In the Sage docs they say that the function canonicalize_radical() chooses a branch based on its behavior at infinity. In their own words this means that
Assumptions are not taken into account during the transformation. This may result in a branch choice inconsistent with your assumptions.
Which, simply put, is a bad thing. A simple example of an answer that would be
assume(x<0)
sqrt(x^2).canonicalize_radical()
which returns x, and has the behavior with assume(x>0). At least we would expect the behavior to be different for the different assumptions.
Using Sage, is there a way of consistently doing this in a way that's always consistent with the assumptions? My main goal here is simplification, so maybe there's another function to do that that I'm not aware of.
There is a long correspondence regarding this on the Sage Trac server and on the Sage developer list and the Maxima developer list. That is why it is now called canonicalize_radical; the Maxima documentation is very clear that this is canonical in some class of expressions but not of functions (multivalued or not). That probably isn't the answer you are looking for, but the Maxima folks make it sound like that is the best one can hope for because of branch cuts.
There are many other simplification routines, also in Sympy and other parts of Sage ...
Related
At first this problem seems trivial: given two ontologies, which term in ontology A best refers to a term in ontology B.
But its simplicity is deceptive: this problem is extremely hard and has currently lead to thousands of academic publications, without any consensus on how to solve this problem.
Naively, one would expect that simply looking at the term "Heart Attack" in both ontologies would suffice.
However, ontologies almost never encode the same phrase.
In simple cases "Heart Attack" might be coded as "Heart Attacks", or "Heart attack (non-fatal)", but in more complicated cases it might only be coded as "Myocardial infarction".
In other cases it is even more complicated, for example dealing with compound (composed) terms.
More importantly, simply matching the term (or string) ignores the "ontological structure".
What if "Heart Attack" in ontology A is coded as caused-by high blood pressure, whereas in ontology B it might be coded as withdrawl-from-trial-non-fatal.
In this case it might be valid to match the two terms, but not trivially so.
And this assumes the equivalent term exists at all.
It's a classical problem called Semantic/Ontology Matching, Alignment, or Harmonization. The research out there involves lexical similarity, term usage in free text, graph homomorphisms, curated mappings (like MeSH/WordNet), topic modeling, and logical inference (first- or higher-order logic). But which is the most user friendly and production ready solution, that can be integrated into a Java(/Clojure) or Python app? I've looked at Ontology matching: A literature review but they don't seem to recommend anything ... any suggestions or experiences?
Have a look at http://oaei.ontologymatching.org/2014/results/ . There were several tracks open for matchers to be sent in and be evaluated. Not every matcher participates in every track. So you might want to read the track descriptions and pick one that seems to be the most similar to your problem. For example if you don't have to deal with multiple languages you probably don't have to check the MultiFarm track. After that check the results by having a look at Recall, Precision and F-Measure and decide for yourself. You also might want to check out some earlier years.
I am currently reading "Artificial Intelligence: A modern Approach". Though the terminology factored, structured and atomic representation is confusing what do these mean exactly?
In relation with programming...
Thanks
I'm not thrilled with the lines that Russell and Norvig draw, but: Generally, when you're using AI techniques to solve a problem, you're going to have a programmed model of the situation. Atomic/factored/structured is a qualitative measure of how much "internal structure" those models have, from least to most.
Atomic models have no internal structure; the state either does or does not match what you're looking for. In a sliding tile puzzle, for instance, you either have the correct alignment of tiles or you do not.
Factored models have more internal structure, although exactly what will depend on the problem. Typically, you're looking at variables or performance metrics of interest; in a sliding puzzle, this might be a simple heuristic like "number of tiles out of place," or "sum of manhatten distances."
Structured models have still more; again, exactly what depends on the problem, but they're often relations either of components of the model to itself, or components of the model to components of the environment.
It is very easy, especially when looking at very simple problems like the sliding tile, to unconsciously do all the hard intelligence work yourself, at a glance, and forget that your model doesn't have all your insight. For example, if you were to make a program to do a graph search technique on the sliding puzzle, you'd probably make some engine that took as input a puzzle state and an action, and generated a new puzzle state from that. The puzzle states are still atomic, but you the programmer are using a much more detailed model to link those inputs and outputs together.
I like explanation given by Novak. My 2 cents is to make clarity on difference between factored vs structured. Here is extract from definitions:
An atomic representation is one in which each state is treated as a
black box.
A factored representation is one in which the states are
defined by set of features.
A structured representation is one in which the states are expressed in form of objects and relations between them. Such knowledge about relations called facts.
Examples:
atomicState == goal: Y/N // Is goal reached?
It is the only question we can ask to black box.
factoredState{18} == goal{42}: N // Is goal reached?
diff( goal{42}, factoredState{18}) = 24 // How much is difference?
// some other questions. the more features => more type of questions
The simplest factored state must have at least one feature (of some type), that gives us ability to ask more questions. Normally it defines quantitative difference between states. The example has one feature of integer type.
11grade#schoolA{John(Math=A-), Marry(Music=A+), Job1(doMath)..} == goal{50% ready for jobs}
The key here - structured representation, allows higher level of formal logical reasoning at the search. See First-Order Logic #berkley for introductory info.
This subject easily confuses a practitioner (especially beginner) but makes great sense for comparing different goal searching algorithms. Such "world" state representation classification logically separates algorithms into different classes. It is very useful to draw lines in academic research and compare apples to apples when reasoning academically.
I'm starting up looking into doing some machine translation of search queries, and have been trying to think of different ways to rate my translation system between iterations and against other systems. The first thing that comes to mind is getting translations of a set of search terms from mturk from a bunch of people and saying each is valid, or something along those lines, but that would be expensive, and possibly prone to people putting in bad translations.
Now that I'm trying to think of something cheaper or better, I figured I'd ask StackOverflow for ideas, in case there's already some standard available, or someone has tried to find one of these before. Does anyone know, for example, how Google Translate rates various iterations of their system?
There is some information here that might be useful as it provides a basic explanation of the BLEU scoring technique that is often used to measure the quality of an MT system by developers.
The first link provides a basic overview of BLEU and the second points out some problems with BLEU in terms of it's limitations.
http://kv-emptypages.blogspot.com/2010/03/need-for-automated-quality-measurement.html
and
http://kv-emptypages.blogspot.com/2010/03/problems-with-bleu-and-new-translation.html
There is also some very specific pragmatic advice on how to develop a useful Test Set at this link: AsiaOnline.Net site in the November newsletter. I am unable to put this link in as there is a limit of two.
I'd suggest refining your question. There are a great many metrics for machine translation, and it depends on what you're trying to do. In your case, I believe the problem is simply stated as: "Given a set of queries in language L1, how can I measure the quality of the translations into L2, in a web search context?"
This is basically cross-language information retrieval.
What's important to realize here is that you don't actually care about providing the user with the translation of the query: you want to get them the results that they would have gotten from a good translation of the query.
To that end, you can simply measure the discrepancy of the results lists between a gold translation and the result of your system. There are many metrics for rank correlation, set overlap, etc., that you can use. The point is that you need not judge each and every translation, but just evaluate whether the automatic translation gives you the same results as a human translation.
As for people proposing bad translations, you can assess whether the putative gold standard candidates have similar results lists (i.e. given 3 manual translations do they agree in results? If not, use the 2 that most overlap). If so, then these are effectively synonyms from the IR perspective.
In our MT Evaluation we use hLEPOR score (see the slides for details)
Questions
I want to classify/categorize/cluster/group together a set of several thousand websites. There's data that we can train on, so we can do supervised learning, but it's not data that we've gathered and we're not adamant about using it -- so we're also considering unsupervised learning.
What features can I use in a machine learning algorithm to deal with multilingual data? Note that some of these languages might not have been dealt with in the Natural Language Processing field.
If I were to use an unsupervised learning algorithm, should I just partition the data by language and deal with each language differently? Different languages might have different relevant categories (or not, depending on your psycholinguistic theoretical tendencies), which might affect the decision to partition.
I was thinking of using decision trees, or maybe Support Vector Machines (SVMs) to allow for more features (from my understanding of them). This post suggests random forests instead of SVMs. Any thoughts?
Pragmatical approaches are welcome! (Theoretical ones, too, but those might be saved for later fun.)
Some context
We are trying to classify a corpus of many thousands of websites in 3 to 5 languages (maybe up to 10, but we're not sure).
We have training data in the form of hundreds of websites already classified. However, we may choose to use that data set or not -- if other categories make more sense, we're open to not using the training data that we have, since it is not something we gathered in the first place. We are on the final stages of scraping data/text from websites.
Now we must decide on the issues above. I have done some work with the Brown Corpus and the Brill tagger, but this will not work because of the multiple-languages issue.
We intend to use the Orange machine learning package.
According to the context you have provided, this is a supervised learning problem.
Therefore, you are doing classification, not clustering. If I misunderstood, please update your question to say so.
I would start with the simplest features, namely tokenize the unicode text of the pages, and use a dictionary to translate every new token to a number, and simply consider the existence of a token as a feature.
Next, I would use the simplest algorithm I can - I tend to go with Naive Bayes, but if you have an easy way to run SVM this is also nice.
Compare your results with some baseline - say assigning the most frequent class to all the pages.
Is the simplest approach good enough? If not, start iterating over algorithms and features.
If you go the supervised route, then the fact that the web pages are in multiple languages shouldn't make a difference. If you go with, say lexical features (bag-o'-words style) then each language will end up yielding disjoint sets of features, but that's okay. All of the standard algorithms will likely give comparable results, so just pick one and go with it. I agree with Yuval that Naive Bayes is a good place to start, and only if that doesn't meet your needs that try something like SVMs or random forests.
If you go the unsupervised route, though, the fact that the texts aren't all in the same language might be a big problem. Any reasonable clustering algorithm will first group the texts by language, and then within each language cluster by something like topic (if you're using content words as features). Whether that's a bug or a feature will depend entirely on why you want to classify these texts. If the point is to group documents by topic, irrespective of language, then it's no good. But if you're okay with having different categories for each language, then yeah, you've just got as many separate classification problems as you have languages.
If you do want a unified set of classes, then you'll need some way to link similar documents across languages. Are there any documents in more that one language? If so, you could use them as a kind of statistical Rosetta Stone, to link words in different languages. Then, using something like Latent Semantic Analysis, you could extend that to second-order relations: words in different languages that don't ever occur in the same document, but which tend to co-occur with words which do. Or maybe you could use something like anchor text or properties of the URLs to assign a rough classification to documents in a language-independent manner and use that as a way to get started.
But, honestly, it seems strange to go into a classification problem without a clear idea of what the classes are (or at least what would count as a good classification). Coming up with the classes is the hard part, and it's the part that'll determine whether the project is a success or failure. The actual algorithmic part is fairly rote.
Main answer is: try different approaches. Without actual testing it's very hard to predict what method will give best results. So, I'll just suggest some methods that I would try first and describe their pros and cons.
First of all, I would recommend supervised learning. Even if the data classification is not very accurate, it may still give better results than unsupervised clustering. One of the reasons for it is a number of random factors that are used during clustering. For example, k-means algorithm relies on randomly selected points when starting the process, which can lead to a very different results for different program runnings (though x-means modifications seems to normalize this behavior). Clustering will give good results only if underlying elements produce well separated areas in the feature space.
One of approaches to treating multilingual data is to use multilingual resources as support points. For example, you can index some Wikipedia's articles and create "bridges" between same topics in different languages. Alternatively, you can create multilingual association dictionary like this paper describes.
As for methods, the first thing that comes to mind is instance-based semantic methods like LSI. It uses vector space model to calculate distance between words and/or documents. In contrast to other methods it can efficiently treat synonymy and polysemy. Disadvantage of this method is a computational inefficiency and leak of implementations. One of the phases of LSI makes use of a very big cooccurrence matrix, which for large corpus of documents will require distributed computing and other special treatment. There's modification of LSA called Random Indexing which do not construct full coocurrence matrix, but you'll hardly find appropriate implementation for it. Some time ago I created library in Clojure for this method, but it is pre-alpha now, so I can't recommend using it. Nevertheless, if you decide to give it a try, you can find project 'Clinch' of a user 'faithlessfriend' on github (I'll not post direct link to avoid unnecessary advertisement).
Beyond special semantic methods the rule "simplicity first" must be used. From this point, Naive Bayes is a right point to start from. The only note here is that multinomial version of Naive Bayes is preferable: my experience tells that count of words really does matter.
SVM is a technique for classifying linearly separable data, and text data is almost always not linearly separable (at least several common words appear in any pair of documents). It doesn't mean, that SVM cannot be used for text classification - you still should try it, but results may be much lower than for other machine learning tasks.
I haven't enough experience with decision trees, but using it for efficient text classification seems strange to me. I have seen some examples where they gave excellent results, but when I tried to use C4.5 algorithm for this task, the results were terrible. I believe you should get some software where decision trees are implemented and test them by yourself. It is always better to know then to suggest.
There's much more to say on every topic, so feel free to ask more questions on specific topic.
do you know any good algorithms that match two strings and then return a percentage in how many percent those two strings match?
And are there some, that work with databases too?
The Levenstein distance is such a measure. It basically tells you how many characters need to be edited, deleted or added, to get from the first to the second string. I'm not sure whether some database systems support that.
But I know for sure that a much more simplified algorithm named Soundex is supported in some database systems.
It depends upon your criteria for similarity. Other people have already referred you to Levenstein distance (edit distance is the same thing). That's usually pretty good, and definitely more language-independent than something like soundex. However, be aware that Levenstein difference does not handle transposition very well. Thus:
Levenstein("copy", "cpoy") == 2
If you're trying to deal with human input, transpositions are fairly common. Whether that's a problem or not depends on your metrics for similarity.
It's been a while, but I believe Postgresql has levenstein() either built-in or available as a contrib C module.
I think the problem you're looking for is called Edit Distance. It is expensive to compute in general, but if you are looking for strings within small edit distance of other strings, it is not so bad. There is more information in the Wikipedia article.
How to best match two strings? Have them go out for coffee, and if they hit it off, dinner and a movie. Or maybe they could do some peer programming? It depends on the strings, really. Even coffee can often be tricky.
Would this be of help? I just ran into it. Comparing Two Strings producing a numeric delta