3 Mathematics

---

3.1 Scalars

xs <- 2 ## Your first scalar
xs  ## Print the scalar

## [1] 2

(xs+1)^2 ## Perform and print a simple calculation

## [1] 9

xs + NA ## often used for missing values

## [1] NA

xs*2

## [1] 4

3.2 Vectors

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  36

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+04

c(1) ## scalars are vectors

## [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 1.0

3.3 Functions

Function of a vector

## Add two to any vector
add2 <- function(x1) {
    x1+2
}
add2(x)

## [1]  2  3  5 12  8

## Generalization
addn <- function(x1,n=2) {
    x1+n
}
addn(x)

## [1]  2  3  5 12  8

addn(x,3)

## [1]  3  4  6 13  9

Function for two vectors

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  81 900 324

Applying the same function over and over again

sapply(1:3, exp)

## [1]  2.718282  7.389056 20.085537

exp(1:3)

## [1]  2.718282  7.389056 20.085537

## mapply takes multiple vectors
mapply(sum, 1:3, exp(1:3) )

## [1]  3.718282  9.389056 23.085537

recursive functions

## For Loop
x <- rep(1, 3)
for(i in 2:length(x) ){
    x[i] <- (x[i-1]+1)^2
}
x

## [1]  1  4 25

r_fun <- function(n){
    x <- rep(1,n)
    for(i in 2:length(x) ){
        x[i] <- (x[i-1]+1)^2
    }
    return(x)
}
r_fun(5)

## [1]      1      4     25    676 458329

Functions can take functions as arguments

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(mean)

## [1] 2

3.4 Matrices

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 
##  0

## 
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         4

y <- apply(x_mat, 1, sum)^2 ## Apply function to each row
## ?apply  #checks the function details
y - sum_squared(x, x) ## tests if there are any differences

## [1]   192   -39 -2304

Many Other Functions

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

##
x_mat1 * x_mat2

##      [,1] [,2] [,3]
## [1,]    8    8    0
## [2,]    9    5   -7

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

Example Calculations

## 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)

## [1] 22

which.min(abs_diff_y)

## x2 
##  2

y[ which.min(abs_diff_y) ]

## x2 
## 25

3.5 Arrays

Generalization of matrices used in spatial econometrics

a <- array(data = 1:24, dim = c(2, 3, 4))
a

## , , 1
## 
##      [,1] [,2] [,3]
## [1,]    1    3    5
## [2,]    2    4    6
## 
## , , 2
## 
##      [,1] [,2] [,3]
## [1,]    7    9   11
## [2,]    8   10   12
## 
## , , 3
## 
##      [,1] [,2] [,3]
## [1,]   13   15   17
## [2,]   14   16   18
## 
## , , 4
## 
##      [,1] [,2] [,3]
## [1,]   19   21   23
## [2,]   20   22   24

a[1, , , drop = FALSE]  # Row 1

## , , 1
## 
##      [,1] [,2] [,3]
## [1,]    1    3    5
## 
## , , 2
## 
##      [,1] [,2] [,3]
## [1,]    7    9   11
## 
## , , 3
## 
##      [,1] [,2] [,3]
## [1,]   13   15   17
## 
## , , 4
## 
##      [,1] [,2] [,3]
## [1,]   19   21   23

a[, 1, , drop = FALSE]  # Column 1

## , , 1
## 
##      [,1]
## [1,]    1
## [2,]    2
## 
## , , 2
## 
##      [,1]
## [1,]    7
## [2,]    8
## 
## , , 3
## 
##      [,1]
## [1,]   13
## [2,]   14
## 
## , , 4
## 
##      [,1]
## [1,]   19
## [2,]   20

a[, , 1, drop = FALSE]  # Layer 1

## , , 1
## 
##      [,1] [,2] [,3]
## [1,]    1    3    5
## [2,]    2    4    6

a[ 1, 1,  ]  # Row 1, column 1

## [1]  1  7 13 19

a[ 1,  , 1]  # Row 1, "layer" 1

## [1] 1 3 5

a[  , 1, 1]  # Column 1, "layer" 1

## [1] 1 2

a[1 , 1, 1]  # Row 1, column 1, "layer" 1

## [1] 1

Apply extends to arrays

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   15

Outer Products yield arrays

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)