Basic Statistics

library("dplyr")
library("ggplot2")
library("gridExtra")

0.1 Some basic R commands

• q(): Exit program
• save.image(): Save the work environment
• ls(): List objects in memory
• example(): Shows example of the selected function.
• edit(): Edit th object data manually, opening a spreadsheet.
• rank(): return the position of a number in ascending order.
• sort(): return the same number array in ascending order.
• order(): return the position of a number in ascending order.
• search(): Search package currently loaded.
• vignette(): View long documentation about a specific packages.
• cat(): Prints the arguments. Useful for producing output in user defined functions.

0.2 Math functions

0.2.1 Logarithm

log_series <- tibble(Sequence = seq(1:100)) %>% 
                mutate(log_sq = log(Sequence))

ggplot(log_series, aes(x = Sequence, y = log_sq)) + geom_line() +
    theme_minimal() + ggtitle("Logarithmic Series") +
  theme(plot.title = element_text(hjust = 0.5))

0.2.2 10 Base Logarithm

log_series <- tibble(Sequence = seq(1:100)) %>% 
                mutate(log_10 = log10(Sequence))

ggplot(log_series, aes(x = Sequence, y = log_10)) + geom_line() +
    theme_minimal() + ggtitle("10 Base Logarithmic Series") +
  theme(plot.title = element_text(hjust = 0.5))

0.2.3 Square root

log_series <- tibble(Sequence = seq(1:100)) %>% 
                mutate(sqrt = sqrt(Sequence))

ggplot(log_series, aes(x = Sequence, y = sqrt)) + geom_line() +
    theme_minimal() + ggtitle("Square root Series") +
  theme(plot.title = element_text(hjust = 0.5))

0.2.4 Trigonometric functions

log_series <- tibble(Sequence = seq(1:50)) %>% 
                mutate(sin = sin(Sequence),
                       cos = cos (Sequence),
                       Tan = tan(Sequence))

ggplot(log_series ) + geom_line(aes(x = Sequence, y = sin),
                                colour = "#ce2e6c") +
                      geom_line(aes(x = Sequence, y = cos),
                                colour = "#52de97") +
    theme_minimal() + ggtitle("Trigonometric Series") +
  theme(plot.title = element_text(hjust = 0.5))

0.3 Matrix operations

col1 <- c(2,1,0)
col2 <- c(1,3,1)
col3 <- c(1,1,2)

mat_a <- cbind(col1,col2,col3)

# transpose matrix

t(mat_a)

##      [,1] [,2] [,3]
## col1    2    1    0
## col2    1    3    1
## col3    1    1    2

## Matrix multiplication

t(mat_a) %*% mat_a

##      col1 col2 col3
## col1    5    5    3
## col2    5   11    6
## col3    3    6    6

#inverse of matrix

solve(mat_a)

##            [,1]       [,2]       [,3]
## col1  0.5555556 -0.1111111 -0.2222222
## col2 -0.2222222  0.4444444 -0.1111111
## col3  0.1111111 -0.2222222  0.5555556

0.4 Descriptive Statistics

0.4.1 Mean

The mean is the most representative value in a list of numbers(calculated with mean() function):

\[\frac{\sum_{i=1}^n x_{i}}{n}\]

0.4.2 Median

The median is the value that separates the higher half from the lower half in a list of numbers. median() function in R applied to a list of numbers.

If the count of the total numbers is an odd number, the formula for the median is the following:

\[ \frac{n+1}{2} \] With even numbers:

\[ \frac{\frac{n}{2}+[\frac{n}{2}+1]}{2} \]

0.4.3 Variance

It measures how far a set of random numbers are spread out from their average value. var() function in R.

\[ Var(X) = \frac{\sum_{i=1}^n(X_{i}-\mu_{x})^2}{n-1} \]

0.4.4 Standard Deviation

square root of the variance. sd() function to perform it.

\[ \sigma_{x} = \sqrt{Var(X)} \] ### Mode

The mode is the most frequently value in a set of data.

0.4.5 Range

The range is the total amplitude of a list of numbers. range() function is appropiate to it.

\[max(x) -min(x)\]

0.4.6 Standard error of the mean

\[\frac{\sigma}{\sqrt{n}}\]

0.4.7 Variation Coefficient

the variation coefficient is the percentage of dispersion from the total average, it’s a way to compare variance between variables with differente scale.

\[\frac{\sigma}{\mu}\]

var_coeff <- function(data){
  return(sd(data)/mean(data))
}

my_mode <- function(x) {
  ux <- unique(x)
  ux[which.max(tabulate(match(x, ux)))]
}

0.4.8 Covariance

Covariance is the measure of the joint variability of two random variables. cov() is the function used to calculate it.

\[ cov(X,Y) = \frac{\sum_{i=1}^n (X_{i}-\mu_{x})(Y_{i}-\mu_{y})}{n-1} \]

0.4.9 Correlation

The correlation measures the strength of the relatioship between variables. cor() is the R function used to calculate it.

\[ \rho(X,Y) = \frac{cov(X,Y)}{\sigma_{x}\sigma_{y}} \]

0.5 Probability

Most Known probability distributions.

0.5.1 Binomial

set.seed(123)

random_binomial_dist <- tibble(
                    sequence = 1:10000,
                    Value = rbinom(n = 10000,
                                  size = 1000,
                                  prob = 0.4)
                              )
  

bin1 <- ggplot(random_binomial_dist, aes(x = Value)) +
      geom_histogram(binwidth = 2, colour = "#ffffff",
                     fill = "#3c9d9b") +
      ggtitle("Random Binomial distribution") +
      theme_minimal() + theme(plot.title = element_text(hjust = 0.5))

## CDF, dentity and  quantile distribution function

#curve(dbinom(size = 1000,x, prob = 0.5),-1,100)
#qbinom()

## Cumulative distribution function

 q <- 1:10000
 binomial_series <- q

 for (i in q) {

 binomial_series[i] <- pbinom(size = 10000, q = q[i], prob =  0.5)

 }

 random_binomial_dist <- cbind(random_binomial_dist, binomial_series)


 bin2 <-ggplot(random_binomial_dist, aes(y = binomial_series, x = sequence)) +
       geom_line(colour = "#730068") +
       ggtitle("Cumulative Binomial distribution") +
       theme_minimal() + theme(plot.title = element_text(hjust = 0.5))

 grid.arrange(bin1,bin2, nrow = 1)

0.5.2 Exponential

random_exponential_dist <- tibble(
                              sequence = 1:10000,
                              Value = rexp(n = 10000,
                                           rate = 2)
                                  )

exp1 <- ggplot(random_exponential_dist, aes(x = Value)) +
      geom_histogram(binwidth = 0.05, colour = "#ffffff",
                     fill = "#caadde") +
      ggtitle("Random Exponential distribution") +
      theme_minimal() + theme(plot.title = element_text(hjust = 0.5))

 q <- seq(0,1,by = 0.0001)
 exp_series <- q
 
 for (i in q) {
   
 exp_series[i] <- pexp( q[i], rate =  2)
   
 }
 
 exp_series <- exp_series[-1]
 
 random_exponential_dist <- cbind(random_exponential_dist,
                                  exp_series)
 
 
 exp2 <-ggplot(random_exponential_dist, aes(y = exp_series, x = sequence)) +
       geom_line(colour = "#730068") +
       ggtitle("Cumulative Exponential distribution") +
       theme_minimal() + theme(plot.title = element_text(hjust = 0.5))
 
 grid.arrange(exp1,exp2, nrow = 1)

0.5.3 Normal - Gaussian

random_normal_dist <- tibble(
                              sequence = 1:10000,
                              Value = rnorm(n = 10000,
                                            mean = 0,
                                            sd = 10)
                                  )


norm1 <- ggplot(random_normal_dist, aes(x = Value)) +
      geom_histogram(binwidth = 10, colour ="#ffffff",
                     fill = "#c70d3a") +
      ggtitle("Random Normal distribution") +
      theme_minimal() 


 q <- seq(-5,5, by = 0.001)[-1]
 norm_series <- q
 
 for (i in q) {
   
 norm_series[i] <- pnorm(q[i])
   
 }
 
 
 
 random_normal_dist <- cbind(random_normal_dist,
                                  norm_series)
 
 
 norm2 <-ggplot(random_normal_dist, aes(y = norm_series, x = sequence)) +
       geom_line(colour = "#730068") +
       ggtitle("Cumulative Normal distribution") +
       theme_minimal() + theme(plot.title = element_text(hjust = 0.5))
 
 grid.arrange(norm1,norm2, nrow = 1)

0.5.4 Geometric

random_geom_dist <- tibble(
                              Value = rgeom(n = 10000,
                                            prob = 0.2)
                                  )

ggplot(random_geom_dist, aes(x = Value)) +
      geom_histogram(binwidth = 1, colour = "#ffffff",
                     fill = "#32dbc6") +
      ggtitle("Random Geometric distribution") +
      theme_minimal() 

0.5.5 HyperGeometric

random_hyper_dist <- tibble(
                              Value = rhyper(100000,
                                             100,
                                             80,
                                             70)
                                  )

ggplot(random_hyper_dist, aes(x = Value)) +
      geom_histogram(binwidth = 1, colour ="#ffffff",
                     fill = "#ed0cef") +
      ggtitle("Random HyperGeometric distribution") +
      theme_minimal() 

0.5.6 Log-Normal

random_lognormal_dist <- tibble(
                              Value = rlnorm(10000)
                                  )

ggplot(random_lognormal_dist, aes(x = Value)) +
      geom_histogram(binwidth = 1, colour = "#ffffff",
                     fill = "#4f81c7") +
      ggtitle("Random Log-Normal distribution") +
      theme_minimal() 

0.5.7 Poisson

random_poisson_dist <- tibble(
                              Value = rpois(n = 10000,
                                            lambda = 10)
                                  )

ggplot(random_poisson_dist, aes(x = Value)) +
      geom_histogram(binwidth = 1, colour = "#ffffff",
                     fill = "#d6e4aa") +
      ggtitle("Random Poisson distribution") +
      theme_minimal() 

0.5.8 Chi-Squared

random_chi2_dist <- tibble(
                              Value = rchisq(n = 10000,
                                            df = 10)
                                  )

ggplot(random_chi2_dist, aes(x = Value)) +
      geom_histogram(binwidth = 0.7, colour ="#ffffff",
                     fill = "#eda593") +
      ggtitle("Random Chi-Squared distribution") +
      theme_minimal() 

0.5.9 Student

random_student_dist <- tibble(
                              Value = rt(n = 10000,
                                            df = 10)
                                  )

ggplot(random_student_dist, aes(x = Value)) +
      geom_histogram(binwidth = 0.5, colour = "#ffffff",
                     fill = "#a0c334") +
      ggtitle("Random Student distribution") +
      theme_minimal() 

0.5.10 Uniform

random_uniform_dist <- tibble(
                              Value = runif(n = 10000,
                                            min = 0,
                                            max = 100)
                                  )

ggplot(random_uniform_dist, aes(x = Value)) +
      geom_histogram(binwidth = 1, colour = "#ffffff",
                     fill = "#979797") +
      ggtitle("Random Uniform distribution") +
      theme_minimal() 

0.5.11 Fisher - Snedecor

random_fisher_dist <- tibble(
                              Value = rf(n = 10000,
                                            df1 = 10,
                                            df2 = 400)
                                  )

ggplot(random_fisher_dist, aes(x = Value)) +
      geom_histogram(binwidth = 0.1 , colour = "#ffffff",
                      fill = "#1089ff") +
      ggtitle("Random fisher distribution") +
      theme_minimal() 

0.5.12 Beta

random_beta_dist <- tibble(
                              Value = rbeta(n = 10000,
                                            shape1 = 0.1,
                                            shape2 = 0.8)
                                  )

ggplot(random_beta_dist, aes(x = Value)) +
      geom_histogram(binwidth = 0.1, colour = "#ffffff",
                     fill = "#ed5107") +
      ggtitle("Random Beta distribution") +
      theme_minimal() 

0.5.13 Weibull

random_weibull_dist <- tibble(
                              Value = rweibull(n = 10000,
                                            shape = 0.4,
                                            scale = 2)
                                  )

ggplot(random_weibull_dist, aes(x = Value)) +
      geom_histogram(binwidth = 20, colour = "#ffffff",
                     fill = "#02383c") +
      ggtitle("Random Weibull distribution") +
      theme_minimal()