The Uncertainty Quantification Toolbox

The intent of subproblem B is to explore answers to the resource allocation problem where the objective is to identify key parameters according to their impact on the p-boxes of dependent variables. In part B1, you are being asked to assess the error that results from refining the epistemic space of the uncertainty model. The error metric was left open intentionally and it will therefore require proposing metrics that evaluate the difference between two p-boxes. One of these two p-boxes is given by the propagation of the uncertainty model corresponding to the epistemic space provided. The other one corresponds to a refined epistemic space. In contrast, subproblems B2 and B3 provide specific error metrics.

By “refinement of the uncertainty model” we mean the reduction of the epistemic space from a set A to a set B, where B is contained by A.

For the sake of the argument, let x be the parameters of a random variable. Given n observations and an assumption for the functional form
of the random variable, confidence intervals for x can be readily calculated. Think of these intervals as A above. Given additional
observations, a new set of improved confidence intervals B can be obtained using all observations available. If the observations are informative, we
will expect B to be smaller than A, thus better. The requirement of B being a subset of A is used to facilitate calculations.

The task at hand is to determine which physical parameters corresponding epistemic space A we want to refine, without having any additional observations yet. Note there are infinitely many B sets that can be constructed based on A. Ideally, one would like to desensitize the resulting ranking from the particular construction assumed (e.g., assuming that B is the left half of A will likely give you a ranking that differs from the one you obtain by assuming that B is the
right half). The requested ranking will be used to determine which parameters we want to get an improved model for. We need to do this
without knowing what the resulting B will actually be. This is the basis for using the word “hope” in the problem statement.

Please note that B1, B2, and B3 subproblems each have two parts. A generic ranking question, followed by a question on error associated with assigning a parameter as a fixed constant. The general rankings requested should be based on a generic B (i.e., B is an unknown subset of A). In contrast, the questions regarding parameters assuming a fixed constant value refer to the physical parameters themselves, e.g. p_1=0.5, and not to their epistemic space B being a point.

Yes it does, however it is important to know that the output of p_to_x1.m is deterministic. If changing the state of the random number generator is undesirable for your approach, please consider the following pseudo code for a workaround.

% create a randstream for your application
stream = RandStream(‘mt19937ar’,’Seed’,0); % Lots of options on how to create a stream
RandStream.setGlobalStream(stream); % set this as the global stream

% lots of code here, e.g.,
u1 = rand(1,5)
u1 = rand(1,5)

stream = RandStream.getGlobalStream; % at anytime retrieve the previously created stream
savedState = stream.State; % grab the stream state

% call challenge functions
h1 = p_to_x1(…..)

RandStream.setGlobalStream(stream);
stream.State = savedState;

% continue coding using your previous, now interrupted stream

This should provide an uninterrupted stream. There are probably other (better) ways to do this. See the test code below to check if this provides the desired functionality.

Sample code follows:

% cut and paste into Matlab
stream = RandStream(‘mt19937ar’,’Seed’,0);
RandStream.setGlobalStream(stream);

u1 = rand(1,5)
u2 = rand(1,5)
u3 = rand(1,5)

% intentionally reset stream to test if working
stream = RandStream(‘mt19937ar’,’Seed’,0);
RandStream.setGlobalStream(stream);

u1s = rand(1,5) % this should match the u1 vector above.
u2s = rand(1,5) % this should match the u2 vector above.

stream = RandStream.getGlobalStream;
savedState = stream.State;

% call challenge function that changes the stream
h1 = p_to_x1(0.1:0.1:0.5)

RandStream.setGlobalStream(stream);
stream.State = savedState;

u3s = rand(1,5) % this should match the u3 vector above.

See the response here.

A question was submitted to us asking about the data provided and whether there was measurement uncertainty or noise added. Answer: The functions that map inputs to outputs, as well as the 25 observations, might be contaminated by noise. These observations however, correspond to the “perfect” uncertainty model in the sense that their corresponding epistemic   Read more …

A question was submitted to us asking about the lower and upper bounds for the design parameters used in the Robust Design problem defined in Subproblem E. Answer: The design parameters are free to assume any finite value, and any sign. However, note that excessively large values will yield large values of “g”. A typical   Read more …

A question was submitted to us asking about the beta distributions used in the challenge problem. These distributions are classical beta distributions and are defined by the matlab functions: betapdf, betacdf, betarnd, betainv, etc. The beta distribution is defined on the support [0,1].

You can download the data and model from the “Download Software” page.   The models will be distributed as Matlab mex files. These are computer platform, and to a lesser degree Matlab version specific executable files. The the data files will be distributed as  Matlab mat files. All data and models are open to the   Read more …