z. 95 Transport 2013
Jakub Montewka
Aalto University, Dept. of Applied Mechanics, Research Group on Maritime Risk and Safety Espoo, FINLAND
Hilary Sinclair
Memorial University Newfoundland, CANADAMODELLING A SHIP PERFORMANCE
IN DYNAMIC ICE: PART II, TRANSFORMING
INFORMATION INTO KNOWLEDGE
The manuscript delivered: May 2013
Abstract: Although ice navigation has received substantial attention over recent decades, there is still
no known modelling technique to predict ship’s speed in a dynamic ice field. This paper introduces probabilistic, data-driven models that predict a ship’s speed and the situations where a ship is probable to get stuck in ice based on the joint effect of ice features, such as the thickness and concentration of level ice, ice ridges, rafted ice, and ice compression. To develop the models, the data from the Automatic Identification System about the performance of a selected ship was used, an ice forecast model was utilized to deliver information about the ice field and the links between the ice conditions and ship movements were established using Bayesian learning algorithms. The case study presented in this paper considers a single and unassisted trip of an ice-strengthened bulk carrier between two Finnish ports in the presence of challenging ice conditions, which varied in time and space. The obtained results show good prediction power of the models, which is on average 80% for predicting the ship’s speed and above 90% for predicting cases where a ship may get stuck in ice.
We expect this new approach to facilitate the safe and effective route selection problem for ice-infested waters where the ship performance is reflected in the objective function.
Keywords: Winter navigation; Ship performance in ice; Bayesian Networks
1. INTRODUCTION
Ship performance in ice has been given a lot of attention in the recent years, leading to the development of semi-empirical methods that estimate ship resistance in ice and tools that simulate ship transit in ice, see for example (Naegle 1980; LaPrairie et al. 1995; Kotovirta et al. 2009; Su et al. 2010; Lubbad & Løset 2011). However the ice conditions
which are modeled are often limited to the level ice and ice channel, in select cases, the effect of ice ridges is taken into account, see for example (Riska et al. 1997; Juva & Riska 2002). But the effect of ice compression on ship’s speed has not been researched in-depth, thus it is usually expressed in a qualitative manner, see for example (Mulherin et al. 1996). Although these methods have been utilized for optimizing shipping routes in ice-covered waters, see (Kotovirta et al. 2009), there are still numerous issues which need further studies, for instance: the effect of ice compression on ship performance, or the quantification of the joint effect of ice conditions (level ice, ridges, compression, the relative angle at which ice reacts on a ship) that can bring a ship to a halt. Moreover, suggestions have been made to move towards probabilistic models.
Therefore another modeling technique can be adopted leading to an event-oriented model, which reflects the conditions (ice features) under which an event of interest occurs (ship proceeding with very low speed or a ship getting stuck in ice). This type of modeling does not provide an insight into the physics of the process of ice breaking, but simply quantifies the joint effect of various ice features on ship’s speed. There exist event-oriented models estimating ship’s speed in given ice conditions which are based on the full scale measurements, but their focus is either on a single ice feature affecting ship’s speed, as the ice thickness see, (Haas et al. 1999); or the model relies on a subjective interpretation of existing ice conditions, based on visual observation carried out from a ship, see (ENFOTEC Technical Services Inc. et al. 1996). Therefore the joint effect of the relevant ice features on ship performance is not addressed.
This paper introduces a probabilistic framework for predicting performance of a ship navigating in ice, meaning the probability for a ship to attain certain speed, considering the following parameters: thickness and concentration of various types of ice such as, level ice, ridged ice, and rafted ice. Ice compression and its relative direction with respect to a ship are taken into account as well. The structure of the models, which are developed here, and the probabilistic relations among the input parameters are defined by applying two Bayesian learning algorithms, called PC and Naïve Bayes. Finally, the obtained framework has been cross-validated and the final results show very good convergence with the real world conditions. Data used to develop the models are described in a companion paper of ours.
The remainder of the paper is organized as follows: Section 2 presents the main idea behind learning from the data. In Section 3 process of model development is elaborated, whereas in Section 4 the model is validated and the results are presented. Section 5 discusses the models, whereas Section 6 concludes the paper and summarizes its main findings.
2. LEARNING FROM THE DATA
The approach taken towards development of the probabilistic models presented in this paper utilizes techniques of Bayesian learning from data. For this purpose, two learning algorithms were used to find correlation and causation in the data to develop probabilistic models capable of forecasting ship performance in ice. These models first determine and
quantify the relations between all the analyzed variables and second they quantify the joint effects of ice features on ship’s speed, allowing probabilistic analysis of ship performance in ice. From the engineering perspective, quantification of the joint effect of various ice features on ship performance is a novel approach, and seems more realistic than the sum of individual components, see for example (Külaots et al. 2013; Riska et al. 1997; Kotovirta et al. 2009).
The models presented here are able to quantify the joint effect of various ice features on ship’s speed, which in turn makes it possible to determine the sea areas where a ship may experience significant speed reduction. The analyzed datasets, representing speed of ship navigating in changing ice conditions is illustrated in Figure 1. The negative values of speed depicted there represent the instants where the ship goes astern after being brought to a halt in difficult ice conditions. To extract the information from the available datasets, we applied a BBN, which suits the purposes, being an established technique for drawing inferences in the presence of uncertainty or limited data. Moreover, the BBN and its learning techniques make it possible to combine the information extracted from the data with the existing background knowledge and understanding of the analyzed phenomena and relations between certain factors. This is realized at the stage of definition of the model’s variables, where the relations between variables can be specified, i.e. forced or forbidden. For instance, a relation between level ice and ship’s speed can be forced, as this is supported by formulae based on physical laws. All other relations which are proven to exist can be incorporated into a model; likewise the information about lack of certain relations which are proven not to exist can be imposed. Moreover the temporal order of variables can be defined at this stage, meaning that the BBN, to some extent, is able to capture also dynamics of a system that has been analyzed.
There are two ways of learning Bayesian networks: Constraint Search-Based Learning (CSBL) and Bayesian Learning (BL). In the first case, an algorithm searches the data for independence relations to determine the causal relations, and in the second case the space of models is searched over and each model found is scored using the posterior probability of the model given the data, and the model which gets the highest score is adopted, see for example (Cooper & Herskovits 1992; Spirtes et al. 2000). In this paper two learning algorithms were utilized and their results compared. The PC algorithm belongs to the CSBL group and the second known as Naïve Bayes (NB) represents the BL group. Intuitively, the CSBL technique is preferred, as it finds the independences in the data and establishes the causal links between variables. Moreover, all existing knowledge on the causality can be incorporated in the CSBL approach, whereas the BL techniques do not allow this. For the description of the algorithms and discussion about they usability the reader is referred to the following publications (Friedman et al., 1997; Cheng & Greiner, 1999; Acid et al., 2004; Darwiche, 2009).
Fig. 1. Time series of the analysed parameters
3. MODEL DEVELOPMENT
For the purpose of this paper, a model is developed with the use of the GeNie software package developed at the Decision Systems Laboratory, University of Pittsburgh, which offers a wide range of learning algorithms, depending on available data and model requirements, see for example (Druzdzel 1999).
The process of model development starts with defining the scope of the analysis, which is followed by relevant data acquisition and organization, as described in our companion paper. There are twelve variables considered in the models, five of them are three-state variables whereas the others have two states. The variables are discretized into states with the use of hierarchical method imposing background knowledge about the process of ship navigation. The list of all variables with their state is gathered in Table 1.
Once the variables are discretized the selected learning algorithms are applied, and the obtained models are cross-validated. The cross-validation is used in the cases where data was used to learn the model structure; part of the data was used to learn the network, and a part to validate it. The cross-validation provides an estimate of the predictive power of a model with respect to a selected hypothesis, which is tantamount to evaluating the probabilities for false positives (FP) and false negatives (FN) for a two-state variable.
Then, the model that passes this test undergoes the behavior analysis tests, which address the following question: does the model behavior predict the behavior of the system being modeled? An analyst who performs the scenario walk-throughs and checks whether the model prediction is consistent with the existing knowledge about the physics of the phenomena, which is being analyzed, and the general knowledge and understanding of the modeled system usually does the latter analysis. The final stages of model development are devoted to performing the sensitivity analysis and the value-of-information analysis. The
sensitivity analysis helps to identify those variables and parameters in a model that are critical. The value of information communicates the distribution of uncertainty among model parameters; it also identifies the variables, which are the most informative with respect to the model output.
The analyses performed in this study have failed to deliver a single unified model, which simultaneously predicts the speed of the ship and situations where a ship gets stuck in ice. Therefore, two separate models have been developed and are presented in the following section; one predicting each event. The models are obtained by applying two different types of learning algorithms, thus, the first model predicting the ship’s speed, called model A, considers the causality discovered in the data; whereas the second model, called model B, disregards the causality, and it searches for the best fit to determine the relations between the situations where ship is stuck in ice and the surrounding ice conditions. However the learning algorithm applied to develop this model assumes that the model’s variables are independent. Model A is rather complex, it encompasses 12 variables, which are connected with 29 arcs, having in total 29 states, and producing large number of conditional probabilities (4415) This makes it impossible to illustrate all parameters and their states in the paper. Consequently, the description and presentation of the model is limited to its qualitative part – model structure – only, see Figure 2. The second model is less complex as it contains only 56 conditional probabilities, thus both parts of the model, namely qualitative and quantitative, can be presented here, see Figure 3 and Table 2.
Fig. 2. Probabilistic model predicting ship’s speed in ice, developed with the use of PC learning algorithm and incorporated background knowledge (model A)
Fig. 3. Probabilistic model for predicting a ship getting stuck in ice, developed with the use of Naïve Bayes learning algorithm (model B)
Table 1
Variables included in the models and their states
No Variable’s name State A State B State C
1 Ship’s speed [kn] <5 5-10 >10
2 Ship stuck in ice YES NO -
3 Level ice thickness [m] 0.3-0.4 0.4-0.5 -
4 Level ice concentration [%] <25 25-75 >75 5 Ridged ice thickness [m] 2.5-3.0 3.0-3.5 -
6 Ridged ice concentration [%] <5 5-15 -
7 Rafted ice thickness [m] 0.3-0.4 0.4-0.5 -
8 Rafted ice concentration [%] <5 5-10 -
9 Wind speed [m/s] 3.5-5.0 5.0-6.5 -
10 Compression level 0-1 1-2 2-3
11 Relative direction of compression [deg] 0-45 45-135 135-180 12 Relative direction of wind [deg] 0-45 45-135 135-180
Table 2
Conditional probability tables for model B
Variable’s name Variable’s states Output variable ship
stuck in ice
No Yes
Ship’s speed [kn] <5 0.10 0.98
5-10 0.26 0.01
>10 0.64 0.01
Level ice thickness [m] 0.3-0.4 0.49 0.34
0.4-0.5 0.51 0.66
Level ice concentration [%] <25 0.32 0.01
25-75 0.25 0.01
>75 0.43 0.98
Ridged ice thickness [m] 2.5-3.0 0.46 0.34
3.0-3.5 0.54 0.66
Ridged ice concentration [%] <5 0.43 0.01
5-15 0.57 0.99
Rafted ice thickness [m] 0.3-0.4 0.42 0.01
0.4-0.5 0.58 0.99
Rafted ice concentration [%] <5 0.29 0.41
5-10 0.71 0.59 Wind speed [m/s] 3.5-5.0 0.23 0.66 5.0-6.5 0.77 0.34 Compression level 0-1 0.61 0.01 1-2 0.24 0.73 2-3 0.16 0.26
Relative direction of compression [deg] 0-45 0.55 0.65 (0deg – wind from the bow, 180deg - wind from
the stern) 45-135 0.36 0.34
135-180 0.09 0.01
Relative direction of wind [deg] 0-45 0.32 0.37 (notation same as above) 45-135 0.66 0.63
135-180 0.02 0.01
4. MODEL VALIDATION
This section describes results of three analyses considered relevant: cross-validation, model behaviour test and the value of information analysis. The results of cross-validity analyses – see tables 3 and 4 - show relatively good prediction power for the model A, which is expressed as the probability of delivering the right answer by the model, which
for the variable ship speed varies between 0.75 and 0.93. In the case of the model B, its predictive power is even higher, 0.9 and 1.0 depending on the hypothesis adopted.
The results of model behaviour analysis are depicted in Figure 4, where the outcome of the model is a two-state variable (ship gets stuck in ice), which is set to one of state at a time (yes/no) and the values of all explanatory variables are obtained from the model by back propagation of this evidence.
The value-of-information analysis identifies the most informative variables, with respect to the output variable, determining the variables among which the probability mass of the output is scattered. For this purposes the concept of Shannon entropy - H(X) - is utilized. In a model, where the outcome variable is conditionally dependent on a number of parental variables, the conditional entropy H(X|Y) needs to be applied. This is a measure of the uncertainty of X given an observation of Y. The results of the value-of-information analysis with respect to the models’ output are gathered in tables 5 and 6, where the actual value of entropy and the values of conditional entropies are shown for all the variables included in the models. Additionally the maximum entropy that a given variable can take is shown. This together with the actual entropy informs where on the entropy scale the model is located. The zero entropy means that the model has no uncertainty, and its outcome is fully predictable, therefore each run of the model does not deliver any new information. Whereas the maximum entropy corresponds to a model which is not predictable, and to gain new information a model needs to be run. Each run of the model deliver certain amount of information which is described by the entropy. The maximum entropy situation is when a realization of a certain process can take n states, and the probabilities for the occurrence of any of these states are equal - 1/n.
Table 3
Results of the cross-validation of the model A - predicting ship’s speed in ice
Ship’s speed [kn]
AIS
data
Model prediction
Below 5 Between 5-10 Above 10
Below 5 0.78 0.22 0.00
Between 5-10 0.21 0.75 0.04
Above 10 0.01 0.06 0.93
Table 4
Results of the cross-validation of the model B - predicting the situation where a ship gets stuck in ice
Ship stuck in ice [yes/no]
AIS data Model prediction No Yes No 0.90 0.10 Yes 0.00 1.00
Table 5
Results of the value-of-information analysis – model A
Variable’s name max H(X) H(X) H(X|Y) H(X|Y)/H(X)
Ship’s speed [kn] 1.580 1.071 -
Compression level - 0.084 8%
Level ice concentration [%] - 0.071 7%
Ridged ice concentration [%] - 0.044 4%
Rafted ice thickness [m] - 0.035 3%
Relative direction of wind [deg] - 0.025 2.5% Rafted ice concentration [%] - 0.015 1.5% Relative direction of compression [deg] - 0.011 1%
Wind speed [m/s] - 0.007 <1%
Ridged ice and level ice thickness [m] - 0.005 <1%
Table 6
Results of the value-of-information analysis – model B
Variable’s name max H(X) H(X) H(X|Y) H(X|Y)/H(X)
Ship stuck in ice 1.000 0.117 -
Ship’s speed [kn] - 0.052 38%
Compression level - 0.022 16%
Level ice concentration [%] - 0.018 13%
Ridged ice concentration [%] - 0.012 9%
Rafted ice thickness [m] - 0.012 9%
Wind speed [m/s] - 0.010 7.5%
Relative direction of compression [deg] - 0.002 1.5% Level ice and ridged ice thickness [m] - 0.001 <1% Rafted ice concentration [%] - <0.001 <1% Relative direction of wind [deg] - <0.001 <1%
5. DISCUSSION
The results, which are obtained in the course of validation analyses, reveal that the models that are developed here shows rather good agreement with the recorded data and general understanding of the analyzed processes.
Model A tends to overestimate the modelled parameter - ship’s speed –for the lowest speed category (below 5kn), where in 22% of the cases the model classifies the speed wrongly, assigning it to the higher speed category (between 5-10kn). This means that the model may deliver results, which are too optimistic for a ship. In other cases the model tends to slightly underestimate the ship’s speed. The accuracy of the model is 78%, 75% and 93% for the low, medium and high-speed categories respectively, see Table 3. The model has problems with proper estimation of the speed category in the locations where ship speed fluctuates significantly, and better accuracy than 78% cannot be attained with the presented set of variables.
The predictive power of the model B with respect to hypothesis stating that the ship gets stuck is very high whereas in the case of alternative hypothesis it is burdened with some inaccuracy, meaning that the model predicts more cases of ship being stuck in ice than obtained from the records. We found three reasons for this:
1. The model does take into account navigation in ice channel, which took place at least two times during approaching harbour of destination. It is visible in the recorded data, where the recorded ship’s speed is above 10 kn, while the model predicts the speed belonging to the lowest category – see model A in Figure 5. The effect of ice channel navigation is also recognized in the graph where the model predicts the ship being stopped in ice where she is underway; these are depicted in Figure 5. However this type of navigation was rather minor, as a significant level of compression was observed, and the ice channels could not remain open for a long time.
2. The model tends to classify the cases where the ice conditions are challenging and the ship’s speed drops below 4 kn as “ship stuck in ice”, which in the reality is not always the case. According to the recorded data, in some instant of time the ship’s speed dropped dramatically from 8 kn to 3 kn, marked in Figure 5 as “speed drop”, however the ship managed to speed up and continue for some time before she eventually stopped. This means that under challenging ice conditions the model overpredicts. 3. The ice hindcast model HELMI is insightful for the ice conditions in geographical
scale, but the exact conditions at the ship location may fluctuate beyond the HELMI output. The HELMI data is an uncertain estimate of the local conditions, given the spatio-temporal grid of 1nmx1nmx1hour. Despite these drawbacks, HELMI model is one of the best choices for delivering the ice forecast for the analyzed sea area. When it comes to the behaviour analysis of the models, one variable in model B, called wind speed, is found not to behave as expected, otherwise the variables react as expected when the outcome variable is set to either of its state. It means that the model delivers higher probability for a ship to get stuck in the presence of lower wind speed. This is not coherent with the available knowledge, as the higher wind speed may create higher resistance as a ship is pressed against the ice which increases friction. But it also shows that the effect of wind is not dominant in this case; as if this variable is removed from the model, model’s accuracy does not change, and there are other variables having a stronger influence, see Table 6.
Analyzing the value-of-information of model A’s, we fund its entropy rather high (67% of its theoretical, maximum entropy), meaning that by running the model, a large amount of information is gained. This comes from the fact, that the model’s outcome has three states, with the following probabilities: 0.25; 0.32; 0.44 for each, meaning that the states are not so far from being distributed evenly. The conditional entropy informs how much information about the model outcome can be gained by knowing a variable in the model. Analyzing Table 5, we can conclude that there is not much to be gained by observing one or even two variables, this can be explained by the complexity of the analyzed process which to be described needs the whole set of variables to be present. In the case of model B, the entropy is relatively small (11% of its theoretical, maximum entropy), which means that the outcome of the model is predictable. This should not be surprising, as the outcome of the model has two states with the following probabilities: 0.97 and 0.03. Such an imbalance suggests the probable result of the model, before running the model, thus the amount of new information gained by running the model is relatively small. However, by
learning about certain variables a significant amount of new information about the model outcome can be gained. This means, that the model is explained to a large degree by five variables, meaning ship’s speed, ice compression level, level ice concentration, ridged ice concentration and rafted ice thickness, see Table 6.
Fig. 5. Model A’s (left) and B’s (right) predictions versus measurements obtained from AIS
6. CONCLUSIONS
In this paper we have introduced two probabilistic models predicting ship performance in ice, meaning ship’s speed and the situations where a ship is probable to get stuck in ice. We assume that the knowledge about ice conditions in the moment of making the prediction comes from the ice forecast only, and information about location of ice channels, which make the ice navigation easier, is not available at this time instant. This holds in the presence of ice compression or ice drift, where the ice channels tend to close rapidly.
The models were developed with the use of the techniques of Bayesian learning and good predictions have been found. Notwithstanding all the assumptions, the results
obtained are promising as they can help to understand the joint effect of ice features on ship’s speed in addition to the conditions associated with ships getting stuck in ice.
The models presented in this paper feature several novelties, first they predict the ship performance in a probabilistic fashion, with the use of full scale data, second the models consider the joint effect of various ice features on ship performance, finally the ice compression has been taken into account. We expect this new approach to facilitate the optimal route selection problem for ice-infested waters where the ship performance is reflected by an objective function.
The models, which are presented here, are valid only for a specific ship (ice going bulk carrier), and the specific ice hindcasting model (HELMI). Further work should focus on analysing the performance of ships of various types and ice classes and incorporating them to the models.
ACKNOWLEDGMENTS
The work presented here has been financially supported by FP7 project SAFEWIN on “Safety of winter navigation in dynamic ice” (www.safewin.org).
The probabilistic models introduced in this paper were created using the GeNie modelling environment developed at the Decision Systems Laboratory, University of Pittsburgh – available from http://genie.sis.pitt.edu/.
The Merenkulun säätiö – the Maritime Foundation - from Helsinki is thanked for the travel grant.
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MODELOWANIE OSIGÓW STATKU NAWIGUJCEGO W DYNAMICZNYM POLU LODOWYM: CZ II, PRZEKSZTACENIE INFORMACJI W WIEDZ
Streszczenie: Pomimo, i egluga w lodach pozostaje tematem wielu opracowa naukowych, tematyka
modelowania zachowania statku w dynamicznym polu lodowym, zwaszcza w obecnoci zjawiska kompresji pokrywy lodowej, pozostaje wci kwesti otwart. W artykule przedstawiono dwa probabilistyczne modele, pierwszy szacujcy prdko statku w polu lodowym oraz drugi okrelajcy warunki lodowe w których statek moe spodziewa si cakowitej utraty prdkoci (moe utkn w lodzie). Modele stworzono w oparciu o informacje uzyskane z systemu AIS dostarczajcego dane o pozycji oraz pooeniu statku w odstpie kilkunastu sekund, wykorzystano take szczegóowe informacje o pokrywie lodowej (stopie koncentracji oraz grubo pokrywy lodowej, zwaów lodowych, nawarstwionego lodu oraz poziom kompresji pokrywy lodowej), pochodzce z numerycznego modelu pogody HELMI, opracowanego w Finskim Instytucie Meteorologicznym. W celu okrelenia zalenoci pomidzy zachowaniem statku a warunkami lodowymi wykorzystano Sieci Bayesa oraz dwa typy algorytmów uczenia maszynowego z danych. Uzyskane modele charakteryzuja si wysokim poziomem prognozowania, 80% dla modelu prognozujcego prdko statku oraz 90% dla modelu prognozujcego sytuacje utknicia w lodzie. Przedstawiona analiza dotyczy pojedynczego przejcia statku masowego posiadajcego wysok klas lodow, pomidzy dwoma portami w Finlandii.
Przedstawione podejcie moe by wykorzystane przy rozwizywaniu problemu wyboru trasy optymalnej w dynamicznym polu lodowym.
