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| | See code at [[ToyProblem_cmr.m]] |
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| =Problem Description= | | =Problem Description= |
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| Problem description: http://files.charlesmartinreid.com/VUQ_Toy_Problem.pdf | | <s> |
| | Problem description: http://files.charlesmartinreid.com/ExperimentalDesign/VUQ_Toy_Problem.pdf |
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| =Inputs and Outputs= | | =Inputs and Outputs= |
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| Imagine constructing a response surface for the observable <math>Y_{P,z1}</math> as a function of only one variable, <math>z_1</math>. This response surface <math>Y_{P,z1}\left( z_1 \right)</math> is simply the concentration <math>Y_P</math> as a function of location. As <math>z_1</math> is varied, different concentrations <math>Y_P</math> are observed - just as different concentrations are observed when the mixing length is changed, or when the reaction rate constant is changed. Just because the input parameter <math>z_i</math> is intuitively easier to connect to the observable <math>Y_P</math> doesn't mean that it can't be treated as an input variable! | | Imagine constructing a response surface for the observable <math>Y_{P,z1}</math> as a function of only one variable, <math>z_1</math>. This response surface <math>Y_{P,z1}\left( z_1 \right)</math> is simply the concentration <math>Y_P</math> as a function of location. As <math>z_1</math> is varied, different concentrations <math>Y_P</math> are observed - just as different concentrations are observed when the mixing length is changed, or when the reaction rate constant is changed. Just because the input parameter <math>z_i</math> is intuitively easier to connect to the observable <math>Y_P</math> doesn't mean that it can't be treated as an input variable! |
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| ==Dealing with Multimodal Variables== | | =See Also= |
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| Sometimes, when constructing response surfaces, modal variables appear. Modal variables are variables that have multiple modes, or distinct sets of values. There are two variations of modal variables:
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| ===1 uncertainty range (sampled with N parameter values)===
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| These types of modal variables have a single range of uncertainty assigned to them, but the values within that range of uncertainty are discrete. In order to sample the parameter within the range of uncertainty, the parameter must be sampled at distinct, discrete values.
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| For example, if I am using the discrete ordinates model (DOM) for radiation calculations, the DOM requires a number of ordinate directions. This is a discrete value with distinct sets of values - e.g. 3, 6, 8, 24, etc.
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| Each discrete value in this case composes a single range of uncertainty. Using the DOM example, that range of uncertainty would be <math>[3, 24]</math>.
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| ===N uncertainty ranges===
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| The other type of modal variables have several ranges of uncertainty assigned to them, with no restriction on values within that range of uncertainty being discrete or distinct. Essentially this can be thought of as a bimodal uncertainty distribution, where the two modes are distinct. Each mode can be sampled as usual, the only sticking point is that there is more than 1, and that they are distinct.
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| This case provides an excellent example. The variable <math>\dot{m}</math> is a modal variable - the two modes are 1.0 and 2.0 - but each mode also has a range of uncertainty, namely <math>5%</math> each.
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| ===How to Deal===
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| Multimodal variables can be dealt with in two ways:
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| '''Method 1: Separate Response Surfaces for Each Mode'''
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| The first way is to create a separate response surface for each distinct mode. This method works for both types of modal variables ([[Example_Problem_for_Experimental_Design#1_uncertainty_range_.28sampled_with_N_parameter_values.29|1 uncertainty range represented by N distinct values]], and [[Example_Problem_for_Experimental_Design#N_uncertainty_ranges|N uncertainty ranges]]). This method is illustrated in the figures below. Each distinct mode (gray region) has its own computed response surface (blue dotted line), distinct from the response surface of the other modes.
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| Of course, if the variable type is [[Example_Problem_for_Experimental_Design#1_uncertainty_range_.28sampled_with_N_parameter_values.29|1 uncertainty range represented by N distinct values]], then there is no uncertainty range for each mode, and each gray region is, in fact, a delta function. As mentioned above, this means that the input variable is eliminated as a response surface parameter.
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| If the variable type is [[Example_Problem_for_Experimental_Design#N_uncertainty_ranges|N uncertainty ranges]], then each uncertainty range is sampled as usual, and each response surface is constructed as usual.
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| {|class="wikitable"
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| |[[Image:ModalResponses1_true.png|250px]]
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| |[[Image:ModalResponses2_modes.png|250px]]
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| |[[Image:ModalResponses3_modalresponses.png|250px]]
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| |An example of a "true" response, which is unknown to the modeler.
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| |The modeler is only interested in distinct regions of the input parameter <math>x</math> (shown in gray). The remaining regions are left out of the response surface.
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| |The response surfaces actually obtained by the user (blue dotted line). There is a separate response surface obtained by the user (2 distinct blue lines) for each mode (gray region).
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| |}
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| '''Method 2: Single Response Surface (Ignore Modes)'''
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| A second way is to create a ''single'' response surface. This is typically only possible with [[Example_Problem_for_Experimental_Design#N_uncertainty_ranges|N uncertainty ranges]] type of problems, because the parameter value is continuous, but it is only certain regions that are of interest. This approach is illustrated below.
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| Essentially, this approach does away with ''any'' special treatment of modes.
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| {|class="wikitable"
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| |[[Image:ModalResponses1_true.png|250px]]
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| |[[Image:ModalResponses2_modes.png|250px]]
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| |[[Image:ModalResponses4_fullresponse.png|300px]]
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| |-
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| |(see above, image repeated for clarity)
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| |(see above, image repeated for clarity)
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| |An example of the second approach, in which the modeler constructs a single response surface, essentially ignoring the modes of the input parameter <math>x</math>.
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| |}
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| =Analysis of Results and Construction of Response Surface=
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| The ultimate reason for sampling the function is to construct a response surface, and in order to construct a response surface, some kind of generalized linear model will have to be used.
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| ==Model Classification==
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| ===Generalized Linear Models===
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| "Generalized linear models" are linear models that can account for arbitrary numbers of inputs and outputs. These models assume errors are Gaussian, use statistics and create statistical models for data analysis.
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| General linear model information:
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| * http://en.wikipedia.org/wiki/Multivariate_regression_model
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| * Nelder and Wedderburn (1972)
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| ===Multiple Linear Regression===
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| "Multiple linear regression" is a model for one response variable ("y"), and multiple predictor variables ("X").
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| Linear regression information:
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| * http://en.wikipedia.org/wiki/Linear_regression
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| * http://www.mathworks.com/help/toolbox/stats/mvregress.html
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| ===Multivariate Linear Regression===
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| "Multivariate linear regression" broadens multiple linear regression to account for more than one response variable ("Y").
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| Multivariate regression/analysis information:
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| * http://en.wikipedia.org/wiki/Multivariate_analysis
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| ===Polynomial Models (Univariate)===
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| Polynomial models can be used to fit a univariate function of a single input paramter, e.g. <math>y(x)</math>
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| This can be done using the following Matlab functions:
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| * polyfit
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| ** http://www.mathworks.com/help/techdoc/ref/polyfit.html
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| ** Fits a polynomial of a given degree to a set of inputs x and outputs y
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| * polyval
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| ** http://www.mathworks.com/help/techdoc/ref/polyval.html
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| ** Evaluates the value of a given polynomial model at given input variable value or values
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| * polyconf
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| ** http://www.mathworks.com/help/toolbox/stats/polyconf.html
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| ** Can be used to construct confidence intervals for polynomial models
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| | * [[Experimental Design Lecture]] |
| | * [[Response Surface Methodology]] |
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| ===Response Surface Models (Multivariate)===
| | {{ExperimentalDesign}} |
See code at ToyProblem_cmr.m
Problem Description
Problem description: http://files.charlesmartinreid.com/ExperimentalDesign/VUQ_Toy_Problem.pdf
Inputs and Outputs
There are several (polynomial) response surfaces being fit for the Monte Carlo simulations. This is because there is one response surface for each output or observable. (Technically, these are all part of one large multivariate response surface, but it is easier to think about them as independent response surfaces).
The variables that are included in the response surface analysis are:
- $ z_i $ (for $ i=1,2,3 $) - the location of measurement of axial concentrations
- $ \dot{m} = \dot{m}_1 = \dot{m}_2 $ - the mass flowrate of the inlet streams of A and B
- $ L_{mix} $ - mixing length (parameter for the mixing model)
- $ k(T) $ - reaction rate for the reaction $ A + B \rightarrow^{k} P $
Variables: I/U Map
| Variable Name
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Input value (I)
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Uncertainty (U)
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Logarithmic Scale?
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| $ z_1 $
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$ 0.5 m $
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$ \pm 0.02 m $
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no
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| $ z_2 $
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$ 1.5 m $
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$ \pm 0.02 m $
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no
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| $ z_3 $
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$ 2.5 m $
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$ \pm 0.02 m $
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no
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| $ \dot{m}_1 $
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$ 1.0 $
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$ \pm 0.05 $
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no
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$ 2.0 $
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$ \pm 0.10 $
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| $ \dot{m}_2 $
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$ 1.0 $
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$ \pm 0.05 $
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no
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$ 2.0 $
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$ \pm 0.10 $
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| $ k(T) $
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$ 1 $
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$ 10^{0 \pm 2} $
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yes
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| $ L_{mix} $
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$ 0.3-3.0 $
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yes
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Response Surfaces
Product at Exit Response Surface
This response surface maps the response of the mass fraction of product at the exit, $ y_{P,exit} $.
This response surface is a function of several variables:
$ y_{P,exit} = y_{P,exit} \left( \dot{m}, k, L_{mix} \right) $
However, the axial location of measurement of product $ z_i $ is not considered, because it does not affect the measurement of P at the exit.
Product at Axial Location Response Surfaces
This response surface maps the response of the mass fraction of product at several axial locations, $ y_{P,z1}, y_{P,z2}, y_{P,z3} $.
This response surface is a function of all variables:
$ y_{P,zi} = y_{P,exit} \left( \dot{m}, k, L_{mix}, z_i \right) $
(Of note is that only the corresponding $ z_i $ will be a response surface independent variable, since other $ z_i $ values have no affect).
NOTE: It is easy to get confused about why a parameter dealing with the model output, like the location at which the observable is actually observed, can be part of the input. However, given some thought, it is easy to see how this is an input variable.
Imagine constructing a response surface for the observable $ Y_{P,z1} $ as a function of only one variable, $ z_1 $. This response surface $ Y_{P,z1}\left( z_1 \right) $ is simply the concentration $ Y_P $ as a function of location. As $ z_1 $ is varied, different concentrations $ Y_P $ are observed - just as different concentrations are observed when the mixing length is changed, or when the reaction rate constant is changed. Just because the input parameter $ z_i $ is intuitively easier to connect to the observable $ Y_P $ doesn't mean that it can't be treated as an input variable!
See Also
| charlesmartinreid.com : Experimental Design |
|---|
| | Lecture Set | | | Introduction Lecture | | | | Example Problem | | | | Regression Methods | | | | Monte Carlo |
MC Method | |  | | MC Implementation | | | MC Results | |
| | | Composite Design |
CD Method | |  | | CD Implementation | | | CD Results | |
| | | Charles Martin Reid:Copyrights |
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