Tuesday, 16 April 2019

President's Trophy - A Curse by Design


There are lots of ways the NHL rewards failure and punishes excellence, like the player draft, ever shrinking salary caps, and the half-win award for participating in overtime, but even the way in which playoff pairings are decided has a perverse incentive.


There are three rewards to doing well in the NHL regular season:
-         1. Going to the playoffs,
-          2. A favorable first round pairing in said playoffs,
-          3. Home team advantage for playoff games more often.

Here I argue that the first-round pairings are not as favorable as they could be. 

Tuesday, 9 April 2019

Natural Language Processing in R: Edit Distance

These are the notes for the second lecture in the unit on text processing. Some useful ideas like exact string matching and the definitions of characters and strings are covered in the notes of Natural Language Processing in R: Strings and Regular Expressions

Edit distance, also called Levenshtein distance, is a measure of the number of primary edits that would need to be made to transform one string into another. The R function adist() is used to find the edit distance.

adist("exactly the same","exactly the same") # edit distance 0 
adist("exactly the same","totally different") # edit distance 14


Natural Language Processing in R: Strings and Regular Expressions.

In this post, I go through a lesson in natural language processing (NLP), in R. Specifically, it covers how strings operate in R, how regular expressions work in the stringr package by Hadley Wickham, and some exercises. Included with the exercises are a list of expected hang-ups, as well as an R function that can quickly check the solutions.

This lesson is designed for a 1.5-2 hour class for senior undergrads.

Contents:
  • Strings in R
    • Strings can be stored and manipulated in a vector
    • Strings are not factors
    • Escape sequences
    • The str_match() function
  • Regular expressions in R
    • Period . means 'any'
    • Vertical line | means 'or'
    • +, *, and {} define repeats
    • ^ and $ mean 'beginning with' and 'ending with'
    • [] is a shortcut for 'or'
    • hyphens in []
    • example: building a regular expression for phone numbers 
  • Exercises
    • Detect e-mail addresses
    • Detect a/an errors
    • Detect Canadian postal codes

Sunday, 7 April 2019

Writing R documentation, simplified


A massive part of statistical software development is the documentation. Good documentation is more than just a help file, it serves as commentary on how the software works, includes use cases, and cites any relevant sources.

One cool thing about R documentation is that it uses a system that allows it to be put into a variety of different formats while only needing to be written once.

Monday, 1 April 2019

Bingo analysis, a tutorial in R



I'm toying with the idea of writing a book about statistical analyses of classic games. The target audience would be mathematically interested laypeople, much like Jeffrey Rosenthal's book Struck by Lightning ( https://www.amazon.ca/Struck-Lightning-Jeffrey-S-Rosenthal/dp/0006394957 ).

The twist would be that chapter would contain step-by-step R code or Python code so that the reader could do the same analysis and make changes based on their own questions. Material would like this post on Bingo, as well as my previous post on Snakes and Ladders ( https://www.stats-et-al.com/2017/11/snakes-and-ladders-and-transition.html ).

There would also be some work on chess variants, othello, poker, and possibly go, mahjong, and pente. Tied to each analysis could be light lessons on statistics. This Bingo analysis involves Monte Carlo style simulation, as well as notes on computing expected values, CDFs and PDFs.

Tuesday, 26 March 2019

Dataset - The Giant Marmots of Moscow


Stat 403/640/890 Analysis Assignment 3: Polluted Giant Marmots
Due Wednesday, April 3rd
Drop off in the dropbox by the stats workshop, or hand in in class.


For this assignment, use the Marmots_Real.csv dataset.

Main goal: The giant marmots of Moscow have a pollution problem. Find a model to predict the pollutant concentration (mg per kg) in the local population without resorting to measuring it directly. (It turns out that measuring this pollutant requires some invasive measures like looking at bone marrow).

The dataset Marmots_real.csv has the data from 60 such marmots, including many variables that are easier measure:

Variable Name
Type
Description
Species
Categorical, Unordered
One of five species of giant marmot
Region
Categorical, Unordered
One of five regions around Moscow where the subject is captured
Age
Numerical, Continuous
Age in years
Pos_x
Numerical, Continuous
Longitude, recoded to (0,1000), of capture
Pos_y
Numerical, Continuous
Latitude, recoded to (0,1000), of capture
Long_cm
Numerical, Continuous
Length nose to tail in cm
Wide_cm
Numerical, Continuous
Width between front paws, outstretched
Sex
Binary
M or F
Lesions
Numerical, Count
Number of skin lesions (cuts, open sores) found upon capture
Injured
Binary
0 or 1, 1 if substantial injury was observed upon capture.
Teeth_Condition
Categorical, Ordered
Condition of teeth upon capture, listed as Very Bad, Bad, Average, or Good.
Weight
Numerical, Continuous
Mass of subject in 100g
Antibody
Numerical, Continuous
Count of CD4 antibody in blood per mL
Pollutant
Numerical, Continuous
mg/kg of selenium found in bone marrow

There are no sampling weights. There is no missing data. There should be little to no convergence or computational issues with this data.


Assignment parts:
1) Build at least three models for pollutant and compare them (e.g. r-squared, general parsimony). Be sure to try interactions and polynomial terms.

Select one to be your ‘model to beat’.

2) Check the diagnostics of your model to beat. Specifically, normality of residuals, influential outliers, and the Variance Inflation Factor. Comment.

3) Try a transformation of the response in your model to beat, and see if you can improve the r-squared.

4) Try a PCA-based model and see if it comes close to you model to beat.

5) Take your ‘model to beat’ and add some terms to it. Call this the ‘full model’, and use that as a basis for model selection using stepwise and the AIC criterion. Is the stepwise-produced model better (r-squared, AIC) than your ‘model to beat’?

6) If you haven’t already, try a random effect of something appropriate, and see if it beats the AIC of the stepwise model. Use the AIC() function to see the AIC of most models.




Useful sample code:

######## Preamble / Setup
## Load the .csv file into R. Store it as 'dat'
dat = read.csv("marmots_real.csv")
dat$region = as.factor(dat$region)

library(car) # for vif() and boxcox()
library(MASS) # for stepAIC()
library(ks) # for kde()
library(lme4) # lmer and glmer



##### Try some models, with interactions
### Saturated. Not enough DoF
mod = lm(antibody ~ species*region*age*weight + long_cm, data=dat)
summary(mod)

### Another possibility, enough DoF, but efficient?
mod = lm(antibody ~ species + region + age*weight*long_cm, data=dat)
summary(mod)
vif(mod)
plot(mod)
AIC(mod)

### With polynomials
mod = lm(pollutant ~ age + long_cm + wide_cm + I(sqrt(age)) + I((long_cm*wide_cm)^3), data=dat)
summary(mod) ## High r-sq and little significance? How?
vif(mod) ## Oh, that's how.


#### Model selection,
mod_full = lm(antibody ~ species + region + age*weight*long_cm + I(log(wide_cm)) + lesions, data=dat)

### Stepwise selection based on AIC
stepAIC(mod_full)

### What if we do a BIC penalty
## Try without trace=FALSE so we can see what's going on.
stepAIC(mod_full, k=log(nrow(dat)))





######### Transformations
### Start with the classic tranforms

### Another possibility, enough DoF, but efficient?
mod = lm(sqrt(pollutant) ~ species + region + age*weight*long_cm, data=dat)
summary(mod)

mod = lm(log(pollutant) ~ species + region + age*weight*long_cm, data=dat)
summary(mod)

                  
### Box-cox to find the range of best ones through
boxcox(pollutant ~ species + region + age*weight*long_cm, data = dat,
       lambda = seq(-2, 3, length = 30))
                       
         
boxcox(antibody ~ species + region + age*weight*long_cm, data = dat,
       lambda = seq(-2, 3, length = 30))
                       
### Anything above the 95% line is perfectly fine. Anything close is probably fine too.
### Reminder:
### Lambda = -1 is 1/x (inverse) transform
### lambda = 0 is log tranform
### Lambda = 1/2 is sqrt tranform
### Lambda = 1 is no tranform
### Lambda = 2 is square transform





#################
### MANOVA
### First, Are the two responses related?
cor(dat$antibody, dat$pollutant)
plot(dat$antibody, dat$pollutant)

### Start with the simple ANOVAs
mod_anti = lm(antibody ~ species + region + age*weight*lesions, data=dat)
mod_poll = lm(pollutant ~ species + region + age*weight*lesions, data=dat)
aov_anti = anova(mod_anti)
aov_anti
summary(aov_anti)


aov_poll = anova(mod_poll)
aov_poll
summary(aov_poll)

### Now try the multiple ANOVA
aov_mult = manova(cbind(antibody, pollutant) ~ species + region + age*weight*lesions)
aov_mult
summary(aov_mult) ### Your job: Make a model that balances simplicity with fit.
                        ## Residual standard errors: Lower = better fit



###################                   
# PCA

### convert the relevant categorical variables
dat$teeth_num = as.numeric(factor(x = dat$teeth_condition, levels=c("Very Bad","Bad","Average","Good")))
dat$sex_num = as.numeric(factor(x = dat$sex, levels=c("F","M")))

PCA_all = prcomp( ~ age + weight + lesions + long_cm + wide_cm + injured + teeth_num + sex_num,
     data = dat,
     scale = TRUE)

summary(PCA_all)
plot(PCA_all, type="lines")


### Add the Principal components to the marmots dataset         
dat = cbind(dat, PCA_all$x)
head(dat)

### Try a few models of the responses using the PCAs
mod_PCA1 = lm(antibody ~ PC1 + PC2 + PC3 + PC4 + PC5 + PC6, data=dat)
summary(mod_PCA1)

mod_PCA2 = lm(antibody ~ PC1 * PC2 * PC3 , data=dat)
summary(mod_PCA2)

mod_PCA3 = lm(antibody ~ PC1 + PC2 + PC3 , data=dat)
summary(mod_PCA3)

mod_PCA4 = lme(pollutant ~ PC1 + PC2 + PC3 + (1|region), data=dat)
summary(mod_PCA4)  ### Why non-zero correlations? Adjustments for region were made




                       
                       
### Fixed vs Random

marmots$region = as.factor(marmots$region)

summary(lm(pollutant ~ region, data=dat))
summary(lmer(pollutant ~ (1|region), data=dat))
summary(lmer(pollutant ~ PC1 + PC2 + (age|region), data=dat))

summary(lmer(pollutant ~  age + (1|region), data=dat))$logLik ### Higher LogLik is better
summary(lmer(pollutant ~  age + (1|region), data=dat))$AICtab ### Lower REML (AIC calculated by REML) is better

### Compare the $AICtab value to the result from stepAIC

Saturday, 23 March 2019

What makes a good data dictionary?


A data dictionary is a guide, external to the dataset in question, that explains what each variable is in a human-readable format. In R, the programming equivalent of a data dictionary is the result of the str() function, which will show the first few values of each variable, the format (e.g. numerical, string, factor), and other important information (e.g. the first few levels of the factor).

A data dictionary may include software-specific features, but it should still be value to anyone using the dataset, regardless of the software they are using.

Other than that, what makes a good data dictionary?