Prediction Analysis Based on Logistic Regression Modelling

Abstract

The chapter aims to show an application of logistic regression modelling for prediction analysis in the offshore industry. The different variables shown in dynamic positioning incident reports are analysed and processed using logistic regression modelling. The results of the models are then analysed, showing which data influence the loss of positioning and human errors and how the model can be interpreted. Afterwards, and based on the obtained models, operational limits can be proposed to reduce downtimes and thus improve the safety of the operations and the productivity of the offshore operations when using dynamic positioning systems.

1. Introduction

A dynamic positioning (DP) system is a piece of automation in which data from wind, currents and ship motions are taken from different sensors. After analysing them, a signal is sent to thrusters and rudders to compensate for those movements. This system seeks two main goals depending on the nature of the operations in progress: maintaining a given position or moving a vessel along a pre-set track.

The DP system has been in use for many decades, and its applications are primarily used in the offshore industry. The complexity and high accuracy requested for the different offshore operations make the dynamic positioning system a valuable tool for this sector.

However, rarely does such a sophisticated automated system always performs smoothly. The study of the incidents reported by vessels is vital to discover any failures that could be corrected and to improve the safety of DP operations.

The International Marine Contractors Association (IMCA) is, without any doubt, one of the most prolific authors to the cause of safety in DP operations. They have published different recommendations to the industry, along with guidelines for operations, sensors, and personnel. It is also important to mention the collection of DP incidents that IMCA has published since 1994. The high volume of DP incidents reported anonymously, and carefully published by IMCA, has been the base of this research.

In this chapter, the focus is set on the research of the incidents reported to IMCA from 2011 until 2015. During this period, the reports presented by IMCA were of the event-tree type, showing information regarding water depth, the configuration of the DP system and meteorological information. Before this period, the event trees lacked some of these data; and since 2015, the reports presented were just a sample and did not include all the incidents reported.

However, the reports collected can contribute to understanding the common patterns that can be found in the different incidents, thus finding an interpretation that could help improve the safety of these operations. Periods of downtime can mean a considerable amount of money loss. In some cases, the incident can lead to catastrophic consequences, leading to the loss of the ship and even pollution of the environment.

In this context, the research presented in this chapter aims to propose a mathematical model using logistic regression, which could help predict under which conditions an incident can occur. Furthermore, the condition of the incident can be determined beforehand, and as such, the likelihood of having an incident ending in an excursion can be modelled.

At the same time, and knowing that perhaps the human error is the easiest of all mistakes to correct, the model will be determined once again taking into account whether a human error was the cause of the incident or not, and the resulting models will be compared.

2. State of the art

2.1 DP elements

Any DP system always has seven segments or elements, namely: controller, DP console, Dynamic Positioning Operator (DPO), position reference systems (PRS), motion reference units (MRU), propulsion and power supply. Each element will be described in this section.

The central element is the controller, composed of computers or processors, which sets a two-way communication with all other DP elements via the vessel network.

The system is controlled with the help of the DP console, which contains operational controls, buttons, screens and a manual joystick.

The DP console is controlled by a Dynamic Positioning Operator (DPO) who should be fully certified to conduct DP operations.

To acquire information on the position of the rig, PRS are used. In DP drilling operations, several PRS provide additional accuracy [1]. Usually, a drilling rig will select dual differential global navigation satellite system (DGNSS) and hydro-acoustic positioning references (HPR), usually of the long-baseline type. Taut wires are only used in shallow waters, as they are not available in deep water [2].

The motions of the vessel are monitored with different sensors. The yawing is monitored with the help of one or more gyrocompasses which send information about the heading. Different MRU help send information about surge and sway.

The wind and current are also monitored for course and speed, and this information is sent to the controller. There are different wind sensors in different positions onboard the rig to avoid errors due to windscreens, turbulences provoked by structures, and other obstacles.

With all this information, the controller can predict the movement on the vessel and send a proper command to the propellers and thrusters (pitch, revolutions per minute, azimuth, rudder angle) to counter-rest the forces and maintain the rig in the desired position.

A vital part of the DP system is the power supply. Diesel, alternators, switchboards, cabling, propulsion motors and power-management form part of the power system related to the DP operations [3].

2.2 The use of dynamic positioning systems in drilling operations

Drilling operations take place over a wellhead. The primary purpose of the DP system is to maintain the position of the drilling vessel so that the riser/stack angle containing the drill string is close to zero, compensating for currents or tidal flow if necessary [4]. This angle is the one measured between the riser (on the top) and the wellhead or lower marine riser package (LMRP) [3]. This function is known as riser angle or riser follow mode. The DPO monitors the riser difference angle through sensors located around the LMRP. A watch circle system is created so the DPO can monitor the movements of the vessel. When the rig is moving, different levels of alarm are set to ensure the safety of the operations at all times [4].

The main risk in any DP operation is losing position (which is known in drilling operations as an excursion) during operations. Therefore, the DPO should react in a short time to correct or mitigate the consequences of this loss [5].

To maintain the position of the drilling riser, the system consists of a closed-loop control function that receives information from different sensors that measure wind, currents, heading and position. It sends a command to the propulsion units to counter rest the forces that, according to the information, tend to take the vessel out of position.

The desired position is input by the DPO, who supervises the operation in the Human Machine Interface (HMI), also known as the DP console. The DPO operator is a certified officer of the watch who has followed a training and certification scheme to cover this board position [6].

Finding which variables and in which way and measure they affect an incident having a human cause can help focus on the riskiest situations and improve the safety of drilling operations. From the results obtained, it would be possible to propose operational limits to improve the safety of drilling operations.

DP drilling incidents have been the object of different academic research. In 2011, Haibo Chen [7] published a paper where he introduced the safety of DP operations based on a barrier model. Previously the same research team had already published an article about the safety of such units [8].

The most interesting approach to the human factors in DP incidents has been proposed by Chae [9], while formal safety assessment was applied to them [10]. Dong [11] focused his research on the incidents that had taken place during offshore loading operations. Overgard [12] also researched the human element during DP incidents.

3. Objectives

There are two main objectives in this chapter.

The first main objective of this paper is to find the mathematical expression that determines the probability of an incident ending in an excursion during DP drilling operations.

With the developed model obtained from this research, drilling companies and other authorities can review their management manuals and propose some effective measures to reduce the probability of loss of position while conducting DP drilling operations. The excellent results obtained by the presented model avail the reliability of this technique.

The second main objective would be to determine whether the model remains the same when there is a human error or not, that is, to determine if the human error can alter the proposed model.

4. Methodology

The first step for applying a regression modelling technique is to have a database with different variables. The variables do not need to follow a normal distribution in the database, which is a condition for other prediction techniques.

In our example, data was gathered from the IMCA station-keeping incidents corresponding to 2011 to 2015. The cases that took place while drilling operations were in progress were selected, 50 in total.

The data described in the event tree was carefully read, and a database was developed, including the following variables, as shown in Table 1.

Variable	Description
Year	In which year did the incident happen.
Water-depth (in metres)	Indicates the water depth at which the drilling operations took place.
Number of thrusters online	The number of thrusters that were online in the DP system.
Number of thrusters stand-by	The number of thrusters that are not online in the DP system, but which are ready to be selected at any time.
Number of generators online	The number of generators that were online in the DP system
Number of generators stand-by	The number of generators that are not online in the DP system, but which are ready to be selected at any time.
Bus tie	Whether it was open or closed.
DGNSS	The number of DGNSS systems that are selected in the DP system
HPR	The number of hydroacoustic systems that are selected in the DP system
Taut wire	The number of taut wires in use during the operations
Inertia system	The number of inertia systems in use during the drilling operations
Gyros	The number of gyros that were in use during the drilling operations
MRU	The number of MRU that were in use during the drilling operations
Wind sensors	The number of wind sensors that were in use during the drilling operations
Wind force	The force in knots of the wind blowing when the incident happened
Wind direction	The direction of the wind in degrees
Current speed	The speed of the current in knots when the incidents happened
Current direction	The direction of the current in degrees
Wave height	The height of the waves in metres
Visibility	The visibility that there was while the incident happened.
Main cause	The leading cause, as defined by the classification presented by IMCA
Secondary cause	The secondary cause, if present, as defined by the classification presented by IMCA
Excursion	Whether an excursion took place or not.

Table 1.

Variables from the database.

Some of these variables had to be treated to be used in this research. Thus, the following variables were obtained, shown in Table 2.

Variable	Description
Percentage of thrusters online	The number of thrusters online divided by the total number of thrusters online and stand-by.
Percentage of generators online	The number of generators online, divided by the total number of generators online and stand-by
Visibility	This variable was categorised using the following criteria: Poor when the visibility is less than two nautical miles, Moderate between 2 and 5 nautical miles, Good above five nautical miles [13].
Human cause	When either the main or secondary causes have a human origin, then 1 is inserted, for the rest of the cases, 0 is inserted indicating no human cause.
Period	The first period is from 2011 till 2013. The second period is from 2014 to 2015.

Table 2.

Variables created from existing variables.

Once the database was created, some missing values were observed for some of the variables. These cases were eliminated to uniform the sample without distorting the values by performing bootstrapping.

A descriptive statistic of each variable is performed before researching the binary logistic regression models.

4.1 Binary logistic regression model

The binary logistic regression technique will provide the probability that a given variable, called the dependent variable, will have a given value based on the values of the other variables, called independent variables.

For our example, the dependent variable will be an excursion. The excursion will have a value of zero if there is no loss of position and a value of 1 if there is an excursion.

The rest of the variables will be considered independent variables. These variables can be quantitative or categorical. In our example, except for the variables water depth, percentage of thrusters online and percentage of generators online, which are all quantitative, the rest of the independent variables are categorical. Due to this, when using a statistical program, it manipulates its values internally to produce as many variables as there are categories minus one. For example, Wind sensors have five categories, and the program produces four variables: Windsensors (i), i = 1, 2, 3, 4. These new variables are dichotomic: the value 1 indicates the presence of a quality, and the value 0 its absence.

The statistical program (in our example, SPSS), considering the values for each case in the independent variables, calculates the probability of excursion for each of them. As this probability varies between 0 and 1, the closer to 0 will mean the most negligible probability of excursion, and the closer to 1 will mean a more significant probability of excursion. Thus, each case is assigned a probability p. This is important to interpret the coefficients in the regression. There has been a recodification, and no information has been lost.

The choice of the variables is made by the selected method: Forward Wald. This method is based on adding or removing variables from the model by using two statistics: the score of Rao and the Wald statistic.

The score of Rao allows to compare for each independent variable Xj the null hypothesis: Ho = Bj = 0; that is, the regression coefficient B associated with the variable in the model is null. The variable that presents the minimum associated p-value provided it is always less than 0.05, for the proposed independent variable will be selected to enter the model.

Also, for the Wald statistic, the null hypothesis can be compared Ho: Bj = 0, but in this case, it is for the independent values that are already selected and have entered the model.

A variable with a p-value associated with the Wald statistic bigger than 0.1 will be eliminated, as this is by default the option of the program.

According to the criteria exposed above, there will be several steps in which independent variables will be entered and eliminated.

At step 0, only the constant is introduced to the model. For this constant, it is essential to measure B (the regression coefficient), the estimated standard error in the estimation (SE), the Wald statistic and its degrees of freedom (df) and the associated p-value. When this p-value is less than 0.1, the constant is considered to be significant.

All the independent variables are out of the model at this step. One variable has to be selected to enter the model in step 1. The variable with the smaller p-value associated with the score of Rao, provided it is less than 0.05, will be selected. It should be considered that the variables created from a categorical variable should be considered as a whole.

If two or more variables have the same p-value, the score should then be considered, choosing the variable with the bigger score to enter the model in Step 1.

Once the variable enters the model, we should study the Wald statistic, given by:

$$\mathit{\text{Wald}}={\left(B/\mathit{SE}\right)}^2$$E1

If its p-value is above 0.1 (output value, POUT), then the corresponding variable would be eliminated (as a whole in the case of the categorical variables). It is always eliminated before the new variable is selected.

After this, another variable would be selected (or not) to enter the model in the next step. Suppose no variable can be selected due to the p-values of the score of Rao. In that case, the process is terminated, and the model is presented with a mathematical formula, given as:

$$Z=B_1{X}_1+\dots +B_q{X}_q+B_0$$E2

being q the number of independent variables, and B the regression coefficients of the independent variables included in the model.

This model would explain the probability of the dependent value to be 1, that is, the possibility of an incident having a loss of position. The parameters that must be estimated are the regression coefficients B₀, B₁, …, Bq.

The column SE presents the standard error for estimating these coefficients, which is necessary for calculating the Wald statistic.

From here, the probability p of a case having an excursion is given by:

$$p=1/\left(1+e^{\left(-Z\right)}\right)$$E3

So the probability p for each case can be obtained. When the value p is less than 0.5, it will indicate that the model classifies this case in the first group (not having excursion), and when the value is bigger than 0.5, then the model predicts the case to have an excursion:

Moreover, the probability of not having any loss of position is:

$$q=1-p$$E4

Furthermore, the relative ratio is defined as:

$$\begin{array}{rcl}p/q& =& \left(1/\left(1+e^{\left(-Z\right)}\right)\right)/\left(1-1/\left(1+e^{\left(-Z\right)}\right)\right)\\ & =& \left(1/\left(1+e^{\left(-Z\right)}\right)\right)/\left(\left(1+e^{\left(-Z\right)}-1\right)/\left(1+e^{\left(-Z\right)}\right)\right)=1/e^{\left(-Z\right)}=e^{\left(Z\right)}\end{array}$$E5

Then, the mean relative ratio can be obtained. According to the definition of relative ratio, the i-th incident will be more likely to occur if P/Q > 1.0, while another incident will be more prone to be associated with not having an excursion when this ratio P/Q < 1.0.

4.2 Goodness of fit

It is not enough to give the model, as the goodness of fit must be checked to decide whether the model is good or not.

We have estimated the possibility of an incident having or not an excursion, but this does not necessarily need to be real. According to the model, the case can have a more significant possibility of belonging to the first group (no excursion) and yet belong to the second group (excursion). It is a bigger problem when the probabilities are close to 0.5. In this case, there is an error, the difference between the observed probability and the estimated probability E_i = p observed_i – p estimated_i, where p_i = can take the values 1 or 0, depending on whether the case belongs or not to the second group.

Evidencing the goodness of fit is checking how probable the obtained results for the estimated model are. It is based on comparing the number of cases that belong to the second group (excursion = yes) with the expected number if the model is valid. This expected number is the product of the total of cases in the sample by the estimated probability of belonging to the second group.

The statistic -2Log Likelihood (abbreviated -2LL) is used for this fit. When the -2LL results in low values, the likelihood is significant; the closer to zero, the bigger the likelihood.

Also, the following statistic can be used to compare the observed probabilities with the estimated from the model:

$$Z^2=\frac{{\sum }_{i=1}^n{E}_i^2}{{\mathit{\text{pestimated}}}_i\left(1-{\mathit{\text{pestimated}}}_i\right)}$$E6

They both follow a chi-square distribution with n-2 degrees of freedom under the hypothesis that the model adjusts to the observed data. It shows the percentage of correctly classified cases after the model has been defined.

When the percentage of correctly classified cases is high, it is expected to provide good results when predicting whether any incident will have an excursion or not.

5. Results

5.1 Descriptive statistics

All 42 cases were included in the analysis. There were no missing cases. 13 had a position loss from these cases, meaning 31% of the total.

5.1.1 Water depth

The mean water depth is 1409 ± 112 metres, the minimum 37 metres and the maximum 2838 metres. The distribution of this variable does not follow the normality.

5.1.2 Percentage of thrusters online

The mean usage of thrusters is 93 ± 2 per cent, with a minimum of 50% and a maximum of 100%. The distribution is not normal.

5.1.3 Percentage of generators online

The mean percentage of generators is 65 ± 3%, with a minimum of 33.33% and a maximum of 100%. This distribution is not normal.

5.1.4 DGNSS

In 3 cases (7%), there is only 1 DNGSS online, in 27 cases (64%), there are 2 DGNSS in function), in 7 cases (17%), there are 3 DGNSS online, and in 5 cases, there are 4 DGNSS working online (12%). The mean value is 2.33 ± 0.79. This distribution is not normal.

5.1.5 HPR

In 4 cases (10%), there was no HPR functioning; in 17 cases, there was 1 HPR working (40%), and in 21 cases, there were 2 HPRs in function (50%). Thus, the mean is set at 1.40 ± 0.67. This distribution is not normal.

5.1.6 Taut wire

There are 38 cases in which the taut wire is not used (91%), while in 3 cases, there is one taut wire in use (7%), and in 1 case, two taut wires were being used (2%). The mean is 0.12 ± 0.395.

5.1.7 Inertia system

In 40 cases (95%), this system is not used, while in 2 cases (5%), they are using it. The mean is then 0.05 ± 0.216. This distribution is not normal.

5.1.8 Gyros

In 1 case, only two gyros were used (2.5%), in 40 cases, three gyros were used (95%), and in 1 case, four gyros were used (2.5%). The mean value is 3 ± 0.221.

5.1.9 MRU

In 4 cases, there were 2 MRUs in use (9.5%), while in 38 cases, there were 3 MRUs in use (90.5%). The mean value is 2.90 ± 0.297.

5.1.10 Wind sensors

In 1 case, there was only one wind sensor online (2%), in 10 cases, there were two wind sensors (24%), in 26 cases, there were three wind sensors (62%), and in 5 cases, there were four wind sensors (12%). The mean is 2.83 ± 0.660.

5.1.11 Wind force

The mean wind force is 16 ± 1.88 knots, the minimum one and the maximum 55 knots metres. The distribution of this variable does not follow the normality.

5.1.12 Current speed

The mean current speed is 1.9 ± 0.23 knots. The minimum is 0.3, and the maximum is 6 knots. This distribution is not normal.

5.1.13 Wave height

The mean wave height is 1.88 ± 0.3 metres. The minimum is 0.1, and the maximum is 9.5 metres. This distribution is not normal.

5.1.14 Visibility

In 2 cases, the visibility was poor (5%), in 4 cases, it was moderate (9%), and in 36 cases, the visibility was good (86%).

5.1.14.1 Human cause

The human nature of the cause is considered when either the main cause or the secondary cause is human. There are 33 cases (78.6%) without human cause, while 9 cases (21.4%) were having a human error origin. The incidents end in an excursion in 10 cases (76.9%) when there is no human cause, and in 3 cases (23.1%) when there is a human cause. When there is not a loss of position, incidents without a human cause occur in 23 cases (79.3%), and incidents with a human cause happen in 6 cases (20.7%).

5.2 Binary logistic regression model: Dependent variable: excursion

As a previous stage, the variables are introduced in the model one by one to check their significance for explaining the answer. These variables are listed in Table 3.

Causal factor	B	Wald	p-value	Odds Ratio (Exp(B))	IC 95%
					lower	upper
Water depth	−0.001	4.645	0.031	0.999	0.998	1.000
Percentage of thrusters	Not in the equation
Percentage of generators	0.047	8.057	0.005	1.048	1.015	1.082
DGNSS	Not in the equation
HPR	—	0.000	1.000	—	—	—
Taut wire	—	0.000	1.000	—	—	—
Inertia system	Not in the equation
Gyros	Not in the equation
MRUs	−22.373	0.000	0.999	0.000	0.000	—
Wind sensors	—	5.389	0.145	—	—	—
Wind force	0.078	5.084	0.024	1.081	1.010	1.156
Force Beaufort	Not in the equation
Wind direction	Not in the equation
Current speed	Not in the equation
Current direction	Not in the equation
Wave height	0.437	3	0.081	1.549	0.947	2.532
Visibility ordinal	Not in the equation

Table 3.

Individual results in step 1 for each independent variable when the forward (Wald) binary regression model is performed, being excursion the dependent variable.

Considering these results, the following variables are selected to enter the model: Water-depth, percentage of generators online, Wind force and Wave height.

In step 0, when the variables are not yet in the equation, the more significant (having the smallest p-value) is the percentage of generators, so this is the variable that enters the equation in step 1. After this, in step 2, the variable water-depth is also included in the equation. The different statistics can be observed in Table 4.

Variables in the equation	B	S.E.	Wald	df	Sig.	Exp(B)	95% C.I.for EXP(B)
							Lower	Upper
Water depth (m)	−0.001	0.001	4.498	1	0.034	0.999	0.998	1.000
Percentage of generators online	0.051	0.018	7.732	1	0.005	1.052	1.015	1.091
Constant	−2.641	1.378	3.676	1	0.055	0.071

Table 4.

Variables in the equation in step 2. All variables are in step 2.

The following expression defines the model:

$$Z=-2.641-0.001\bullet \mathit{\text{Waterdepth}}+0.051\bullet \mathit{\text{Perc}}.\mathit{\text{ofgenerators}}$$E7

The mean ratio can then be expressed as:

$$\frac{p}{q}=e^{-2.64}\bullet e^{-0.001\bullet \mathit{\text{Waterdepth}}}\bullet e^{0.051\bullet \mathit{\text{Percentageofgenerators}}}$$E8

Alternatively, using the Odds Ratio (column Exp(B):

$$\frac{p}{q}=e^{-2.64}\bullet {0.999}^{\mathit{\text{Waterdepth}}}\bullet {1.052}^{\mathit{\text{Percentageofgenerators}}}$$E9

In Table 5, the values obtained from the binary regression model, Z, P and P/Q, for each incident can be found.

Incident	Water-depth	Percentage of generators	Z	P	P/Q	P estimated	P observed	E
1	338	66.67	0.42117	0.603763187	1.523743	1	0	1
2	1700	50.00	–1.79100	0.142950164	0.166793	0	0	0
3	1860	50.00	–1.95100	0.124444359	0.142132	0	0	0
4	744	33.33	–1.68517	0.156412089	0.185413	0	0	0
5	750	66.67	0.00917	0.502292484	1.009212	1	0	1
6	1656	50.00	–1.74700	0.148425985	0.174296	0	0	0
7	1656	50.00	–1.74700	0.148425985	0.174296	0	0	0
8	1656	62.50	–1.10950	0.247964116	0.329724	0	0	0
9	1900	50.00	–1.99100	0.120151107	0.136559	0	0	0
10	1782	50.00	–1.87300	0.133194979	0.153662	0	0	0
11	1782	50.00	–1.87300	0.133194979	0.153662	0	0	0
12	1782	50.00	–1.87300	0.133194979	0.153662	0	0	0
13	1782	50.00	–1.87300	0.133194979	0.153662	0	0	0
14	1782	57.14	–1.50886	0.181107803	0.221162	0	0	0
15	2465	50.00	–2.55600	0.072024433	0.077615	0	0	0
16	1718	50.00	–1.80900	0.140759028	0.163818	0	0	0
17	1233	42.86	–1.68814	0.156020605	0.184863	0	0	0
18	1340	50.00	–1.43100	0.19294292	0.239070	0	0	0
19	1700	100.00	0.75900	0.681136584	2.136139	1	0	1
20	1700	100.00	0.75900	0.681136584	2.136139	1	0	1
21	1250	50.00	–1.34100	0.207345657	0.261584	0	0	0
22	880	50.00	–0.97100	0.274681226	0.378704	0	0	0
23	2460	100.00	–0.00100	0.499750000	0.999000	0	0	0
24	2090	50.00	–2.18100	0.101469718	0.112929	0	0	0
25	1710	100.00	0.74900	0.678960765	2.114884	1	0	1
26	1300	50.00	–1.39100	0.199248161	0.248826	0	0	0
27	1437	50.00	–1.52800	0.178286498	0.216969	0	0	0
28	2838	37.50	–3.56650	0.027478187	0.028255	0	0	0
29	450	50.00	–0.54100	0.367954987	0.582166	0	0	0
30	315	100.00	2.14400	0.895106767	8.533503	1	1	0
31	798	50.00	–0.88900	0.291316235	0.411067	0	1	–1
32	2506	66.67	–1.74683	0.148447474	0.174326	0	1	–1
33	2118	42.86	–2.57314	0.070887218	0.076296	0	1	–1
34	38	66.67	0.72117	0.672864607	2.056838	1	1	0
35	108	66.67	0.65117	0.657274071	1.917783	1	1	0
36	959	100.00	1.50000	0.817574476	4.481689	1	1	0
37	854	100.00	1.60500	0.832716044	4.977860	1	1	0
38	1850	100.00	0.60900	0.647712655	1.838592	1	1	0
39	2100	100.00	0.35900	0.588798341	1.431897	1	1	0
40	37	100.00	2.42200	0.918489603	11.26837	1	1	0
41	1710	100.00	0.74900	0.678960765	2.114884	1	1	0
42	54	50.00	–0.145	0.463813380	0.865022	0	1	–1

Table 5.

Values obtained from the binary regression model, Z, P and P/Q, for each incident.

The goodness of fit is given by the -2LL statistic and the percentage of correctly classified cases. This statistic has a value of 42.732 for Step 1 and 37.510 for Step 2. This indicates that the goodness of fit is improved in Step 2 of the model.

Out of the 42 valid cases, 29 were not ending in an excursion, while 13 had a loss of position.

In Step 1, after the variable percentage of generators was included in the equation, it was obtained that from the 29 cases without excursion, according to the model, there are 25 cases correctly classified (86.2% of the total) and that from the ones having a loss of position, seven are correctly classified (53.8% of the total). There are 25 + 7 = 32 cases out of 42 that are correctly classified, representing 76.2% of the studied incidents.

In Step 2, when the variable water-depth is included in the equation, for the incidents not having excursion, the number of correctly-classified cases is maintained, and for the cases with excursion, there is an improvement in the number of correctly classified cases, which are now 8 (61.5% of the total). In this second step, there are now 33 cases correctly classified, which means 78.6% of the studied incidents. There has been an evident improvement in the model with the addition of the variable water depth.

Figure 1 graphically shows the model predictions for loss of position for different values of water-depth and percentage of generators.

Figure 1.

Prediction chart showing the trends for wind force and percentage of generators according to the prediction model, for cases with no human cause.

The relative ratio can show the prediction for loss of position for the different main causes, as shown in Figure 2. The dashed line allows us to appreciate better those mean values above 1, which show a higher likelihood of having a loss of position. The main causes that are more prone to end in an excursion are environmental, computer and human.

Figure 2.

Mean relative ratio for each main cause group.

The distribution of the mean relative ratio among the human cause or not of the incident is shown in Figure 3. The dashed line shows the value 1; above this value, the incidents are more prone to have a loss of position according to the prediction model. In this case, the incidents without a human cause have a bigger likelihood to end in a loss of position.

Figure 3.

Distribution of the mean relative ratio among the existence or not of a human cause for the incident.

5.3 Model stratified by human cause

Of the 42 selected cases, 9 have a main or secondary cause with a human origin, and 33 have no evidence of human causality.

There are no significant changes in the means of the variables when they are split into the subgroups human cause no and human cause yes, except for the variable percentage of thrusters, where it can be observed that the mean is 97.46 ± 1.48% when there is no human cause, and 74.54 ± 6.89% when there is a human cause.

5.3.1 No human cause

The 33 cases where there is no human cause are selected.

In the preliminary stage, the variables are introduced in the model one by one to check their significance for explaining the answer. The variables are presented in Table 6.

Causal factor	n	B	Wald	p-value	Odds Ratio (Exp(B))	IC 95%
						lower	upper
Water-depth	33	Not in the equation
Percentage of thrusters	33	Not in the equation
Percentage of generators	33	0.039	4.757	0.029	1.04	1.004	1.078
DGNSS	33	Not in the equation
HPR	33	Not in the equation
Taut wire	33	Not in the equation
Inertia system	33	Not in the equation
Gyros	33	Not in the equation
MRUs	33	22.547	0.000	0.999	6192653647	0.000	—
Wind sensors	33	Not in the equation
Windforce	33	0.090	4.368	0.037	1.095	1.006	1.191
Force Beaufort	33	Not in the equation
Wind direction	33	Not in the equation
Current speed	33	Not in the equation
Current direction	33	Not in the equation
Wave height	33	0.393	2.641	0.104	1.481	0.922	2.378
Visibility ordinal	33	Not in the equation

Table 6.

Individual results in step 1 for each independent variable when the forward (Wald) binary regression model is performed, being excursion the dependent variable and selection variable human cause = 0 (no human cause).

Considering these results, the following variables are selected to enter the model: Percentage of generators and Wind force.

In step 0, when the variables are not yet in the equation, the more significative (with less p-value) is wind force (score 7.085, p-value 0.008), so this is the variable that enters the equation in step 1. After this, in step 2, the variable water depth is also included in the equation (score 5.436, p-value 0.02). The different statistics obtained in Step 2 can be observed in Table 7.

Variables in the equation	B	S.E.	Wald	df	Sig.	Exp(B)	95% C.I.for EXP(B)
							Lower	Upper
Percentage of generators online	0.051	0.024	4.344	1	0.037	1.052	1.003	1.104
Windforce	0.120	0.061	3.876	1	0.049	1.128	1.001	1.271
Constant	−6.223	2.371	6.889	1	0.009	0.002

Table 7.

Variables in the equation in step 2.

The following expression defines the model:

$$Z=-6.223+0.051\bullet \mathit{\text{Perc}}.\mathit{\text{ofgenerators}}+0.12\bullet \mathit{\text{Windforce}}$$E10

The relative mean ratio of excursion can then be expressed as:

$$\frac{p}{q}=e^{-6.223}\bullet e^{0.051\bullet \mathit{\text{Percentageofgenerators}}}\bullet e^{0.12\bullet \mathit{\text{Windforce}}}$$E11

Alternatively, using the Odds Ratio (column Exp(B):

$$\frac{p}{q}=e^{-6.223}\bullet {1.052}^{\mathit{\text{Percentageofgenerators}}}\bullet {1.128}^{\mathit{\text{Windforce}}}$$E12

In Table 8, the values obtained from the binary regression model, Z, P and P/Q, for each incident can be found.

Incident	Percentage of generators	Wind force	Z	P	P/Q	P estimated	P observed	E
1	66.67	25.00	0.17717	0.544177004	1.193834	1	0	1
4	33.33	5.00	–3.92317	0.019394704	0.019778	0	0	0
5	66.67	15.00	–1.02283	0.264476518	0.359576	0	0	0
6	50.00	12.00	–2.233	0.096825971	0.107206	0	0	0
7	50.00	22.00	–1.033	0.262502905	0.355938	0	0	0
8	62.50	19.00	–0.7555	0.319624061	0.469776	0	0	0
9	50.00	10.00	–2.473	0.07777279	0.084331	0	0	0
10	50.00	4.00	–3.193	0.039429997	0.041049	0	0	0
11	50.00	19.00	–1.393	0.198929256	0.248329	0	0	0
12	50.00	15.00	–1.873	0.133194979	0.153662	0	0	0
13	50.00	12.00	–2.233	0.096825971	0.107206	0	0	0
14	57.14	18.00	–1.14886	0.24069737	0.316998	0	0	0
16	50.00	9.00	–2.593	0.069590289	0.074795	0	0	0
17	42.86	12.00	–2.59714	0.069322712	0.074486	0	0	0
18	50.00	2.00	–3.433	0.0312799	0.03229	0	0	0
19	100.00	4.00	–0.643	0.3445687	0.525713	0	0	0
20	100.00	13.00	0.437	0.607543959	1.548056	1	0	1
21	50.00	8.00	–2.713	0.0622106	0.066337	0	0	0
23	100.00	5.00	–0.523	0.372151001	0.592740	0	0	0
24	50.00	8.00	–2.713	0.0622106	0.066337	0	0	0
25	100.00	11.00	0.197	0.549091337	1.217744	1	0	1
27	50.00	6.00	–2.953	0.049594915	0.052183	0	0	0
29	50.00	16.00	–1.753	0.14766921	0.173253	0	0	0
30	100.00	50.00	4.877	0.992437784	131.2364	1	1	0
31	50.00	45.00	1.727	0.849028285	5.623757	1	1	0
32	66.67	11.20	–1.47883	0.185604206	0.227904	0	1	–1
33	42.86	18.00	–1.87714	0.132717725	0.153027	0	1	–1
34	66.67	26.00	0.29717	0.573750554	1.346044	1	1	0
35	66.67	18.00	–0.66283	0.340104178	0.515391	0	1	–1
37	100.00	55.00	5.477	0.995835559	239.1282	1	1	0
38	100.00	13.00	0.437	0.607543959	1.548056	1	1	0
39	100.00	7.00	–0.283	0.42971844	0.75352	0	1	–1
41	100.00	1.00	–1.003	0.268351995	0.366777	0	1	–1

Table 8.

Values obtained from the binary regression model, Z, P, and P/Q, for each incident without human cause.

The goodness of fit is given by the -2LL statistic and the percentage of correctly classified cases. This statistic has a value of 33.453 for Step 1 and 28.147 for Step 2. This indicates that the goodness of fit is improved in Step 2 of the model.

Out of the 33 valid cases, 23 were not ending in an excursion, while ten had a loss of position.

In Step 1, after the variable wind force was included in the equation, it was obtained that from the 23 cases without excursion, according to the model, all of them were correctly classified (100% of the total) and that from the ones having a loss of position, four are correctly classified (40% of the total). There are 23 + 4 = 27 cases out of 33 that are correctly classified, representing 66.7% of the studied incidents.

In Step 2, when the variable percentage of generators is included in the equation, for the incidents not having excursion, the number of correctly-classified cases has become 20, representing 87% of the total. There is an improvement in the number of correctly classified cases for the excursion cases, which are now 5 (50% of the total). In this second step, there are now 25 cases correctly classified, which means 75.8% of the studied incidents. Although, there has been an evident downgrade in the prediction of the model with the addition of the variable percentage of generators, the prediction for the cases with loss of position has improved.

In Figure 4, it can be seen how the model predicts a loss of position for more significant values of wind force and of the percentage of generators.

Figure 4.

Prediction chart showing the trends for wind force and percentage of generators according to the prediction model, for cases with no human cause.

The relative ratio can show the prediction for loss of position for the different main causes, as shown in Figure 5. The dashed line allows us to appreciate better those mean values above 1, which show a higher likelihood of having a loss of position. The main causes that are more prone to end in an excursion, according to this new model, are environmental and references.

Figure 5.

Mean relative ratio for each main cause group, for the model obtained for the incidents without human cause, obtaining the likelihood of a loss of position.

5.3.2 Human cause

The 9 cases where there is no human cause are selected.

In the preliminary stage, the variables are introduced in the model one by one to check their significance for explaining the answer. The variables are presented in Table 9.

Causal factor	n	B	Wald	p-value	Odds Ratio (Exp(B))	IC 95%
						lower	upper
Water depth	9	−0.006	0.912	0.34	0.994	0.981	1.007
Percentage of thrusters	9	Not in the equation
Percentage of generators	9	1.341	0	0.999	3.824	0	—
DGNSS	9	Not in the equation
HPR	9	—	0	1	—	—	—
Taut wire	9	Not in the equation
Inertia system	9	Constant value
Gyros	9	Constant value
MRUs	9	Constant value
Wind sensors	9	Not in the equation
Windforce	9	Not in the equation
Force Beaufort	9	Not in the equation
Wind direction	9	Not in the equation
Current speed	9	Not in the equation
Current direction	9	Not in the equation
Wave height	9	Not in the equation
Visibility ordinal	9	Not in the equation

Table 9.

Individual results in step 1 for each independent variable when the forward (Wald) binary regression model is performed, being excursion the dependent variable and selection variable human cause = 1 (human cause).

Considering these results, only the variable water depth is selected to enter the model.

In step 0, when the variable is not yet in the equation, it is considered significative (score 5.248, p-value 0.022), so it enters the equation in step 1. The different statistics obtained in Step 1 can be observed in Table 10.

Variables in the equation	B	S.E.	Wald	df	Sig.	Exp(B)	95% C.I.for EXP(B)
							Lower	Upper
Waterdepth	–0.006	0.007	0.912	1	0.340	0.994	0.981	1.007
Constant	5.592	6.426	0.757	1	0.384	268.187

Table 10.

Variables in the equation in Step 1. It can be observed that the independent variable, with a p-value of 0.34, cannot be considered to have any relation with the loss of position or not.

It can be observed that the p-value associated with the Wald statistic is bigger than 0.1, which means that this variable does not explain the model with the desired significance, and so it must be rejected.

Out of the nine valid cases, six were not ending in an excursion, while three were losing position.

In Step 1, after the variable water-depth was included in the equation, it was obtained that from the 6 cases without excursion, according to the model, there were 5 cases correctly classified (83.3% of the total) and that from the ones having a loss of position, two are correctly classified (66.7% of the total). There are 5 + 2 = 7 cases out of 9 that are correctly classified, representing 77.8% of the studied incidents.

6. Discussion

The first approximation to the regression model was to include the variables one by one to determine which variables could explain the answer.

It was interesting that the categorical variables did not explain the model. However, the data analysis could suggest that, had the sample been more prominent, they could have influenced the result. This study aimed not to distort the sample by performing a bootstrapping, not only because of the possible distortion of the sample but also because of the complication implied when the variables do not follow a normal distribution. The sample size is considered to be representative of the period of study.

With the first approach, the variables that could explain the probability of an excursion are determined to be: water-depth, percentage of generators online, Wind force and Wave height. The first two variables belong to the DP system configuration, and the last two are related to meteorological conditions.

However, when all of them are entered into the model, we obtained that only the first two explain the answer, while the other two could not improve the model already created. Although, this could give an idea of the less importance of the meteorological variables when explaining the excursion, it should not be forgotten that the meteorology, and especially the wind force (which creates waves with a height that is proportionally correlated to the force in knots), can also influence the probability of a unit having a loss of position while performing DP drilling operations.

The two selected independent variables, water-depth and percentage of generators, can explain the probability of losing position. This possibility will increase when the water depth has small values and the percentage of generators has significant values. These results are very interesting from the operator’s point of view, as the lower values of water depths have traditionally been a common drilling ground where DP was not necessary, and other methods were used to achieve the position keeping. This could partly explain the problems of DP station keeping incidents when the drilling operations take place in shallow waters.

Studying the mean relative ratios for each main cause group, it is interesting to note that the model can explain environmental-, computer- and human-caused incidents more precisely than other causes. Within the group of human causes, the incidents without a human cause have a better prediction using the proposed model.

In general, this model correctly classifies 79% of the incidents, which is considered to be a very good prediction overall.

When the data is split into subgroups defined by the existence or not of a human cause, it can be seen how the mean percentage of thrusters is significantly more prominent for the cases where there is no human cause and smaller when there is a human cause. However, it does not explain whether the probability of an excursion is bigger or smaller for any of the subgroups. However, it can suggest that it would be a significant variable when the dependent variable is used to determine the possibility of a human error.

Always taking the general regression model from above into account, the regression model for the subgroup without human cause proposes the percentage of generators and wind force as variables that could explain the model. Wave height is not significant (although with a p-value of 0.104, it could be said that it is at the edge of being significant), and the water depth does not even enter the equation.

The fact that water depth was not even entering the equation when considered individually suggests it does not influence the probability of excursion when the cause is not human.

When studying the cases in the subgroup human cause, we obtain that the only variable that could explain the model is water depth. However, its p-value is bigger than 0.1, after entering the equation and because of this, it is rejected. In the iteration of the model, it can be seen how when this variable is included, the percentage of cases that are correctly classified decreases for the first group (no excursion). At the same time, the percentage of correctly-classified cases improves for group 2 (excursion). Probably the comparatively small number of cases (only 9 out of the 42 cases in total) contribute to the decision of rejecting this variable from the equation. Nonetheless, it suggests that this variable can be expected to be added to the model when a bigger sample is studied.

7. Conclusions

The purpose of this chapter was to determine the mathematical expression that explains the possibility of a loss of position during DP drilling operations.

With a sample of 42 incidents from 2011 till 2015, it was determined that the mathematical expression for the binary logistic regression model is shown in Eq. 9. The loss of position of an incident depends on the water depth and the percentage of generators used.

With this model, it can be determined that the probability of loss of position will increase when the water-depth has small values and the percentage of generators has bigger values.

Having considered that the percentage of cases correctly classified by the model which takes into account both variables percentage of generators online and water-depth is high (78.6%), it is expected to provide excellent results when predicting whether any incident will have a loss of position or not.

Once this model has been determined, the secondary objective of this paper was to find and compare the mathematical expression, taking into account whether the nature of the cause leading to the incident was human or not.

With a sample of 33 incidents without human cause, it was determined that the mathematical expression for determining the probability of loss of position is given in Eq. 13. This model can determine that the probability of having an excursion increases as the percentage of generators and the wind force have bigger values.

The percentage of cases correctly classified by the model according to this sample is high (75.8%), and it can be expected to provide excellent results when predicting whether an incident that has no human origin will have a loss of position or not.

The sample of the incidents with human cause was relatively small (9 cases only), so no independent variable could explain the model within a confidence interval of 10%. However, the variable water-depth, which appears above in the general model, and does not appear in the model for the cases without human cause, can be suspected to explain the model, although it will be necessary to perform further research on a bigger sample to obtain significant results.

Conflict of interest

“The authors declare no conflict of interest.”

Nomenclature

-2LL	-2LogLikelihood, statistic used to define the goodness of fit of a model
B	regression coefficients of the independent variables in the model
DGNSS	Differencial Global Navigation Satellite System
DP	dynamic positioning
DPO	dynamic positioning operator
E	Error between the estimated probability and the observed probability
HMI	Human Machine Interface
HPR	Hydro-acoustic Positioning References
IMCA	International Marine Contractors Association
LMRP	lower marine riser package
MRU	motion reference units
p	probability that the dependent variable will obtain a value 1
p/q	Mean relative ratio of the probability of the variable to obtain a value of 1
POUT	output value
PRS	position reference systems
q	probability that the dependent variable will obtain a value 0
SE	standard error
Wald	Statistic used to determine whether a variable is contributing to defining the model or not
Z	formula defining the model