Introduction to the Course; Basics of Data

36-350
25 August 2014

Agenda

• Course overview and mechanics
• Built-in data types
• Built-in functions and operators
• First data structures: Vectors and arrays

Why good statisticians learn to program

• Independence: Otherwise, you rely on someone else having given you exactly the right tool

• Honesty: Otherwise, you end up distorting your problem to match the tools you have

• Clarity: Making your method something a machine can do disciplines your thinking and makes it public; that's science

How this class will work

• No programming knowledge presumed
• Some stats. knowledge presumed
• General programming mixed with data-manipulation and statistical inference
• Class will be very cumulative
• Keep up with the readings and assignments!

Mechanics

• Two lectures a week: concepts, methods, examples
• Lab to try stuff out and get fast feedback (10%)
• HW weekly to do longer and more complex things (30%)
• Mid-term project (2 weeks) (20%)
• Final group project (1 month) (40%)

Assignments, office hours, class notes, grading policies, useful links on R: http://www.stat.cmu.edu/~cshalizi/statcomp/14

Blackboard for a grade-book and for turning in homework only, check the class website for everything else

The class in a nutshell: Functional programming

2 sorts of things (objects): data and functions

• Data: things like 7, “seven”, $$7.000$$, the matrix $$\left[ \begin{array}{ccc} 7 & 7 & 7 \\ 7 & 7 & 7\end{array}\right]$$

• Functions: things like $$\log{}$$, $$+$$ (two arguments), $$<$$ (two), $$\mod{}$$ (two), mean (one)

A function is a machine which turns input objects (arguments) into an output object (return value), possibly with side effects, according to a definite rule

Programming is writing functions to transform inputs into outputs

Good programming ensures the transformation is done easily and correctly

Machines are made out of machines; functions are made out of functions, like $$f(a,b) = a^2 + b^2$$

The route to good programming is to take the big transformation and break it down into smaller ones, and then break those down, until you come to tasks which the built-in functions can do

Before functions, data

Different kinds of data object

All data is represented in binary format, by bits (TRUE/FALSE, YES/NO, 1/0)

• Booleans Direct binary values: TRUE or FALSE in R
• Integers: whole numbers (positive, negative or zero), represented by a fixed-length block of bits
• Characters fixed-length blocks of bits, with special coding; strings = sequences of characters
• Floating point numbers: a fraction (with a finite number of bits) times an exponent, like $$1.87 \times {10}^{6}$$, but in binary form
• Missing or ill-defined values: NA, NaN, etc.

Operators

• Unary - for arithmetic negation, ! for Boolean
• Binary usual arithmetic operators, plus ones for modulo and integer division; take two numbers and give a number
7+5

[1] 12

7-5

[1] 2

7*5

[1] 35

7^5

[1] 16807

7/5

[1] 1.4

7 %% 5

[1] 2

7 %/% 5

[1] 1


The R Console

Basic interaction with R is by typing in the console, a.k.a. terminal or command-line

You type in commands, R gives back answers (or errors)

Menus and other graphical interfaces are extras built on top of the console

Operators cont'd.

Comparisons are also binary operators; they take two objects, like numbers, and give a Boolean

7 > 5

[1] TRUE

7 < 5

[1] FALSE

7 >= 7

[1] TRUE

7 <= 5

[1] FALSE

7 == 5

[1] FALSE

7 != 5

[1] TRUE


Boolean operators

Basically “and” and “or”:

(5 > 7) & (6*7 == 42)

[1] FALSE

(5 > 7) | (6*7 == 42)

[1] TRUE


(will see special doubled forms, && and ||, later)

More types

typeof() function returns the type

is.foo() functions return Booleans for whether the argument is of type foo

as.foo() (tries to) “cast” its argument to type foo — to translate it sensibly into a foo-type value

typeof(7)

[1] "double"

is.numeric(7)

[1] TRUE

is.na(7)

[1] FALSE

is.na(7/0)

[1] FALSE

is.na(0/0)

[1] TRUE


Why is 7/0 not NA, but 0/0 is?

is.character(7)

[1] FALSE

is.character("7")

[1] TRUE

is.character("seven")

[1] TRUE

is.na("seven")

[1] FALSE

as.character(5/6)

[1] "0.833333333333333"

as.numeric(as.character(5/6))

[1] 0.8333

6*as.numeric(as.character(5/6))

[1] 5

5/6 == as.numeric(as.character(5/6))

[1] FALSE


(why is that last FALSE?)

Data can have names

We can give names to data objects; these give us variables

A few variables are built in:

pi

[1] 3.142


Variables can be arguments to functions or operators, just like constants:

pi*10

[1] 31.42

cos(pi)

[1] -1


Most variables are created with the assignment operator, <- or =

approx.pi <- 22/7
approx.pi

[1] 3.143

diameter.in.cubits = 10
approx.pi*diameter.in.cubits

[1] 31.43


The assignment operator also changes values:

circumference.in.cubits <- approx.pi*diameter.in.cubits
circumference.in.cubits

[1] 31.43

circumference.in.cubits <- 30
circumference.in.cubits

[1] 30


Using names and variables makes code: easier to design, easier to debug, less prone to bugs, easier to improve, and easier for others to read

Avoid “magic constants”; use named variables you will be graded on this!

Named variables are a first step towards abstraction

The workspace

What names have you defined values for?

ls()

[1] "approx.pi"               "circumference.in.cubits"
[3] "diameter.in.cubits"

objects()

[1] "approx.pi"               "circumference.in.cubits"
[3] "diameter.in.cubits"


Getting of variables:

rm("circumference.in.cubits")
ls()

[1] "approx.pi"          "diameter.in.cubits"


First data structure: vectors

Group related data values into one object, a data structure

A vector is a sequence of values, all of the same type

x <- c(7, 8, 10, 45)
x

[1]  7  8 10 45

is.vector(x)

[1] TRUE


c() function returns a vector containing all its arguments in order

x[1] is the first element, x[4] is the 4th element
x[-4] is a vector containing all but the fourth element

vector(length=6) returns an empty vector of length 6; helpful for filling things up later

weekly.hours <- vector(length=5)
weekly.hours[5] <- 8


Vector arithmetic

Operators apply to vectors “pairwise” or “elementwise”:

y <- c(-7, -8, -10, -45)
x+y

[1] 0 0 0 0

x*y

[1]   -49   -64  -100 -2025


Recycling

Recycling repeat elements in shorter vector when combined with longer

x + c(-7,-8)

[1]  0  0  3 37

x^c(1,0,-1,0.5)

[1] 7.000 1.000 0.100 6.708


Single numbers are vectors of length 1 for purposes of recycling:

2*x

[1] 14 16 20 90


Can also do pairwise comparisons:

x > 9

[1] FALSE FALSE  TRUE  TRUE


Note: returns Boolean vector

Boolean operators work elementwise:

(x > 9) & (x < 20)

[1] FALSE FALSE  TRUE FALSE


To compare whole vectors, best to use identical() or all.equal():

x == -y

[1] TRUE TRUE TRUE TRUE

identical(x,-y)

[1] TRUE

identical(c(0.5-0.3,0.3-0.1),c(0.3-0.1,0.5-0.3))

[1] FALSE

all.equal(c(0.5-0.3,0.3-0.1),c(0.3-0.1,0.5-0.3))

[1] TRUE


Functions on vectors

Lots of functions take vectors as arguments:

• mean(), median(), sd(), var(), max(), min(), length(), sum(): return single numbers
• sort() returns a new vector
• hist() takes a vector of numbers and produces a histogram, a highly structured object, with the side-effect of making a plot
• Similarly ecdf() produces a cumulative-density-function object
• summary() gives a five-number summary of numerical vectors
• any() and all() are useful on Boolean vectors

Vector of indices:

x[c(2,4)]

[1]  8 45


Vector of negative indices

x[c(-1,-3)]

[1]  8 45


(why that, and not 8 10?)

Boolean vector:

x[x>9]

[1] 10 45

y[x>9]

[1] -10 -45


which() turns a Boolean vector in vector of TRUE indices:

places <- which(x > 9)
places

[1] 3 4

y[places]

[1] -10 -45


Named components

You can give names to elements or components of vectors

names(x) <- c("v1","v2","v3","fred")
names(x)

[1] "v1"   "v2"   "v3"   "fred"

x[c("fred","v1")]

fred   v1
45    7


note the labels in what R prints; not actually part of the value

names(x) is just another vector (of characters):

names(y) <- names(x)
sort(names(x))

[1] "fred" "v1"   "v2"   "v3"

which(names(x)=="fred")

[1] 4


Take-Aways

• We write programs by composing functions to manipulate data
• The basic data types let us represent Booleans, numbers, and characters
• Data structure let us group related values together
• Vectors let us group values of the same type
• Use variables rather a profusion of magic constants
• Name components of structures to make data more meaningful

Peculiarites of floating-point numbers

The more bits in the fraction part, the more precision

The R floating-point data type is a double, a.k.a. numeric back when memory was expensive, the now-standard number of bits was twice the default

Finite precision $$\Rightarrow$$ arithmetic on doubles $$\neq$$ arithmetic on $$\mathbb{R}$$.

0.45 == 3*0.15

[1] FALSE

0.45 - 3*0.15

[1] 5.551e-17


Often ignorable, but not always

• Rounding errors tend to accumulate in long calculations
• When results should be $$\approx 0$$, errors can flip signs
• Usually better to use all.equal() than exact comparison
(0.5 - 0.3) == (0.3 - 0.1)

[1] FALSE

all.equal(0.5-0.3, 0.3-0.1)

[1] TRUE


Peculiarities of Integers

Typing a whole number in the terminal doesn't make an integer; it makes a double, whose fractional part is 0

is.integer(7)

[1] FALSE


This looks like an integer

as.integer(7)

[1] 7


To test for being a whole number, use round():

round(7) == 7

[1] TRUE