Content uploaded by Lokukaluge Prasad Perera
Author content
All content in this area was uploaded by Lokukaluge Prasad Perera on Aug 07, 2021
Content may be subject to copyright.
Content uploaded by Lokukaluge Prasad Perera
Author content
All content in this area was uploaded by Lokukaluge Prasad Perera on Aug 13, 2020
Content may be subject to copyright.
Ocean Engineering 209 (2020) 107478
0029-8018/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Contents lists available at ScienceDirect
Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
A dual linear autoencoder approach for vessel trajectory prediction using
historical AIS data
Brian Murray ∗, Lokukaluge Prasad Perera
UiT The Arctic University of Norway, Tromsø, Norway
ARTICLE INFO
Keywords:
Maritime situation awareness
Trajectory prediction
Collision avoidance
Machine learning
Autoencoder
AIS
ABSTRACT
Advances in artificial intelligence are driving the development of intelligent transportation systems, with the
purpose of enhancing the safety and efficiency of such systems. One of the most important aspects of maritime
safety is effective collision avoidance. In this study, a novel dual linear autoencoder approach is suggested to
predict the future trajectory of a selected vessel. Such predictions can serve as a decision support tool to
evaluate the future risk of ship collisions. Inspired by generative models, the method suggests to predict the
future trajectory of a vessel based on historical AIS data. Using unsupervised learning to facilitate trajectory
clustering and classification, the method utilizes a cluster of historical AIS trajectories to predict the trajectory
of a selected vessel. Similar methods predict future states iteratively, where states are dependent upon the
prior predictions. The method in this study, however, suggests predicting an entire trajectory, where all states
are predicted jointly. Further, the method estimates a latent distribution of the possible future trajectories of
the selected vessel. By sampling from this distribution, multiple trajectories are predicted. The uncertainties
of the predicted vessel positions are also quantified in this study.
1. Introduction
As more advanced technologies are introduced into transportation
systems, the opportunity to enhance the safety of these systems in-
creases. Increased computational power in conjunction with advances
in artificial intelligence, and the ubiquity of sensor data, allow for
new methods to be implemented across a wide number of sectors.
Some argue that an industrial revolution is taking place, naming it
Industry 4.0 (Hermann et al.,2016). The automotive industry is an
example of a sector in which such technological advances are embraced
and integrated into existing systems. The shipping industry, however,
has historically been more conservative in adopting new technologies,
often relying on older, but proven systems. Nonetheless, advances are
being made, with some arguing that shipping is also undergoing a
technological revolution, Shipping 4.0 (Rødseth et al.,2015).
1.1. Maritime situation awareness
An essential aspect of Shipping 4.0 is arguably implementing mod-
ern technologies to enhance the safety of maritime operations. Effective
collision avoidance strategies are an integral part of maintaining safe
operations. The efficacy of such strategies relies on the degree of
situation awareness of the navigator. Situation awareness was defined
in Endsley et al. (2003) as ‘‘being aware of what is happening around
∗Corresponding author.
E-mail address: brian.murray@uit.no (B. Murray).
you and understanding what that information means to you now and in the
future’’.Endsley (1995) separated situation awareness into three levels:
1. Perception of the elements in the current situation
2. Comprehension of the current situation
3. Projection of the future status
Maritime situation awareness largely relates to obstacle detection
and prediction of close-range encounter situations. Such obstacles will
primarily be other vessels, referred to as target vessels in an en-
counter situation. It is, therefore, essential for a navigator to have
an adequate degree of situation awareness in order to conduct effec-
tive collision avoidance maneuvers. A thorough review of collision
avoidance methods can be found in Tam et al. (2009).
The main challenge for a navigator is determining the risk of
collision based on their degree of situation awareness. In light of
Endsley’s definition, one can classify level one situation awareness as
relating to obstacle detection, and level two an evaluation of the current
collision risk. Perera and Guedes Soares (2015) addressed collision
risk detection and quantification techniques with respect to integrating
modern technologies. The study discussed the concept of e-Navigation
as introduced by the International Maritime Organization. e-Navigation
aims to utilize maritime information by electronic means to enhance
https://doi.org/10.1016/j.oceaneng.2020.107478
Received 12 November 2019; Received in revised form 31 March 2020; Accepted 30 April 2020
Ocean Engineering 209 (2020) 107478
2
B. Murray and L.P. Perera
Nomenclature
𝐚Arbitrary AIS Parameter Vector
𝐀Set of AIS Data
𝐛Backward Trajectory Feature Vector
𝐂Data Cluster
𝑑Euclidean Distance
𝐞Eigenvector
𝐄Eigenvector Matrix
𝐟Forward Trajectory Feature Vector
𝐡Latent Feature Vector
𝐇Distribution of Latent Features
𝑘Number of Nearest Neighbors
𝑙Number of Eigenvectors
𝐿Number of Data Points in Selected Trajec-
tory
𝑁Number of Trajectories
𝑀Number of Models in Mixture Model
𝐩Selected Vessel Position Vector
𝐏Distribution of Selected Vessel Position
𝐬Vessel State Vector
𝐒Scatter Matrix
𝑇Time Horizon [min]
𝑣Speed over Ground [m/s]
𝑤Interpolation Weight
𝑥UTM x-coordinate [m]
𝑦UTM y-coordinate [m]
𝜦Eigenvalue Matrix
𝝁Mean Vector
𝜋Prior Distribution
𝜎Standard Deviation
𝜮Covariance Matrix
𝜒Course over Ground [◦]
Subscripts
0 Initial State
𝑏Backward Trajectory
𝑏𝑐 Between-class
𝑤𝑐 Within-class
𝑓Forward Trajectory
𝑖Sample Number
𝑗 𝑗th State
𝑚Model Number in GMM
𝑝Trajectory Prediction
𝑠Selected Vessel
𝛿Maximum Offset
Superscripts
̂Estimated Parameter/State
the safety of maritime operations. It was argued that integrated bridge
systems are the main focus. Here, relevant information relating to ship
navigation should be properly integrated in order to provide decision
support to navigators. As such, Perera and Guedes Soares (2015) argued
that the best navigation tools possible should be available on board the
vessel to aid the navigator in identifying high risk situations. Based
on this risk evaluation, adequate collision avoidance maneuvers can
be conducted that adhere to the COLREGS as outlined in Perera et al.
(2010).
A wide range of technologies are currently adopted to aid in pro-
viding situation awareness to navigators, including radar, conning and
ECDIS (Electronic Chart Display and Information System). Radar sys-
tems facilitated by ARPA (Automatic Radar Plotting Aid) and the ECDIS
are essential in aiding navigators to determine the risk of collision.
Generally, the future state of a target vessel is estimated based on
calculations of constant course and speed values. These estimates can
then be used by the navigator to estimate collision risk parameters
relating to the closest point of approach (CPA), such as the time
(TCPA) and distance (DCPA). Based on this information, a navigator
can make a decision with respect to a potential collision situation.
However, predicting collision situations far in advance, i.e. level three
of Endsley’s situation awareness model, will be the focus area of this
study.
1.2. Vessel trajectory prediction
Predicting ship behavior as in Perera (2017) can provide decision
support to navigators to make appropriate collision avoidance maneu-
vers. Advanced techniques, e.g. Perera et al. (2012), where extended
Kalman filters were utilized to estimate ship trajectories, can further
enhance the situation awareness of navigators. Such methods, however,
are only useful for prediction horizons in the order of seconds to
minutes. As such, they will only aid navigators in cases in which close-
range encounter situations are imminent. As a result, it was suggested
in Perera and Murray (2019) to introduce an advanced ship predictor.
This study focused on methods to provide autonomous vessels with
adequate situation awareness. However, such methods are also relevant
for use in decision support to ship navigators. In this approach, a
local and global scale ship predictor were suggested. At a local scale,
techniques such as those outlined in Perera (2017) can be utilized to
aid in short term trajectory predictions in order to aid in effective
collision avoidance maneuvers once a collision is deemed imminent. On
the global scale however, long term trajectory predictions, on the scale
of 5–30 min, are conducted. Such predictions aim to prevent close-
range encounter situations from occurring at all. Such predictions are,
however, not straight forward, as the future intentions of the vessel are
unknown, and may potentially be complex.
1.2.1. AIS based vessel trajectory prediction
One method to conduct vessel trajectory predictions on a global
scale is to utilize historical AIS (Automatic Identification System) data.
By exploiting AIS data, insight into historical ship behavior can be
gained. Multiple ship parameters relating to historical ship movement
are stored in databases, available for use. Such parameters include the
position, speed and course over ground. Recently, there has been a
significant increase in research into exploiting AIS data for maritime
situation awareness. A number of studies have focused on evaluat-
ing grouping trajectories together to gain insight into maritime pat-
terns. Aarsæther and Moan (2009) for instance utilized computer vision
techniques to group trajectories and subsequently calculate statistics
for each traffic pattern. Zhang et al. (2018) also utilized AIS data via
a data driven approach that compressed and clustered trajectories to
extrapolate the general behavior patterns of vessels traveling along the
same route. Subsequently, given a starting point, the Ant Colony Algo-
rithm was utilized to output an optimal route to the destination. Zhang
and Meng (2019) also presented a data driven method to determine a
probabilistic ship domain based on AIS data. Such ship domains can
subsequently be utilized for collision risk assessment. A comprehensive
review of various methods to exploit AIS data for maritime navigation
was presented in Tu et al. (2017).
Of primary interest for this study, however, is the work done to
predict the future trajectory of a vessel that can be utilized in a global
scale ship predictor. As such, the aforementioned methods are of lim-
ited usefulness. Ristic et al. (2008) utilized a particle filter to predict the
future behavior of vessels using historical AIS data, but the predicted
Ocean Engineering 209 (2020) 107478
3
B. Murray and L.P. Perera
future positions had a large uncertainty associated with them, making
the method of limited use with respect to collision avoidance decisions
and actions. A number of studies also have focused on clustering
historical trajectories, and subsequently classifying a vessel to one of
these groups. Pallotta et al. (2013) for instance presented the TREAD
(Traffic Route Extraction and Anomaly Detection) methodology that
clustered all historical trajectories in a specific region to identify traffic
routes and subsequently classify a partial trajectory to one of these
routes for anomaly detection. The method also addressed assessing
the probability of a position along a route. Pallotta et al. (2014)
further expanded upon the TREAD methodology by predicting the
vessel position along a route using the Ornstein–Uhlenbeck stochastic
process. The TREAD technique, however, clustered entry points, way-
points and stationary points of trajectories within a defined region. In
this respect, the trajectory through the entire region was utilized to
group similar trajectories together. This can result in trajectories with
large differences between sub-trajectories being clustered together. For
predictions in the order of hours, this is not an issue, and the outlined
method is quite effective. For collision avoidance purposes, a higher fi-
delity prediction is required that requires more discrimination between
trajectories. Other studies on clustering and classification include Zhao
and Shi (2019) which clustered trajectories by using dynamic time
warping and the Douglas Peucker algorithm, in addition to Zhou et al.
(2019) which clustered using 𝑘-means, and subsequently classified ship
behavior. Methods relying on dynamic time warping and way-point
based clustering will cluster trajectories based on similar spatial behav-
ior, but be invariant with respect to time. As such, trajectories that have
similar spatial shapes will be grouped together despite various behavior
being observed at different relative times. This may be detrimental to
a subsequent trajectory prediction, in that the clustering capability is
restricted to the shapes of trajectories, irrespective of their duration and
potential differences in sub-trajectories.
Mazzarella et al. (2015) also presented a trajectory prediction ap-
proach using AIS data, via a Bayesian network approach with a particle
filter. This method was designed for predictions in the order of hours,
and as such of limited use with respect to collision avoidance. Other
methods include Hexeberg et al. (2017), where a Single Point Neighbor
Search method was presented based on historical AIS data. The method
does not involve any clustering or classification steps, and as such
suffers when handling branching. Dalsnes et al. (2018) built upon this
work and provided multiple predictions using a prediction tree. This
approach allows for a probability estimate of a future prediction to be
estimated using a Gaussian mixture model. These methods, however,
do not utilize the relationship between data points, as future states
are based solely on the neighborhood of previous states which may
not have any relationship to the prior predicted states. This will have
a negative effect on the accuracy. Rong et al. (2019) also presented
a probabilistic trajectory prediction method using a Gaussian Process
model. This method, in addition to predicting the future position of
a vessel, gave an uncertainty estimate associated with the prediction.
The method had good results for the regular trajectories investigated
off the coast of Portugal, but did not address how to deal with more
complex traffic situations and trajectories, which likely will degrade
the outcome.
1.3. Generative models
The method utilized in this study takes an alternative approach
to those that come before. It is inspired by a field of deep learning
known as generative models (Foster,2019), widely adopted in the field
of machine learning. Such models have recently gained a high degree
of popularity due to the powerful generative ability of deep learning
models. One such general model is the autoencoder. An autoencoder is
a type of neural network, with its most simple form being a multi-layer
perceptron (Bourlard and Kamp,1988). The objective of an autoen-
coder is to reconstruct the data fed into it, essentially copying its input
Fig. 1. Autoencoder architecture with the encoder illustrated by the blue dashed box,
and the decoder by the green dashed box-. (For interpretation of the references to color
in this figure legend, the reader is referred to the web version of this article.)
to its output. Such techniques are, however, not extensively applied in
the maritime domain. Some studies have looked into applying these
approaches in the maritime domain, e.g. Perera and Mo (2018), where
autoencoders were suggested as a tool to compress data to facilitate
more effective maritime data transmission. Autoencoders are consid-
ered to have two parts: an encoder function 𝑓(𝐱)that produces the code,
𝐡, shown in (1), and a decoder function, 𝑔(𝐡), that reconstructs the data
from the code shown in (2). An integral part of an autoencoder is the
internal hidden layer, 𝐡, that represents the code space, often referred
to as the latent representation of the data.
𝐡=𝑓(𝐱)(1)
̂
𝐱=𝑔(𝐡)(2)
For an autoencoder to be useful, it must provide a form of func-
tionality other than mapping the input to the output. Undercomplete
autoencoders (Goodfellow et al.,2016), i.e. where 𝐡has a smaller
dimension than 𝐱, provide a bottleneck in the code space through which
the network can learn a meaningful latent representation of the data.
The mapping function of the input data to the code space, 𝑓(𝐱), can be
thought of as a data compression operation, or parameter reduction.
The encoder strives to create a meaningful latent representation that
preserves as much information as possible, such that the decoder
has adequate information to reconstruct the data. As such, when an
autoencoder is trained on a dataset, it will adapt such that the encoder
preserves the most important information in the dataset.
Traditionally, autoencoders have been utilized for dimensionality
reduction and feature generation (Goodfellow et al.,2016). In this case,
the latent representation can be utilized for data visualization or to
generate more relevant features for further processing. Additionally,
once an autoencoder is trained, data can be compressed and stored.
Subsequently, it can be decoded for later use. Such applications are
often very useful. However, the generative capabilities of autoencoders
have recently also gained interest. Alternatively to encoding and de-
coding the data, one can solely utilize the decoder in order to generate
new data. This is done by sampling a data point in the code space, and
subsequently running a forward pass through the decoder to reconstruct
the data. In this manner, one can interpolate between existing data
points in the code space to generate new samples.
The variational autoencoder (Kingma and Welling,2014;Rezende
et al.,2014) is a popular type of generative model. A variational
autoencoder is a probabilistic version of an autoencoder where the
Ocean Engineering 209 (2020) 107478
4
B. Murray and L.P. Perera
network learns a probability distribution of the reconstructed data
based on a learned distribution over the code. In this manner, there
is a continuous distribution in the code space that can be sampled
from. Kingma and Welling (2014) investigated the use of a variational
autoencoder and presented a figure illustrating generated images from
a variational autoencoder trained on the MNIST dataset of handwritten
digits. The figure illustrates the interpolation of the digits with a 2-
D latent code. Each axis along the figure can be thought of as one
dimension in the latent code. It is evident that as one moves around
within the code space, the digits morph from one digit to another. The
latent representation is able to capture the most important differences
in the data along the respective axes. As such, one can generate a
new image simply by interpolating within the code space generated by
training the autoencoder.
Such generative capabilities can be extended to virtually any
dataset, where an autoencoder is trained, and based on the latent distri-
bution of the data, can generate new data samples from the distribution.
As such, if an autoencoder is trained on a cluster of trajectories, it
should be able to generate a new trajectory by interpolating in the
latent space.
1.4. Contribution
The objective of this study is to provide an architecture that can
support collision avoidance actions by providing situational awareness
to navigators or autonomous agents. As a result, the architecture differs
from that of similar studies with respect to its design. To aid in
situation awareness, a method is suggested to provide a global scale
ship predictor that estimates the future 30 min trajectory of a selected
vessel with a high degree of fidelity. As opposed to a number of other
studies, the approach in this study is designed to run live, i.e. without
any pre-trained models. A ship in any region, given an adequate density
of historical AIS data can, therefore, utilize the developed architecture.
In the suggested approach, relevant historical ship trajectories are ex-
tracted from an AIS database, that represent the possible future 30 min
behavior of a selected vessel. This dataset comprises only relevant data
with respect to the observed state of a selected vessel for the purpose of
trajectory prediction, and as such provides the basis for the remainder
of the prediction methodology. Inherent differences in behavior are
described by these trajectories, which in turn represent the possible
modes of the future 30 min behavior the selected vessel may belong
to. Therefore, the trajectory representation differs from other methods
that evaluate entire trajectories for a region. The representation in this
study provides higher fidelity predictions as a result.
In order to discover clusters of similar trajectories, other approaches
utilize trajectory representations that introduce invariance with respect
to time, e.g. dynamic time warping, or point based techniques using
waypoints. These techniques are effective for clustering trajectories of
similar shapes together. For the purpose of this study however, it is
of interest to discover all possible trajectory modes that represent the
future 30 min behavior of the selected vessel, not just trajectories of
similar shapes for the region. As such, trajectories should not be in-
variant with respect to time. Therefore, by representing each trajectory
by vectors of equal length containing the future 30 min of trajectory
data, the representations will be sensitive to the time at which various
behavior is observed. Such a representation will, therefore, be more
sensitive to modes within primary ship routes. Discovering these modes
will provide a much better basis for a subsequent trajectory prediction
for collision avoidance purposes, as the prediction must be as accurate
as possible. This study suggests to cluster compressed trajectories via
Gaussian Mixture Models to an unspecified number of clusters, each
representing a mode of future behavior, and is shown to have good
performance for the purpose of the study.
Once a selected vessel is classified to a given cluster of historical
AIS behavior, this data is used directly in the dual linear autoen-
coder prediction architecture. This architecture differs significantly
from other methods, which generally predict future states in manner
such that they are predicated upon previous predicted states. In this
study, it is suggested to predict entire trajectories, i.e. all future states
are predicted jointly. A novel trajectory prediction technique inspired
by generative models is, therefore, suggested using a dual linear au-
toencoder approach. In this approach, a latent representation of the
possible future behavior of the selected vessel is calculated. The latent
representation can be viewed as an encoded version of the data. Using
this distribution, the encoded representation of the selected vessel’s
future behavior is estimated by interpolating between the encoded data
points. By decoding the estimate of the latent representation of the
future trajectory, an entire trajectory is predicted by a single matrix
multiplication operation. Other methods predict an average of the
behavior in the cluster, i.e. the average of the distribution, whereas the
method suggested in this study will estimate the most likely sample.
As such, the prediction is discrete, and can provide more accurate
predictions than other methods in which the behavior is averaged out.
The prediction accuracy will also be enhanced for clustering
schemes that are able to identify ship modes with a high degree of
fidelity, as clusters that contain multiple ship modes will result in
the prediction averaging out the behavior between modes due to the
interpolation. As a result, the overall architecture of the study allows
for higher fidelity predictions than other methods. Additionally, the
study provides a method to estimate the distribution of the selected
vessel’s future trajectory latent representation. This is to account for
uncertainty in the estimate, and by decoding samples from this distri-
bution, a region of uncertainty for the predicted position at a given
prediction horizon can be evaluated. The suggested architecture also
utilizes linear autoencoders. Therefore, it allows for fast predictions
as they are facilitated by calculating eigenvectors, and conducting
subsequent matrix multiplications. As such, there is no training of
a deep neural network. This architecture is, therefore, ideal for live
predictions, as the calculations involved in the prediction itself will
be fast. This approach in this study, therefore, provides a method to
conduct live predictions of higher fidelity with respect to collision
avoidance purposes on a global scale than other methods, as well as an
effective method to quantify the uncertainty of the predicted positions.
2. Methodology
In this section, the methodology utilized to predict the future tra-
jectory of a selected vessel is outlined. The objective of the method
is to accurately predict the future trajectory of a selected vessel, and
provide an uncertainty estimate with respect to the predicted positions.
The overall architecture of the method is illustrated in Fig. 2. The
method can be separated into three modules. The first is the trajectory
clustering module, where groupings of similar historical trajectories are
discovered. It is assumed that the future trajectory of a selected vessel
can be inferred based on the historical trajectories of other vessels
in the region. As such, the selected vessel is classified to one of the
discovered clusters in the trajectory classification module. Based on
the cluster of trajectories to which the selected vessel is classified, a
trajectory prediction is conducted in the trajectory prediction module.
This is achieved via a novel dual linear autoencoder approach. In this
approach, two linear autoencoders are utilized. The forward linear au-
toencoder provides a latent representation of the historical trajectories
that can be used to infer the future trajectory of the selected vessel.
The backward linear autoencoder provides a latent representation of
the prior behavior of the historical trajectories. Based on a similarity
measure evaluated in the latent space of the backward linear autoen-
coder, a latent interpolation is conducted to estimate the forward latent
representation of the selected vessel. Subsequently, this estimate can be
decoded, resulting in a trajectory prediction.
Ocean Engineering 209 (2020) 107478
5
B. Murray and L.P. Perera
Fig. 2. Prediction architecture, with the dual linear autoencoder block within the
dashed lines.
2.1. Unsupervised trajectory clustering and classification
In this section, the methodology involved in clustering historical
AIS trajectories and classifying the trajectory of a selected vessel is
outlined. This work in this section builds upon preliminary work de-
scribed in Murray and Perera (2019). The reader is, therefore, referred
to Murray and Perera (2019) for further details. It can be argued that
investigating the historical behavior of vessels in a particular geograph-
ical region can provide insight into the future behavior of a vessel
observed in that region. However, historical vessel trajectories will
have a high degree of variation. This variation is due to the existence of
multiple traffic routes, as well as the characteristics of the vessel with
respect to the speed it will traverse along a given route. It is, therefore,
of interest to identify groupings of similar trajectories, such that specific
traffic behavior can be identified. Once such groupings are identified,
a selected vessel can be classified as belonging to a given group. In
this manner, a subsequent trajectory prediction can be conducted on an
enhanced data set, where the data used for prediction will likely have
a high degree of similarity to that of the selected vessel. This can be
thought of as advanced form of preprocessing of the AIS data, such that
subsequent trajectory predictions will have a higher degree of accuracy.
Grouping such data can be conducted via a technique from the field
of machine learning known as clustering. This is a form of unsupervised
learning, where labels for the data are unavailable. Clustering has as its
goal to discover underlying groupings in the data, i.e. identify clusters
of data. Once the historical vessel trajectories have been clustered, the
observed trajectory can be used to classify the selected vessel to one of
the discovered clusters.
2.1.1. Trajectory extraction
The initial state of a selected vessel is defined in (3). This state
represents the observed parameters of the selected vessel available via
the on-board sensor suite of the own-ship. The parameters in this state
provide the basis for the selection of relevant historical ship trajectories
for a subsequent prediction of the selected vessel’s future trajectory.
𝐬0→[𝑥0, 𝑦0, 𝜒0, 𝑣0](3)
The method first identifies historical AIS data points with a high
degree of similarity to 𝐬0. In essence, this means that it is desirable
to identify ships that were at a similar position, with a similar course
and speed, at some point in history. In order to achieve this, an
initial cluster 𝐂0is created. 𝐂0is defined to be a rectangular cluster
orthogonal to 𝜒0, with a height of 𝛿𝐻parallel to 𝜒0, and a width 𝛿𝑊
orthogonal to 𝜒0.𝑧′is the rotated space with the orthogonal vectors
in the original space as basis vectors. 𝐂0is defined according to the
following equation (Murray and Perera,2019):
𝐂0={𝐚𝑖∈𝐀∶ (|𝑥𝑧′𝑖−𝑥𝑧′0|≤𝛿𝑊∧|𝑦𝑧′𝑖−𝑦𝑧′0|≤𝛿𝐻)
∧(|𝜒𝑖−𝜒0|≤𝜒𝛿∧|𝑣𝑖−𝑣0|≤𝑣𝛿)} (4)
Additionally, data points that do not match the ship type of the
selected vessel are removed. 𝐂0will, however, likely contain multiple
data points from the same trajectory. As such, unique trajectories are
identified, and the most similar point to 𝐬0in each unique trajectory
determined. 𝐂0is then updated by filtering out all data points other
than these most similar points. In this manner, 𝐂0only contains one
data point per trajectory.
Once the initial clustering phase is completed, a forward and back-
ward trajectory extraction operation is conducted. This entails that for
all trajectories in 𝐂0, the forward trajectory from the corresponding
point in 𝐂0is extracted. This can be thought of as the future trajectory
defined in relation to the point in 𝐂0. The length of the extracted
forward trajectory is defined based on the desired prediction horizon,
𝑇𝑝. For instance, if a 30 min prediction is desired, 30 min of the forward
trajectory will be extracted. Similarly, the backward, i.e. past, trajec-
tory from its corresponding point in 𝐂0of a length corresponding to 𝑇𝑏
into the past is extracted. Both the forward and backward trajectories
are subsequently interpolated at 30 s intervals for comparative analysis.
As such, each trajectory will have 𝐿= 2 × 𝑇𝑝entries, where each entry
can be used to compare positions at a given time instance defined from
the origin of the trajectory (see Fig. 3).
2.1.2. Trajectory clustering
One of the objectives of extracting the forward trajectories is to
provide a dataset upon which one can identify possible future routes
that the selected vessel may follow. It is, therefore, desirable to group,
or cluster, these trajectories such that each possible route can be eval-
uated individually, as there may be many possible future routes that
the selected vessel may follow. This is conducted by first generating
Ocean Engineering 209 (2020) 107478
6
B. Murray and L.P. Perera
Fig. 3. 𝐂0illustrated as a rectangular red box. The forward and backward extracted
trajectories illustrated as green and orange lines respectively. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of
this article.)
features for each trajectory by flattening the positional data given in
UTM-coordinates, i.e. [𝑥, 𝑦], into a feature vector.
Clustering is a technique that groups data points based on some
similarity measure, i.e. data points that are closer to one another in
some n-dimensional space are more likely to be considered part of the
same cluster. If the dimensionality of the space is large however, the
clustering algorithm may suffer due to the curse of dimensionality dis-
cussed in Steinbach et al. (2004). One aspect of curse of dimensionality
relates to points getting lost in the space due to large distances between
points with respect to certain dimensions. This can make clustering
in a high dimensional space challenging. Dimensionality reduction is,
therefore, conducted for each trajectory via the Karhunen–Loéve (KL)
transform (Karhunen,1946) in (5), where the dimensionality is reduced
from 2𝐿to 𝑙.
𝐡𝐟=𝐄𝑇𝐟(5)
where 𝐡𝐟∈R𝑙×1,𝐟∈R2𝐿×1 and 𝐄∈R2𝐿×𝑙
𝜮=𝐄𝜦𝐄𝑇(6)
where 𝜮∈R2𝐿×2𝐿and 𝜦∈R2𝐿×2𝐿
The next step is to cluster the forward trajectories. This is conducted
using Gaussian Mixture Model (GMM) clustering via the Expectation
Maximization (EM) algorithm. A Gaussian Mixture Model (Reynolds
et al.,2000) assumes that data is comprised of a mixture of 𝑀different
Gaussian distributions, each with their own mean vector 𝝁𝑚, covariance
matrix 𝜮𝑚and prior distribution 𝝅𝑚. Each data point representing a
forward trajectory will be clustered to the distribution of the highest
probability. The EM algorithm updates the underlying parameters until
a model of best fit is discovered. The assumed number of underlying
distributions, 𝑀, is also varied to discover the most likely mixture. For
more details on GMM clustering of trajectories, see Murray and Perera
(2019).
2.1.3. Trajectory classification
Once the forward trajectories have been clustered, it is desirable
to classify the selected vessel to one of the discovered clusters. One
method to achieve this is to investigate the backward trajectories.
Assuming that the past behavior of the selected vessel is available
for a period corresponding to 𝑇𝑏, one can compare the past behavior
of the selected vessel to the backward trajectories extracted from 𝐂0.
The aforementioned backward and forward trajectories are in fact
one single trajectory, but the forward trajectories are the section cor-
responding to the future behavior, and the backward trajectory the
past behavior. In the classification module, the extracted backward
trajectories are assigned the class labels of their corresponding forward
trajectories, discovered in the clustering module. These are utilized to
classify the observed trajectory of the selected vessel to one of these
classes. In order for the classification process to be as effective as
possible, it is of interest to generate optimal features to represent the
backward trajectories. This is achieved via Linear Discriminant Analysis
(LDA) (Fisher,1936). LDA requires that the data points are labeled,
and as such, the backward trajectories are given the labels of the
corresponding forward trajectories. The transformation is conducted
via (7). Subsequently, a classifier of choice can be utilized to classify
the transformed backward trajectory of the selected vessel to one of the
clusters. This will yield the most likely future route that the selected
vessel will follow. In this study, a 𝑘NN classifier is utilized.
𝐡𝐛=𝐄𝑇𝐛(7)
where 𝐡𝐛∈R𝑙×1,𝐛∈R2𝐿×1 and 𝐄∈R2𝐿×𝑙
𝐒−1
𝐰𝐜𝐒𝐛𝐜 =𝐄𝛬𝐄𝑇(8)
where 𝐒−1
𝐰𝐜𝐒𝐛𝐜 ∈R2𝐿×2𝐿and 𝜦∈R𝑙×𝑙.
2.2. Dual linear autoencoder
This study introduces a novel dual linear autoencoder trajectory
prediction method that is further described in this section. The motiva-
tion is to predict the future trajectory of a selected vessel. The method
is inspired by the generative models addressed in Section 1.3. If one
can create a latent distribution of possible future trajectories, one can
then interpolate between existing trajectories in the latent space, and
generate a new trajectory that corresponds to the selected vessel.
Autoencoders generally have non-linear activation functions. How-
ever, the linear autoencoders investigated in this study do not have
non-linear activation functions in the network, and as such the encoder
and decoder functions will simply be linear transformations of the data.
Consider a 2-layer linear autoencoder as illustrated in Fig. 1. Let the
encoder function be described by (9) and the decoder function by (10).
If the network is trained using the mean squared error shown in (11),
as the loss function 𝐽, the minimum reconstruction error is shown to
be achieved if 𝐕=𝐖and 𝐜=𝝁, where the columns of 𝐖span
the orthonormal basis spanned by the eigenvectors of the covariance
matrix 𝜮of the dataset (Goodfellow et al.,2016). The columns of
𝐖are ordered by the magnitude of their corresponding eigenvalues.
One recognizes that the encoder function 𝑓(𝐱)is in fact the same as
the KL-transform for the case of a linear autoencoder. This allows for
efficient calculations, as the covariance matrix and its corresponding
eigenvectors and eigenvalues can easily be calculated, significantly sav-
ing computation time compared to training a network. The eigenvectors
calculated here capture the directions in which there is the greatest
degree of variation in the data. Data can, therefore, be compressed and
reconstructed as a linear combination of the projections of that data
onto a subspace spanned by the top 𝑙eigenvectors with the largest
eigenvalues.
𝐡=𝑓(𝐱) = 𝐖𝑇(𝐱−𝝁)(9)
̂
𝐱=𝑔(𝐡) = 𝐕𝐡 +𝐜(10)
𝐽=𝐸[‖𝐱−̂
𝐱‖2](11)
The basis of the method is to train two linear autoencoders. All for-
ward trajectories belonging to the class of the selected vessel are input
to the forward linear autoencoder. In the latent representation, i.e. the
code space, one can then interpolate between existing data points, and
in this manner predict the latent representation of the selected vessel’s
future trajectory. If one then runs a forward pass through the decoder,
i.e. (10), one will get a full trajectory prediction, at the cost of a matrix
multiplication operation. One can in theory move about the latent
space and generate new trajectories in a similar manner to the MNIST
Ocean Engineering 209 (2020) 107478
7
B. Murray and L.P. Perera
digits in Kingma and Welling (2014). The underlying distribution of
possible future trajectories would then be visualized, where moving in
one dimension or another represents the most variation in the possible
future trajectories. The interpolation, however, depends on a similarity
measure of the backward trajectories to the backward, i.e. observed,
trajectory of the selected vessel. This is facilitated via the backward
linear autoencoder.
2.2.1. Forward linear autoencoder
The forward linear autoencoder has as its goal to create a meaning-
ful latent representation of the extracted forward trajectories. However,
training an autoencoder on all the forward trajectories will yield a
latent representation that describes the greatest variations in the data,
i.e. between all clusters of trajectories. This may for instance yield
predictions where a data point is interpolated between clusters, and in
fact represents an unrealistic data point that is not part of the original
distribution. If one, however, considers solely the cluster of trajectories
that the selected vessel has been classified to, one now has a subset of
trajectories that are highly similar to each other, where interpolation
between points should be meaningful. As such, training an autoencoder
on this subset of data will allow it to learn a latent representation that
describes this specific cluster. Decoding a data point from this latent
representation will, therefore, yield a trajectory prediction of higher
fidelity. The encoder and decoder functions are shown in (12) and
(13) respectively, where 𝐄𝑓is the matrix of the subset of the top 𝑙
eigenvectors of the covariance matrix of the forward trajectories.
𝐡𝑓=𝐄𝑇
𝑓𝐟(12)
̂
𝐟=𝐄𝑓𝐡𝑓(13)
2.2.2. Backward linear autoencoder
The success of the trajectory prediction technique relies on the
interpolation in the latent space of the forward linear autoencoder.
Given that the future trajectory is unknown, one must infer the latent
representation of the selected vessel in the forward latent representa-
tion. It is, therefore, suggested to investigate the backward trajectories
of the classified cluster in comparison to the backward trajectory of
the selected vessel. By identifying the degree of similarity between all
backward trajectories in the cluster, and the backward trajectory of the
selected vessel, one can interpolate in the latent space of the forward
linear autoencoder, using the similarity of the backward trajectories as
weights.
It is suggested in this study to utilize a linear autoencoder to
evaluate the similarity. In the same manner as the forward linear
autoencoder, the backward linear autoencoder will learn a meaningful
latent representation that describes the variation in the underlying
trajectory data. In this lower dimensional latent space, the distance
from the encoded selected vessel trajectory to all other trajectories
can be measured. Conducting such a similarity measure in this space
will yield better results due to the same challenges relating to curse of
dimensionality (Steinbach et al.,2004) as those addressed in 2.1.2. The
encoder and decoder functions are shown in (14) and (15) respectively,
where 𝐄𝑏is the matrix of the subset of top 𝑙eigenvectors of the
covariance matrix of the backward trajectories.
𝐡𝑏=𝐄𝑇
𝑏𝐛(14)
̂
𝐛=𝐄𝑏𝐡𝑏(15)
2.2.3. Latent interpolation
Since there is no explicit mapping function from the latent space of
the backward autoencoder to the latent space of the forward autoen-
coder, a similarity-based mapping approach is suggested. The architec-
ture of the suggested method is visualized in Fig. 4. The figure shows
how the backward trajectories are mapped to a latent representation,
𝐡𝑏, in orange, as are the forward trajectories in green to the latent
space 𝐡𝑓.𝐞represents the coordinate systems of the latent spaces.
The backward trajectory of the selected vessel is illustrated as the
solid red line, and represents the information available of the past
behavior of the selected vessel. This information is then encoded in the
backward latent representation, 𝐡𝑏,𝑠, as the red data point. The goal of
the mapping operation is to map to the corresponding red data point
in the latent representation of the forward trajectories.
The mapping function can be considered an interpolation between
the data points of the encoded forward trajectories. The similarity
between the encoded backward trajectory of the selected vessel 𝐡𝑏,𝑠,
and all the backward trajectories is calculated as the Euclidean distance
according to (16). One common form of interpolation for multivariate
data is inverse distance weighting. An interpolation scheme is presented
in Shepard (1968) with a weighting function according to (17), and
the interpolated value calculated according to (18). The equation in-
terpolates within a neighborhood, such that the 𝑘nearest data points
are found, and the interpolated value is calculated on a subset of
neighboring data. In this manner, the interpolated value is not as
affected by outliers, and, therefore, more likely to be closer to the true
value.
𝑑𝑖=‖𝐡𝑏,𝑖 −𝐡𝑏,𝑠‖(16)
𝑤𝑖=1
𝑑𝑖
(17)
̂
𝐡𝑓,𝑠 =∑𝑘
𝑖=1 𝑤𝑖𝐡𝑓,𝑖
∑𝑘
𝑖=1 𝑤𝑖
(18)
2.2.4. Decoded trajectory prediction
Subsequent to the latent interpolation operation, the future trajec-
tory of the selected vessel can be decoded, i.e. predicted, according to
(19). Once this is completed, ̂
𝐟𝑠must be reshaped to a matrix containing
the spatial data (𝑥, 𝑦) as its columns. The prediction is subsequently
updated such that the offset between the true initial position (𝑥0, 𝑦0)
and the predicted initial position (̂𝑥0, ̂𝑦0)is subtracted from all the
entries of the prediction to account for minor offsets that occur due
to the approximation inherent in the latent interpolation. This yields a
trajectory prediction for the selected vessel at 30 s intervals, up to the
desired prediction horizon, 𝑇𝑝. One can evaluate each row of the matrix
as the predicted vessel position, ̂
𝐩𝑗, in each vessel state, 𝐬𝑗, where each
state is separated by 30 s.
̂
𝐟𝑠=𝐄𝑓̂
𝐡𝑓,𝑠 (19)
2.2.5. Uncertainty estimate of predicted position
The trajectory prediction gives a single prediction. However, the
outlined method does not give a measure of uncertainty related to
the predicted position at each time interval. A method is therefore
suggested to achieve this utilizing the linear autoencoder architecture
previously introduced. Some uncertainty can be attributed to the re-
construction loss that results from reducing the dimensionality in the
autoencoders, but the primary source is the uncertainty associated with
the latent interpolation.
It is, therefore, suggested to create a distribution in the latent space
of the encoded forward trajectories, i.e. 𝐇𝑓,𝑠 , that can account for
some of the interpolation error. If one considers the neighborhood of
̂
𝐡𝑓,𝑠 , one can investigate the uncertainty with respect to the 𝑘nearest
neighbors in the latent representation of the backward trajectories,
𝐡𝑏. The method suggests to assume that ̂
𝐡𝑓,𝑠 is the mean of a normal
distribution according to (20), with a weighted unbiased covariance
according to (21). These weights correspond to those in (17). In this
manner, the distribution will reflect the relevant importance of each
latent forward trajectory representation, based on their weights from
the backward trajectory similarity measure.
𝐇𝑓,𝑠 ∼(̂
𝐡𝑓,𝑠 ,𝜮ℎ,𝑠 )(20)
𝜮ℎ,𝑠 =∑𝑘
𝑖=1 𝑤𝑖(𝐡𝑓,𝑖 −̂
𝐡𝑓,𝑠 )(𝐡𝑓 ,𝑖 −̂
𝐡𝑓,𝑠 )𝑇
∑𝑘
𝑖=1 𝑤𝑖−∑𝑘
𝑖=1 𝑤2
𝑖
∑𝑘
𝑖=1 𝑤𝑖
(21)
Ocean Engineering 209 (2020) 107478
8
B. Murray and L.P. Perera
Fig. 4. Illustration of the latent representations in the dual linear autoencoder architecture. The interpolation between the known latent representation of the backward trajectory
of the selected vessel to the unknown forward latent representation is also illustrated. (For interpretation of the references to color in this figure legend, the reader is referred to
the web version of this article.)
This, however, only yields a form of uncertainty with respect to the
latent representation of the selected vessel’s forward trajectory. What
is of interest, however, is the uncertainty of the predicted position at
various vessel states. To achieve this, it is suggested to run a Monte
Carlo simulation (Raychaudhuri,2008) that samples from the latent
normal distribution in (20) to approximate the distribution of the
trajectory predictions. Each sample from 𝐇𝑓,𝑠 is decoded according to
(19), yielding a full trajectory prediction. In the same manner as in
Section 2.2.4, the sampled predictions are also updated based on the
offset of the true and predicted initial positions. This correction will
be greater for samples further away from the true value of 𝐡𝑓,𝑠 , but
it is assumed to have limited effect on the predictions with respect
to estimating the uncertainty. After the samples are decoded, the
distribution of the decoded trajectory positions can be evaluated at
each time instance. This can be viewed as the distribution 𝐏𝑗of the
predicted position ̂
𝐩𝑗for each vessel state 𝐬𝑗, where each state is that
of the selected vessel at 30 s intervals. The distribution of the position in
each state can further be assumed to be normally distributed according
to (22), where the mean and covariance are calculated based on the
sampled predictions. As such, uncertainty measures can be calculated
with respect to the standard deviation of the distribution.
𝐏𝑗∼(𝝁𝑗,𝜮𝑗)(22)
3. Results and discussion
To evaluate the method, 100 random data points were selected
from a dataset of historical AIS data in the region surrounding the
city of Tromsø, Norway. The dataset corresponds to that collected from
January 1st, 2017 to January 1st, 2018. Each data point represents a
selected vessel state, that will be initialized as the initial state, 𝐬0, of that
vessel. The aforementioned trajectory prediction methodology is then
utilized to predict the future 30 min of each selected vessel’s trajectory,
such as to evaluate the performance of the method. The true future
trajectories of the selected vessels can be thought of as the test dataset
for each respective prediction. The remainder of the AIS data is then
the training dataset utilized to conduct the predictions. In this manner,
the method predicts the future trajectory of 100 different vessels, and
the accuracy can be evaluated based on the true trajectory of the vessel.
A value of 𝑙= 3 is utilized for the latent representation of both 𝐡𝑏and
𝐡𝑓. Additionally, the 100 most similar vessels to each selected vessel,
i.e. 𝑘= 100, is utilized for the latent interpolation.
3.1. Classification accuracy
The input to the trajectory prediction module is the set of extracted
trajectories corresponding to the output of the classification module.
As such, the method relies on the accuracy of this classification, as
an incorrect classification will result in a prediction with respect to
a cluster of ship behavior that does not match the selected vessel. In
this study, a value of 𝑘= 7 was utilized for the 𝑘NN classifier. For
the results presented in this section, 67% of the selected vessels were
classified correctly.
In many cases, however, the incorrect classification can be at-
tributed an incorrect behavior mode, i.e. cluster. These modes can be
along the correct route, but may for instance traverse further to one
side of the lane, or have variations in the speed profile along the
route. Predictions with respect to these modes, despite being incorrect,
can nonetheless result in reasonably accurate predictions. This is due
to the clustering algorithm identifying multiple modes that are quite
similar. One should note that this can be seen as a situation where some
data clusters can overlap each other. As such, the 33% of incorrectly
classified cases in this study likely includes many cases in which the
selected vessel was classified to an incorrect mode along the correct
route, i.e. a similar trajectory mode.
Additionally, the success of the classification depends on the com-
plexity of the discovered clusters. It is on the one hand desirable
for the clustering algorithm to discover as many trajectory modes as
Ocean Engineering 209 (2020) 107478
9
B. Murray and L.P. Perera
Fig. 5. Trajectory prediction with correct classification. Contours are illustrated for
𝑇𝑃= 30. (For interpretation of the references to color in this figure legend, the reader
is referred to the web version of this article.)
possible, as this can enhance the accuracy of the subsequent trajectory
prediction. On the other hand, an increase in the number of clusters
will provide a more difficult classification task. This will be further
complicated due to overlapping clusters as mentioned previously. Each
cluster is multi-dimensional, and classifying in this space can be chal-
lenging. However, the focus of the study is the dual linear autoencoder
prediction technique.
It should be noted that the trajectory prediction methodology utiliz-
ing a dual linear autoencoder as described in this study can be utilized
based on any previous clustering and classification technique. However,
it does require that trajectories are extracted utilizing the methods
described in Section 2.1.1, such that trajectories can be encoded and
decoded properly. This trajectory extraction process can, however, also
be conducted after an alternative clustering and classification regime
has been utilized.
3.2. Trajectory prediction
Fig. 5 illustrates an example of a trajectory prediction for one of the
randomly selected vessels in the dataset. All presented predictions are
evaluated with 𝐩0as the origin of the coordinate system to more easily
evaluate the distances involved. In the case of Fig. 5, the algorithm
classified the selected vessel to the correct cluster of trajectories. The
green dotted line represents the predicted trajectory of the selected
vessel, and the red dotted line represents the true trajectory of the
selected vessel. It is clear that for this case, the trajectory prediction
was quite accurate. Additionally, an estimate of the uncertainty of the
position at a 30 min prediction horizon is illustrated. Each black dot
illustrated represents a decoded sample from the normal distribution of
𝐇𝑓,𝑠 in (20). Utilizing these predictions, a normal distribution 𝐏𝑗, was
fit to the predicted positions for each state according to (22). Based
on this distribution, the 1𝜎,2𝜎and 3𝜎contours could be evaluated,
and are visualized in the figure. Such contours can be evaluated at
any prediction horizon, but only the results for the 30 min prediction
horizons are illustrated in this section. Fig. 6 shows a prediction of
a more complex trajectory, showing that the method is also able to
successfully reconstruct more complex trajectories.
In certain cases, however, the classification is incorrect. This, in
some cases, can result in a vessel prediction along an incorrect route,
resulting in a degree of error with respect to the predicted position. An
example of such a case is illustrated in Fig. 7.
Fig. 6. Prediction of complex trajectory. Contours are illustrated for 𝑇𝑃= 30.
Fig. 7. Trajectory prediction with incorrect classification. Contours are illustrated for
𝑇𝑃= 30.
3.3. Prediction accuracy
The performance of the method is most effectively measured based
on the accuracy of the prediction with respect to the true vessel
position. The predicted position error is calculated as the distance from
the mean, 𝝁𝑗, of the distribution 𝐏𝑗, and the true position in that state,
i.e. 𝐩𝑗. Additionally, the error is presented as a percentage of the true
distance traveled by the selected vessel. This is due to various vessels
having traveled different distances during the course of 30 min. In this
manner, one can compare the error irrespective of the distance traveled.
Fig. 8 illustrates the median error of all 100 predictions as a function
of time, i.e. the desired prediction horizon. The overall error for all
selected vessels is evaluated, where the median error for a 30 min
prediction is found to be 2.5%. If one looks only at the vessels that were
incorrectly classified, it is evident that they result in a higher degree
of error, where the median error at a prediction horizon of 30 min is
9.6% of the distance traveled. If one solely investigates the correctly
classified vessels, however, the accuracy of the prediction increases
significantly, with a median position error of 1.6% for a prediction
horizon of 30 min. This illustrates the importance of correctly classi-
fying the selected vessel, as the predictions are discrete with respect
to each class of trajectories. An incorrect classification results in the
prediction being conducted on a cluster corresponding to a different
mode of ship behavior than that of the selected vessel. Fig. 9 illustrates
a box plot of the positional error at 5 min intervals for the correctly
classified vessels. The green bars correspond to the median values in
Fig. 8. It is clear that the variance of the error increases as a function of
Ocean Engineering 209 (2020) 107478
10
B. Murray and L.P. Perera
Fig. 8. Median position error of 100 randomly selected vessels as a function of the
prediction horizon. (For interpretation of the references to color in this figure legend,
the reader is referred to the web version of this article.)
Fig. 9. Box plot of error for correctly classified vessels.
time. Generally, it appears that the method has good performance when
vessels are correctly classified for prediction horizons up to 30 min.
3.4. Position uncertainty estimate
As previously mentioned, a normal distribution of the predicted
position 𝐏𝑗in each state is evaluated according to (22). The resultant
1𝜎,2𝜎and 3𝜎contours can be utilized to give a measure of uncertainty
relating to the prediction in each state. It is desirable for this uncer-
tainty to be as small as possible whilst still capturing the behavior of
the selected vessel. If the uncertainty is too small, however, it may
not include the true future position of the selected vessel. Allowing
for too much uncertainty on the other hand, is not desirable either.
It is conceivable to extend the region of uncertainty such that the true
future position of the selected vessel will always be included within
the contours. In such cases, however, the usefulness with respect to
maritime situation awareness will be degraded, as there will be a risk
of collision for a very large area. This increases the likelihood of the
navigator needing to take action in cases where the true risk of collision
is low.
The degree of uncertainty is dependent upon two factors: the power
of the clustering algorithm, and the number of vessels included in the
latent interpolation. If the clustering algorithm is able to discover a
group of trajectories with very specific behavior, the uncertainty of
the prediction will decrease as the variance of the behavior within the
Fig. 10. Mean error for values of 𝑘used in interpolation.
cluster is limited. In essence, all the historical trajectories utilized in the
prediction are almost the same, causing all predictions to have similar
values when sampling from the latent space. Similarly, if one only
selects a limited number of vessels to evaluate the latent representation
of the selected vessel in (20), the prediction will be restricted to
the behavior of these historical trajectories. Increasing the number of
trajectories will increase the variance of the behavior, and contribute to
a larger region of uncertainty. The effect of this was investigated, where
the number of similar vessels, 𝑘, utilized in (20), was varied and the
position error evaluated as a percentage of the distance traveled in the
same manner as described in Section 3.3.Fig. 10 illustrates the mean
error for various values of 𝑘as a function of the prediction horizon for
the 100 randomly selected vessels. It is evident that increasing the value
of 𝑘contributes to an increase in error, but that this error converges as
𝑘increases.
It should be noted that the probability contours solely relate to the
probability of the predicted future position, and as such are entirely
dependent upon the model developed in this study. They provide a
measure of uncertainty with respect to the predicted positions, where
the true position should fall within the region enclosed by the contours.
The predictions are based on historically similar vessels, whose behav-
ior do not necessarily match that of the selected vessel. The assumption
that the future trajectory of a selected vessel depends on its past
trajectory is, therefore, a limitation of the method. The uncertainty of
the predictions can, therefore, be thought of as describing the variance
of the historical behavior, where it is likely that the vessel will fall
somewhere within the specified region. The method is designed to
identify the most similar trajectories. However, the data may be dom-
inated by specific vessel behavior that has a higher frequency. If more
similar historical trajectories have a lower frequency, the data will be
dominated by the less similar trajectories of the highest frequency. This
effect will, however, be somewhat ameliorated due to the weights in
(17), as the more similar trajectories will have higher weights when
calculating ̂
𝐡𝑓,𝑠 and 𝜮ℎ,𝑠 .
The percentage of the correctly classified vessels whose true position
after 30 min was within the regions bounded by the corresponding 𝜎-
contours was also investigated. This was conducted in order to evaluate
the uncertainty measure’s ability to capture the true position of the
selected vessel. The results are shown in Table 1. The values are
estimated such that the 3𝜎contour includes the points inside the 2𝜎
contour which again includes the points inside the 1𝜎contour. 75% of
all true vessel positions were captured by the 3𝜎contour for the tested
selected vessels. Cases in which the true positions did not reside within
the contours were, therefore, investigated.
Fig. 11 illustrates one such case. The prediction appears to be
quite accurate, with the true and predicted trajectories nearly exactly
Ocean Engineering 209 (2020) 107478
11
B. Murray and L.P. Perera
Fig. 11. Prediction of ferry trajectory. 𝑇𝑃= 30.
Fig. 12. Uncertainty contours for ferry trajectory prediction.
aligned. It is evident, however, that the uncertainty contours are not
visible. Fig. 12 illustrates a close up of the predicted position after
30 min. Here one can see that the predicted final position, illustrated
as the largest red dot, falls outside the uncertainty contours. When
investigating the scale involved, one can see that the true and predicted
positions in fact reside less than 20 m from each other. The uncertainty
ellipses are extremely small, and therefore reflect a very high certainty
of the model with respect to the predicted position. Fig. 13 illustrates
the cluster of trajectories utilized to conduct the prediction. It is discov-
ered that this cluster corresponds to a ferry, where the final position
after 30 min is at one of its ports. As such, the data upon which the
prediction is determined is concentrated about this position. An offset
of 20 m can be accounted for by error inherent in the AIS data, in
addition to the orientation of the vessel when in a port.
There are multiple ferries in the region surrounding Tromsø. When
investigating all tested vessels with a ship type of ‘‘Passenger Vessel’’,
it was found that the performance with respect to the uncertainty
measures was degraded, despite the predictions being quite accurate.
The percentage of these vessels is also shown in Table 1, in addition to
results for all vessels except those labeled as ‘‘Passenger Vessel’’. It is
clear that the performance increases in this case. The performance of
other vessel types is, however, likely affected by vessels with similar
effects, where the cluster of underlying data is too similar to allow for
a large uncertainty measure, whilst still providing accurate predictions.
3.5. Running time
As the algorithm is intended to run live, it is of interest to investigate
the running time of the method. To support the discussion, the authors
Fig. 13. Cluster used for ferry trajectory prediction.
Table 1
Percentage of true 30 min selected vessel position inside corresponding 𝜎-contour.
𝜎All vessels Passenger vessels Without passenger
vessels
1 37% 11% 41%
2 63% 22% 70%
3 75% 33% 82%
have evaluated the running time of the method for the cases in this
study. All evaluations have been run on a 2.30 GHz CPU and 16 GB
ram. In the following sections, the running times are evaluated in two
parts. The first addresses the running time of the trajectory extraction
algorithm, and the other the classification, clustering and prediction
algorithms.
The data utilized in this study consisted of approximately 15 million
data points. These data points are input to the algorithm without
any pre-processing. In the trajectory extraction step, relevant historical
trajectories need to be extracted from the raw AIS data. Once this is
conducted, the data will be available for the period a prediction is
required. Fig. 14 illustrates the running time of the extraction of tra-
jectory data. This is visualized as a function of the number of relevant
trajectories extracted. A third order regression was applied to the data
to visualize the relationship between the running time and the number
of trajectories. It is clear that the running time increases with the
number of extracted trajectories. It appears that for most cases, the tra-
jectories were extracted within two minutes. The trajectory extraction
process only needs to be conducted once, and subsequent predictions
can utilize the previously extracted data. The algorithm utilized to
extract the trajectories from the raw data has not been optimized in
the current implementation, however. As such, the running time of
the extraction phase can likely be significantly improved through opti-
mization. Additionally, in a future system utilized for vessel trajectory
prediction, a more advanced computer would be utilized to conduct the
prediction. Furthermore, speed can be increased by pre-processing data
for regions such that whole trajectories are available for extraction,
instead of raw data points that require trajectories to be created.
Nonetheless, the extraction times evaluated in the implementation in
this study are reasonable for the outlined purposes.
Of most interest to the study is arguably the performance of the
clustering, classification and prediction algorithms. Fig. 15 illustrates
the individual algorithm running times in addition to the total running
time, i.e. the sum of the clustering, classification and prediction running
times. These are again plotted as a function of the number of extracted
trajectories with a third order regression. It is clear that all algorithms
are quite fast. The classification was virtually instantaneous for all
cases, and the dual linear autoencoder trajectory prediction took less
than one second for all cases. The clustering algorithm dominates
Ocean Engineering 209 (2020) 107478
12
B. Murray and L.P. Perera
Fig. 14. Trajectory extraction running time.
Fig. 15. Algorithm running times.
the total running time, where most cases took between one and two
seconds. However, the overall total running time for all algorithms
was nonetheless quite low, with the worst case being just below four
seconds, and the majority of the evaluated cases below two seconds.
This is considered to be acceptable for the purposes of this study. With a
more advanced computer, and optimized implementation, the running
time would likely be even lower.
4. Conclusion
A linear version of the autoencoder is implemented in this study,
and it is shown that it can predict complex trajectories with a high
degree of accuracy. Training the linear version of the autoencoder
utilized in this study is also less computationally demanding than
deeper autoencoders. Compared to methods that predict future states
conditioned upon their prediction of the previous state, this method
draws upon the generative ability of autoencoders to predict entire
trajectories. Generative models have been shown to have good per-
formance in creating new data points that belong to the distribution
of the training data. By interpolating in the latent space of historical
trajectories, the method in this study is able generate an entirely new
trajectory. The method is, however, dependent on the ability to cluster
the trajectories. If one were to apply the same method to all historical
trajectories, as opposed to a cluster, one may end up interpolating
between clusters. As such, a subsequent prediction will result in an
unrealistic trajectory that does not belong to the distribution of the
historical data.
Applying the method on a single cluster, however, will increase
the ability to describe subtle differences between trajectories, thereby
enhancing the subsequent prediction. Also, given that all trajectories
within the same cluster are quite similar, the likelihood of generating a
trajectory that does not belong to the original distribution is unlikely.
Additionally, as the method generates an entire trajectory, and not
iterative states conditioned upon the previous prediction, prediction
errors will not propagate as a function of time. The error will, therefore,
be related to the error of the entire trajectory. By evaluating multiple
trajectory predictions, however, one can estimate the degree of uncer-
tainty of the prediction, and this uncertainty can be modeled using the
outlined method in this study.
The approach suggested in this study provides an effective method
to predict the future trajectories of ocean going vessels. Specifically, the
method provides the basis for an advanced ship predictor on a global
scale. This ship predictor will aid in providing situation awareness
to navigators, in that the future trajectory of potential target vessels
can be predicted far in advance. Based on a subsequent evaluation
of the collision risk, simple corrective measures can be conducted
to prevent close-range encounter situations from arising. If effective,
such a method will increase the safety associated with maritime op-
erations. Such situation awareness can also potentially be extended to
autonomous vessels, which can make system level intelligent decisions
based on input from the outlined approach.
Future work will include investigating deep learning methodologies
that introduce nonlinearity, and how increases in the complexity of the
model can potentially increase the performance of the predictions. In
addition, further work will be conducted on integrating such methods
into an advanced ship predictor to provide situation awareness to
navigators.
CRediT authorship contribution statement
Brian Murray: Conceptualization, Methodology, Software, Formal
analysis, Writing - original draft, Visualization. Lokukaluge Prasad
Perera: Conceptualization, Writing - review & editing, Supervision.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgments
This work was supported by the Norwegian Ministry of Education
and Research and the MARKOM-2020 project, a development project
for maritime competence established by the Norwegian Ministry of
Education and Research in cooperation with the Norwegian Ministry
of Trade and Industry. The authors would also like to express gratitude
to the Norwegian Coastal Administration for providing access to their
AIS database.
References
Aarsæther, K.G., Moan, T., 2009. Estimating navigation patterns from AIS. J. Navig. 62
(04), 587–607. http://dx.doi.org/10.1017/S0373463309990129.
Bourlard, H., Kamp, Y., 1988. Auto-association by multilayer perceptrons and singular
value decomposition. Biol. Cybernet. 59 (4–5), 291–294. http://dx.doi.org/10.
1007/BF00332918.
Dalsnes, B.R., Hexeberg, S., Flåten, A.L., Eriksen, B.-O.H., Brekke, E.F., 2018. The
neighbor course distribution method with Gaussian mixture models for AIS-based
vessel trajectory prediction. In: 2018 21st International Conference on Information
Fusion. FUSION, IEEE, pp. 580–587.
Endsley, M.R., 1995. Toward a theory of situation awareness in dynamic systems. Hum.
Factors: J. Hum. Factors Ergon. Soc. 37 (1), 32–64. http://dx.doi.org/10.1518/
001872095779049543.
Endsley, M.R., Bolté, B., Jones, D.G., 2003. Designing for Situation Awareness : An
Approach to User-Centered Design. Taylor & Francis.
Ocean Engineering 209 (2020) 107478
13
B. Murray and L.P. Perera
Fisher, R.A., 1936. The use of multiple measurements in taxonomic problems. Ann.
Eugen. 7 (2), 179–188. http://dx.doi.org/10.1111/j.1469-1809.1936.tb02137.x.
Foster, D., 2019. Generative Deep Learning. O’Reilly Media, Inc..
Goodfellow, I., Bengio, Y., Courville, A., 2016. Deep Learning. MIT Press, http://www.
deeplearningbook.org.
Hermann, M., Pentek, T., Otto, B., 2016. Design principles for industrie 4.0 scenarios.
In: 2016 49th Hawaii International Conference on System Sciences. HICSS, IEEE,
pp. 3928–3937. http://dx.doi.org/10.1109/HICSS.2016.488.
Hexeberg, S., Flaten, A.L., Eriksen, B.-O.H., Brekke, E.F., 2017. AIS-based vessel
trajectory prediction. In: 2017 20th International Conference on Information Fusion.
FUSION, IEEE, http://dx.doi.org/10.23919/ICIF.2017.8009762.
Karhunen, K., 1946. Zur spektraltheorie stochastischer prozesse. Ann. Acad. Sci. Fenn.
37.
Kingma, D.P., Welling, M., 2014. Auto-encoding variational bayes. In: Proceedings of
the International Conference on Learning Representations. ICLR, http://arxiv.org/
abs/1312.6114.
Mazzarella, F., Arguedas, V.F., Vespe, M., 2015. Knowledge-based vessel position
prediction using historical AIS data. In: 2015 Sensor Data Fusion: Trends, Solutions,
Applications. SDF, IEEE, pp. 1–6. http://dx.doi.org/10.1109/SDF.2015.7347707.
Murray, B., Perera, L.P., 2019. An AIS-based multiple trajectory prediction approach for
collision avoidance in future vessels. In: 38th International Conference on Ocean,
Offshore & Arctic Engineering, Vol. 7B-2019. OMAE 2019, American Society of
Mechanical Engineers (ASME), http://dx.doi.org/10.1115/OMAE2019-95963.
Pallotta, G., Horn, S., Braca, P., Bryan, K., 2014. Context-enhanced vessel predic-
tion based on ornstein-uhlenbeck processes using historical AIS traffic patterns
: Real-world experimental results. In: Information Fusion (FUSION), 2014 17th
International Conference on. July, pp. 1–7.
Pallotta, G., Vespe, M., Bryan, K., 2013. Vessel pattern knowledge discovery from AIS
data: A framework for anomaly detection and route prediction. Entropy 15 (12),
2218–2245. http://dx.doi.org/10.3390/e15062218.
Perera, L.P., 2017. Navigation vector based ship maneuvering prediction. Ocean Eng.
138, 151–160. http://dx.doi.org/10.1016/j.oceaneng.2017.04.017.
Perera, L.P., Carvalho, J.P., Guedes Soares, C., 2010. Autonomous guidance and
navigation based on the COLREGs rules and regulations of collision avoidance.
Adv. Ship Des. Pollut. Prev. 205–216. http://dx.doi.org/10.1201/b10565-26.
Perera, L.P., Guedes Soares, C., 2015. Collision risk detection and quantification in
ship navigation with integrated bridge systems. Ocean Eng. 109, 344–354. http:
//dx.doi.org/10.1016/j.oceaneng.2015.08.016.
Perera, L.P., Mo, B., 2018. Ship performance and navigation data compression and
communication under autoencoder system architecture. J. Ocean Eng. Sci. 3 (2),
133–143. http://dx.doi.org/10.1016/j.joes.2018.04.002.
Perera, L.P., Murray, B., 2019. Situation awareness of autonomous ship navigation
in a mixed environment under advanced ship predictor. In: 38th International
Conference on Ocean, Offshore & Arctic Engineering, Vol. 7B-2019. OMAE 2019,
American Society of Mechanical Engineers (ASME), http://dx.doi.org/10.1115/
OMAE2019-95571.
Perera, L.P., Oliveira, P., Guedes Soares, C., 2012. Maritime traffic monitoring based
on vessel detection, tracking, state estimation, and trajectory prediction. IEEE
Trans. Intell. Transp. Syst. 13 (3), 1188–1200. http://dx.doi.org/10.1109/TITS.
2012.2187282.
Raychaudhuri, S., 2008. Introduction to Monte Carlo simulation. In: 2008 Winter
Simulation Conference. IEEE, pp. 91–100. http://dx.doi.org/10.1109/WSC.2008.
4736059.
Reynolds, D.A., Quatieri, T.F., Dunn, R.B., 2000. Speaker verification using adapted
Gaussian mixture models. Digit. Signal Process. 10 (1–3), 19–41.
Rezende, D.J., Mohamed, S., Wierstra, D., 2014. Stochastic backpropagation and
approximate inference in deep generative models. In: Xing, E.P., Jebara, T.
(Eds.), Proceedings of the 31st International Conference on Machine Learning. In:
Proceedings of Machine Learning Research, vol. 32, (2), PMLR, Bejing, China, pp.
1278–1286, http://proceedings.mlr.press/v32/rezende14.html.
Ristic, B., Scala, B.L., Morelande, M., Gordon, N., 2008. Statistical analysis of motion
patterns in AIS data: Anomaly detection and motion prediction. In: 2008 11th
International Conference on Information Fusion. pp. 40–46. http://dx.doi.org/10.
1109/ICIF.2008.4632190.
Rødseth, Ø.J., Perera, L.P., Mo, B., 2015. Big data in shipping - challenges and
opportunities. In: 15th International Conference on Computer and IT Applications
in the Maritime Industries. COMPIT ’16, pp. 361–373.
Rong, H., Teixeira, A., Guedes Soares, C., 2019. Ship trajectory uncertainty prediction
based on a Gaussian process model. Ocean Eng. 182, 499–511. http://dx.doi.org/
10.1016/J.OCEANENG.2019.04.024.
Shepard, D., 1968. A two-dimensional interpolation function for irregularly-spaced data.
In: Proceedings of the 1968 23rd ACM National Conference. ACM ’68, ACM, New
York, NY, USA, pp. 517–524. http://dx.doi.org/10.1145/800186.810616.
Steinbach, M., Ertöz, L., Kumar, V., 2004. The challenges of clustering high dimensional
data. In: New Directions in Statistical Physics. Springer, Berlin, Heidelberg, pp.
273–309.
Tam, C., Bucknall, R., Greig, A., 2009. Review of collision avoidance and path
planning methods for ships in close range encounters. J. Navig. 62 (03), 455.
http://dx.doi.org/10.1017/S0373463308005134.
Tu, E., Zhang, G., Rachmawati, L., Rajabally, E., Huang, G.-B., 2017. Exploiting AIS
data for intelligent maritime navigation: A comprehensive survey from data to
methodology. IEEE Trans. Intell. Transp. Syst. 1–24. http://dx.doi.org/10.1109/
TITS.2017.2724551.
Zhang, L., Meng, Q., 2019. Probabilistic ship domain with applications to ship
collision risk assessment. Ocean Eng. 186, http://dx.doi.org/10.1016/j.oceaneng.
2019.106130.
Zhang, S.-k., Shi, G.-y., Liu, Z.-j., Zhao, Z.-w., Wu, Z.-l., 2018. Data-driven based
automatic maritime routing from massive AIS trajectories in the face of disparity.
Ocean Eng. 155, 240–250. http://dx.doi.org/10.1016/j.oceaneng.2018.02.060.
Zhao, L., Shi, G., 2019. A trajectory clustering method based on Douglas–Peucker com-
pression and density for marine traffic pattern recognition. Ocean Eng. 456–467.
http://dx.doi.org/10.1016/j.oceaneng.2018.12.019.
Zhou, Y., Daamen, W., Vellinga, T., Hoogendoorn, S.P., 2019. Ship classification based
on ship behavior clustering from AIS data. Ocean Eng. 176–187. http://dx.doi.org/
10.1016/j.oceaneng.2019.02.005.
