**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**.

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**.


The following expression defines the model:

#### **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.*


#### **Table 4.**

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

$$Z = -2.641 - 0.001 \bullet Water depth + 0.051 \bullet perc\,\text{of\\$}\text{perxators} \tag{7}$$

The mean ratio can then be expressed as:

$$\frac{p}{q} = e^{-2.64} \bullet e^{-0.001 \bullet Water depth} \bullet e^{0.051 \bullet Percentage of generators} \tag{8}$$

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

$$\frac{p}{q} = e^{-2.64} \text{\*} 0.999^{Waterdepth} \text{\*} \mathbf{1.052}^{Percentage of generators} \tag{9}$$

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

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



#### **Table 5.**

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

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.

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.

*Prediction Analysis Based on Logistic Regression Modelling DOI: http://dx.doi.org/10.5772/intechopen.103090*

#### **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.*

#### **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.

#### **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.

#### **Figure 3.**

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

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**.

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**.

The following expression defines the model:

$$Z = -6.223 + 0.051 \bullet \text{Per. of generators} + 0.12 \bullet \text{Wind force} \tag{10}$$

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

$$\frac{p}{q} = e^{-6.223} \bullet e^{0.051 \bullet \text{Percentage of generators}} \bullet e^{0.12 \bullet \text{Wind force}} \tag{11}$$

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

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#### **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).*


**Table 7.**

*Variables in the equation in step 2.*

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

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


#### **Table 8.**

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