Vector embeddings part 1: Word2vec with Gensim

Since the advent of neural networks, vector embeddings for text processing have gained traction in both scientific and applied text classification problems, for example in text sentiment analysis. Using (pre-trained) embeddings has become a de facto standard for attaining a high rating in scientific sentiment analysis contests such as SemEval. However, vector embeddings are finding their way into many other fascinating applications as well.

In the current post, I want to provide a intuitive, easy to understand introduction to the subject. I will touch upon both vector embeddings in general, and applied to text analysis in particular. Furthermore, I’ll provide some quite simple code with Python and Gensim for generating vector embeddings from text. In further posts, I will apply vector embeddings to transforming categorical to continuous variables and generating vector definitions in N-dimensional space for entities.

I’ll cycle through the following points:

  • What are vector embeddings?
  • How do you use vector embeddings?
  • Why do you need vector embeddings?
  • How do you generate vector embeddings?
  • How does the neural network optimize the vector matrix?
  • A practical example (with code): word vectorization for tweets.

What are vector embeddings?

In a nutshell, a vector embedding for a certain item (a word, or a person, or any other object you can think of) is a mathematical definition for that specific item. This definition is comprised of a set of float numbers between -1 and 1, which is called a vector.

A vector is a specific location in N-dimensional space, in comparison to the zero-point in that space. So, a vector embedding for an item means, that the item is represented by a vector, embedded in a space with as many dimensions as there are numbers in the vector. Every position in the vector, indicates the location of the item on a certain axis; the first number represents the X-axis, the second represents the Y-axis, and so forth.

To keep it simple, let’s say the vector embedding for the word ‘bird’ equals [0.6, -0.2, 0.6]. Therefore, ‘bird’ is represented in a specific spot in 3-dimensional space. Simple, right?

Like the following image:

However, it gets more complicated when you start adding dimensions. As a rule of thumb, 50 dimensions is a usual vector length for representing things like grocery items, or people, or countries. For language-related tasks, somewhere in the range of 200-300 dimensions is more common. Our human brains are just not equipped for imagining anything beyond 3 to 4 dimensions, so we’ll have to trust the math.

Lastly, a constraint that is put on vector embeddings, is that they should always add up to 1.

How do you use vector embeddings?

Of course, a list of float numbers on its own does not mean anything. It only becomes interesting once you start factoring in that you could encode multiple items in high-dimensional space. Naturally, items that are closer to each other can be thought of as more similar or compatible. Consequently, items that are far away, are dissimilar. Items that are neither far away nor close by (think of it like a 90 degree angle, just in high-dimensional space) do not really have any discernible relationship to each other. A mathematical term for this is ‘orthogonal’.

The measure that is used for checking whether two vectors are similar / close, is called cosine similarity. Cosine similarity can be computed by calculating the dot product between two vectors. It goes like this;  Given two vectors [A, B, C] and [D, E, F], the numbers in each position are multiplied by each other; similarity equals A times D plus B times E plus C times F. This outcome is called a scalar in linear algebra. Subsequently, the dot product equals the sum of the previously calculated scalar.

The trick here is that for each position in the two vectors that are closer to each other within the -1 to 1 space, the result of the multiplication is higher. Therefore, the dot product between two vectors increases as they resemble each other more.

To test this out, let’s add two more words:

‘bird’                   = [0.6, -0.2, 0.6]

‘wing’                 = [0.7, -0.3, 0.6]

‘lawnmower’    = [-0.2, 0.8, 0.4]

Now what we have here, is actually our first vector matrix, which is just a fancy name for a set of vectors.

Intuitively, ‘bird’ and ‘wing’ should be quite similar, so have a high dot product. Let’s try it out:

Dot([0.7, -0.3, 0.6], [0.6, -0.2, 0.6]) = 0.42 + 0.06 + 0.36 = 0.86

On the opposite, ‘bird’ and ‘lawnmower’ shouldn’t really be related in any way, so the dot product should be close to 0:

Dot([0.6, -0.2, 0.6], [-0.2, 0.8, 0.4]) = -0.12 + -0.16 + 0.24 = 0.06

When you start adding lots of words to your vector matrix, it will start becoming a word cloud in N-dimensional space, in which certain clusters of words can be recognized. Furthermore, vector embeddings do not only represent bilateral relationships between items, but also capture more complicated relationships; In our language use, we have concepts of how words relate to each other. We understand that a leg is to a human, what a paw is to a cat. If our model is correctly trained on sufficient data, these types of relationships should be available.

Why do you need vector embeddings?

In any language-related machine learning task, your results will be best when using an algorithm that actually speaks the language that you’re aiming at analysing. In other words, you might want to use some technique that somehow captures the meanings of the words and sentences you’re processing.

Now, all ML algorithms require numerical INPUT. So, a requirement for an algorithm to capture language meaning, is that it should be numerical; every word should be represented by either a number or a series of numbers.

Generally, for transforming categorical entities such as words to numerical representations, one would apply one-hot encoding; for each of the amount of entities, which is called the cardinality of the variable, a separate column is generated, filled with 1s and 0s, like this:

Country Country_Greece Country_Chile Country_Italy
Greece 1 0 0
Chile 0 1 0
Italy 0 0 1
Chile 0 1 0

Essentially, one-hot encoding already transforms entities to vectors; each entity is represented by a list of numbers. However, this transformation has a number of drawbacks;

  • If you apply one-hot encoding to a text corpus, you’ll need to create a massive amount of extra columns. That’s both very inefficient and complicated to analyse and interpret. Text datasets often contain thousands of unique words. In other words; the created vector is too large to be useful.
  • Due to the Boolean nature of this transformation, each entity is equally as similar to all other entities. No possibility exists for somehow putting more similar items close to each other. Yet, we also want to capture meanings of words. Therefore,  we need a certain way of representing whether two words are closely related or not.

Embedding entities as vectors solves both these problems. Summarized, the method lets you encode every word in your dataset as a short list of float numbers, which are all between 1 and -1. Given that two words are similar, their respective lists resemble each other more.

How do you generate vector embeddings?

As I described before, similarity is essential to understanding vector embeddings. In the case of actually generating the embeddings from scratch, we’ll need a measure for similarity. Usually, in text analysis, we derive that from word co-occurrence in a text corpus. Essentially, words that are close together often within sentences, are theorized to be quite similar. The opposite is true for dissimilar words; they are never close together in text.

This is where neural networks come in: you use the neural network to either make two vectors more similar to each other each time the underlying entities occur together, or to make two vectors for items more dissimilar given that they don’t occur together.

Firstly, we need to transform our dataset to a usable format. Since the neural network will require some number to be predicted, we need a binary output variable. all similar word combinations will be accompanied with a 1, and all dissimilar combinations will be accompanied by a 0, like this;

First word Second word Binary output
‘bird ‘wing’ 1
‘bird’ ‘lawnmower’ 0
‘wing’ ‘lawnmower’ 0

These word combinations, accompanied by the output variable, are fed to the neural network. Essentially, the neural network predicts whether a certain combination of words should be accompanied by a 1 or a 0, and if it’s wrong compared to the real value, subsequently adjusts the weights in its internal vector matrix to make a more correct prediction next time that specific combination comes around.

How does the neural network optimize the vector matrix?

I’ll try to keep it simple. The neural network is trained by feeding it lots of bilateral word combinations, accompanied by 0s and 1s. Once a certain word combination enters the model, the two respective word vectors are retrieved from the vector matrix. At first, the numbers in the vectors are randomly generated, close to 0.

Subsequently, the dot product for those specific two vectors is calculated. Based on the outcome, the model will predict either a 0 (non-combination) or a 1 (actual combination). A sigmoid transformation is applied to the dot product of each combination, so that the outcome of the model is either 0, for a prediction of a non-combination, or a 1, for the prediction of a true combination.

The learning of the model, as with all neural networks, happens through backpropagation; after feeding a word combination to the network, the model prediction is compared with the actual outcome value. If those don’t match, this is propagated back to the embedding layer, where the embeddings are adjusted accordingly. If two words occur together frequently as a true combination, their vector embeddings will be adjusted to resemble each other more. The opposite happens for false combinations; those embeddings are adjusted so that they resemble each other less.

After training, we end up with an embedding matrix containing optimized embeddings for all words in the embedding matrix. We discard the sigmoid layer so we can examine dot products between different words.

A practical example (with code): word vectorization for tweets.

So, neural network vector embeddings can be challenging to understand. However, there are some libraries available that do all the heavy lifting in terms of programming; I’ll be using Gensim.

Some library imports:

import pandas as pd
import numpy as np
import csv
from gensim.models import word2vec
import nltk
import re
import collections

Firstly, we need some textual data. I chose for a set of 1.6 million tweets that were provided with a sentiment analysis competition on Kaggle. The data were provided in .csv format. Let’s load them. I’ll be keeping only a subset, for quick training.

##open kaggle file. We only need the tweet text, which is in 5th position of every line
with open('sentiment140\kaggle_set.csv', 'r') as f:
    r = csv.reader(f)
    tweets = [line[5] for line in r]
tweets = tweets[:400000]

Here’s what they look like.

print(tweets[:10])

#["@switchfoot http://twitpic.com/2y1zl - Awww, that's a bummer.  You shoulda got David Carr of Third Day to do it. ;D", "is upset that he can't update his Facebook by texting it... and might cry as a result  School today also. Blah!", '@Kenichan I dived many times for the ball. Managed to save 50%  The rest go out of bounds', 'my whole body feels itchy and like its on fire ', "@nationwideclass no, it's not behaving at all. i'm mad. why am i here? because I can't see you all over there. ", '@Kwesidei not the whole crew ', 'Need a hug ', "@LOLTrish hey  long time no see! Yes.. Rains a bit ,only a bit  LOL , I'm fine thanks , how's you ?", "@Tatiana_K nope they didn't have it ", '@twittera que me muera ? ']

It appears that some data cleaning is in order.

##clean the data. We'll use nltk tokenization, which is not perfect. 
##Therefore I remove some of the things nltk does not recognize with regex
emoticon_str = r"[:=;X][oO\-]?[D\)\]pP\(/\\O]"
tweets_clean = []

counter = collections.Counter()
for t in tweets:
    t = t.lower()
    ##remove all emoticons
    t = re.sub(emoticon_str, '', t)
    ##remove username mentions, they usually don't mean anything
    t = re.sub(r'(?:@[\w_]+)', '', t)
    ##remove urls
    t = re.sub(r"http\S+", '', t)
    ##remove all reading signs
    t = re.sub(r'[^\w\s]','',t)
    ##tokenize the remainder
    words = nltk.word_tokenize(t)
    tweets_clean.append(words)

That’s better:

print(tweets_clean[:10])
##[['awww', 'thats', 'a', 'bummer', 'you', 'shoulda', 'got', 'david', 'carr', 'of', 'third', 'day', 'to', 'do', 'it', 'd'], ['is', 'upset', 'that', 'he', 'cant', 'update', 'his', 'facebook', 'by', 'texting', 'it', 'and', 'might', 'cry', 'as', 'a', 'result', 'school', 'today', 'also', 'blah'], ['i', 'dived', 'many', 'times', 'for', 'the', 'ball', 'managed', 'to', 'save', '50', 'the', 'rest', 'go', 'out', 'of', 'bounds'], ['my', 'whole', 'body', 'feels', 'itchy', 'and', 'like', 'its', 'on', 'fire'], ['no', 'its', 'not', 'behaving', 'at', 'all', 'im', 'mad', 'why', 'am', 'i', 'here', 'because', 'i', 'cant', 'see', 'you', 'all', 'over', 'there'], ['not', 'the', 'whole', 'crew'], ['need', 'a', 'hug'], ['hey', 'long', 'time', 'no', 'see', 'yes', 'rains', 'a', 'bit', 'only', 'a', 'bit', 'lol', 'im', 'fine', 'thanks', 'hows', 'you'], ['nope', 'they', 'didnt', 'have', 'it'], ['que', 'me', 'muera']]

Now building the actual model is super simple:

##now for the word2vec
##list of lists input works fine, you could train in batches as well if your set is too large for memory. 
model = word2vec.Word2Vec(tweets_clean, iter=5, min_count=30, size=300, workers=1)

And check out the results. The embeddings generate some cool results; you can clearly see that the most similar words to a certain word, are actually almost identical in semantic meaning:

print(model.wv.similarity('yellow','blue'))
print(model.wv.similarity('yellow','car'))
print(model.wv.similarity('bird','wing'))

#0.725584
#0.25973
#0.484771

print(model.wv.most_similar('car'))
print(model.wv.most_similar('bird'))
print(model.wv.most_similar('face'))

#[('truck', 0.7357473373413086), ('bike', 0.6849836707115173), ('apt', 0.6751911640167236), ('flat', 0.666846752166748), ('room', 0.6251213550567627), ('garage', 0.6184542775154114), ('van', 0.6047226190567017), ('house', 0.5993092060089111), ('license', 0.5938838720321655), ('passport', 0.5903675556182861)]
#[('squirrel', 0.7508804202079773), ('spider', 0.7459014654159546), ('cat', 0.7174966931343079), ('kitten', 0.7117133140563965), ('nest', 0.7008006572723389), ('giant', 0.6956182718276978), ('frog', 0.6906625032424927), ('rabbit', 0.685787558555603), ('mouse', 0.6779413223266602), ('hamster', 0.6754754781723022)]
#[('mouth', 0.6230158805847168), ('lip', 0.5884073972702026), ('butt', 0.5822890996932983), ('finger', 0.579236626625061), ('smile', 0.5790724754333496), ('eye', 0.5769941806793213), ('cheek', 0.5746228098869324), ('skin', 0.5726771354675293), ('arms', 0.5703110694885254), ('neck', 0.5571259260177612)]

That’s it for today! I’ll be elaborating more on how to generate vector embeddings by defining neural networks with Tensorflow and Keras, in a future post.

How to convert Categorical Data to Continuous

One of the main aspects of preparing your dataset for statistics, machine learning, or other advanced analyses, is understanding exactly with which datatypes you’re dealing, and subsequently transforming them to desired datatypes for analysis.

In this post, I’ve gathered a number of common and less common methods from machine learning and statistics. These are, to my mind, adjacent fields that often deal with exactly the same problems, just with different terminology. Therefore, regardless of your field, elect the most suitable method. The list does probably not encompass all available transformations. Yet, I try to touch upon at least all of the common techniques.

Because the topic is enormous, I’ll try to provide intuitive, concise descriptions for each transformation. I intend to elaborate on some of the more advanced methods (e.g. vector embeddings) in future posts.

For the sake of clarity; in this article, I’ll use the word categorical as synonym for nominal and ordinal variables, and I’ll use the term continuous as a synonym for ratio and interval variables. Other common denominations are discrete  and numerical, respectively. Furthermore, a variable, or column, contains some characteristics of cases, or lines. These are different between cases, otherwise you would have no reason for maintaining the variable in your data. All these different characteristics are denominated levels through the entire article. The amount of unique levels in a variable, is called its cardinality.

In this post, I first describe data types. Subsequently, I touch upon the following transformations:

  • One-hot encoding
  • Binary encoding, or dummy variables
  • Contrasts
  • Transformation to dichotomous
  • Counting levels
  • Ranking based on count
  • Vector embeddings
  • Ignoring transformation altogether

First though, we should go through all data types available.

Data Types

Different sorts of information are often split into the following hierarchy.

  • Nominal data: A nominal variable contains different levels that are not more or less than each other; there is no hierarchy present. For example, one could think of car brands; there’s no clear hierarchy of which brand is better than the other.
  • Ordinal data: This data type contains different levels, in which a clear hierarchy is established. For these types of variables, it’s often difficult to define whether the distances between different levels are equally large. For example, car X may be better than car Y, and car Z is better than both. However, is car Z better than car Y by the same amount as car Y is better than car X?

Often, it’s quite difficult to distinguish between both aforementioned data types. For example, I mentioned that different car brands contain no hierarchy. Your opinion on the other hand may be that some brands are better than others, and that car brand is therefore an ordinal variable. Within statistics, nominal and ordinal are usually treated equally; for an analysis, they mean the same thing.

  • Ratio data: A ratio variable contains different levels, between which a very clear hierarchy can be established. Also, the distances between those levels are equal. For example, one’s net worth. You may be worth millions, or you may be millions in debt. There is always a clear concept that the difference between owning $1000 and $2000, is exactly the same as the difference between $10000 and $11000.
  • Interval data: This data type is exactly the same as the previous one, bar one exception. An interval variable has an absolute starting point, in other words, a zero. One’s salary is an example of this. One may earn anything between $0.01 and millions, but the general agreement is that a salary cannot be negative; you cannot earn less than $0 over a period of time.

As before, the difference between ratio and interval is only a slight nuance. Therefore, they’re usually treated as being equal.

Within statistics and machine learning, in almost any algorithm, variables are expected to be either in interval or ratio. Therefore, in many situations, one might want to change the datatype of one or more variables from categorical to continuous.

In order to get to a mathematical formula to predict / explain some output variable, the assumption of equal distances between levels needs to be met. Now is may seem that there are analyses which can have categorical variables as input. One of those is a method widely used in social sciences, named analysis of variance (ANOVA). This analysis requires categorical variables as input, and continuous variables as output. However, in the background, it transforms all categorical inputs to continuous with one-hot encoding. Also, some analyses do exist that use both categorical inputs and outputs, such as the chi-square test of independence. Yet, even chi-square transforms your categorical levels to counts of how often they occur, which is in essence continuous information.

Therefore, you might want to take full control of the data types in your set. Now that you understand data types, here are the most common methods for transforming categorical variables (at least the ones that I know of).

One-hot encoding

One-hot encoding, and very similarly creating dummy variables, may be the most widespread method for categorical to continuous transformation. As I mentioned before, some analyses even do it automatically without you noticing.

What essentially happens is this; your categorical variable contains K levels. Therefore, K new variables are created. On each of those new variables, cases that have the corresponding variable are set to 1, and all other levels are set to 0. For example:

Person ID Hair_colour Red Black Blonde
1 Red 1 0 0
2 Red 1 0 0
3 Blonde 0 0 1
4 Black 0 1 0
5 Black 0 1 0

As mentioned before, the Hair colour variable with three levels is split into three binary dummy variables, that all encode a specific colour.

Note. Dummy encoding is common in statistics, and slightly different from one-hot encoding; K – 1 new variables are created, and one level is set to 0 on all of those. Like this:

Person ID Hair_colour Red Black
1 Red 1 0
2 Red 1 0
3 Blonde 0 0
4 Black 0 1
5 Black 0 1

One-hot encoding is a very elegant transformation; it captures all available information efficiently. Nevertheless, it’s not suited to all situations. It works well when a variable contains a relatively small amount of levels; say, hair colour or blood type. When the amount of levels on the other hand is large, one-hot encoding becomes an unmanageable wildgrowth of variables. Given a variable that contains items bought in a supermarket, dummy variables are useless. A supermarket may sell thousands of different products, resulting in thousands of variables in your set. You lose both oversight on your data, and efficiency of analysis in the process.

Yet maybe the largest drawback to one-hot encoding is the lack of information on relationships between levels. With supermarket items, we know that apples and pears are very similar, but detergent is not like those items. Nevertheless, items are encoded in a way, that they are equally similar to each other.

Binary encoding

This transformation is quite similar to one-hot encoding. However, it is better suited to variables with high cardinality. In simple terms, binary encoding is just the transformation of the number that a certain level has, to its binary representation. For example, the number 2 could also be written as 010. It has the characteristic that all the levels are equally distant to each other, like one-hot. This can be advantageous or not, depending on your data. Let’s see:

Car_color Binary_1 Binary_2 Binary_3
Blue 0 0 0
Red 0 0 1
Grey 0 1 0
Black 0 1 1
Yellow 1 0 0
Purple 1 0 1
White 1 1 1

In this case, the cardinality determines the amount of columns required. As you see in the previous example, binary with three places supports up to seven levels, thereafter you’d go to four places.

Binary encoding works better than one-hot encoding on variables with high cardinality. Nevertheless, it has some drawbacks; like one-hot encoding, this technique does not capture relationships between levels in any way.

For one-hot encoding and binary transformation, many adjacent methods, such as hashing, are to be discussed. Yet, for an introduction, these two suffice.

Contrasts

Within linear regression analysis, it is common to have an independent variable with more than two levels, predicting a continuous dependent variable. One should never apply linear regression to a categorical variable with over two levels, because you are not certain that those levels are equally distant from each other (the condition for an interval variable). You don’t even know whether the levels are in the right order. It would be like trying to fit a straight line through a random order of car colors.

Therefore, you it’s possible to transform the independent variable to a set of contrasts; you define which levels should be linearly compared to each other, with dichotomous tests. You do this based on your hypotheses; which comparisons are relevant for your specific analysis? Because the amount of contrasts is always K – 1, you need to be selective regarding which comparisons should be made.

Treatment contrasts

Within experimental research, you often want to compare some treatment groups to your baseline, or control group. Let’s say, you’re assessing the effectiveness of two drug treatments A and B for symptoms of depression. Your baseline group of patients receives a placebo, and your two experimental groups receive the drug treatments, respectively. Your variable has three levels, therefore you get to specify two contrasts. You’d specify the following dichotomous comparisons, i.e. treatment contrasts:

  • Compare the severity of depression for treatment A against the baseline group;
  • Compare the severity of depression for treatment B against the baseline group.

Usually, regression models let you specify the following matrix for this. Each contrast embodies a dichotomous comparison.

Variable Contrast_1 Contrast_2
Baseline 0 0
Treatment A 1 0
Treatment B 0 1

As you can see, it’s very similar to dummy coding, and the variable that the baseline, is never set to 1 in the contrasts.

Helmert contrasts

Another common one is the Helmert contrast. Again, choosing for this one depends on which question you’re trying to answer. The basic functionality of the contrasts is this; the mean of every level is compared to the mean of the previous levels.

This is best demonstrated with an example; say, you are testing how drug treatment X affects depression symptoms. You take four measurements (denoted as M1 through M4) over the entire treatment. You want to know whether patients experience less symptoms at each measurement. Four levels, so three contrasts to be defined. Helmert contrasting dictates the following comparisons:

  • Compare the mean of M2 to that of M1;
  • Compare the mean of M3 to the grand mean of M1 and M2;
  • Compare the mean of M4 to the grand mean of M1, M2, and M3.

This is captured in the following matrix, with each comparison embodied in a contrast, respectively:

Variable Contrast_1 Contrast_2 Contrast_3
M1 -1 -1 -1
M2 1 -1 -1
M3 0 2 -1
M4 0 0 3

There are many more contrast designs such as sum contrasts and orthogonal contrasts, but for an intuitive understanding of the method, I’ll stick to these two. Using contrasts can be quite advantageous; the method allows you to exactly specify which questions should be answered, and therefore forces you to instil prior knowledge into the model. This completely opposes the usual expectation-free manner in which inputs and outputs are linked in many machine learning applications. Nevertheless, contrasts are quite a niche tool for regression, and are difficult to define once your research question spans more than four or five dichotomous comparisons.

Transformation to dichotomous

Variables that only contain two levels, can be used as continuous, even if they contain categories. This is because only two levels need to be compared. Distance between levels only matters given that you would have three levels or more. For example, if the detail level of your flight class variable is 0 (economy) or 1 (business) you may enter it into a linear regression without transformation.

Alternatively, you can aggregate a categorical variable that has more than two levels, to binary. Let’s say that your data contains a variable with levels that are car brands. You want to transform that variable to continuous, and notice that the people in your dataset only drive German or Japanese cars. Then, you’d be able to encode the car brand someone drives as either 0 (German) or 1 (Japanese). Another case in which transformation to binary may be beneficial, is when you only have three or four levels in your variable. You may merge the most similar ones, so that only two levels remain.

The advantage of applying this transformation is that it’s very fast. Both generating the variable and running analyses with it is very efficient. However, you might lose a lot of information; what if you’re predicting reliability, and some German brands turn out to be reliable, and others don’t? That information will be lost with aggregation. Nonetheless, this method may be beneficial in some situations.

Counts

Just counting the amount of occurrences of each level in the data, is actually a method that is used quite often in statistics. For example, in the chi-square test of independence I previously mentioned.

Say, you have supermarket data on individual sales of products, and you want to know what each customer has bought over a period of time. You might structure the data like this, eliminating any need for dummy or binary coding:

Customer Toilet paper Detergent Chilli sauce
Joe 20 14 50
Anne 13 16 17
Pedro 0 40 20
Jen 50 60 3

The major drawback here, is that aggregating your data to a more dense format always leads to information loss. For example, any sequence in which customers bought the product, cannot be analysed from this set. Nevertheless, counting often suffices in situations where aggregation is inevitable, and the cardinality of the continuous variable is quite low.

Ranking

The technique of ranking your levels, is a slightly more advanced method than just counting them.

This method rests upon the following premise: Levels of a categorical variable do not have any numerical meaning. However, if we are able to order them in some meaningful manner, they do may categorical value. So, how often each level occurs in the categorical variable, is a numerical representation for that level. Therefore, if we rank levels by how often they occur, our transformation should work. Naturally, the higher the count, the higher the rank that should be given to the level. This method should to some extent satisfy the condition for continuous variables that levels which are closer to each other, are more similar.

In the example below, the color blonde occurs the least often, and receives rank 1. Black hair occurs 3 times, and is assigned rank 2. Red occurs most times, and is therefore ranked 3.

Person_ID Hair_color Rank_score
1 Blonde 1
2 Red 3
3 Black 2
4 Red 3
5 Red 3
6 Black 2
7 Black 2
8 Red 3

However, applying the aforementioned may lead to the situation in which at least two levels occur equally as often in the variable. You can’t just assign them the same rank, which would mean that those levels are indistiguishable. We need a method to differentiate between them, because otherwise we don’t satisfy the condition that there should be K ranks for K levels. This paper actually has a clever mathematical solution for that.

Of course, the obvious disadvantage here is having a large number of levels with the same count. Yet, your analysis may actually benefit from ranking your variables, as the authors of the previously mentioned paper point out.

Vector embeddings

Simply put, every level of your variable receives a vector, or list of numbers, of length X. That vector represents the location of a specific level in X-dimensional space, usually between 50 and 300. Moreover, levels that are more similar to each other, are closer together in embedding space.

Think of it like GPS coordinates of cities on a map; cities that are closer to each other have more similar coordinates. And because two cities close together are likely in the same country, they may resemble each other very much. Nevertheless, vector embeddings are often computed in up to 300 dimensions, instead of two-dimensional maps.

Vector embeddings are most commonly used for transforming text to a usable mathematical transformation. A large text corpus often contains thousands of unique words, rendering most transformation techniques useless. Moreover, you might want to capture some of the rich semantic meanings of words that are present in text format.

For that purpose, you can use neural networks. What basically happens under the hood, is this; each unique word is assigned a vector. Your neural network model is then taught to predict whether two random words occur close to each other or not. As your network becomes proficient at this, it pushes co-occurring words closer to each other in vector space, and words that are never close, away from each other.

This results in an embedding space that actually captures all the meanings of words in relation to each other, in a mathematical way. Of course, our brains limited to three dimensions won’t understand anything of it.

Not only text is suitable for embeddings. Any data in which combinations of certain levels of the variable occur, is sufficient for generating embeddings. For example, if you’re working with product sales information from a supermarket, your data contains patterns on which items are bought together often, and which aren’t. Based on the aforementioned product combinations, you can produce vectors that group similar products together. These vector embeddings actually capture some type of mathematical meaning of a product. For example, it’s very unlikely that two brands of milk are often bought together. You either choose one or the other. However, those two milk products will probably be encoded as very similar vectors. This is because although they do not occur together and therefore have no direct relationship, their relationships to other products are very likely to be similar. If you buy milk, you might buy flour, regardless of the brand of milk. Thus, the different milk products will be close in vector space.

Creating vector embeddings is, to my opinion, the most elegant way to transform some categorical information to continuous. This is because it not only encodes each available level fully, but vector embeddings also contain information on how different levels of your variable are related to each other.

On the other hand, vector embeddings are only applicable in very limited situations. You need some information on which combinations do or do not occur between levels. This is most easily found in text, but as I mentioned before, sales data also works, as do Wikipedia links, or social media mentions. You’ll need to be creative in how to acquire combination information if you’d like to use vector embeddings.

Or, just ignore the transformation

The last, and probably very incorrect option available, is to just avoid any transformation and use your categorical variable as continuous. In this case, you’d transform all the different levels in your variable to the numbers 1 to K, in random order. You basically ignore any ranking or clustering within the variable. Although I do not have any experience with this, and it goes against basic assumptions of any advanced analysis, I think that in some situations, it may actually work well.

For example, if you’re trying to apply linear regression with an independent variable with three levels, and you have some knowledge on the hierarchy within those levels (so an ordinal variable) the regression function may fit well.

In my opinion, the most important thing to remember here is that, given that you’re going to avoid any transformation, this should be a well-founded decision; without a solid indication that it may actually improve your analysis, don’t do it.

Conclusion

So there you go, all common available methods for transforming categorical data to continuous. Spending some time on finding the right one for your feature engineering or statistical analysis, may actually mean a large increase in performance. Now go apply them!

P = NP: A Million Dollar Problem

In 2000, the Clay Math Institute awarded one million dollars each for seven important, long-standing math problems. Of these, only one has been solved since (the Poincaré conjecture). I will discuss another one, the P = NP problem. This is arguably one of the most relevant and famous problems in modern computer science. Solving the question cures cancer, obliterates all existent computer privacy, instantly beats everyone at chess and Tetris, reconstructs the full history of species from DNA, and would most likely create general mayhem on earth.

I stumbled upon this problem while reading The Master Algorithm by Pedro Domingos. He provides a very intuitive description of the problem: ‘P and NP are the two most important classes of problems in computer science. [..] A problem is in P if we can solve it efficiently, and it’s in NP if we can efficiently check its solution. The famous P = NP question is whether every efficiently checkable problem is also efficiently solvable’.

Now if we want to solve the problem and win the lottery, we need to dig a little deeper than that. As Domingos writes, the two most important classes of problems in computer science are P and NP. We designate any problem that can be solved in polynomial time to be P (polynomial), and with NP (non-deterministic polynomial) we mean any problem that can be solved in at least exponential time but for which the solution can be checked in polynomial time. The catch here is, if you find a way to efficiently solve one of the NP problems, you solve all of them because at their core, they are all the same. This is called NP-completeness.

For understanding what the aforementioned sentences mean, we need to grasp exactly what computer scientists mean when they talk about time. For solving problems, they do not use the normal, linear time that we know, but correct for the amount of elements in solving a problem, namely N. In this case, time is linear only for very simple problems. For example, let’s say you need to find the highest number in a list (or select for any other condition). Then you need to iterate through the entire list only once. This is a linear problem, in the sense that every added element (to the length of the list) adds only one extra computation/extra time unit. In the graph, it’s the O(n) line.

However, many problems in computer science are in polynomial time. This means that the amount of computations needed to solve the problem is the amount of elements raised to some power (for example N-squared or N3; O(n^2) in Figure 1). One notable problem in P is that of exactly figuring out what the maximal profit is in the case of multiple cost functions, called linear programming. For example, a car manufacturer disposes of one factory, a certain amount of employees and a certain amount of raw materials. The company can choose between building two car models that both have a different required amount of raw materials, labor, and retail price. Within the boundaries of available resources, you’d compute all possible combinations of amounts for both car models (so compute the full solution space), and go with the most profitable, or optimal, solution. For more info, check this page.

However, problems in NP need more than that amount of computing to be solved. The solution space is way larger than that of exponential problems. The official definition is a problem that can be solved in polynomial time by a non-deterministic Turing machine (Cook, 2000).

An ‘easier’ way to look at NP problems is this: They require at the very least an amount of computations that is equal to some number raised to N, sometimes even a factorial of N. So for example if N = 20, at least 220 computations are needed (Figure 1: O(2^n)) . This means that the number of computations increases so fast with N, that we can only compute exact solutions for very small problems. To illustrate this, I’ll use an example from MIT: If a computer takes a second to perform a computation with 100 elements in the case the algorithm is in linear time (N = amount of computations), that same algorithm will take ~3 hours if the amount of computations is equal to N3 (polynomial time), and will take 300 quintillion years if the computation time is equal to 2N (exponential time). This example should provide some insight into the timescale which is required for solving NP problems with large Ns. As a result, we can currently only solve very simple NP problems with only a few components.

However, NP problems are usually easily checkable in polynomial time. For example, computing the optimal solution for a game of Tetris requires a ridiculously large amount of computations, but it is very easy to see that one has solved Tetris. Same for Sudoku; solving is difficult, checking the solution for mistakes is easy.

Another way of looking at many NP problems is that for most NP problems, one needs to compute all possible combinations of the included elements to find the optimal solution. This is way worse even than 2N; in this case the amount of possible combinations is (N-1)!, which is, in the case of 9 elements, 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1=40320 (~1 hour), and in the case of 100 elements, too large to compute on my laptop. The point here is that the time increase for each added element is so large, that only very simple computations can be made with current methods.

An example of an NP problem is that of the travelling salesman. Let’s say that the salesman needs to visit a number of cities (N), that all have different distances from each other, and he wants to visit them in the most efficient route possible. In that case, he needs to compute distance to be travelled for all possible travel itineraries, which equals (N-1)!. That is, even for 10 cities, 362880 possibilities. In the case of the following picture, with 50 cities, that is approximately 6082818 * 1086 possibilities.

The aforementioned example is an example of a sub-class of NP problems, namely NP-hard problems. These are problems that cannot be solved in polynomial time and whose solutions cannot be checked in polynomial time. In the aforementioned example, this implicates that, even if you were able to have a computer put out a solution for optimal travel, you would never be sure if the computer got it right. Another example of an NP-hard problem is chess; if you have a computer find the optimal move to make, how would you even know that the computer got it right?

Here, I’ll recap NP-completeness; even though the travelling salesman problem, sudoku, cracking encryption and reverse engineering gene sequences seem not even remotely similar, they’re all the same problem at heart. They all consist of finding a unique, perfect solution, in which with current methods, iterating through all possible solutions is required. Therefore, solving one means solving them all. You’ll need to prove that problems that can only be solved in exponential or factorial time, perfect solutions exist as well in polynomial time (all relative to N). In terms that we can all understand, this implies that for receiving the million dollars, you need to prove that you can find the perfect combination of elements for a problem, without going through all possible combinations.

Now the aforementioned does not mean that scientists have not found solutions to any of the NP problems in this essay. For example, if you look at the map of the United States, it is likely that you could draw a line between all the cities that is going to be very close to the optimal solution. This is one way scientists approach NP problems; through heuristics. These are decision models that do not require computing all possible solutions, but that use a set of rules to approximate the solution. For example in the travelling salesman problem, one could set the rule that the computer/algorithm needs to move to the closest next city. This might in some cases result in an approximate solution. However, for things like cracking 50-digit passwords and reconstructing species from gene codes, this will not get us very far.

It is not very likely that mathematicians will ever find evidence for P = NP. For example. In a 2002 study, MIT researchers found that 61 computer scientists thought that P probably is not equal to NP, and 9 thought that is does (6). However, some of those told the researcher that they just said that to be controversial. Time has proven them right; no solution has arisen in the 16 years since. Thus, the general consensus us that P is not equal to NP, but no evidence has been found for this statement either.

Then why does anyone still look for the solution to P = NP? Firstly, many breakthroughs in computer science come from someone searching a solution and accidentally finding ways to make algorithms more efficient or powerful. Secondly, looking makes one understand one of the major limitations of computer science and how to deal with it. Thirdly, it provides insight into some major problems in modern society. And lastly, it’s just interesting. So have a go!

I refer to this video for another quite intuitive explanation of P=NP.