Lecture 2: Vector and Matrix Operations

class: center, middle, inverse, title-slide

.title[
# Lecture 2: Vector and Matrix Operations
]
.subtitle[
## Filtering, Recycling, Vectorization
]
.author[
### Robin Liu
]
.institute[
### UCSB
]
.date[
### 2022-06-22
]

---

class: inverse, middle, center
# Three Important R Techniques
Filtering, Recycling, and Vectorization

---
# Filtering
Recall that we can subset a vector by a logical vector

```r
x <- 1:10
subs <- rep(c(F, T), each = 5)
```
Quick quiz: what is the result of `x[subs]`?
<div class="countdown" id="timer_62b35104" style="right:0;bottom:0;" data-warnwhen="0">
<code class="countdown-time"><span class="countdown-digits minutes">01</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code>
</div>
--

Often we want to select values from `x` that satisfy some property.
The above can be returned as follows:

```r
x[x > 5]
```

```
## [1]  6  7  8  9 10
```
---
# Filtering
## How did this work?

```r
x
```

```
##  [1]  1  2  3  4  5  6  7  8  9 10
```

```r
x > 5
```

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

```r
x[x > 5]
```

```
## [1]  6  7  8  9 10
```

We *filtered* `x` based on its values.

---
# Filtering
## My gas bill in dollars for every month in order in 2021 is given:
### 46, 33, 39, 37, 46, 30, 48, 32, 49, 35, 30, and 48.
Create a vector called `gas_bill`.

1. Verify that all months are accounted for (12 entries total).
1. Find the number of months where the gas bill exceeded $40. How does this work?
1. Find the percentage of months where the gas bill exceeded $40.
1. Find the total of how much I paid in gas over all months where the gas bill exceeded $40.

<div class="countdown" id="timer_62b35184" style="right:0;bottom:0;" data-warnwhen="0">
<code class="countdown-time"><span class="countdown-digits minutes">05</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code>
</div>
---
# Recycling
In most cases, two vectors must have the same length in order to be operated on:

```r
c(2, 4) + c(6, 3)
```

```
## [1] 8 7
```

For many operations between two vectors, the elements in the shorter vector
are *recycled* so that the lengths of the two vectors match.

```r
1:4 + c(2, 3)
```

```
## [1] 3 5 5 7
```

```r
1:4 + c(2, 3, 2, 3)
```

```
## [1] 3 5 5 7
```
---
# Recycling
### A warning may appear when...

```r
1:4 + 1:3
```

```
## Warning in 1:4 + 1:3: longer object length is not a multiple of shorter object
## length
```

```
## [1] 2 4 6 5
```

```r
1:4 + c(1, 2, 3, 1)
```

```
## [1] 2 4 6 5
```

You are warned against possible mistakes.
---
# Recycling
## My gas bill in dollars for every month in order in 2021 is given:
### 46, 33, 39, 37, 46, 30, 48, 32, 49, 35, 30, and 48.

SoCal Gas overcharged me $3 on odd numbered months (January, March, …) and undercharged me 
$7 on even numbered months (February, April, …).

1. Update `gas_bill` to reflect my true gas costs in 2021
2. What is the difference between what I paid and what I actually owe?

<div class="countdown" id="timer_62b350a7" style="right:0;bottom:0;" data-warnwhen="0">
<code class="countdown-time"><span class="countdown-digits minutes">05</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code>
</div>
---
# Recycling
### Adding a vector to a number.

```r
x <- 1:10
x + 6
```

```
##  [1]  7  8  9 10 11 12 13 14 15 16
```
Did recycling occur?

---
# Vectorization
A function `$f$` is *vectorized* if applying `$f$` to a vector `$\vec x = (x_1, x_2,\dotsc, x_n)$`
results in a vector of `$f$` applied to each element of `$\vec x$`.

`$f(\vec x) = (f(x_1), f(x_2), \dotsc, f(x_n))$`.
- Vector in, vector out
--

- Vectorization is *extremely fast*
--

- Later we will discuss loops. You should always look for a vectorized solution before reaching for a loop in R.

Many operations in R are already vectorized

```r
(x <- (5:10)^2)
```

```
## [1]  25  36  49  64  81 100
```

```r
log(x)
```

```
## [1] 3.218876 3.583519 3.891820 4.158883 4.394449 4.605170
```
---
# Vectorization

1. Find the sum of the square roots of every integer from 1 to 1000.
1. Find the product of the natural log of every integer from 100 to 200.
1. Find the sum of the integers between 1 and 100 whose square is between 300 and 500.

<div class="countdown" id="timer_62b35106" style="right:0;bottom:0;" data-warnwhen="0">
<code class="countdown-time"><span class="countdown-digits minutes">10</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code>
</div>
---

class: inverse, middle, center
# Matrices, Arrays, and Lists
Data with dimensionality
---
# Data naturally have dimensions
Often times, a vector (1-D) is not enough.

.center[![:scale 70%](Lec2_files/drone.png)]
https://trucvietle.me/r/tutorial/2017/01/18/spatial-heat-map-plotting-using-r.html
---
# Matrices
## Create a matrix in several ways
.pull-left[

```r
(x <- seq(-5, 5, 2))
```

```
## [1] -5 -3 -1  1  3  5
```

```r
x_mx <- matrix(x, nrow=2, ncol=3)
x_mx
```

```
##      [,1] [,2] [,3]
## [1,]   -5   -1    3
## [2,]   -3    1    5
```
]
--
.pull-right[

```r
rbind(c(-5, -1, 3), c(-3, 1, 5))
```

```
##      [,1] [,2] [,3]
## [1,]   -5   -1    3
## [2,]   -3    1    5
```

```r
cbind(c(-5, -3), c(-1, 1), c(3, 5))
```

```
##      [,1] [,2] [,3]
## [1,]   -5   -1    3
## [2,]   -3    1    5
```
]

---
# Matrices
## A matrix is just an atomic vector...
except endowed with extra information: the *dimension*

.pull-left[

```r
(x <- 1:4)
```

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

```r
dim(x)
```

```
## NULL
```

```r
is.matrix(x)
```

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

--
.pull-right[

```r
dim(x) <- c(2, 2)
x
```

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

```r
dim(x)
```

```
## [1] 2 2
```

```r
is.matrix(x)
```

```
## [1] TRUE
```
]

---
# Matrices
Because matrices are just vectors; recycling, filtering, and vectorization
rules apply to them.

Subsetting a matrix is different: an index is provided to each dimension.
.pull-left[

```r
(x <- matrix(1:4, nrow=2, ncol=2))
```

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

```r
x[1, 1]
```

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

```r
x[2, 1]
```

```
## [1] 2
```
]
.pull-right[
**Omitting** an index returns all items in that dimension

```r
x[, 2]
```

```
## [1] 3 4
```

```r
x[1, ]
```

```
## [1] 1 3
```
]
---
# Matrices

```r
(x <- matrix(1:9, nrow=3, ncol=3))
```

```
##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9
```
What is the result of `x[1:2, -3]`?

<div class="countdown" id="timer_62b351de" style="right:0;bottom:0;" data-warnwhen="0">
<code class="countdown-time"><span class="countdown-digits minutes">01</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code>
</div>
---
# Matrices
Create a `$4\times 5$` matrix `mx` of integers 1 through 17 (inclusive).
Print the matrix.

1. Did filtering, recycling, or vectorization occur?
2. Subset `mx` to create the following matrix:

```
##      [,1] [,2] [,3]
## [1,]    1    5   17
## [2,]    4    8    3
```
3. Set all entries of `mx` greater than 10 to zero. Did filtering, recycling, or vectorization occur?

---
# Matrices
Like vectors, rows and columns of a matrix can be named.

```r
scores <- matrix(c(7, 6, 9, 8, 8, 9, 10, 5, 7), nrow = 3, ncol = 3)
colnames(scores) <- c("Anna", "Joe", "Carl")
rownames(scores) <- c("midterm1", "midterm2", "final")
scores
```

```
##          Anna Joe Carl
## midterm1    7   8   10
## midterm2    6   8    5
## final       9   9    7
```

.pull-left[

```r
scores["midterm2", -3]
```

```
## Anna  Joe 
##    6    8
```
]
--
.pull-right[
What is the output?

```r
scores[scores[,3] > 5, ]
```
]
<div class="countdown" id="timer_62b3507c" style="right:0;bottom:0;" data-warnwhen="0">
<code class="countdown-time"><span class="countdown-digits minutes">00</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">30</span></code>
</div>

---
# Image manipulation demo

```r
install.packages("pixmap")
library(pixmap)
```

```r
mtrush1 <- read.pnm("mtrush1.pgm")
str(mtrush1)
```

```
## Formal class 'pixmapGrey' [package "pixmap"] with 6 slots
##   ..@ grey    : num [1:194, 1:259] 0.278 0.263 0.239 0.212 0.192 ...
##   ..@ channels: chr "grey"
##   ..@ size    : int [1:2] 194 259
##   ..@ cellres : num [1:2] 1 1
##   ..@ bbox    : num [1:4] 0 0 259 194
##   ..@ bbcent  : logi FALSE
```
---
# Image manipulation demo
`mtrush1@grey` is a matrix containing pixel intensities.

What is the intensity of the pixel at row 70, column 120?

Blur out Teddy Roosevelt's face at `$84\colon163 \times 135\colon177$`
![](Lec2_files/figure-html/unnamed-chunk-27-1.png)
---
# Arrays
Higher (>2) dimensional vectors are called *arrays*

```r
rubiks <- array(1:27, c(3, 3, 3))
dim(rubiks)
```

```
## [1] 3 3 3
```

```r
hypercube <- array(1:16, c(2, 2, 2, 2))
dim(hypercube)
```

```
## [1] 2 2 2 2
```

--
.center[
![](Lec2_files/rubiks.png)
[![](Lec2_files/dali.png)](https://en.wikipedia.org/wiki/Crucifixion_%28Corpus_Hypercubus%29)
![:scale 25%](Lec2_files/tesseract.gif)
]
---
# Arrays
.pull-left[

```r
rubiks
```

```
## , , 1
## 
##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9
## 
## , , 2
## 
##      [,1] [,2] [,3]
## [1,]   10   13   16
## [2,]   11   14   17
## [3,]   12   15   18
## 
## , , 3
## 
##      [,1] [,2] [,3]
## [1,]   19   22   25
## [2,]   20   23   26
## [3,]   21   24   27
```
]
.pull-right[

```r
hypercube
```

```
## , , 1, 1
## 
##      [,1] [,2]
## [1,]    1    3
## [2,]    2    4
## 
## , , 2, 1
## 
##      [,1] [,2]
## [1,]    5    7
## [2,]    6    8
## 
## , , 1, 2
## 
##      [,1] [,2]
## [1,]    9   11
## [2,]   10   12
## 
## , , 2, 2
## 
##      [,1] [,2]
## [1,]   13   15
## [2,]   14   16
```
]
---
# Application of Arrays
### Color images are often represented as a 3D array

.center[![](Lec2_files/channels.png)]

---
# Lists
### A **list** is also a vector, but is not atomic.
Usually "vector" means "atomic vector".
Recall that atomic vectors can only hold one data type.
Lists are also vectors, but can hold many different data types.

For the rest of the class,
I will say "vector" to mean "atomic vector" and "list" to mean "list".
But in the R literature, lists are vectors.

.center[![](Lec2_files/listhier.png)]

---
# Lists

```r
x <- list(first = 1:10, second = "cat", third = c(T, F))
```
Single brackets `[ ]` return a list while double brackets `[[ ]]` return the element.

```r
x[1]
```

```
## $first
##  [1]  1  2  3  4  5  6  7  8  9 10
```

```r
x[[1]]
```

```
##  [1]  1  2  3  4  5  6  7  8  9 10
```

Getting an element by name:

```r
x$first
```

```
##  [1]  1  2  3  4  5  6  7  8  9 10
```

---
# Lists
### Is it possible to create a vector of vectors?

```r
c(1:10, 66, 3:7)
```
---
# Checking the structure
- Is `x` an atomic vector? `is.atomic(x)`
- Is `x` a list? `is.list(x)`
- Is `x` a matrix? `is.matrix(x)`
- Is `x` an array? `is.array(x)`

.pull-left[
But beware:

```r
x <- rbind(1:2, 3:4)
is.atomic(x)
```

```
## [1] TRUE
```

```r
is.matrix(x)
```

```
## [1] TRUE
```

```r
is.array(x)
```

```
## [1] TRUE
```
]
.pull-right[
Sometimes better to check the *class*

```r
class(x)
```

```
## [1] "matrix" "array"
```
]
---
# AVOID is.vector
`is.vector` does not do what you think it does.

---
# Summary
- Practice filtering, recycling, and vectorization.
- Practice subsetting matrices.