This document discusses using operators in APL to perform statistical analysis. It proposes defining operators that take statistical functions or distributions as left operands and relations as right operands. This reduces the number of functions needed compared to other languages. Examples of operators include probability, criticalValue, and hypothesis. Sample data can be represented as raw values, frequencies, or summary statistics, making them interchangeable for the operators. The TamingStatistics namespace implements this approach in APL.

1. Stephen Mansour, PhD
University of Scranton and The Carlisle Group
Dyalog ’14 Conference, Eastbourne, UK

2.  Many statistical software packages out there:
Minitab, R, Excel, SPSS
 Excel has about 87 statistical functions. 6 of
them involve the t distribution alone:
T.DIST T.INV
T.DIST.RT T.INV.2T
T.DIST.2T T.TEST
 R has four related functions for each of 20
distributions resulting in a total of 80
distribution functions alone

3. Defined Operators!
 How can we exploit operators to reduce the
explosive number of statistical functions?
 Let’s look at an example . . .

4.  Typical attendance is about 100 delegates
with a standard deviation of 20.
 Assume next year’s conference centre can
support up to130 delegates.
 What are the chances that next year’s
attendance will exceed capacity?

5. =1-NORM.DIST(130,100,20,TRUE)
Now let’s use R-Connect in APL:
+#.∆r.x 'pnorm(⍵,⍵,⍵,⍵)' 130 100 20 0
Wouldn’t it be nice to enter:
100 20 normal probability > 130
100 20 (normal probability >) 130

6. normal probability < 1.64
100 20 normal probability between 110 130
5 0.5 binomial probability = 2
7 tDist criticalValue < 0.05
5 chiSquare randomVariable 13
mean confidenceInterval X
(SEX='F') proportion hypothesis ≥ 0.5
GROUPA mean hypothesis = GROUPB
variance theoretical binomial 5 0.2

7.  Summary Functions
◦ Descriptive Statistics
 Probability Distributions
◦ Theoretical Models
 Relations

8.  Summary functions are of the form:
𝑦 = 𝑓 𝑥1, 𝑥2, … 𝑥 𝑛
 They produce a single value from a vector.
 Structurally they are equivalent to g/ where g is a
scalar function and the right argument is a simple
numeric vector.
 A statistic is a summary function of a sample; a
parameter is a summary function of a population.

9.  Examples
◦ Measures of central tendency:
mean, median, mode
◦ Measures of Spread
variance, standard deviation, range , IQR
◦ Measures of Position
min, max, quartiles, percentiles
◦ Measures of shape
skewness, kurtosis

10.  Probability Distributions are functions defined
in a natural way when they are called without
an operator:
◦ Discrete: probability mass function
◦ Continuous: density function
 Left argument is parameter list
 Right argument can be any value taken on by
the distribution.
 Probability Distributions are scalar with
respect to the right argument.

11. Discrete
Distributions
Parameter List
uniform a - lower bound (default 1), b - upper bound.
binomial n - Sample size, p - probability of success
poisson λ - average number of arrivals per time period
negativeBinomial n - number of success, p - probability of success
hyperGeometric m - number of successes , n - sample size ,
N - Population size
multinomial V - List of Values (default 1 thru n),
P - List of probabilities totaling 1

12. Continuous Distributions Parameter List
normal μ - theoretical mean (default 0); σ - standard deviation
(default 1)
exponential λ - mean time to fail
rectangular (continuous
uniform)
a - lower bound (default 0), b - upper bound (default 1)
triangular a - lower bound, m - most common value,
b - upper bound
chiSquare df - degrees of freedom
tDist (Student) df - degrees of freedom
fDist df1 - degrees of freedom for numerator,
df2 - degrees of freedom for denominator

13.  Relational functions are dyadic functions
whose range is {0,1}
 1=relation is satisfied, 0 otherwise.
 Examples:
< ≤ = ≥ > ≠ ∊
between←{¯1=×/×⍺∘.-⍵}

14.  By limiting the domain of an operator to one
of the previously-defined functional
classifications, we can create an operator to
perform statistical analysis.
 For a dyadic operator, each operand can be
limited to a particular (but not necessarily the
same) functional classification.

15. Operator Left Operand Right Operand
probability Distribution Relation
criticalValue Distribution Relation
confidenceInterval Summary N/A
hypothesis Summary Relation
goodnessOfFit Distribution N/A
randomVariable Distribution N/A
theoretical Summary Distribution
running Summary N/A

16.  Most functions and operators can easily be
written in APL.
 Internals not important to user
 R interface can be used if necessary for
statistical distributions.
 Correct nomenclature and ease of use is
critical.

17. A sample can be represented by raw data, a
frequency distribution, or sample statistics.
The following items are interchangeable as
arguments to the limited domain operators
above:
 Raw data: Vector
 Frequency Distribution: Matrix
 Summary Statistics: PropertySpace

18. Matrix: Frequency
Distribution
Namespace: Sample
Statistics
D
2 0 3 4 3 1 0 2 0 4
⎕←FT←frequency D
0 3
1 1
2 2
3 2
4 2
mean D
1.9
variance D
2.5444
PS←⎕NS ''
PS.count←10
PS.mean←1.9
PS.variance←2.544

19.  )LOAD TamingStatistics
◦ All APL version
 )LOAD TamingStatisticsR
◦ Third party – Must install R (Free)

20.  There are many statistical packages out there;
some, like R can be used with APL
 Operator syntax is unique to APL
 R can be called directly from APL using
RCONNECT, but APL operator syntax is easier
to understand.