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Recently I am working with fluid modeling with Modelica, but I come across a lot of divergence problems of nonlinear equations, like in the following screenshot.
So I am considering if it is possible to use the min/max/nominal attributes of variables to improve the model's convergence, especially when a user comes across the nonlinear solver failure. According to the answer of this question on StackOverflow, min/max attributes won't help convergence, and based on the Modelica Specification 4.8.6, nomial attributes are used to determine appropriate tolerances or epsilons, or may be used for scaling.
So my question is:
If I meet this kind of divergence problem caused by the nonlinearity of my model, how could I help the compiler to get convergence better and quicker?
Someone might suggest better start values of variables used as state variables, but when I am dealing with large models, I am not sure how to find the specific state variables of which I should modify the start values.
Chapter 2.6.13 "Online diagnostics for non-linear systems" in manual 1B and following in the manual should help. You can e.g. list states that dominates error: usually these states are a good hint where to start your improvements.
Adding to the answer by Imke Krueger.
If the models fail after 2917 s one possibility is that the solution was diverging before that, with e.g., enthalpy decreasing further and further until the model has left the valid regions.
Assuming it happened fairly slowly it is best to plot the states and other variables in that components. Additionally the states dominating the error as indicated in the answer by Imke Krueger and see if any of them seem to diverge.
If it happened more quickly:
Log events and check whether something important like a flow reversal just happened before that time.
Disable equidistant output, as it is possible that the model diverged between two output points.
An eigenvalue-based analysis of the Jacobin at time = 0 provides a ranking of state-variables from most significant to the least one. That could be a heuristic to examine the influence of start variables of most significant state-variables.
What could be also helpful is to conduct a similar analysis a little time before the problem occurs.
Also there is a possibility to compute dynamic parameter sensitivities of state variables (before the problem occurs) w.r.t. start values, see e.g. https://github.com/Mathemodica/DerXP for a suggested approach. This gives you a hint which start values significantly influences the values of state variables.
Hello guys I'd like to know the answer to the question that is the titled named by.
For example if I have physical system described in differential equation(s), how should I know when I should use step, pulse or ramp generator?
What exactly does it do?
Thank you for your answers.
They are mostly the remnants of the classical control times. The main reason why they are so famous is because of their simple Laplace transform terms. 1,1/s and 1/s^2. Then you can multiply these with the plant and you would get the Laplace transform of the output.
Back in the day, what you only had was partial fraction expansion and Laplace transform tables to get an idea what the response would look like. And today, you can basically simulate whatever input you like. So they are not really neeeded which is the answer to your question.
But since people used these signals so often they have spotted certain properties. For example, step response is good for assessing the transients and the steady state tracking error value, ramp response is good for assessing (reference) following error (which introduces double integrators) and so on. Hence, some consider these signals as the characteristic functions though it is far from the truth. Especially, you should keep in mind that, just because the these responses are OK, the system is not necessarily stable.
However, keep in mind that these are extremely primitive ways of assessing the system. Currently, they are taught because they are good for giving homeworks and making people acquainted with Simulink etc.
They are used to determine system characteristics. If you are studying a system of differential equations you would want to know different characteristics from the response of the system from these kinds of inputs since these inputs are the very fundamental ones. For example a system whose output blows up for a pulse input is unstable, and you would not want to have such a system in real life(except in rare situations). It's too difficult for me to explain it all in an answer, you should start with this wiki page.
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I find this question a little tricky. Maybe someone knows an approach to answer this question. Imagine that you have a dataset(training data) which you don't know what it is about. Which features of training data would you look at in order to infer classification algorithm to classify this data? Can we say anything whether we should use a non-linear or linear classification algorithm?
By the way, I am using WEKA to analyze the data.
Any suggestions?
Thank you.
This is in fact two questions in one ;-)
Feature selection
Linear or not
add "algorithm selection", and you probably have three most fundamental questions of classifier design.
As an aside note, it's a good thing that you do not have any domain expertise which would have allowed you to guide the selection of features and/or to assert the linearity of the feature space. That's the fun of data mining : to infer such info without a priori expertise. (BTW, and while domain expertise is good to double-check the outcome of the classifier, too much a priori insight may make you miss good mining opportunities). Without any such a priori knowledge you are forced to establish sound methodologies and apply careful scrutiny to the results.
It's hard to provide specific guidance, in part because many details are left out in the question, and also because I'm somewhat BS-ing my way through this ;-). Never the less I hope the following generic advice will be helpful
For each algorithm you try (or more precisely for each set of parameters for a given algorithm), you will need to run many tests. Theory can be very helpful, but there will remain a lot of "trial and error". You'll find Cross-Validation a valuable technique.
In a nutshell, [and depending on the size of the available training data], you randomly split the training data in several parts and train the classifier on one [or several] of these parts, and then evaluate the classifier on its performance on another [or several] parts. For each such run you measure various indicators of performance such as Mis-Classification Error (MCE) and aside from telling you how the classifier performs, these metrics, or rather their variability will provide hints as to the relevance of the features selected and/or their lack of scale or linearity.
Independently of the linearity assumption, it is useful to normalize the values of numeric features. This helps with features which have an odd range etc.
Within each dimension, establish the range within, say, 2.5 standard deviations on either side of the median, and convert the feature values to a percentage on the basis of this range.
Convert nominal attributes to binary ones, creating as many dimensions are there are distinct values of the nominal attribute. (I think many algorithm optimizers will do this for you)
Once you have identified one or a few classifiers with a relatively decent performance (say 33% MCE), perform the same test series, with such a classifier by modifying only one parameter at a time. For example remove some features, and see if the resulting, lower dimensionality classifier improves or degrades.
The loss factor is a very sensitive parameter. Try and stick with one "reasonnable" but possibly suboptimal value for the bulk of the tests, fine tune the loss at the end.
Learn to exploit the "dump" info provided by the SVM optimizers. These results provide very valuable info as to what the optimizer "thinks"
Remember that what worked very well wih a given dataset in a given domain may perform very poorly with data from another domain...
coffee's good, not too much. When all fails, make it Irish ;-)
Wow, so you have some training data and you don't know whether you are looking at features representing words in a document, or genese in a cell and need to tune a classifier. Well, since you don't have any semantic information, you are going to have to do this soley by looking at statistical properties of the data sets.
First, to formulate the problem, this is more than just linear vs non-linear. If you are really looking to classify this data, what you really need to do is to select a kernel function for the classifier which may be linear, or non-linear (gaussian, polynomial, hyperbolic, etc. In addition each kernel function may take one or more parameters that would need to be set. Determining an optimal kernel function and parameter set for a given classification problem is not really a solved problem, there are only useful heuristics and if you google 'selecting a kernel function' or 'choose kernel function', you will be treated to many research papers proposing and testing various approaches. While there are many approaches, one of the most basic and well travelled is to do a gradient descent on the parameters-- basically you try a kernel method and a parameter set , train on half your data points and see how you do. Then you try a different set of parameters and see how you do. You move the parameters in the direction of best improvement in accuracy until you get satisfactory results.
If you don't need to go through all this complexity to find a good kernel function, and simply want an answer to linear or non-linear. then the question mainly comes down to two things: Non linear classifiers will have a higher risk of overfitting (undergeneralizing) since they have more dimensions of freedom. They can suffer from the classifier merely memorizing sets of good data points, rather than coming up with a good generalization. On the other hand a linear classifier has less freedom to fit, and in the case of data that is not linearly seperable, will fail to find a good decision function and suffer from high error rates.
Unfortunately, I don't know a better mathematical solution to answer the question "is this data linearly seperable" other than to just try the classifier itself and see how it performs. For that you are going to need a smarter answer than mine.
Edit: This research paper describes an algorithm which looks like it should be able to determine how close a given data set comes to being linearly seperable.
http://www2.ift.ulaval.ca/~mmarchand/publications/wcnn93aa.pdf
I'm quite new with this topic so any help would be great. What I need is to optimize a neural network in MATLAB by using GA. My network has [2x98] input and [1x98] target, I've tried consulting MATLAB help but I'm still kind of clueless about what to do :( so, any help would be appreciated. Thanks in advance.
Edit: I guess I didn't say what is there to be optimized as Dan said in the 1st answer. I guess most important thing is number of hidden neurons. And maybe number of hidden layers and training parameters like number of epochs or so. Sorry for not providing enough info, I'm still learning about this.
If this is a homework assignment, do whatever you were taught in class.
Otherwise, ditch the MLP entirely. Support vector regression ( http://www.csie.ntu.edu.tw/~cjlin/libsvm/ ) is much more reliably trainable across a broad swath of problems, and pretty much never runs into the stuck-in-a-local-minima problem often hit with back-propagation trained MLP which forces you to solve a network topography optimization problem just to find a network which will actually train.
well, you need to be more specific about what you are trying to optimize. Is it the size of the hidden layer? Do you have a hidden layer? Is it parameter optimization (learning rate, kernel parameters)?
I assume you have a set of parameters (# of hidden layers, # of neurons per layer...) that needs to be tuned, instead of brute-force searching all combinations to pick a good one, GA can help you "jump" from this combination to another one. So, you can "explore" the search space for potential candidates.
GA can help in selecting "helpful" features. Some features might appear redundant and you want to prune them. However, say, data has too many features to search for the best set of features by some approaches such as forward selection. Again, GA can "jump" from this set candidate to another one.
You will need to find away to encode the data (input parameters, features...) fed to GA. For finding a set of input paras or a good set of features, I think binary encoding should work. In addition, choosing operators for GA to reproduce offsprings is also important. Yet GA needs to be tuned, too (early stopping which can also be applied to ANN).
Here are just some ideas. You might want to search for more info about GA, feature selection, ANN pruning...
Since you're using MATLAB already I suggest you look into the Genetic Algorithms solver (known as GATool, part of the Global Optimization Toolbox) and the Neural Network Toolbox. Between those two you should be able to save quite a bit of figuring out.
You'll basically have to do 2 main tasks:
Come up with a representation (or encoding) for your candidate solutions
Code your fitness function (which basically tests candidate solutions) and pass it as a parameter to the GA solver.
If you need help in terms of coming up with a fitness function, or encoding of candidate solutions then you'll have to be more specific.
Hope it helps.
Matlab has a simple but great explanation for this problem here. It explains both the ANN and GA part.
For more info on using ANN in command line see this.
There is also plenty of litterature on the subject if you google it. It is however not related to MATLAB, but simply the results and the method.
Look up Matthew Settles on Google Scholar. He did some work in this area at the University of Idaho in the last 5-6 years. He should have citations relevant to your work.
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A professor asked me to help making a specification for a college project.
By the time the students should know the basics of programming.
The professor is a mathematician and has little experience in other programming languages, so it should really be in MATLAB.
I would like some projects ideas. The project should
last about 1 to 2 months
be done individually
have web interface would be great
doesn't necessary have to go deep in maths, but some would be great
use a database (or store data in files)
What kind of project would make the students excited?
If you have any other tips I'll appreciate.
UPDATE: The students are sophomores and have already studied vector calculus. This project is for an one year Discrete Mathematics course.
UPDATE 2: The topics covered in the course are
Formal Logic
Proofs, Recursion, and Analysis of Algorithms
Sets and Combinatorics
Relations, Functions, and Matrices
Graphs and Trees
Graph Algorithms
Boolean Algebra and Computer Logic
Modeling Arithmetic, Computation, and Languages
And it'll be based on this book Mathematical Structures for Computer Science: A Modern Approach to Discrete Mathematics by Judith L. Gersting
General Suggestions:
There are many teaching resources at The MathWorks that may give you some ideas for course projects. Some sample links:
The MATLAB Central blogs, specifically some posts by Loren that include using LEGO Mindstorms in teaching and a webinar about MATLAB for teaching (note: you will have to sign up to see the webinar)
The Curriculum Exchange: a repository of course materials
Teaching with MATLAB and Simulink: a number of other links you may find useful
Specific Suggestions:
One of my grad school projects in non-linear dynamics that I found interesting dealt with Lorenz oscillators. A Lorenz oscillator is a non-linear system of three variables that can exhibit chaotic behavior. Such a system would provide an opportunity to introduce the students to numerical computation (iterative methods for simulating systems of differential equations, stability and convergence, etc.).
The most interesting thing about this project was that we were using Lorenz oscillators to encode and decode signals. This "encrypted communication" aspect was really cool, and was based on the following journal article:
Kevin M. Cuomo and Alan V. Oppenheim,
Circuit Implementation of Synchronized Chaos with Applications
to Communications, Physical Review
Letters 71(1), 65-68 (1993)
The article addresses hardware implementations of a chaotic communication system, but the equivalent software implementation should be simple enough to derive (and much easier for the students to implement!).
Some other useful aspects of such a project:
The behavior of the system can be visualized in 2-D and 3-D plots, thus exposing the students to a number of graphing utilities in MATLAB (PLOT, PLOT3, COMET, COMET3, etc.).
Audio signals can be read from files, encrypted using the Lorenz equations, written out to a new file, and then decrypted once again. You could even have the students each encrypt a signal with their Lorenz oscillator code and give it to another student to decrypt. This would introduce them to various file operations (FREAD, FWRITE, SAVE, LOAD, etc.), and you could even introduce them to working with audio data file formats.
You can introduce the students to the use of the PUBLISH command in MATLAB, which allows you to format M-files and publish them to various output types (like HTML or Word documents). This will teach them techniques for making useful help documentation for their MATLAB code.
I have found that implementing and visualizing Dynamical systems is great
for giving an introduction to programming and to an interesting branch of
applied mathematics. Because one can see the 'life' in these systems,
our students really enjoy this practical module.
We usually start off by visualizing a 1D attractor, so that we can
overlay the evolution rule/rate of change with the current state of
the system. That way you can teach computational aspects (integrating the system) and
visualization, and the separation of both in implementation (on a simple level, refreshing
graphics at every n-th computation step, but in C++ leading to threads, unsure about MATLAB capabilities here).
Next we add noise, and then add a sigmoidal nonlinearity to the linear attractor. We combine this extension with an introduction to version control (we use a sandbox SVN repository for this): The
students first have to create branches, modify the evolution rule and then merge
it back into HEAD.
When going 2D you can simply start with a rotation and modify it to become a Hopf oscillator, and visualize either by morphing a grid over time or by going 3D when starting with a distinct point. You can also visualize the bifurcation diagram in 3D. So you again combine generic MATLAB skills like 3D plotting with the maths.
To link in other topics, browse around in wikipedia: you can bring in hunter/predator models, chaotic systems, physical systems, etc.etc.
We usually do not teach object-oriented-programming from within MATLAB, although it is possible and you can easily make up your own use cases in the dynamical systems setting.
When introducing inheritance, we will already have moved on to C++, and I'm again unaware of MATLAB's capabilities here.
Coming back to your five points:
Duration is easily adjusted, because the simple 1D attractor can be
done quickly and from then on, extensions are ample and modular.
We assign this as an individual task, but allow and encourage discussion among students.
About the web interface I'm at a loss: what exactly do you have in mind, why is it
important, what would it add to the assignment, how does it relate to learning MATLAB.
I would recommend dropping this.
Complexity: A simple attractor is easily understood, but the sky's the limit :)
Using a database really is a lot different from config files. As to the first, there
is a database toolbox for accessing databases from MATLAB. Few institutes have the license though, and apart from that: this IMHO does not belong into such a course. I suggest introducing to the concept of config files, e.g. for the location and strength of the attractor, and later for the system's respective properties.
All this said, I would at least also tell your professor (and your students!) that Python is rising up against MATLAB. We are in the progress of going Python with our tutorials, but I understand if someone wants to stick with what's familiar.
Also, we actually need the scientific content later on, so the usefulness for you will probably depend on which department your course will be related to.
A lot of things are possible.
The first example that comes in mind is to model a public transportation network (the network of your city, with underground, buses, tramways, ...). It is represented by a weighted directed graph (you can use sparse matrix to represent it, for example).
You may, for example, ask them to compute the shortest path from one station to another one (Moore-dijkistra algorithm, for example) and display it.
So, for the students, the several steps to do are:
choose an appropriate representation for the network (it could be some objects to represent the properties of the stations and the lines, and a sparse matrix for the network)
load all the data (you can provide them the data in an XML file)
be able to draw the network (since you will put the coordinates of the stations)
calculate the shortest path from one point to another and display it in a pretty way
create a fronted (with GUI)
Of course, this could be complicated by adding connection times (when you change from one line to another), asking for several options (shortest path with minimum connections, take in considerations the time you loose by waiting for a train/bus, ...)
The level of details will depend on the level of the students and the time they could spend on it (it could be very simple, or very realist)
You want to do a project with a web interface and a database, but not any serious math... and you're doing it in MATLAB? Do you understand that MATLAB is especially designed to be used for "deep math", and not for web interfaces or databases?
I think if this is an intro to a Discrete Mathematics course, you should probably do something involving Discrete Mathematics, and not waste the students' time as they learn a bunch of things in that language that they'll never actually use.
Why not do something involving audio? I did an undergraduate project in which we used MATLAB to automatically beat-match different tunes and DJ mix between them. The full program took all semester, but you could do a subset of it. wavread() and the like are built in and easy to use.
Or do some simple image processing like finding Waldo using cross-correlation.
Maybe do something involving cryptography, have them crack a simple encryption scheme and feel like hackers.
MATLAB started life as a MATrix LAB, so maybe concentrating on problems in linear algebra would be a natural fit.
Discrete math problems using matricies include:
Spanning trees and shortest paths
The marriage problem (bipartite graphs)
Matching algorithms
Maximal flow in a network
The transportation problem
See Gil Strang's "Intro to Applied Math" or Knuth's "Concrete Math" for ideas.
You might look here: http://www.mathworks.com/academia/student_center/tutorials/launchpad.html
on the MathWorks website. The interactive tutorial (second link) is quite popular.
--Loren
I always thought the one I was assigned in grad school was a good choice-a magnetic lens simulator. The math isn't completely overwhelming so you can focus more on learning the language, and it's a good intro to the graphical capabilities (e.g., animating the path of an off-axis electron going through the lens).
db I/O and fancy interfaces are out of place in a discrete math course.
my matlab labs were typically algorithm implementations, with charts as output, and simple file input.
how hard is the material? image processing is really easy in matlab, can you do some discrete 2D filtering? blurs and stuff. http://homepages.inf.ed.ac.uk/rbf/HIPR2/filtops.htm