As a fun and simple exercise with text processing in R, we'll build a simple model that generates random character strings which mimic the roughly 6,000 ticker symbols listed on the AMEX, NASDAQ, and NYSE. A basic tool for generating random text patterned off of an input source text

*is the Markov chain*. Markov chains are the engine behind gibberish generators, a briefly popular internet meme in the 90's.

We start with the base of all real ticker symbols. Using the TTR ("Technical Trading Rules") library, it's easy to get the full list for the AMEX, NASDAQ, and NYSE.

library(TTR) listings <- stockSymbols()

On November 17, 2013, TTR found 6,462 ticker symbols. The 10 largest symbols by market cap at that time were:

listings <- listings[order(listings$MarketCap,decreasing=TRUE),] Symbol Name MarketCap IPOyear Exchange AAPL Apple Inc. 472354352358 1980 NASDAQ XOM Exxon Mobil Corporation 416188308487 NA NYSE GOOG Google Inc. 345299382446 2004 NASDAQ MSFT Microsoft Corporation 315895470257 1986 NASDAQ GE General Electric Company 275192436800 NA NYSE JNJ Johnson & Johnson 266315572747 NA NYSE WMT Wal-Mart Stores, Inc. 256994921591 1970 NYSE CVX Chevron Corporation 230896151941 NA NYSE PG Procter & Gamble Company (The) 230614695048 NA NYSE WFC Wells Fargo & Company 229351244873 NA NYSE

Ticker symbols pulled down from TTR range from 1 to 8 characters in length and make use of the capital letters A through Z and the hyphen "-". In ticker symbols that contains hyphens, the characters following a hyphen appear to be the "behind the dot" codes that provide additional information on the asset type. To keep things simple, we'll just strip off all this additional info:

symbols <- unique(gsub("-.*","",listings$Symbol))

This leaves us with a list of 5,594 symbols. Let's get the distribution of symbol lengths for our symbol list:

lengths <- nchar(symbols) pLength <- prop.table(table(lengths)) print(round(pLength,4)) lengths 1 2 3 4 5 0.0039 0.0387 0.4572 0.4698 0.0305

Let's now explain a very simple Markov chain model for text strings. A word $w$ of length $l$ is a combination of letters $c_1, c_2,...,c_n$, which we naturally write as $w=c_1c_2...c_n$. If the first $n-1$ letters of $w$ are known, we write the probability that the $n$-th letter is $c_n$ as $p(c_n|c_1c_2...c_{n-1})$.

A Markov chain model of $p(c_n|c_1c_2...c_{n-1})$ allows us to assume that $$p(c_n|c_1c_2...c_{n-1})=p(c_n | c_{n-1}).$$ That is, we can assume that the $n$-th letter of the word only depends on the letter that directly precedes it. The Markov assumption often allows us to make huge simplifications to probability models and computations.

As an example, suppose that ticker symbols are generated by a probabilistic process. If we have a three letter ticker symbol with its first two letters being "WM", we can meaningfully quantify the probability that the third letter is a "T". Under a Markov model, $p(T|WM)=p(T|M)$. Now there is a little bit of ambiguity in the expression $p(T|M)$. Is this the probability that $T$ directly follows an $M$ anywhere in a symbol or is the probability that $T$ is the third letter given $M$ is the second? Here we'll assume that the position information is relevant and we'll compute $$p(c_3=T|WM) = p(c_3=T | c_2 = M).$$

With our sample of symbols, here is a way to approximate $p(T|WM)$ in R using the Markov model:

sum(grepl("^.MT",symbols[lengths==3]))/sum(grepl("^.M",symbols[lengths==3])) [1] 0.08391608Unpacking the syntax, grepl is regular expression matching and returns TRUE when a match is found. The expression "^.MT" matches all the symbols whose first letter is anything and whose second and third letter are "MT" while "^.M" matches to anything with second letter "M". The sums count the number of TRUE evaluations.

With the preliminaries out of the way, let's demonstrate a function which generates a random symbol based on the Markov chain model:

# Create a random symbol of length n based on a length 1 Markov Chain newSymbol <- function(n=3){ symbolSet <- symbols[lengths==n] s <- "" for(i in 1:n){ # Match s in the first i characters of symbolSet pattern <- paste0("^",substr(s,i-1,i-1)) match <- grepl(pattern,substr(symbolSet,i-1,i-1)) # Distribution of next letter nxtLetterDist <- substr(symbolSet,i,i)[nxtLetterDist!="" & match] # Sample the next character from the distribution nextChar <- sample(nxtLetterDist,1) # Append to the string s <- paste0(s,nextChar) } return(s) }

Here is an example of the generator in action:

> replicate(10, newSymbol(3)) [1] "EMS" "ZTT" "NBS" "BTG" "MAG" "HAM" "PLK" "ALX" "CFM" "FLS"

How can we test that the symbol generator is really making symbols that are similar to the actual ticker symbols? Well, one way is to inspect how the generator performs relative to a completely random symbol generator. Since there are 26 letters available to use in a valid ticker symbol, a completely random generator would use any given letter with probability 1/26. So, for example, the distribution of the second character of completely random generated symbol should be a uniform distribution on the letters A-Z with uniform probability 1/26. For the Markov chain model, we would hope that the second character of a generated random symbol would have a probability distribution similar to the distribution of the second character of the set of actual ticker symbols.

The following snippet inspects the distribution of the second character of symbols of length 3 for natural symbols, Markov generated symbols, and completely random symbols.

# Generate a bunch of symbols with the Markov chain symbolsA <- unique(replicate(10000, newSymbol(3))) # Build data frame for second character distributions require(ggplot2) require(reshape2) Natural <- prop.table(table(substr(symbols[lengths==3],2,2))) Markov <- prop.table(table(substr(symbolsA,2,2))) secondChar<-as.data.frame(t(rbind(Natural,Markov))) secondChar$character <- rownames(secondChar) secondChar <- melt(secondChar, id=c("character")) # Build the distribution plot ggplot(secondChar, aes(x=character,y=value)) + geom_point(aes(color=variable), alpha=.75,size=5)+ ylab("Probability")+xlab("Character")+ ggtitle("Ticker Symbols: Second Character Distribution")+ geom_hline(yintercept=1/26, size=1,color="black")

Here is the plot: In this figure, the black line represents the uniform probability distribution corresponding to a completely random generation of length 3 ticker symbols. The blue and red circles compare the Markov generator model to the true distribution of the second characters in length 3 ticker symbols.

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