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xs <- 2 # Make your first scalar
xs # Print the scalar
## [1] 22 Mathematics
2.1 Objects
In R: scalars, vectors, and matrices are different kinds of “objects”.
These objects are used extensively in data analysis
- scalars: summary statistics (average household income).
- vectors: single variables in data sets (the household income of each family in Vancouver).
- matrices: multiple variables in data sets (the age and education level of every person in class).
Vectors are probably your most common object in R, but we will start with scalars (which are treated as a special case in R).
Scalars.
Make your first scalar
Perform simple calculations and see how R is doing the math for you
Code
xs + 2
## [1] 4
xs*2 # Perform and print a simple calculation
## [1] 4
(xs+1)^2 # Perform and print a simple calculation
## [1] 9
xs + NA # often used for missing values
## [1] NANow change xs, predict what will happen, then re-run the code.
Vectors.
Make Your First Vector
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x <- c(0,1,3,10,6) # Your First Vector
x # Print the vector
## [1] 0 1 3 10 6
x[2] # Print the 2nd Element; 1
## [1] 1
x+2 # Print simple calculation; 2,3,5,8,12
## [1] 2 3 5 12 8
x*2
## [1] 0 2 6 20 12
x^2
## [1] 0 1 9 100 36Apply Mathematical calculations “elementwise”
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x+x
## [1] 0 2 6 20 12
x*x
## [1] 0 1 9 100 36
x^x
## [1] 1.0000e+00 1.0000e+00 2.7000e+01 1.0000e+10 4.6656e+04See that scalars are vectors
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c(1)
## [1] 1
1:7
## [1] 1 2 3 4 5 6 7
seq(0,1,by=.1)
## [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Matrices.
Matrices are also common objects
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x1 <- c(1,4,9)
x2 <- c(3,0,2)
x_mat <- rbind(x1, x2)
x_mat # Print full matrix
## [,1] [,2] [,3]
## x1 1 4 9
## x2 3 0 2
x_mat[2,] # Print Second Row
## [1] 3 0 2
x_mat[,2] # Print Second Column
## x1 x2
## 4 0
x_mat[2,2] # Print Element in Second Column and Second Row
## x2
## 0There are elementwise calculations
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x_mat+2
## [,1] [,2] [,3]
## x1 3 6 11
## x2 5 2 4
x_mat*2
## [,1] [,2] [,3]
## x1 2 8 18
## x2 6 0 4
x_mat^2
## [,1] [,2] [,3]
## x1 1 16 81
## x2 9 0 4
x_mat + x_mat
## [,1] [,2] [,3]
## x1 2 8 18
## x2 6 0 4
x_mat*x_mat
## [,1] [,2] [,3]
## x1 1 16 81
## x2 9 0 4
x_mat^x_mat
## [,1] [,2] [,3]
## x1 1 256 387420489
## x2 27 1 4And you can also use matrix algebra
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x_mat1 <- matrix(2:7,2,3)
x_mat1
## [,1] [,2] [,3]
## [1,] 2 4 6
## [2,] 3 5 7
x_mat2 <- matrix(4:-1,2,3)
x_mat2
## [,1] [,2] [,3]
## [1,] 4 2 0
## [2,] 3 1 -1
tcrossprod(x_mat1, x_mat2) #x_mat1 %*% t(x_mat2)
## [,1] [,2]
## [1,] 16 4
## [2,] 22 7
crossprod(x_mat1, x_mat2)
## [,1] [,2] [,3]
## [1,] 17 7 -3
## [2,] 31 13 -5
## [3,] 45 19 -7
# x_mat1 * x_mat22.2 Functions
Functions are applied to objects
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# Define a function that adds two to any vector
add_2 <- function(input_vector) {
output_vector <- input_vector + 2 # new object defined locally
return(output_vector) # return new object
}
# Apply that function to a vector
x <- c(0,1,3,10,6)
add_2(x)
## [1] 2 3 5 12 8
# notice 'output_vector' is not available hereThere are many many generalizations
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add_vec <- function(input_vector1, input_vector2) {
output_vector <- input_vector1 + input_vector2
return(output_vector)
}
add_vec(x,3)
## [1] 3 4 6 13 9
add_vec(x,x)
## [1] 0 2 6 20 12
sum_squared <- function(x1, x2) {
y <- (x1 + x2)^2
return(y)
}
sum_squared(1, 3)
## [1] 16
sum_squared(x, 2)
## [1] 4 9 25 144 64
sum_squared(x, NA)
## [1] NA NA NA NA NA
sum_squared(x, x)
## [1] 0 4 36 400 144
sum_squared(x, 2*x)
## [1] 0 9 81 900 324Functions can take functions as arguments
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fun_of_seq <- function(f){
x <- seq(1,3, length.out=12)
y <- f(x)
return(y)
}
fun_of_seq(mean)
## [1] 2
fun_of_seq(sd)
## [1] 0.6555548You can apply functions to matrices
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sum_squared(x_mat, x_mat)
## [,1] [,2] [,3]
## x1 4 64 324
## x2 36 0 16
# Apply function to each matrix row
y <- apply(x_mat, 1, sum)^2
# ?apply #checks the function details
y - sum_squared(x, x) # tests if there are any differences
## [1] 196 21 160 -375 52There are many possible functions you can apply
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# Return Y-value with minimum absolute difference from 3
abs_diff_y <- abs( y - 3 )
abs_diff_y # is this the luckiest number?
## x1 x2
## 193 22
#min(abs_diff_y)
#which.min(abs_diff_y)
y[ which.min(abs_diff_y) ]
## x2
## 25There are also some useful built in functions
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m <- matrix(c(1:3,2*(1:3)),byrow=TRUE,ncol=3)
m
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 2 4 6
# normalize rows
m/rowSums(m)
## [,1] [,2] [,3]
## [1,] 0.1666667 0.3333333 0.5
## [2,] 0.1666667 0.3333333 0.5
# normalize columns
t(t(m)/colSums(m))
## [,1] [,2] [,3]
## [1,] 0.3333333 0.3333333 0.3333333
## [2,] 0.6666667 0.6666667 0.6666667
# de-mean rows
sweep(m,1,rowMeans(m), '-')
## [,1] [,2] [,3]
## [1,] -1 0 1
## [2,] -2 0 2
# de-mean columns
sweep(m,2,colMeans(m), '-')
## [,1] [,2] [,3]
## [1,] -0.5 -1 -1.5
## [2,] 0.5 1 1.5Loops.
Applying the same function over and over again
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exp_vector <- vector(length=3)
for(i in 1:3){
exp_vector[i] <- exp(i)
}
# Compare
exp_vector
## [1] 2.718282 7.389056 20.085537
c( exp(1), exp(2), exp(3))
## [1] 2.718282 7.389056 20.085537store complicated example
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complicate_fun <- function(i, j=0){
x <- i^(i-1)
y <- x + mean( j:i )
z <- log(y)/i
return(z)
}
complicated_vector <- vector(length=10)
for(i in 1:10){
complicated_vector[i] <- complicate_fun(i)
}recursive loop
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x <- vector(length=4)
x[1] <- 1
for(i in 2:4){
x[i] <- (x[i-1]+1)^2
}
x
## [1] 1 4 25 676Logic.
Basic Logic
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x <- c(1,2,3,NA)
x > 2
## [1] FALSE FALSE TRUE NA
x==2
## [1] FALSE TRUE FALSE NA
any(x==2)
## [1] TRUE
all(x==2)
## [1] FALSE
2 %in% x
## [1] TRUE
is.numeric(x)
## [1] TRUE
is.na(x)
## [1] FALSE FALSE FALSE TRUEThe “&” and “|” commands are logical calculations that compare vectors to the left and right.
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x <- 1:3
is.numeric(x) & (x < 2)
## [1] TRUE FALSE FALSE
is.numeric(x) | (x < 2)
## [1] TRUE TRUE TRUE
if(length(x) >= 5 & x[5] > 12) print("ok")Advanced Logic.
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x <- 1:10
cut(x, 4)
## [1] (0.991,3.25] (0.991,3.25] (0.991,3.25] (3.25,5.5] (3.25,5.5]
## [6] (5.5,7.75] (5.5,7.75] (7.75,10] (7.75,10] (7.75,10]
## Levels: (0.991,3.25] (3.25,5.5] (5.5,7.75] (7.75,10]
split(x, cut(x, 4))
## $`(0.991,3.25]`
## [1] 1 2 3
##
## $`(3.25,5.5]`
## [1] 4 5
##
## $`(5.5,7.75]`
## [1] 6 7
##
## $`(7.75,10]`
## [1] 8 9 10Code
xs <- split(x, cut(x, 4))
sapply(xs, mean)
## (0.991,3.25] (3.25,5.5] (5.5,7.75] (7.75,10]
## 2.0 4.5 6.5 9.0
# shortcut
aggregate(x, list(cut(x,4)), mean)
## Group.1 x
## 1 (0.991,3.25] 2.0
## 2 (3.25,5.5] 4.5
## 3 (5.5,7.75] 6.5
## 4 (7.75,10] 9.0see https://bookdown.org/rwnahhas/IntroToR/logical.html
2.3 Arrays
Arrays are generalization of matrices. They are often used in spatial econometrics, and are a very efficient way to store numeric data with the same dimensions.
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a <- array(data = 1:24, dim = c(2, 3, 4))Code
a
a[1, , , drop = FALSE] # Row 1
#a[, 1, , drop = FALSE] # Column 1
#a[, , 1, drop = FALSE] # Layer 1
a[ 1, 1, ] # Row 1, column 1
#a[ 1, , 1] # Row 1, "layer" 1
#a[ , 1, 1] # Column 1, "layer" 1
a[1 , 1, 1] # Row 1, column 1, "layer" 1Apply extends to arrays
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apply(a, 1, mean) # Row means
## [1] 12 13
apply(a, 2, mean) # Column means
## [1] 10.5 12.5 14.5
apply(a, 3, mean) # "Layer" means
## [1] 3.5 9.5 15.5 21.5
apply(a, 1:2, mean) # Row/Column combination
## [,1] [,2] [,3]
## [1,] 10 12 14
## [2,] 11 13 15Outer products yield arrays
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x <- c(1,2,3)
x_mat1 <- outer(x, x) # x %o% x
x_mat1
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 2 4 6
## [3,] 3 6 9
is.array(x_mat) # Matrices are arrays
## [1] TRUE
x_mat2 <- matrix(6:1,2,3)
outer(x_mat2, x)
## , , 1
##
## [,1] [,2] [,3]
## [1,] 6 4 2
## [2,] 5 3 1
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 12 8 4
## [2,] 10 6 2
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 18 12 6
## [2,] 15 9 3
# outer(x_mat2, matrix(x))
# outer(x_mat2, t(x))
# outer(x_mat1, x_mat2)