DOE :The Standard Full 2nd Order Polynomial Model, The Kriging Model, Non-Parametric Regression Model, Neural Network Model, Sparse Grid model

Introducing the Standard Full 2nd Order Polynomial Model

The standard full 2nd order polynomials are the starting point for many design points. This model is based on a modified quadratic formulation; thus, each output parameter or variable is a quadratic function of the input parameters or variables. This method will result in satisfactory results when changes in output parameters or variables are made smoothly or gently.

In this model, write the output parameter function in terms of input variables as follows; So that the function f is a quadratic polynomial function:

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Introducing the Kriging Model

The kriging model is a multidimensional interpolation that combines a polynomial model similar to one of the standard response levels considered as a global model of the design space, plus a specific local deviation so that this model can intercept design points.

In this model, write the output parameter function in terms of input variables as follows; So that function f is a quadratic polynomial function (expressing the general behavior of the model), and function z is a term of deviation or turbulence (expressing the local behavior of the model):

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Since the kriging model fits the contact surfaces at all design points, the goodness of fit criterion will always be appropriate. The kriging model will have better results than the standard response surface model; whenever the output parameters are stronger and nonlinear. One of the disadvantages of this model is that fluctuations can occur on the response surface.

Figure 1 shows the behavioral pattern of the function related to the kriging model. As we can see from the figure, the behavior of the estimated function consists of a general function (f) combined with a local function (z).

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In the kriging model, it is also possible to apply refinements to the design points. This model can determine the accuracy of contact surfaces and can also determine the points needed to increase accuracy. In this model, modify the design points (refinement type)  manually and automatically.

The refinement section of this model, like the previous model, has configuration options. Application of parts of the maximum number of refinement points, crowding distance separation percentage, output variable combinations, and several refinement points as in the previous model Is. The maximum predicted relative error option is also used to define the maximum percentage of relative error indicated during the process.

Verification points can also be activated to detect and determine the quality of the response surface. This mode is recommended when creating the response level using the kriging model. The operating mechanism of this type of point is that it compares the estimated values ​​of the desired output parameter and the actual values ​​observed from the same output parameter in different positions of the design space.

Verification points can also be defined automatically and manually for software. Also, if checkpoints are activated, these points will be added to the goodness of fittable.

Introduction of Non-Parametric Regression Model

The non-parametric regression model tends to be a general class of support vector method (RSM) techniques. The basic idea of ​​this model is that the tolerance or tolerance of the epsilon (Ԑ) forms a narrow envelope around the output response surface and extends around it. This envelope space should include all the design sample points or most of these points.

We create instability regression to estimate the regression function directly; this means that instability regression can examine the effect of one or more independent variables on a dependent variable without considering a particular function to establish the relationship between the independent and dependent variables.

Figure 2 shows the behavioral pattern of the function related to the nonparametric regression model. As we can see from the figure, the behavior of the estimated function consists of a principal response surface function with a margin of tolerance on either side.

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In general, the features of the non-parametric regression model are:

  • Suitable for non-linear responses.
  • Used when the results are so-called noisy; when the number of results is enormous, this model can approximate the design points by considering a tolerance limit (Ԑ).
  • It usually has a low computational speed.
  • It recommends to use only when the goodness of fit criterion of the quadratic response level model does not reach the desired level.
  • In some specific problems, such as lower-order polynomials, fluctuations may occur between the design points of the test environment.

Introduction of Neural Network Model

The neural network model represents a mathematical technique based on natural neural networks in the human brain. The structure of this neural network model is such that each of the input parameters (inputs) are connected to weights by arrows, which determine the active or inactive hidden functions. And hidden functions are the threshold functions that are connected or disconnected to the output function based on a set of their input parameters; finally, each time the process is repeated, these weight functions are adjusted to minimize the error between the response levels or the same output functions with the design points or the same inputs.

Figure 3 shows the behavioral algorithm of the cellular network model. This algorithm consists of input parameters, hidden functions, and output functions.

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In general, the features of the Neural Network model are:

  • Successful for high nonlinear responses.
  • The control of this algorithm is very restrictive.
  • Seventy percent of the design points are learning points and thirty percent are checking points.
  • Used when input parameters and the number of design points for each parameter are significant.

Introduction of Sparse Grid Model

The sparse grid model is a kind of adaptive response surface; it can constantly correct itself automatically. This model usually requires more design points than other methods of creating response surfaces and is used when the model simulation solution process is fast. This model will be usable when the sparse grid initialization design method is used to generate design points in the experiment design. The capability of this model is that it modifies the design points only in the directions that are required; for this reason, it needs fewer design points to achieve the same quality response level. This model is also suitable for cases that involve multiple discontinuities.

Figure 4 shows the behavioral pattern of a distributed network model and how it is hierarchically interpolated. According to the figure below, the first cell is located at the top left and has a design point. We see the cells ‘ changes and the design points by following the cells in the horizontal and vertical directions. The peak symbol indicates the interpolation of a design point from two design points on either side of the cell. The bottom symbol indicates the division of a cell into several other cells at the location of the generated design points.

Now, if we follow the path of change of a cell and the design points in it in the horizontal direction, we see that first, we have a cell with a design point in the middle of it. That cell from the design point in the horizontal direction to The two semicircles is divided, and the new design points of these two semicircles are placed on its borders. Interpolation is made between both cell borders, and new design points are created in the middle of these borders. As a result, it creates two design points in the middle of the two cells again. The new cell is divided from the two design points created into two other half-cells in the horizontal direction. The same procedure continues in the horizontal direction. According to the figure, it does all the said steps horizontally, with the same procedure in the vertical direction.

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