PARAMETRIC MODEL DEVELOPMENT The parametric study of ship designs requires

models that relate form, characteristics, and performance to the fundamental dimensions, form coefficients, and parameters of the design. Various techniques can be used to develop these models. In pre-computer days, data was graphed on Cartesian, semi-log, or log-log coordinates and if the observed relationships could be represented as straight lines in these coordinates linear (y = a₀ + a₁x), exponential (y = abx_{), and geometric (y = ax}b_{) models, respectively,}

were developed. With the development of statistical computer software, multiple linear regression has become a standard tool for developing models from data for similar vessels. More recently, Artificial Neural Networks (ANN) have begun to be used to model nonlinear relationships among design data. This Section provides an introduction to the development of ship models from similar ship databases using multiple linear regression and neural networks.

11.5.1 Multiple Linear Regression Analysis Regression analysis is a numerical method which can be used to develop equations or models from data when there is no or limited physical or theoretical basis for a specific model. It is very useful in developing parametric models for use in early ship

design. Effective capabilities are now available in personal productivity software, such as Microsoft Excel.

In multiple linear regression, a minimum least squares error curve of a particular form is fit to the data points. The curve does not pass through the data, but generalizes the data to provide a model that reflects the overall relationship between the dependent variable and the independent variables. The effectiveness (goodness of fit) of the modeling can be assessed by looking at the following statistical measures:

R = coefficient of correlation which expresses how closely the data clusters around the regression curve (0≤R≤1, with 1 indicating that all the data is on the curve).

R2 _{= coefficient of determination which}

expresses the fraction of the variation of the data about its mean that is captured by the regression curve (0≤R2_{≤1, with 1 indicating}

that all the variation is reflected in the curve). SE = Standard Error which has units of the

dependent variable and is for large n the standard deviation of the error between the data and the value predicted by the regression curve.

The interpretation of the regression curve and Standard Error is illustrated in Figure 11.26 where for an example TEU capacity is expressed as a function of Cubic Number CN. The regression curve will provide the mean value for the population that is consistent with the data. The Standard Error yields the standard deviation σ for the normal distribution (in the limit of large n) of the population that is consistent with the data.

The modeling process involves the following steps using Excel or a similar program:

1. select independent variables from first principles or past successful modeling; 2. observe the general form of the data on a

scatter plot,

3. select a candidate equation form that will model the data most commonly using a linear, multiple linear, polynomial, exponential, or geometric equation, 4. transform the data as needed to achieve a

linear multiple regression problem (e.g. the exponential and geometric forms require log transformations), 5. regress the data using multiple linear

regression,

6. observe the statistical characteristics R, R2_{, and SE,}

7. iterate on the independent variables, model form, etc. to provide an acceptable fit relative to the data quality.

Fi gure 11.25 - Sampl e Seakeepi n g Predi cti on Program (SPP) Output (40)

pdf TEU TEU o mean µ CN regression curve CN o Standard Error σ Normal probability density function

Figure 11.26 - Probabilistic Interpretation of Regression Modeling

.

Numerous textbooks and software user's manuals can be consulted for further guidance and instructions if the reader does not currently have experience with multiple linear regression.

11.5.2 Neural Networks

An Artificial Neural Network (ANN) is a numerical mapping between inputs and output that is modeled on the networks of neurons in biological systems (62, 63). An ANN is a layered, hierarchical structure consisting of one input layer, one output layer and one or more hidden layers located between the input and output layers. Each layer has a number of simple processing elements called neurons (or nodes or units). Signal paths with multiplicative weights w interconnect the neurons. A neuron receives its input(s) either from the outside of the network (i.e., neurons in the input layer) or from the other neurons (those in the input and hidden layers). Each neuron computes its output by its transfer (or activation) function and sends this as input to other neurons or as the final output from the system. Each neuron can also have a bias constant b included as part of its transfer function. Neural networks are effective at extracting nonlinear relationships and models from data. They have been used to model ship parametric data (64, 65) and shipbuilding and shipping markets (66).

A typical feedforward neural network, the most commonly used, is shown schematically in Figure 11.27. In a feedforward network the signal flow is only in the forward direction from one layer to the next from the input to the output. Feedforward neural networks are commonly trained by the supervised learning algorithm called backpropagation. Backpropagation uses a gradient decent technique to adjust the weights and biases of the neural network in a backwards, layer-by-layer manner. It adjusts the weights and biases until the vector of the neural network outputs for the corresponding vectors of

training inputs approaches the required vector of training outputs in a minimum root mean square (RMS) error sense. The neural network design task involves selection of the training input and output vectors, data preprocessing to improve training time, identification of an effective network structure, and proper training of the network. The last issue involves a tradeoff between overtraining and under training. Optimum training will capture the essential information in the training data without being overly sensitive to noise. Li and Parsons (67) present heuristic procedures to address these issues.

The neurons in the input and output layers usually have simple linear transfer functions that sum all weighted inputs and add the associated biases to produce their output signals. The inputs to the input layer have no weights. The neurons in the hidden layer usually have nonlinear transfer functions with sigmoidal (or S) forms the most common. Neuron j with bias b_j and n inputs each with signal x_i and weight w_ij will have a linearly combined activation signal z_j as follows:

n

z_j =

Σ

w_ij x_i + b_j [85] i=1

A linear input or output neuron would just have this z_j as its output. The most common nonlinear hidden layer transfer functions use the exponential logistic function or the hyperbolic tangent function, respectively, as follows:

y_j = (1+ e–zj₎ –1 _[86]

y_{j = tanh(zj}) = (ezj – e–zj_{)/( e}zj _{+ e}–zj_{) [87]}

These forms provide continuous, differentiable nonlinear transfer functions with sigmoid shapes.

One of the most important characteristics of neural networks is that they can “learn” from their training experience. Learning provides an adaptive capability that can extract nonlinear parametric relationships from the input and output vectors without the need for a mathematical theory or explicit modeling. Learning occurs during the process of weight and bias adjustment such that the outputs of the neural network for the selected training inputs match the corresponding training outputs in a minimum RMS error sense. After training, neural networks then have the capability to generalize; i.e., produce useful outputs from input patterns that they have never seen before. This is achieved by utilizing the information stored in the weights and biases to decode the new input patterns.

Figure 11.27 - Schematic of (4x4x1) Feedforward Artificial Neural Network Theoretically, a feedforward neural network can

approximate any complicated nonlinear relationship between input and output provided there are a large enough number of hidden layers containing a large enough number of nonlinear neurons. In practice, simple neural networks with a single hidden layer and a small number of neurons can be quite effective. Software packages, such as the MATLAB neural network toolbox (68), provide readily accessible neural network development capabilities.

11.5.3 Example Container Capacity Modeling The development of parametric models using Multiple Linear Regression Analysis and Artificial Neural Networks will be illustrated through the development of models for the total (hull plus deck) TEU container capacity of hatch covered cellular container vessel as a function of LBP, B, D, and V_k. A mostly 1990’s dataset of 82 cellular container ships ranging from 205 to 6690 TEU was used for this model development and testing. To allow a blind model evaluation using data not used in the model development, the data was separated into a training dataset of 67 vessels for the model development and a separate test dataset of 15 vessels for the final model evaluation and comparison. The modeling goal was to develop a generalized estimate of the total TEU

The total TEU capacity of a container ship will be related to the overall vessel size and the volume of the hull. Perhaps the most direct approach would be to estimate the total TEU capacity using LBP, B, and D in meters as independent variables in a multiple linear regression model. This analysis was performed using the Data Analysis option in the Tools menu in Microsoft Excel to yield the equation,

TEU = – 2500.3 + 19.584 LBP + 16.097 B + 46.756 D [88]

(n = 67, R = 0.959, SE = 469.8 TEU)

This is not a very successful result as seen by the Standard Error in particular. Good practice should report n, R, and SE with any presented regression equations.

The container block is a volume so it would be reasonable to expect the total TEU capacity to correlate strongly with hull volume, which can be represented by the metric Cubic Number CN = LBP•BD/100. The relationship between the TEU capacity and the Cubic Number for the training set is visualized using the Scatter Plot Chart option in Excel in Figure 11.28. The two variables have a strong linear correlation so either a linear equation or a quadratic equation in CN could provide an effective model. Performing a linear regression analyses yields

TEU = 142.7 + 0.02054 CN [89]

(n = 67, R = 0.988, SE = 254.9 TEU)

which shows a much better Coefficient of Correlation R and Standard Error.

The speed of vessel affects the engineroom size, which competes with containers within the hull volume, but could also lengthen the hull allowing more deck containers. It is, therefore, reasonable to try as independent variables CN and V_{k to see if further} improvement can be achieved. This regression model is as follows:

TEU = – 897.7 + 0.01790 CN + 66.946 V_k [90]

(n = 67, R = 0.990, SE = 232.4 TEU)

which shows a modest additional improvement in both R and SE. Although the relationship between total TEU capacity and CN is highly linear, it is still reasonable to investigate the value of including CN2 as a third independent variable. This multiple linear regression model is as follows:

TEU = – 1120.5 + 0.01464 CN

+ 0.000000009557CN2 + 86.844 V_k [91]

(n = 67, R = 0.990, SE = 229.1 TEU)

which shows, as expected, a small coefficient for CN2 and only a small additional improvement in SE.

To illustrate an alternative approach using simple design logic, the total TEU capacity could be postulated to depend upon the cargo box volume L_cBD. Further, the ship could be modeled as the cargo box, the bow and stern portions, which are reasonably constant fractions of the ship length, and the engine room that has a length which varies as the speed V_k. This logic gives a cargo box length L_{c = L – aL – bVk} and a cargo box volume L_cBD = (L – aL – bV_k)BD = (1 – a)LBD – bBDV_k. Using these as the independent variables with CN in place of LBD yields the alternative regression equation,

TEU = 109.6 + 0.01870 CN + 0.02173 BDV_k

(n = 67, R = 0.988, SE = 256.1 TEU) [92]

which is possibly not as effective as the prior two models primarily because the largest vessels today are able to carry containers both on top of the engine room and on the stern.

Figure 11.28 - Total TEU Capacity versus Metric Cubic Number

For comparison, a (4x4x1) neural network was developed by David J. Singer using inputs LBP, B, D, and V_k and output TEU. The ANN has four linear neurons in the input layer, one hidden layer with four nonlinear hyperbolic tangent neurons, and a single linear neuron output layer. This neural network was trained with the MATLAB Neural Network Toolbox (68) using the 67 training container ships used to develop the linear regression models. This ANN design evaluated nets with 2, 4, 6, 8, and 10 hidden layer neurons with 4 giving the best results. The ANN was trained for 500 through 5000 epochs (training iterations) with 2500 giving the best results.

To evaluate the performance of the regression equations and neural network using data that was not used in their development, the final 15 test ships were used to test the neural network and the five regression equations presented above. They were compared in terms of their RMS relative error defined as,

15

RMS_i = {

Σ

((TEU_j – TEU_ij)/TEU_j)2/15}1/2 [93] j=1

where index i indicates the model and index j indicates the test dataset vessel. A summary of these results is shown in Table 11.X. The most effective regression equation for this test data is equation 90, which had the highest R and nearly the lowest Standard Error. The ANN performed similarly. Note that for this highly linear example, as shown in Figure 11.28, the full capability of the nonlinear ANN is not being exploited. Table 11.X- MAXIMUM AND RMS RELATIVE

ERROR FOR REGRESSIONS AND ANN

In document Empirical Formula (Page 74-79)