Vectors: What are they?

• Remember back to linear algebra…

Examples:

\[ \begin{matrix} \left(\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}\right) & \text{or} & \left(\begin{matrix} 1 & 2 & 3 & 4 \end{matrix}\right) \end{matrix} \]

A row (or column) of data

Vector creation

• Vectors are entered using the c() command
• Any data type is fine, but all elements must be the same type

company <- c("Google", "Microsoft", "Goldman")
company

## [1] "Google"    "Microsoft" "Goldman"

tech_firm <- c(TRUE, TRUE, FALSE)
tech_firm

## [1]  TRUE  TRUE FALSE

earnings <- c(12662, 21204, 4286)
earnings

## [1] 12662 21204  4286

A vector in R is a 1 dimensional collection of 1 or more of the same data type

Special cases for vectors

1:5

## [1] 1 2 3 4 5

seq(from=0, to=100, by=5)

##  [1]   0   5  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80  85  90
## [20]  95 100

rep(1,times=10)

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

rep("hi",times=5)

## [1] "hi" "hi" "hi" "hi" "hi"

Vector math

Works the same as scalars, but applies element-wise

• First element with first element,
• Second element with second element,
• …

earnings  # previously defined

## [1] 12662 21204  4286

earnings + earnings  # Add element-wise

## [1] 25324 42408  8572

earnings * earnings  # multiply element-wise

## [1] 160326244 449609616  18369796

Vector math

Can also use 1 vector and 1 scalar

• Scalar is applied to all vector elements

earnings + 10000  # Adding a scalar to a vector

## [1] 22662 31204 14286

10000 + earnings  # Order doesn't matter

## [1] 22662 31204 14286

earnings / 1000  # Dividing a vector by a scalar

## [1] 12.662 21.204  4.286

Vector math

• From linear algebra, you might remember multiplication being a bit different, as a dot product. That can be done with %*%

# Dot product: sum of product of elements
earnings %*% earnings  # returns a matrix though...

##           [,1]
## [1,] 628305656

drop(earnings %*% earnings)  # Drop drops excess dimensions

## [1] 628305656

• Other useful functions, length() and sum():

length(earnings)  # returns the number of elements

## [1] 3

sum(earnings)  # returns the sum of all elements

## [1] 38152

Naming vectors

earnings

## [1] 12662 21204  4286

names(earnings) <- c("Google",
                     "Microsoft",
                     "Goldman")
earnings

##    Google Microsoft   Goldman 
##     12662     21204      4286

# Equivalently:
names(earnings) <- company
earnings

##    Google Microsoft   Goldman 
##     12662     21204      4286

Selecting and combining vectors

earnings[1]

## Google 
##  12662

earnings["Google"]

## Google 
##  12662

# Each of the above 3 is equivalent
earnings[1:2]

##    Google Microsoft 
##     12662     21204

c1 <- c(1,2,3)
c2 <- c(4,5,6)
c3 <- c(c1,c2)
c3

## [1] 1 2 3 4 5 6

Vector example: Profit margin for tech firms

# Calculating proit margin for all public US tech firms
# 715 tech firms with >1M sales in 2017
summary(earnings_2017)  # Cleaned data from Compustat, in $M USD

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -4307.49   -15.98     1.84   296.84    91.36 48351.00

summary(revenue_2017)  # Cleaned data from Compustat, in $M USD

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
##      1.06    102.62    397.57   3023.78   1531.59 229234.00

profit_margin <- earnings_2017 / revenue_2017
summary(profit_margin)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -13.97960  -0.10253   0.01353  -0.10967   0.09295   1.02655

# These are the worst, midpoint, and best profit margin firms in 2017. Our names carried over :)
profit_margin[order(profit_margin)][c(1,length(profit_margin)/2,length(profit_margin))]

## HELIOS AND MATHESON ANALYTIC                   NLIGHT INC 
##                 -13.97960161                   0.01325588 
##            CCUR HOLDINGS INC 
##                   1.02654899

Practice: Vectors

• This practice explores the ROA of Goldman Sachs, JPMorgan, and Citigroup in 2017
• Do exercise 2 on the supplementary R practice file:
  • R Practice

Matrices: What are they?

• Remember back to linear algebra…

Example:

\[ \left(\begin{matrix} 1 & 2 & 3 & 4\\ 5 & 6 & 7 & 8\\ 9 & 10 & 11 & 12 \end{matrix}\right) \]

A rows and columns of data

Matrix creation

• Matrices are entered using the matrix() command
• Any data type is fine, but all elements must be the same type

columns <- c("Google", "Microsoft", "Goldman")
rows <- c("Earnings","Revenue")

# equivalent: matrix(data=c(12662, 21204, 4286, 110855, 89950, 42254),ncol=3)
firm_data <- matrix(data=c(12662, 21204, 4286, 110855, 89950, 42254),nrow=2)
firm_data

##       [,1]   [,2]  [,3]
## [1,] 12662   4286 89950
## [2,] 21204 110855 42254

Math with matrices

Everything with matrices works just like vectors

firm_data + firm_data

##       [,1]   [,2]   [,3]
## [1,] 25324   8572 179900
## [2,] 42408 221710  84508

firm_data / 1000

##        [,1]    [,2]   [,3]
## [1,] 12.662   4.286 89.950
## [2,] 21.204 110.855 42.254

Matrix math with matrices

• Matrix transposing, \(A^T\), uses t()

firm_data_T <- t(firm_data)
firm_data_T

##       [,1]   [,2]
## [1,] 12662  21204
## [2,]  4286 110855
## [3,] 89950  42254

• Matrix multiplication, \(A~B\), uses %*%

firm_data %*% firm_data_T

##            [,1]        [,2]
## [1,] 8269698540  4544356878
## [2,] 4544356878 14523841157

We won’t use these much, but they can be useful

Matrix naming

• We can name matrix rows and columns, much like we named vector elements
• Use rownames() for rows
• Use colnames() for columns

rownames(firm_data) <- rows
colnames(firm_data) <- columns
firm_data

##          Google Microsoft Goldman
## Earnings  12662      4286   89950
## Revenue   21204    110855   42254

Selecting from matrices

• Select using 2 indexes instead of 1:
  • matrix_name[rows,columns]
  • To select all rows or columns, leave that index blanks

firm_data[2,3]

## [1] 42254

firm_data[,c("Google","Microsoft")]

##          Google Microsoft
## Earnings  12662      4286
## Revenue   21204    110855

firm_data[1,]

##    Google Microsoft   Goldman 
##     12662      4286     89950

Combining matrices

• Matrices are combined top to bottom as rows with rbind()
• Matrices are combined side-by-side as columns with cbind()

# Preloaded: industry codes as indcode (vector)
#     - GICS codes: 40=Financials, 45=Information Technology
#     - See: https://en.wikipedia.org/wiki/Global_Industry_Classification_Standard
# Preloaded: JPMorgan data as jpdata (vector)

mat <- rbind(firm_data,indcode)  # Add a row
rownames(mat)[3] <- "Industry"  # Name the new row
mat

##          Google Microsoft Goldman
## Earnings  12662      4286   89950
## Revenue   21204    110855   42254
## Industry     45        45      40

mat <- cbind(firm_data,jpdata)  # Add a column
colnames(mat)[4] <- "JPMorgan"  # Name the new column
mat

##          Google Microsoft Goldman JPMorgan
## Earnings  12662      4286   89950    17370
## Revenue   21204    110855   42254   115475

Lists: What are they?

• Like vectors, but with mixed types
• Generally not something we will create
• Often returned by analysis functions in R
  • Such as the linear models we will look at next week

# Ignore this code for now...
model <- summary(lm(earnings ~ revenue, data=tech_df))
#Note that this function is hiding something...
model

## 
## Call:
## lm(formula = earnings ~ revenue, data = tech_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16045.0     20.0    141.6    177.1  12104.6 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.837e+02  4.491e+01  -4.091 4.79e-05 ***
## revenue      1.589e-01  3.564e-03  44.585  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1166 on 713 degrees of freedom
## Multiple R-squared:  0.736,  Adjusted R-squared:  0.7356 
## F-statistic:  1988 on 1 and 713 DF,  p-value: < 2.2e-16

Looking into lists

• Lists generally use double square brackets, [[index]]
  • Used for pulling individual elements out of a list
• [[c()]] will drill through lists, as opposed to pulling multiple values
• Single square brackets pull out elements as is
• Double square brackets extract just the element
• For 1 level, we can also use $

model["r.squared"]

## $r.squared
## [1] 0.7360059

model[["r.squared"]]

## [1] 0.7360059

model$r.squared

## [1] 0.7360059

earnings["Google"]

## Google 
##  12662

earnings[["Google"]]

## [1] 12662

#Can't use $ with vectors

Structure of a list

• str() will tell us what’s in this list

str(model)

## List of 11
##  $ call         : language lm(formula = earnings ~ revenue, data = tech_df)
##  $ terms        :Classes 'terms', 'formula'  language earnings ~ revenue
##   .. ..- attr(*, "variables")= language list(earnings, revenue)
##   .. ..- attr(*, "factors")= int [1:2, 1] 0 1
##   .. .. ..- attr(*, "dimnames")=List of 2
##   .. .. .. ..$ : chr [1:2] "earnings" "revenue"
##   .. .. .. ..$ : chr "revenue"
##   .. ..- attr(*, "term.labels")= chr "revenue"
##   .. ..- attr(*, "order")= int 1
##   .. ..- attr(*, "intercept")= int 1
##   .. ..- attr(*, "response")= int 1
##   .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> 
##   .. ..- attr(*, "predvars")= language list(earnings, revenue)
##   .. ..- attr(*, "dataClasses")= Named chr [1:2] "numeric" "numeric"
##   .. .. ..- attr(*, "names")= chr [1:2] "earnings" "revenue"
##  $ residuals    : Named num [1:715] -59.7 173.8 -620.2 586.7 613.6 ...
##   ..- attr(*, "names")= chr [1:715] "1" "2" "3" "4" ...
##  $ coefficients : num [1:2, 1:4] -1.84e+02 1.59e-01 4.49e+01 3.56e-03 -4.09 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:2] "(Intercept)" "revenue"
##   .. ..$ : chr [1:4] "Estimate" "Std. Error" "t value" "Pr(>|t|)"
##  $ aliased      : Named logi [1:2] FALSE FALSE
##   ..- attr(*, "names")= chr [1:2] "(Intercept)" "revenue"
##  $ sigma        : num 1166
##  $ df           : int [1:3] 2 713 2
##  $ r.squared    : num 0.736
##  $ adj.r.squared: num 0.736
##  $ fstatistic   : Named num [1:3] 1988 1 713
##   ..- attr(*, "names")= chr [1:3] "value" "numdf" "dendf"
##  $ cov.unscaled : num [1:2, 1:2] 1.48e-03 -2.83e-08 -2.83e-08 9.35e-12
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:2] "(Intercept)" "revenue"
##   .. ..$ : chr [1:2] "(Intercept)" "revenue"
##  - attr(*, "class")= chr "summary.lm"

Practice: Lists

• In this practice, we will explore lists and how to parse them
• Do exercise 3 on the supplementary R practice file:
  • R Practice

What are data frames?

• Data frames are like a hybrid between lists and matrices

Think of columns as variables, rows as observations

Example of a data frame

library(DT)  # This library is great for including larger collections of data in output
datatable(tech_df[1:20,c("conm","tic","margin")], rownames=FALSE)

How to create data frames

1. On import of data, usually you will get a data frame
2. Using the data.frame() function

df <- data.frame(companyName=company,
                 earnings=earnings,
                 tech_firm=tech_firm)
df

##           companyName earnings tech_firm
## Google         Google    12662      TRUE
## Microsoft   Microsoft    21204      TRUE
## Goldman       Goldman     4286     FALSE

Note: stringsAsFactors=FALSE is no longer needed as of R 4.0.0

Selecting from data frames

• Access like a matrix

df[,1]

## [1] "Google"    "Microsoft" "Goldman"

• Access like a list

df$companyName

## [1] "Google"    "Microsoft" "Goldman"

df[[1]]

## [1] "Google"    "Microsoft" "Goldman"

All are relatively equivalent. Using $ is generally most natural. Using [,] is good for complex references.

Making new columns in a data frame

Suggested method: use $

df$all_zero <- 0
df$revenue <- c(110855, 89950, 42254)
df$margin <- df$earnings / df$revenue
# Custom function for small tables -- see last slide for code
html_df(df)

Alternative method: use cbind() just like with matrices

Sorting data frames

• To sort a vector, we could use the sort()

sort(df$earnings)

## [1]  4286 12662 21204

THIS CAN’T SORT DATA FRAMES

• A column of a data frame is fine, but it can’t sort the whole thing!

Sorting data frames

• To sort a data frame, we use the order() function
  • It returns the order of each element in increasing value
    • 1 is the lowest value
  • Then we pass the new order like we are selecting elements

ordering <- order(df$earnings)
ordering

## [1] 3 1 2

df <- df[ordering,]
df

##           companyName earnings tech_firm all_zero revenue    margin
## Goldman       Goldman     4286     FALSE        0   42254 0.1014342
## Google         Google    12662      TRUE        0  110855 0.1142213
## Microsoft   Microsoft    21204      TRUE        0   89950 0.2357310

Sorting data frames

• Order can sort by multiple levels
  • order(level1,level2,...), where level_ are vectors or data frame columns

# Example of multicolumn sorting:
example <- data.frame(firm=c("Google","Microsoft","Google","Microsoft"),
                      year=c(2017,2017,2016,2016))
example

##        firm year
## 1    Google 2017
## 2 Microsoft 2017
## 3    Google 2016
## 4 Microsoft 2016

# with() allows us to avoiding prepending each column with "example$"
ordering <- order(example$firm, example$year)
example <- example[ordering,]
example

##        firm year
## 3    Google 2016
## 1    Google 2017
## 4 Microsoft 2016
## 2 Microsoft 2017

Subsetting data frames

1. We can use the selecting methods from before
2. We can pass a vector of logical values telling R what to keep
   • This is pretty useful!

df[df$tech_firm,]  # Remember the comma!

##           companyName earnings tech_firm all_zero revenue    margin
## Google         Google    12662      TRUE        0  110855 0.1142213
## Microsoft   Microsoft    21204      TRUE        0   89950 0.2357310

3. We can use the subset() function
   • I don’t recommend this function, as it does not always work
     • There are times where it is useful though

subset(df,earnings < 20000)

##         companyName earnings tech_firm all_zero revenue    margin
## Goldman     Goldman     4286     FALSE        0   42254 0.1014342
## Google       Google    12662      TRUE        0  110855 0.1142213

Practice: Data frames

• This exercise explores the nature of banks’ deposits
  • We will see which of Goldman, JPMorgan, and Citigroup have (since 2010):
    • The least of their assets in deposits
    • The most of their assets in deposits
• Do exercise 4 on the supplementary R practice file:
  • R Practice

Why use logical expressions?

• We just saw an example in our subsetting function
  • earnings < 20000
• Logical expressions give us more control over the data
• They let us easily create logical vectors for subsetting data

df$earnings

## [1]  4286 12662 21204

df$earnings < 20000

## [1]  TRUE  TRUE FALSE

Logical operators

== != > < >= <= ! | &

• Comparing strings is done character by character
  • Be very careful with it

Logical operators

== != > < >= <= ! | &

Logical operators

• Not: !
  • This simply inverts everything
  • !TRUE \(\rightarrow\) FALSE
  • !FALSE \(\rightarrow\) TRUE
• And: &
  • TRUE & TRUE \(\rightarrow\) TRUE
  • TRUE & FALSE \(\rightarrow\) FALSE
  • FALSE & FALSE \(\rightarrow\) FALSE
• Or: | (pipe, same key as ‘\’)
  • Note that | is evaluated after all &s
  • TRUE | TRUE \(\rightarrow\) TRUE
  • TRUE | FALSE \(\rightarrow\) TRUE
  • FALSE | FALSE \(\rightarrow\) FALSE
• You can mix in parentheses for grouping as needed

Examples for logical operators

• How many tech firms had >$10B in revenue in 2017?

sum(tech_df$revenue > 10000)

## [1] 46

• How many tech firms had >$10B in revenue but had negative earnings in 2017?

sum(tech_df$revenue > 10000 & tech_df$earnings < 0)

## [1] 4

• Who are those 4 with high revenue and negative earnings?

columns <- c("conm","tic","earnings","revenue")
tech_df[tech_df$revenue > 10000 & tech_df$earnings < 0, columns]

##                             conm   tic  earnings  revenue
## 35                   CORNING INC   GLW  -497.000 10116.00
## 45  TELEFONAKTIEBOLAGET LM ERICS  ERIC -4307.493 24629.64
## 120        DELL TECHNOLOGIES INC 7732B -3728.000 78660.00
## 214                   NOKIA CORP   NOK -1796.087 27917.49

Other special values

• We know TRUE and FALSE already
  • Note that FALSE can be represented as 0
  • Note that TRUE can be represented as any non-zero number
• There are also:
  • Inf: Infinity, often caused by dividing something by 0
  • NaN: “Not a number,” likely that the expression 0/0 occurred
  • NA: A missing value, usually not due to a mathematical error
  • Null: Indicates a variable has nothing in it
• We can check for these with:
  • is.inf()
  • is.nan()
  • is.na()
  • is.null()

Practice: Subsetting our data frame

• This practice focuses on subsetting out potentially interesting parts of our data frame
  • We will also see which of Goldman, JPMorgan, and Citigroup, in which year, had the lowest earnings since 2010
• Do exercise 5 on the supplementary R practice file:
  • R Practice

Other uses

• Conditional statements (used for programming)

# cond1, cond2, etc. can be any logical expression
if(cond1) {
  # Code runs if cond1 is TRUE
} else if (cond2) { # Can repeat 'else if' as needed
  # Code runs if this is the first condition that is TRUE
} else {
  # Code runs if none of the above conditions TRUE
}

• Vectorized conditional statements using ifelse()
  • If else takes 3 vectors and returns 1 vector
    • A vector of TRUE or FALSE
    • A vector of elements to return from when TRUE
    • A vector of elements to return from when FALSE

# Outputs odd for odd numbers and even for even numbers
even <- rep("even",5)
odd <- rep("odd",5)
numbers <- 1:5
ifelse(numbers %% 2, odd, even)

## [1] "odd"  "even" "odd"  "even" "odd"

Help functions

• There are two equivalent ways to quickly access help files:
  • ? and help()
  • Usage to get the help file for data.frame():
    • ?data.frame
    • help(data.frame)
• To see the options for a function, use args()

args(data.frame)

## function (..., row.names = NULL, check.rows = FALSE, check.names = TRUE, 
##     fix.empty.names = TRUE, stringsAsFactors = default.stringsAsFactors()) 
## NULL

A note on using functions

args(data.frame)

## function (..., row.names = NULL, check.rows = FALSE, check.names = TRUE, 
##     fix.empty.names = TRUE, stringsAsFactors = default.stringsAsFactors()) 
## NULL

• The ... represents a series of inputs
  • In this case, inputs like name=data, where name is the column name and data is a vector
• The ____ = ____ arguments are options for the function
  • The default is prespecified, but you can overwrite it
• Options can be very useful or save us a lot of time!
• You can always find them by:
  • Using the ? command
  • Checking other documentation like www.rdocumentation.org
  • Using the args() function

Installing more functions

• R Provides an easy way to install packages without ever leaving R
  • The install.packages() command
  • Can install a single package or a vector of packages

# To install the tidyverse package:
install.packages("tidyverse")

# To install ggplot2, dplyr, and magrittr packages:
install.packages(c("ggplot2", "dplyr", "magrittr"))

• Load packages using library()
  • Need to do this each time you open a new instance of R

# Load the tidyverse package
library(tidyverse)

Pipe notation

Pipe notation is never necessary and not built in to R

• Pipe notation is provided by the magrittr package
  • Part of tidyverse, an extremely popular collection of packages
• Pipe notation is done using %>%
  • Left %>% Right(arg2, ...) is the same as Right(Left, arg2, ...)

Piping can drastically improve code readability

Piping example

Plot tech firms’ earnings vs revenue, >$10B in revenue

library(tidyverse)
library(plotly)

plot <- tech_df %>%
  subset(revenue > 10000) %>%
  ggplot(aes(x=revenue,y=earnings)) + # ggplot comes from ggplot2, part of tidyverse
  geom_point(shape=1, aes(text=sprintf("Ticker: %s", tic)))  # Adds point, and ticker
ggplotly(plot)  # Makes the plot interactive

Piping example: Without piping

library(tidyverse)
library(plotly)

plot <- ggplot(subset(tech_df, revenue > 10000), aes(x=revenue,y=earnings)) +
  geom_point(shape=1, aes(text=sprintf("Ticker: %s", tic)))
ggplotly(plot)  # Makes the plot interactive

Practice: External library usage

• This practice focuses on using an external library
  • We will also see which of Goldman, JPMorgan, and Citigroup, in which year, had the lowest earnings since 2010
• Do exercise 6 on the supplementary R practice file:
  • R Practice

Note: The ~ indicates a formula the left side is the y-axis and the right side is the x-axis

Note: The | tells lattice to make panels based on the variable(s) to the right

Math functions

• sum(): Sum of a vector
• abs(): Absolute value
• sign(): The sign of a number

vector = c(-2,-1,0,1,2)
sum(vector)

## [1] 0

abs(vector)

## [1] 2 1 0 1 2

sign(vector)

## [1] -1 -1  0  1  1

Stats functions

• mean(): Calculates the mean of a vector
• median(): Calculates the median of a vector
• sd(): Calculates the sample standard deviation of a vector
• quantile(): Provides the quartiles of a vector
• range(): Gives the minimum and maximum of a vector
  • Related: min() and max()

quantile(tech_df$earnings)

##         0%        25%        50%        75%       100% 
## -4307.4930   -15.9765     1.8370    91.3550 48351.0000

range(tech_df$earnings)

## [1] -4307.493 48351.000

Make your own functions!

• Use the function() function!
  • my_func <- function(agruments) {code}

Simple function: Add 2 to a number

add_two <- function(n) {
  n + 2
}

add_two(500)

## [1] 502

Slightly more complex function example

mult_together <- function(n1, n2=0, square=FALSE) {
  if (!square) {
    n1 * n2
  } else {
    n1 * n1
  }
}

mult_together(5,6)

## [1] 30

mult_together(5,6,square=TRUE)

## [1] 25

mult_together(5,square=TRUE)

## [1] 25