Statistical Computing, 36-350

Monday August 27, 2018

**Independence**: otherwise, you rely on someone else giving you exactly the right tool**Honesty**: otherwise, you end up distorting your problem to match the tools you have**Clarity**: often, turning your ideas into something a machine can do refines your thinking**Fun**: these were the best of times (the worst of times)

- Instructor: Ryan Tibshirani
- Head TA: Theresa Gebert
- TAs: Tim Barry, Beomjo Park, Aijin Wang
- No real programming knowledge assumed
- Some statistics knowledge assumed
- Focus is R programming language
- Class will be cumulative, so keep up with the material and assignments!

- Monday class period: lecture
- Wednesday and Friday periods: lab
- Quiz due at 11:59pm each Monday, submitted through QNA
- Lab due at 11:59pm each Sunday (end of week), submitted through Canvas
- No midterm, final take-home programming assignment

- Grading breakdown:
- Labs: 60%
- Quizzes: 20%
- Final: 20%

- Class website: http://www.stat.cmu.edu/~ryantibs/statcomp/
- Piazza group: for announcements and discussions
- Canvas: used to collect submissions, and keep track of grades

- R is a programming language for statistical computing
- RStudio is an integrated development environment for R programming

- R Markdown is a markup language for combining R code with text

All 3 are free, and all 3 will be used extensively in this course

It’s on the course website, please read it (actually read it)

Don’t do it, refer to syllabus if you’re unclear about anything

(and if you’re still unclear, come see me)

Please don’t call me “Professor”. Call me:

- Professor Tibshirani
- Professor Tibs
- Ryan

(demo)

*Data types, operators, variables*

Two basic types of things/objects: **data** and **functions**

**Data**: things like 7, “seven”, \(7.000\), and \(\left[ \begin{array}{ccc} 7 & 7 & 7 \\ 7 & 7 & 7\end{array}\right]\)**Functions**: things like`log`

,`+`

(takes two arguments),`<`

(two),`%%`

(two), and`mean`

(one)

A function is a machine which turns input objects, or

arguments, into an output object, or areturn 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 trick to good programming is to take a big transformation and

break it downinto smaller ones, and then break those down, until you come to tasks which are easy (using built-in functions)

At base level, all data can represented in binary format, by **bits** (i.e., TRUE/FALSE, YES/NO, 1/0). Basic data types:

**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**: an integer times a positive integer to the power of an integer, as in \(3 \times 10^6\) or \(1 \times 3^{-1}\)**Missing or ill-defined values**:`NA`

,`NaN`

, etc.

**Unary**: take just one argument. E.g.,`-`

for arithmetic negation,`!`

for Boolean negation**Binary**: take two arguments. E.g.,`+`

,`-`

,`*`

, and`/`

(though this is only a partial operator). Also,`%%`

(for mod), and`^`

(again partial)

`-7`

`## [1] -7`

`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`

These are also binary operators; they take two objects, and give back 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`

Warning: `==`

is a comparison operator, `=`

is not!

These basic ones are `&`

(and) and `|`

(or)

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

`## [1] FALSE`

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

`## [1] TRUE`

`(5 > 7) | (6 * 7 == 42) & (0 != 0)`

`## [1] FALSE`

`(5 > 7) | (6 * 7 == 42) & (0 != 0) | (9 - 8 >= 0)`

`## [1] TRUE`

Note: The double forms `&&`

and `||`

are different! We’ll see them later

- The
`typeof()`

function returns the data 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 such a 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`

`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] 0.8333333`

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

`## [1] 5`

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

`## [1] FALSE`

We can give names to data objects; these give us **variables**. Some variables are built-in:

`pi`

`## [1] 3.141593`

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

`pi * 10`

`## [1] 31.41593`

`cos(pi)`

`## [1] -1`

We create variables with the **assignment operator**, `<-`

or `=`

```
approx.pi = 22/7
approx.pi
```

`## [1] 3.142857`

```
diameter = 10
approx.pi * diameter
```

`## [1] 31.42857`

The assignment operator also changes values:

```
circumference = approx.pi * diameter
circumference
```

`## [1] 31.42857`

```
circumference = 30
circumference
```

`## [1] 30`

- The code you write will be made of variables, with descriptive names
- Easier to design, easier to debug, easier to improve, and easier for others to read
- Avoid “magic constants”; instead use named variables
- Named variables are a first step towards
**abstraction**

What variables have you defined?

`ls()`

`## [1] "approx.pi" "circumference" "diameter"`

Getting rid of variables:

```
rm("circumference")
ls()
```

`## [1] "approx.pi" "diameter"`

```
rm(list=ls()) # Be warned! This erases everything
ls()
```

`## character(0)`

*Data structures*

- A
**data structure**is a grouping of related data values into an object - 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`

- The
`c()`

function returns a vector containing all its arguments in specified order `1:5`

is shorthand for`c(1,2,3,4,5)`

, and so on`x[1]`

would be the first element,`x[4]`

the fourth element, and`x[-4]`

is a vector containing*all but*the fourth element

`vector(length=n)`

returns an empty vector of length *n*; helpful for filling things up later

```
weekly.hours = vector(length=5)
weekly.hours
```

`## [1] FALSE FALSE FALSE FALSE FALSE`

```
weekly.hours[5] = 8
weekly.hours
```

`## [1] 0 0 0 0 8`

Arithmetic operator apply to vectors in a “componentwise” fashion

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

`## [1] 0 0 0 0`

`x * y`

`## [1] -49 -64 -100 -2025`

**Recycling** repeat elements in shorter vector when combined with a longer one

`x + c(-7,-8)`

`## [1] 0 0 3 37`

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

`## [1] 7.000000 1.000000 0.100000 6.708204`

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

`2 * x`

`## [1] 14 16 20 90`

Can do componentwise comparisons with vectors:

`x > 9`

`## [1] FALSE FALSE TRUE TRUE`

Logical operators also 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`

Note: these functions are slightly different; we’ll see more later

Many functions can take vectors as arguments:

`mean()`

,`median()`

,`sd()`

,`var()`

,`max()`

,`min()`

,`length()`

, and`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`ecdf()`

similarly 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`

Boolean vector:

`x[x > 9]`

`## [1] 10 45`

`y[x > 9]`

`## [1] -10 -45`

`which()`

gives the elements of a Boolean vector that are `TRUE`

:

```
places = which(x > 9)
places
```

`## [1] 3 4`

`y[places]`

`## [1] -10 -45`

We can give names to elements/components of vectors, and index vectors accordingly

```
names(x) = c("v1","v2","v3","fred")
names(x)
```

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

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

```
## fred v1
## 45 7
```

Note: here R is printing the labels, these are not additional components of `x`

`names()`

returns another vector (of characters):

```
names(y) = names(x)
sort(names(x))
```

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

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

`## [1] 4`

An **array** is a multi-dimensional generalization of vectors

```
x = c(7, 8, 10, 45)
x.arr = array(x, dim=c(2,2))
x.arr
```

```
## [,1] [,2]
## [1,] 7 10
## [2,] 8 45
```

`dim`

says how many rows and columns; filled by columns- Can have 3d arrays, 4d arrays, etc.;
`dim`

is vector of arbitrary length

Some properties of our array:

`dim(x.arr)`

`## [1] 2 2`

`is.vector(x.arr)`

`## [1] FALSE`

`is.array(x.arr)`

`## [1] TRUE`

`typeof(x.arr)`

`## [1] "double"`

Can access a 2d array either by pairs of indices or by the underlying vector (column-major order):

`x.arr[1,2]`

`## [1] 10`

`x.arr[3]`

`## [1] 10`

Omitting an index means “all of it”:

`x.arr[c(1,2),2]`

`## [1] 10 45`

`x.arr[,2]`

`## [1] 10 45`

`x.arr[,2,drop=FALSE]`

```
## [,1]
## [1,] 10
## [2,] 45
```

Note: the optional third argument `drop=FALSE`

ensures that the result is still an array, not a vector

Many functions applied to an array will just boil things down to the underlying vector:

`which(x.arr > 9)`

`## [1] 3 4`

This happens unless the function is set up to handle arrays specifically

And there are several functions/operators that *do* preserve array structure:

```
y = -x
y.arr = array(y, dim=c(2,2))
y.arr + x.arr
```

```
## [,1] [,2]
## [1,] 0 0
## [2,] 0 0
```

A **matrix** is a specialization of a 2d array

```
z.mat = matrix(c(40,1,60,3), nrow=2)
z.mat
```

```
## [,1] [,2]
## [1,] 40 60
## [2,] 1 3
```

`is.array(z.mat)`

`## [1] TRUE`

`is.matrix(z.mat)`

`## [1] TRUE`

- Could also specify
`ncol`

for the number of columns - To fill by rows, use
`byrow=TRUE`

- Elementwise operations with the usual arithmetic and comparison operators (e.g.,
`z.mat/3`

)

Matrices have its own special multiplication operator, written `%*%`

:

```
six.sevens = matrix(rep(7,6), ncol=3)
six.sevens
```

```
## [,1] [,2] [,3]
## [1,] 7 7 7
## [2,] 7 7 7
```

`z.mat %*% six.sevens # [2x2] * [2x3]`

```
## [,1] [,2] [,3]
## [1,] 700 700 700
## [2,] 28 28 28
```

Can also multiply a matrix and a vector

Row/column sums, or row/column means:

`rowSums(z.mat)`

`## [1] 100 4`

`colSums(z.mat)`

`## [1] 41 63`

`rowMeans(z.mat)`

`## [1] 50 2`

`colMeans(z.mat)`

`## [1] 20.5 31.5`

The `diag()`

function can be used to extract the diagonal entries of a matrix:

`diag(z.mat)`

`## [1] 40 3`

It can also be used to change the diagonal:

```
diag(z.mat) = c(35,4)
z.mat
```

```
## [,1] [,2]
## [1,] 35 60
## [2,] 1 4
```

Finally, `diag()`

can be used to create a diagonal matrix:

`diag(c(3,4))`

```
## [,1] [,2]
## [1,] 3 0
## [2,] 0 4
```

`diag(2)`

```
## [,1] [,2]
## [1,] 1 0
## [2,] 0 1
```

Transpose:

`t(z.mat)`

```
## [,1] [,2]
## [1,] 35 1
## [2,] 60 4
```

Determinant:

`det(z.mat)`

`## [1] 80`

Inverse:

`solve(z.mat)`

```
## [,1] [,2]
## [1,] 0.0500 -0.7500
## [2,] -0.0125 0.4375
```

`z.mat %*% solve(z.mat)`

```
## [,1] [,2]
## [1,] 1 0
## [2,] 0 1
```

- We can name either rows or columns or both, with
`rownames()`

and`colnames()`

- These are just character vectors, and we use them just like we do
`names()`

for vectors - Names help us understand what we’re working with

A **list** is sequence of values, but not necessarily all of the same type

```
my.dist = list("exponential", 7, FALSE)
my.dist
```

```
## [[1]]
## [1] "exponential"
##
## [[2]]
## [1] 7
##
## [[3]]
## [1] FALSE
```

Most of what you can do with vectors you can also do with lists

- Can use
`[ ]`

as with vectors

- Or use
`[[ ]]`

, but only with a single index`[[ ]]`

drops names and structures,`[ ]`

does not

`my.dist[2]`

```
## [[1]]
## [1] 7
```

`my.dist[[2]]`

`## [1] 7`

`my.dist[[2]]^2`

`## [1] 49`

Add to lists with `c()`

(also works with vectors):

```
my.dist = c(my.dist, 9)
my.dist
```

```
## [[1]]
## [1] "exponential"
##
## [[2]]
## [1] 7
##
## [[3]]
## [1] FALSE
##
## [[4]]
## [1] 9
```

Chop off the end of a list by setting the length to something smaller (also works with vectors):

`length(my.dist)`

`## [1] 4`

```
length(my.dist) = 3
my.dist
```

```
## [[1]]
## [1] "exponential"
##
## [[2]]
## [1] 7
##
## [[3]]
## [1] FALSE
```

Pluck out all but one piece of a list (also works with vectors):

`my.dist[-2]`

```
## [[1]]
## [1] "exponential"
##
## [[2]]
## [1] FALSE
```

We can name some or all of the elements of a list:

```
names(my.dist) = c("family","mean","is.symmetric")
my.dist
```

```
## $family
## [1] "exponential"
##
## $mean
## [1] 7
##
## $is.symmetric
## [1] FALSE
```

`my.dist[["family"]]`

`## [1] "exponential"`

`my.dist["family"]`

```
## $family
## [1] "exponential"
```

Lists have a special shortcut way of using names, with `$`

:

`my.dist[["family"]]`

`## [1] "exponential"`

`my.dist$family`

`## [1] "exponential"`

Creating a list with names:

```
another.dist = list(family="gaussian",
mean=7, sd=1, is.symmetric=TRUE)
```

Adding named elements:

```
my.dist$was.estimated = FALSE
my.dist[["last.updated"]] = "2015-09-01"
```

Removing a named list element, by assigning it the value `NULL`

:

`my.dist$was.estimated = NULL`

- Lists give us a natural way to store and look up data by
*name*, rather than by*position* - A really useful programming concept with many names:
**key-value pairs**, i.e.,**dictionaries**, or**associative arrays** - If all our distributions have components named
`family`

, we can look that up by name, without caring where it is (in what position it lies) in the list

- The classic data table, \(n\) rows for cases, \(p\) columns for variables
- Lots of the really-statistical parts of R presume data frames

- Not just a matrix because
*columns can have different types* - Many matrix functions also work for data frames (e.g.,
`rowSums()`

,`summary()`

,`apply()`

)

```
a.mat = matrix(c(35,8,10,4), nrow=2)
colnames(a.mat) = c("v1","v2")
a.mat
```

```
## v1 v2
## [1,] 35 10
## [2,] 8 4
```

`a.mat[,"v1"] # Try a.mat$v1 and see what happens`

`## [1] 35 8`

```
a.df = data.frame(a.mat,logicals=c(TRUE,FALSE))
a.df
```

```
## v1 v2 logicals
## 1 35 10 TRUE
## 2 8 4 FALSE
```

`a.df$v1`

`## [1] 35 8`

`a.df[,"v1"]`

`## [1] 35 8`

`a.df[1,]`

```
## v1 v2 logicals
## 1 35 10 TRUE
```

`colMeans(a.df)`

```
## v1 v2 logicals
## 21.5 7.0 0.5
```

We can add rows or columns to an array or data frame with `rbind()`

and `cbind()`

, but be careful about forced type conversions

`rbind(a.df,list(v1=-3,v2=-5,logicals=TRUE))`

```
## v1 v2 logicals
## 1 35 10 TRUE
## 2 8 4 FALSE
## 3 -3 -5 TRUE
```

`rbind(a.df,c(3,4,6))`

```
## v1 v2 logicals
## 1 35 10 1
## 2 8 4 0
## 3 3 4 6
```

Much more on data frames a bit later in the course …

So far, every list element has been a single data value. List elements can be other data structures, e.g., vectors and matrices, even other lists:

```
my.list = list(z.mat=z.mat, my.lucky.num=13, my.dist=my.dist)
my.list
```

```
## $z.mat
## [,1] [,2]
## [1,] 35 60
## [2,] 1 4
##
## $my.lucky.num
## [1] 13
##
## $my.dist
## $my.dist$family
## [1] "exponential"
##
## $my.dist$mean
## [1] 7
##
## $my.dist$is.symmetric
## [1] FALSE
##
## $my.dist$last.updated
## [1] "2015-09-01"
```

- We write programs by composing functions to manipulate data
- The basic data types let us represent Booleans, numbers, and characters
- Data structures let us group together related values
- Vectors let us group values of the same type
- Arrays add multi-dimensional structure to vectors
- Matrices act like you’d hope they would
- Lists let us combine different types of data
- Data frames are hybrids of matrices and lists, allowing each column to have a different data type