}$ is for the range):

$\begin{bmatrix}0.00349066&0\\\ 0&0.5\end{bmatrix}$

##  4 Experimenting With Different Noise Covariances

Unlike the Kalman filter, the neural network is not provided with the object’s
motion model, but only with measurements. It is therefore plausible to assume
that measurements with lower covariance, thus closer to the ground truth
values, would result in better testing outcomes for the neural network. The
measurements noise covariance matrix is currently following the stone soap
default setting mentioned above.

###  4.1 Setting the Covariance to 0.001

Setting both values on the diagonal of the measurements covariance matrix to
0.001 (Figure 6):

$\begin{bmatrix}0.001&0\\\ 0&0.001\end{bmatrix}$

Measurements are now less scattered around the ground truth. Ground truth
values remain the same. Data, as previously explained, is divided into 15 rows
for training and 4 for testing after implementing the sliding window.
Repeating the training process with the following inputs:

  * •

Number of hidden layers: 3

  * •

Number of nodes in each hidden layer: 7 & 5 & 4

  * •

Number of iterations: 2000000

  * •

Learning rate: 0.0001

After 2000000 iterations, the cost drops to 0.0102. The predicted output of
the training set is (shown in Table VIII):

Table 8: Training Set NN Output

X | 2.23 | 3.24 | 4.07 | 5.05 | 6.22 | 7.34 | 8.69 | 9.66 | 10.31 | 10.89 | 11.4 | 11.93 | 12.81 | 13.92 | 14.6
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Y | 2.06 | 3.15 | 4.73 | 6.42 | 8.25 | 10.21 | 12.67 | 15.06 | 17.52 | 19.9 | 22.36 | 25.35 | 28.29 | 31.61 | 34.97

Plotting the neural network track obtained from the training set output along
with the ground truth and EKF track yields (Figure 6):

Figure 6: NN Training Output, GT, & EKF

The mean squared error for the training set is: 0.0078
The neural network predictions of the training set are very close to the
ground truth values. Taking a closer look at the plot (Figure. 6). The sum of
the Euclidean distances between the neural network output of the training set
and the ground truth is: 1.142
The sum of the Euclidean distances between the Extended Kalman Filter track
corresponding to the training set and the ground truth is: 9.167.
The results of testing on similar cases, Figures 8, 9, and 10 below show
testing results (tracks generated by the neural network) for two different
cases and for 10 input measurement values (measurements values were not used
in training).

Figure 7: Testing with Ground Truth-1 and Cov-1

Figure 8: Testing with Ground Truth-2 and Cov-1

Figure 9: Testing with Ground Truth-4 and Cov-1

More general and comprehensive results can be summarized in Table IX and
depicted in Figure 10. Results show outcomes of testing with three different
ground truth tracks and five different pairs of covariance values that were
used in generating the measurements values as follows:
cov1=(0.01,0.3),cov2=(0.11,0.5),cov3(0.15,0.7), cov4=(0.2,1) and
cov5=(0.3,1.5). The first covariance value is for the bearing angle and it is
in radians and the second covariance value is for the range and it is in
meters. The sum of Euclidean distances (RMS) values is calculated as a measure
of quality. The RMS measures the difference between the estimated track values
(generated by the Neural Network or the EKF) and the ground truth calculated
over 10 successive inputs of measurements (again, those measurements values
were not used in training):

Table IX: Summary of Results

Figure 10: Histogram for the values in Table IX

##  5 Summary

This work proposes a sliding window neural network for estimating the tracks
of flying objects. The measurements were simulated by stone soup software
based on a Gaussian Noise model. Unlike the Kalman Filter, the proposed neural
network does not use a motion model, as in some cases it is difficult to
decide or select a proper motion model to use. Moreover, linearization of the
motion models can be very costly. Besides, the neural network, once trained,
does not require the complex prediction/update process that the KFs have. The
training was done based on several given ground truth tracks, testing was
based on new samples of measurements. Once training is finished, the inference
(testing) is usually fast and efficient, it is simply a single phase of
multiplications and additions. The results showed that the proposed (no motion
model) system could achieve comparable results to the Extended Kalman Filter
(EKF) (which is an advanced form of Kalman ). Results were close to the (EKF)
results when using low measurements noise covariance values and for most
ground truth tracks. The proposed system can be used in cases when Signal-to-
Noise ratios are high and when it is difficult to find a proper motion model
to use by the EKF. Future work will include incorporating a motion model
within the neural tracker in order to achieve better tracking accuracy and
more reliable performance. It is clear also that proposing another neural
network to estimate the states of the object model will increase the accuracy
of the estimated tracks. In this study, we used only a measurement model to
estimate the tracks with the use of the neural network. Future work will
include a dual neural network (one for the measurements and the other one for
the object dynamics) for learning the estimation of the tracks.

###### Acknowledgements.

This work was partially Funded by the Thales Chair of Excellence project,
Sorbonne Center of Artificial Intelligence, Sorbonne University, Abu Dhabi,
UAE.

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Haya Ejjawihaya.jpeg is a student at the department of Physics and Sciences,
Sorbonne University, Abu Dhabi, and was an intern with Thales/SCAI. Her
interests are in machine learning and drone recognition and tracking. Amal El
Fallah Seghrouchniamal.jpg is the Head of Ai Movement, International AI Center
of Morocco, UM6P, Full Professor@Sorbonne Univ, Paris, France, LIP6 and CNRS,
Member of the COMEST, UNESCO - and General Chair of AAMAS 2020 (Auckland -
NZ). She is the author of more than 100 publications and supervised more than
33 Ph.D. students. Her research interests are in multi-agent systems, AI
systems, and ambient intelligence. Frederic BarbarescoFred.png is a senior
THALES Expert in Artificial Intelligence at the Technical Department of THALES
Land and Air Systems. SMART SENSORS Segment Leader for the THALES Corporate
Technical Department (Key Technology Domain ”Processing, Control, and
Cognition”). THALES representative at the AI Expert Group of ASD (AeroSpace
and Defense Industries Association of Europe). 2014 Aymée Poirson Prize of the
French Academy of Science for the application of science to industry. Ampère
Medal, Emeritus Member of the SEE, and President of the SEE ISIC club
”Information and Communication Systems Engineering”. He is a French MC
representative of European COST Calista. General Chair of several elite and
highly specialized conferences. Guest Editors of Special Issues ”Lie Group
Machine Learning and Lie Group Structure Preserving Integrators”. Author of
more than 200 scientific publications and more than 20 patents.

Raed Abu ZitarRaed.png obtained his Ph.D. in computer engineering from Wayne
State University, USA, in 1993. He is an IEEE member and a senior research
scientist in the context of the Thales Endowed Chair of Excellence, SCAI,
Sorbonne University, Abu Dhabi, UAE. He has been with SCAI/Sorbonne
University, Abu Dhabi, since Feb-2021. He has more than 75 publications and is
engaged in many funded projects along with graduate students’ supervision. His
research interests are in machine learning, neural networks, cyber security,
and stochastic processing.