Computational Statistics Week 1

1 Basic Syntax

Question 1 \begin{aligned} &3^4-\frac{2^6}{4}+\sqrt{225}-\ln \left(e^5\right)+\frac{\left(\lceil 7.2 \rceil \times \lfloor 9.9 \rfloor -(29 \bmod 6)^2\right)}{5} \end{aligned}

3^4 - 2^6/4 + sqrt(225) - log(exp(5)) + (ceiling(7.2) * floor(9.9) - (29%%6)^2)/5

[1] 84.4

Question 2 \begin{aligned} &\sin (\pi / 2)+\cos (\pi / 3)-\tan (\pi / 4)+2^{1.5}-9^{1 / 2}+\frac{5^3-4^4}{3^2+2^5} \end{aligned}

sin(pi/2) + cos(pi/3) - tan(pi/4) + 2^1.5 - 9^(1/2) + (5^3-4^4)/(3^2+2^5)

[1] -2.866695

Question 3 \begin{aligned} &\sqrt{2^{10}+3^6}-\ln \left(\frac{e^8}{e^3}\right)+\frac{\ln (1000)}{\ln (10)}+e^0+\frac{\lfloor 123.99\rfloor-\lceil 45.01\rceil}{7} \end{aligned}

sqrt(2^10+3^6) - log(exp(8)/exp(3)) + (log(1000)/log(10)) + exp(0) + (floor(123.99-ceiling(45.01)))/7

[1] 51.86884

Question 4 \begin{aligned} &\frac{7^5-6^4+5^3-4^2+3}{\frac{2^8-1}{3^3-1}+(10 \bmod 6)+\sqrt{144}}+\frac{\ln \left(e^{12}\right)-\ln \left(e^7\right)}{\lceil 19.01\rceil-\lfloor 3.99\rfloor}-\left(\frac{1+2+3+\cdots+30}{1 \times 2 \times 3 \times \cdots \times 10}\right)^2 \end{aligned}

q4_1 <- (7^5-6^4+5^3-4^2+3) / ((2^8-1)/(3^3-1) + (10 %% 6) + sqrt(144))
q4_2 <- (log(exp(12)) - log(exp(7))) / (ceiling(19.01) - floor(3.99))
q4_3 <- (sum(1:30) / factorial(30))^2

q4_1 + q4_2 - q4_3

[1] 605.6563

Question 5 \begin{aligned} &\int_{-\infty}^{1.96} \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2} d x \end{aligned}

pnorm(q = 1.96, mean = 0, sd = 1)

[1] 0.9750021

Question 6

Given \begin{aligned} &\mathbf{A}=\left[\begin{array}{lll} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 4 & 1 & 1 \end{array}\right] \text { and } \mathbf{B}=\left[\begin{array}{lll} 1 & 2 & 0 \\ 3 & 1 & 4 \\ 2 & 5 & 1 \end{array}\right] \end{aligned}

Then, calculate

\mathbf{C} = \mathbf{A} \mathbf{B}

\mathbf{C}^\top \mathbf{C}

(\mathbf{C}^\top \mathbf{C})^{-1}

A <- matrix(c(2,1,4, 1,0,1, 3,2,1), ncol = 3, nrow = 3)
print(A)

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

B <- matrix(c(1,3,2, 2,1,5, 0,4,1), ncol = 3, nrow = 3)
print(B)

     [,1] [,2] [,3]
[1,]    1    2    0
[2,]    3    1    4
[3,]    2    5    1

C <- A %*% B
print(C)

     [,1] [,2] [,3]
[1,]   11   20    7
[2,]    5   12    2
[3,]    9   14    5

t(C) %*% C

     [,1] [,2] [,3]
[1,]  227  406  132
[2,]  406  740  234
[3,]  132  234   78

solve(t(C) %*% C)

           [,1]        [,2]       [,3]
[1,]  1.0164609 -0.26748971 -0.9176955
[2,] -0.2674897  0.09670782  0.1625514
[3,] -0.9176955  0.16255144  1.0781893

2 Data Structure in R

2.1 Basic Data Types in R

my_integer <- 42
class(my_integer)

[1] "numeric"

my_integer2 <- 42L
class(my_integer2)

[1] "integer"

my_numeric <- 42.5
class(my_numeric)

[1] "numeric"

my_character <- "some text"
class(my_character)

[1] "character"

my_character2 <- "42"
class(my_character2)

[1] "character"

my_logical <- TRUE
class(my_logical)

[1] "logical"

2.2 Data Frame

data <- iris
class(data)

[1] "data.frame"

names(data)

[1] "Sepal.Length" "Sepal.Width"  "Petal.Length" "Petal.Width"  "Species"     

head(data, 10)

2.3 Array and Matrices

x_vec <- 1:24
print(x)

[1] 25 24 30 78 21 30 37

x_mat <- matrix(1:24, nrow=4, ncol=6)
print(x_mat)

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

x_arr <- array (1:24, c(3, 4, 2))
print(x_arr)

, , 1

     [,1] [,2] [,3] [,4]
[1,]    1    4    7   10
[2,]    2    5    8   11
[3,]    3    6    9   12

, , 2

     [,1] [,2] [,3] [,4]
[1,]   13   16   19   22
[2,]   14   17   20   23
[3,]   15   18   21   24

2.4 List

set.seed(123)
n <- 1000
x <- rbinom(n, size = 1, prob = 0.5)
table(x)

x
  0   1 
507 493 

head(x, 30)

 [1] 0 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0

r <- rle(x)
str(r)

List of 2
 $ lengths: int [1:489] 1 1 1 2 1 3 1 1 1 2 ...
 $ values : int [1:489] 0 1 0 1 0 1 0 1 0 1 ...
 - attr(*, "class")= chr "rle"

head(r$lengths, 10)

 [1] 1 1 1 2 1 3 1 1 1 2

head(r[[1]], 10)

 [1] 1 1 1 2 1 3 1 1 1 2

2.5 Factor

gender_vector <- c("Male", "Female", "Female", "Male", "Male")
class(gender_vector)

[1] "character"

class(gender_vector)

[1] "character"

factor_gender_vector <- factor(gender_vector)
factor_gender_vector

[1] Male   Female Female Male   Male  
Levels: Female Male

class(factor_gender_vector)

[1] "factor"

3 Operators, Conditions & Loops

3.1 Relational Operators

TRUE == TRUE

[1] TRUE

"hello" == "goodbye"

[1] FALSE

TRUE == FALSE

[1] FALSE

3 == 2

[1] FALSE

TRUE != TRUE

[1] FALSE

"hello" != "goodbye"

[1] TRUE

TRUE != FALSE

[1] TRUE

3 != 2

[1] TRUE

3 > 5

[1] FALSE

4 <= 4

[1] TRUE

# Alphabetical Order!
"Hello" > "Goodbye"

[1] TRUE

# TRUE coerces to 1
# FALSE coerces to 0
TRUE < FALSE

[1] FALSE

linkedin <- c (16, 9, 13, 5, 2, 17, 14)
linkedin > 10

[1]  TRUE FALSE  TRUE FALSE FALSE  TRUE  TRUE

x <- 17
x > 5 & x < 15

[1] FALSE

y <- 14
y < 5 | y > 15

[1] FALSE

c(TRUE, TRUE, FALSE) & c(TRUE, FALSE, FALSE)

[1]  TRUE FALSE FALSE

3.2 IF-ELSE Statement

Convert Nilai Angka to Nilai Huruf

Nilai Angka	Nilai Huruf
86–100	A
76–85	AB
66–75	B
61–65	BC
56–60	C
41–55	D
0–40	E

convert_score <- function(x){
  x <- round(x,0)
  if (x >= 86 & x <= 100){
    return("A")
  } else if (x >= 76 & x <= 85){
    return("AB")
  } else if (x >= 66 & x <= 75){
    return("B")
  } else if (x >= 61 & x <= 65){
    return("BC")
  } else if (x >= 56 & x <= 60){
    return("C")
  } else if (x >= 41 & x <= 55){
    return("D")
  } else if (x >= 0 & x <= 40){
    return("E")
  } else{
    return(NA)
  }
}

x <- c(85.5)
convert_score(x)

[1] "A"

3.3 WHILE Loop

ctr <- 1
while(ctr <= 7) {
  print(paste("ctr is set to" , ctr))
  ctr <- ctr + 1
}

[1] "ctr is set to 1"
[1] "ctr is set to 2"
[1] "ctr is set to 3"
[1] "ctr is set to 4"
[1] "ctr is set to 5"
[1] "ctr is set to 6"
[1] "ctr is set to 7"

3.4 FOR Loop

cities <- c("New York", "Paris",
            "London", "Tokyo",
            "Rio de Janeiro", "Cape Town")
for (city in cities){
  print(city)
}

[1] "New York"
[1] "Paris"
[1] "London"
[1] "Tokyo"
[1] "Rio de Janeiro"
[1] "Cape Town"

cities <- c("New York", "Paris",
            "London", "Tokyo",
            "Rio de Janeiro", "Cape Town")
for (i in 1:length(cities)){
  print(cities[i])
}

[1] "New York"
[1] "Paris"
[1] "London"
[1] "Tokyo"
[1] "Rio de Janeiro"
[1] "Cape Town"

4 Functions

average <- function(x){
  n <- length(x)
  sumX <- 0
  for (i in 1:n){
    sumX <- sumX + x[i]
    }
  return (sumX/n)
}

x <- c(25,24,30, 78, 21, 30, 37)
average(x)

[1] 35

mean(x)

[1] 35

Question

Create a function for sample variance

S^2=\frac{1}{n-1} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2

sample_var <- function(x){
  n <- length(x)
  
  sumX <- 0
  for (i in 1:n){
    sumX <- sumX + x[i]
  }
  meanX <- sumX /n
  
  varX <- 0
  for (i in 1:n){
    varX <- varX + (x[i]-meanX)^2
  }
  
  return(varX/(n-1))
}

sample_var(x)

[1] 386.6667

var(x)

[1] 386.6667

5 Workspace and Files

5.1 Read CSV files

link_url <- "https://github.com/novrisuhermi/dataset/raw/refs/heads/main/data.csv"

data <- read.csv(link_url, header = TRUE, sep = ",")
head(data, 10)

5.2 Read Excel files

library(readxl)

download.file(
  "https://raw.githubusercontent.com/novrisuhermi/dataset/main/data.xlsx",
  destfile = "data.xlsx",
  mode = "wb"
)

trying URL 'https://raw.githubusercontent.com/novrisuhermi/dataset/main/data.xlsx'
Content type 'application/octet-stream' length 13255 bytes (12 KB)
==================================================
downloaded 12 KB

data2 <- read_excel("data.xlsx")

head(data2, 10)

5.3 Write CSV files

library(summarytools)

summaryData <- descr(data)
print(summaryData)

Descriptive Statistics  
data  
N: 32  

                        am     carb      cyl     disp     drat     gear       hp      mpg     qsec
----------------- -------- -------- -------- -------- -------- -------- -------- -------- --------
             Mean     0.41     2.81     6.19   230.72     3.60     3.69   146.69    20.09    17.85
          Std.Dev     0.50     1.62     1.79   123.94     0.53     0.74    68.56     6.03     1.79
              Min     0.00     1.00     4.00    71.10     2.76     3.00    52.00    10.40    14.50
               Q1     0.00     2.00     4.00   120.65     3.08     3.00    96.00    15.35    16.88
           Median     0.00     2.00     6.00   196.30     3.70     4.00   123.00    19.20    17.71
               Q3     1.00     4.00     8.00   334.00     3.92     4.00   180.00    22.80    18.90
              Max     1.00     8.00     8.00   472.00     4.93     5.00   335.00    33.90    22.90
              MAD     0.00     1.48     2.97   140.48     0.70     1.48    77.10     5.41     1.42
              IQR     1.00     2.00     4.00   205.18     0.84     1.00    83.50     7.38     2.01
               CV     1.23     0.57     0.29     0.54     0.15     0.20     0.47     0.30     0.10
         Skewness     0.36     1.05    -0.17     0.38     0.27     0.53     0.73     0.61     0.37
      SE.Skewness     0.41     0.41     0.41     0.41     0.41     0.41     0.41     0.41     0.41
         Kurtosis    -1.92     1.26    -1.76    -1.21    -0.71    -1.07    -0.14    -0.37     0.34
          N.Valid    32.00    32.00    32.00    32.00    32.00    32.00    32.00    32.00    32.00
                N    32.00    32.00    32.00    32.00    32.00    32.00    32.00    32.00    32.00
        Pct.Valid   100.00   100.00   100.00   100.00   100.00   100.00   100.00   100.00   100.00

Table: Table continues below

 

                        vs       wt
----------------- -------- --------
             Mean     0.44     3.22
          Std.Dev     0.50     0.98
              Min     0.00     1.51
               Q1     0.00     2.54
           Median     0.00     3.33
               Q3     1.00     3.65
              Max     1.00     5.42
              MAD     0.00     0.77
              IQR     1.00     1.03
               CV     1.15     0.30
         Skewness     0.24     0.42
      SE.Skewness     0.41     0.41
         Kurtosis    -2.00    -0.02
          N.Valid    32.00    32.00
                N    32.00    32.00
        Pct.Valid   100.00   100.00

write.csv(summaryData, "summary_data.csv")