How can I do symbolic computations in mathematica? I do not want to give a value to any Variable. Can I use Mathematica to create a set of equations, consisting only of symbolic variables, and then combine them, or solve by one specific variable?
If this were good, it could pretty much replace a paper sheet for doing mathematics.
It would be awesome if you could provide some examples. I hope the question is clear
Related
A motivating issue, implemented in Matlab:
N = 1000;
R = zeros(2*N);
for i=0:N-1
R = R(2:end-1, 2:end-1);
end
For this code timeit() gives a time 2.9793 on my machine. It isn't really great.
By canonical way I mean a discussion that isn't just acceptable, but a performant implementation that respects very large matrices reduced. I would be very appreciative of any answer, referrals to other discussions or literature.
As for language, I am not really a programmer, this question is motivated by a mathematics inquiry and I have encountered performance issues implementing any such reduction process in Matlab. Is there a solution to this in Matlab, or must one delve into the scary depths of C/C++?
One note: One may ask, why not just keep the matrix as is and consider parts of it as needed? To clarify, the reduction process in practice of course depends on the actual (nonzero) values of the elements, e.g. by processing the matrix in 2x2 blocks, and the removal of edge-values is needed to prepare the matrix for then next reduction step.
R(2:end-1, 2:end-1) is the correct way of extracting the part of the array that is all values except the ones at the edges. This requires copying the data, so will take some time. There is no legal way around the copy, and no alternative for extracting a part of the array. (subsref might seems like an alternative, but is the function that is internally called for the given syntax.)
As for illegal ways, you could try James Tursa’s sharedchild from the MATLAB FileExchange. It allows to create an array that references subsets of the data of another array. James is well known in the MATLAB user community as one of the people reverse-engineering the system and bending it to his will. This is solid code. But every version of MATLAB introduces new changes to the infrastructure, so upgrading MATLAB might break your program if you use this code.
You don't need the for loop. If you want to remove L elements from the borders, simply do:
R=R(L+1:end-L, L+1:end-L)
I am surprised you didn't get an error with that code. I think you should end up with an empty matrix at the end of the loop.
I would like to get some helpful instructions about how to use the Q-learning algorithm with function approximation. For the basic Q-learning algorithm I have found examples and I think I did understand it. In case of using function approximation I get into trouble. Can somebody give me an explanation through a short example how it works?
What I know:
Istead of using matrix for Q-values we use features and parameters.
Make approximation with the linear combination of feauters and parameters.
Update the parameters.
I have checked this paper: Q-learning with function approximation
But I cant find any useful tutorial how to use it.
Thanks for help!
To my view, this is one of the best references to start with. It is well written with several pseudo-code examples. In your case, you can simplify the algorithms by ignoring eligibility traces.
Also, to my experience and depending on your use case, Q-Learning might not work very well (sometimes it needs huge amounts of experience data). You can try Fitted-Q value for example, which is a batch algorithm.
I have found automatic differentiation to be extremely useful when writing mathematical software. I now have to work with random variables and functions of the random variables, and it seems to me that an approach similar to automatic differentiation could be used for this, too.
The idea is to start with a basic random vector with given multivariate distribution and then you want to work with the implied probability distributions of functions of components of the random vector. The idea is to define operators that automatically combine two probability distributions appropriately when you add, multiply, divide two random variables and transform the distribution appropriately when you apply scalar functions such as exponentiation. You could then combine these to build any function you need of the original random variables and automatically have the corresponding probability distribution available.
Does this sound feasible? If not, why not? If so and since it's not a particularly original thought, could someone point me to an existing implementation, preferably in C
There has been a lot of work on probabilistic programming. One issue is that as your distribution gets more complicated you start needing more complex techniques to sample from it.
There are a number of ways this is done. Probabilistic graphical models gives one vocabulary for expressing these models, and you can then sample from them using various Metropolis-Hastings-style methods. Here is a crash course.
Another model is Probabilistic Programming, which can be done through an embedded domain specific language, directly. Oleg Kiselyov's HANSEI is an example of this approach. Once they have the program they can inspect the tree of decisions and expand them out by a form of importance sampling to gain the most information possible at each step.
You may also want to read "Nonstandard Interpretations of Probabilistic
Programs for Efficient Inference" by Wingate et al. which describes one way to use extra information about the derivative of your distribution to accelerate Metropolis-Hastings-style sampling techniques. I personally use automatic differentiation to calculate those derivatives and this brings the topic back to automatic-differentiation. ;)
I'm not sure if this is the best place to ask this, but you guys have been helpful with plenty of my CS homework in the past so I figure I'll give it a shot.
I'm looking for an algorithm to blindly combine several dependent variables into an index that produces the best linear fit with an external variable. Basically, it would combine the dependent variables using different mathematical operators, include or not include each one, etc. until an index is developed that best correlates with my external variable.
Has anyone seen/heard of something like this before? Even if you could point me in the right direction or to the right place to ask, I would appreciate it. Thanks.
Sounds like you're trying to do Multivariate Linear Regression or Multiple Regression. The simplest method (Read: less accurate) to do this is to individually compute the linear regression lines of each of the component variables and then do a weighted average of each of the lines. Beyond that I am afraid I will be of little help.
This appears to be simple linear regression using multiple explanatory variables. As the implication here is that you are using a computational approach you could do something as simple apply a linear model to your data using every possible combination of your explanatory variables that you have (whether you want to include interaction effects is your choice), choose a goodness of fit measure (R^2 being just one example) and use that to rank the fit of each model you fit?? The quality of a model is also somewhat subjective in many fields - you could reject a model containing 15 variables if it only moderately improves the fit over a far simpler model just containing 3 variables. If you have not read it already I don't doubt that you will find many useful suggestions in the following text :
Draper, N.R. and Smith, H. (1998).Applied Regression Analysis Wiley Series in Probability and Statistics
You might also try doing a google for the LASSO method of model selection.
The thing you're asking for is essentially the entirety of regression analysis.
this is what linear regression does, and this is a good portion of what "machine learning" does (machine learning is basically just a name for more complicated regression and classification algorithms). There are hundreds or thousands of different approaches with various tradeoffs, but the basic ones frequently work quite well.
If you want to learn more, the coursera course on machine learning is a great place to get a deeper understanding of this.
For my problem it would be best to find a numeric representation of kazakh national ornaments for generating new ones. But other approaches are also fine.
The ornaments essentially consist of combinations of relatively basic ornaments. Usually the ornaments are symmetrical.
Here are few examples of basic elements:
(The images are a bit distorted)
And this is an example of a more complex ornament:
How could I encode an ornament's representation in as few numbers as possible? So that I could write a program that would generate an ornament, given some sequence of numbers
Any ideas are appreciated.
As I write this, I have thought that generating images of snowflakes may be somewhat relevant, although it's possibly just a fractal.
You have to realize that your question is actually not how to represent them but how to generate them.
Still you might get some ideas. But don't hold your breath, because it can get complicated
EDIT:
In researching problems such as this you could start with L-systems, this paper seems to convey the idea.
Actually here's an attempt at an answer:
Represent it as a set of grammar rules.
I found this dissertation (read it as book) on image texture generation. Image Texture Tools