Data modeling, restructuring, and massaging

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This is more for overview of my own than for teaching or exercise.

Overview of the math's areas

Arithmetic · 'elementary mathematics' and similar concepts
Set theory, Category theory
Geometry and its relatives · Topology
Elementary algebra - Linear algebra - Abstract algebra
Calculus and analysis
: Information theory · Number theory · Decision theory, game theory · Recreational mathematics · Dynamical systems · Unsorted or hard to sort

Math on data:

  • Statistics as a field
some introduction · areas of statistics
types of data · on random variables, distributions
Virtues and shortcomings of...
on sampling · probability
glossary · references, unsorted
Footnotes on various analyses

  • Other data analysis, data summarization, learning
Data modeling, restructuring, and massaging
Statistical modeling · Classification, clustering, decisions, and fuzzy coding ·
dimensionality reduction ·
Optimization theory, control theory · State observers, state estimation
Connectionism, neural nets · Evolutionary computing
  • More applied:
Formal grammars - regular expressions, CFGs, formal language
Signal analysis, modeling, processing
Image processing notes


NLP data massage / putting meanings or numbers to words

Bag of words / bag of features

The bag-of-words model (more broadly bag-of-features model) use the collection of words in a context, unordered, in a multiset, a.k.a. bag.

In other words, we summarize a document (or part of it) it by appearance or count of words, and ignore things like adjacency and order - so any grammar.

In text processing

In introductions to Naive Bayes as used for spam filtering, its naivety essentially is this assumption that feature order does not matter.

Though real-world naive bayes spam filtering would take more complex features than single words (and may re-introduce adjacenct via n-grams or such), examples often use 1-grams for simplicity - which basically is bag of words, exc.

Other types of classifiers also make this assumption, or make it easy to do so.

Bag of features

While the idea is best known from text, hence bag-of-words, you can argue for bag of features, applying it to anything you can count, and may be useful even when considered independently.

For example, you may follow up object detection in an image with logic like "if this photo contains a person, and a dog, and grass" because each task may be easy enough individually, and the combination tends to narrow down what kind of photo it is.

In practice, the bag-of-features often refers to models that recognize parts of a whole object (e.g. "we detected a bunch of edges of road signs" might be easier and more robust than detecting it fully), and used in a number image tasks, such as feature extraction, object/image recognition, image search, (more flexible) near-duplicate detection, and such.

The idea that you can describe an image by the collection of small things we recognize in it, and that combined presence is typically already a strong indicator (particularly when you add some hypothesis testing). Exact placement can be useful, but often easily secondary.

See also:

N-gram notes

N-grams are contiguous sequence of length n.

They are most often seen in computational linguistics.

Applied to sequences of characters it can be useful e.g. in language identification, but the more common application is to words.

As n-grams models only include dependency information when those relations are expressed through direct proximity, they are poor language models, but useful to things working off probabilities of combinations of words, for example for statistical parsing, collocation analysis, text classification, sliding window methods (e.g. sliding window POS tagger), (statistical) machine translation, and more

For example, for the already-tokenized input This is a sentence . the 2-grams would be:

This   is
is   a
a   sentence
sentence   .

...though depending on how special you do or do not want to treat the edges, people might fake some empty tokens at the edge, or some special start/end tokens.


This article/section is a stub — some half-sorted notes, not necessarily checked, not necessarily correct. Feel free to ignore, or tell me about it.

Note: Skip-grams seem to refer to now two different things.

An extension of n-grams where components need not be consecutive (though typically stay ordered).

A k-skip-n-gram is a length-n sequence where the components occur at distance at most k from each other.

They may be used to ignore stopwords, but perhaps more often they are intended to help reduce data sparsity, under a few assumptions.

They can help discover patterns on a larger scale, simply because skipping makes you look further for the same n. (also useful for things like fuzzy hashing).

Skip-grams apparently come from speech analysis, processing phonemes.

In word-level analysis their purpose is a little different. You could say that we acknowledge the sparsity problem, and decide to get more out of the data we have (focusing on context) rather than trying to smooth.

Actually, if you go looking, skip-grams are now often equated with a fairly specific analysis.

Syntactic n-grams


Words as features - one-hot coding and such

Putting numbers to words

Computers and people and numbers

vector space representations, word embeddings, and more

Just count in a big table
Word embeddings
Static embeddings
Contextual word embeddings
Subword embeddings
The hashing trick (also, Bloom embeddings)
Now we have nicer numbers, but how how I use them?
vectors - unsorted

Moderately specific ideas and calculations


Collocations are statistically idiosyncratic sequences - the math that is often used asks "do these adjacent words occur together more often than the occurrence of each individually would suggest?" - though ideally is a little more refined than that.

This doesn't ascribe any meaning, or comparability, it just tends to signal anything from jargon, various substituted phrases, empty habitual etiquette, and many other things that go beyond purely compositional construction, because why other than common sentence structures would they co-occur so often?

...actually, there are varied takes on how useful collocations are, and why.

latent semantic analysis

Latent Semantic Analysis (LSA) is the application of Singular Value Decomposition on text analysis and search.

random indexing

Topic modeling

Roughly the idea given documents that are about a particular topic, one would expect particular words to appear in the each more or less frequently.

Assuming such documents sharing topics, you can probably find groups of words that belong to those topics.

Assuming each document is primarily about one topic, you can expect a larger set of documents to yield multiple topics, and an assignment of one or more of these topics, so act like a soft/fuzzy clustering.

This is a relatively weak proposal in that it relies on a number of assumptions, but given that it requires zero training, it works better than you might expect when those assumptions are met. (the largest probably being your documents having singular topics).


This article/section is a stub — some half-sorted notes, not necessarily checked, not necessarily correct. Feel free to ignore, or tell me about it.

word2vec is one of many ways to put semantic vectors to words (in the distributional hypothesis approach), and refers to two techniques, using either bag-of-words and skip-gram as processing for a specific learner, as described in T Mikolov et al. (2013), "Efficient Estimation of Word Representations in Vector Space", probably the one that kicked off this dense-vector idea into the interest.

Word2vec amounts could be seen as building a classifier that predicts what word apear in a context, and/or what context appears around a word, which happens to do a decent task of classifying that word.

That paper mentions

  • its continuous bag of words (cbow) variant predicts the current word based on the words directly around it (ignoring order, hence bow(verify))
  • its continuous skip-gram variant predicts surrounding words given the current word.
Uses skip-grams as a concept/building block. Some people refer to this technique as just 'skip-gram' without the 'continuous',

but this may come from not really reading the paper you're copy-pasting the image from?

seems to be better at less-common words, but slower

(NN implies one-hot coding, so not small, but it turns out to be moderately efficient(verify))