R courses

Set Operations in R Definition and Creation of Sets In R, sets are often represented by vectors and manipulated using specialized functions. Although R does not have a native “set” type like some other languages, vectors can be used to represent sets and perform operations on them. Creating Sets:  # Create sets A <- c(1, 2, 3, 4, 5) B <- c(4, 5, 6, 7, 8) Basic Operations Union The union of two sets is a set containing all elements from both sets, with duplicates removed. Function in R: union()  # Calculate the union union_set <- union(A, B) print(union_set) # [1] 1 2 3 4 5 6 7 8 Intersection The intersection of two sets is a set containing only the elements present in both sets. Function in R: intersect()  # Calculate the intersection intersection_set <- intersect(A, B) print(intersection_set) # [1] 4 5 Difference The difference between two sets is a set containing the elements of the first set that are not in the second set. Function in R: setdiff()  # Calculate the difference difference_set_A_B <- setdiff(A, B) print(difference_set_A_B) # [1] 1 2 3 difference_set_B_A <- setdiff(B, A) print(difference_set_B_A) # [1] 6 7 8 Symmetric Difference The symmetric difference between two sets is a set containing elements that are in either of the sets but not in both. Function in R: setxor()  # Calculate the symmetric difference symmetric_difference <- setxor(A, B) print(symmetric_difference) # [1] 1 2 3 6 7 8 Advanced Operations Subset Check Check if one set is a subset of another set. Function in R: all() with match()  # Check if A is a subset of B is_subset <- all(A %in% B) print(is_subset) # [1] FALSE Set Equality Check if two sets are equal, meaning they contain the same elements. Function in R: identical()  # Check if A and B are equal are_equal <- identical(sort(A), sort(B)) print(are_equal) # [1] FALSE Manipulating Sets Adding Elements Add elements to a set by taking the union with a vector of new elements. Example:  # Add elements to a set A <- union(A, c(9, 10)) print(A) # [1] 1 2 3 4 5 9 10  Removing Elements Remove elements from a set by taking the difference. Example:  # Remove elements from a set A <- setdiff(A, c(1, 10)) print(A) # [1] 2 3 4 5 9 Practical Applications Example: Group Management Suppose you manage groups of people and want to find members who are only in one group but not the other.  # Group memberships group1 <- c(“Alice”, “Bob”, “Charlie”) group2 <- c(“Bob”, “Charlie”, “David”) # Members only in group1 only_group1 <- setdiff(group1, group2) print(only_group1) # [1] “Alice” # Members only in group2 only_group2 <- setdiff(group2, group1) print(only_group2) # [1] “David” Summary Union: union(A, B) — Combines all elements from both sets. Intersection: intersect(A, B) — Common elements between both sets. Difference: setdiff(A, B) — Elements in the first set but not in the second. Symmetric Difference: setxor(A, B) — Elements in either set but not in both. Subset Check: all(A %in% B) — Checks if all elements of A are in B.

Linear Algebra Operations on Vectors and Matrices in R Vectors Vector Addition and Subtraction Example:  # Define vectors v1 <- c(1, 2, 3) v2 <- c(4, 5, 6) # Addition v_sum <- v1 + v2 print(v_sum) # [1] 5 7 9 # Subtraction v_diff <- v1 – v2 print(v_diff) # [1] -3 -3 -3  Scalar Multiplication Example:  # Scalar scalar <- 2 # Scalar multiplication v_scaled <- scalar * v1 print(v_scaled) # [1] 2 4 6  Dot Product The dot product (or inner product) of two vectors. Example:  # Dot product dot_product <- sum(v1 * v2) print(dot_product) # [1] 32  Vector Norm The Euclidean norm (or magnitude) of a vector. Example:  # Vector norm norm_v1 <- sqrt(sum(v1^2)) print(norm_v1) # [1] 3.741657 Matrices Matrix Creation Example:  # Create a matrix m <- matrix(1:6, nrow = 2, ncol = 3) print(m) #      [,1] [,2] [,3] # [1,]    1    3    5 # [2,]    2    4    6 Matrix Addition and Subtraction Example:  # Define matrices m1 <- matrix(1:4, nrow = 2, ncol = 2) m2 <- matrix(5:8, nrow = 2, ncol = 2) # Addition m_sum <- m1 + m2 print(m_sum) #      [,1] [,2] # [1,]    6    8 # [2,]    8   10 # Subtraction m_diff <- m1 – m2 print(m_diff) #      [,1] [,2] # [1,]   -4   -4 # [2,]   -4   -4  Scalar Multiplication Example:  # Scalar scalar <- 3 # Scalar multiplication m_scaled <- scalar * m1 print(m_scaled) #      [,1] [,2] # [1,]    3    6 # [2,]    9   12 Matrix Multiplication Example:  # Define matrices m1 <- matrix(1:4, nrow = 2, ncol = 2) m2 <- matrix(5:8, nrow = 2, ncol = 2) # Matrix multiplication m_product <- m1 %*% m2 print(m_product) #      [,1] [,2] # [1,]   19   22 # [2,]   43   50 Transposition Example:  # Matrix transposition m_transposed <- t(m1) print(m_transposed) #      [,1] [,2] # [1,]    1    3 # [2,]    2    4 Determinant The determinant of a square matrix. Example:  # Define a square matrix m <- matrix(c(1, 2, 3, 4), nrow = 2, ncol = 2) # Calculate the determinant determinant <- det(m) print(determinant) # [1] -2  Inverse The inverse of a square matrix. Example:  # Define a square matrix m <- matrix(c(1, 2, 3, 4), nrow = 2, ncol = 2) # Calculate the inverse m_inverse <- solve(m) print(m_inverse) #       [,1] [,2] # [1,] -2.0  1.0 # [2,]  1.5 -0.5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors Example:  # Define a matrix m <- matrix(c(1, 2, 3, 4), nrow = 2, ncol = 2) # Compute eigenvalues and eigenvectors eigen_result <- eigen(m) print(eigen_result$values) # [1]  5.372281  -0.372281 print(eigen_result$vectors) #          [,1]      [,2] # [1,] -0.8245648  0.4159736 # [2,] -0.5657675 -0.9093769 Solving Linear Systems Solving a Linear System To solve Ax=bAx = bAx=b, where AAA is a matrix and bbb is a vector. Example:  # Define matrix A and vector b A <- matrix(c(2, 1, -1, 1, 3, 2, 1, 1, 4), nrow = 3, byrow = TRUE) b <- c(8, 13, 11) # Solve for x x <- solve(A, b) print(x) # [1] 2 3 1 Singular Value Decomposition (SVD) Performing SVD Example:  # Define a matrix m <- matrix(c(1, 2, 3, 4, 5, 6), nrow = 2, ncol = 3) # Perform Singular Value Decomposition svd_result <- svd(m) print(svd_result$u) #           [,1]       [,2] # [1,] -0.3879274 -0.9223658 # [2,] -0.9223658  0.3879274 print(svd_result$d) # [1] 9.508032 0.772869 print(svd_result$v) #           [,1]       [,2]       [,3] # [1,] -0.4286671  0.5663065 -0.7034620 # [2,] -0.5663065  0.8259630  0.0542291 Matrix Decomposition LU Decomposition Example:  # Load the Matrix package install.packages(“Matrix”) library(Matrix) # Define a matrix m <- matrix(c(4, 3, 6, 3, 4, 5, 2, 2, 3), nrow = 3, byrow = TRUE) # Perform LU Decomposition lu_decomposition <- lu(m) print(lu_decomposition$L) #       [,1] [,2] [,3] # [1,]  1.0  0.0  0.0 # [2,]  0.5  1.0  0.0 # [3,]  0.5  0.5  1.0 print(lu_decomposition$U) #       [,1] [,2] [,3] # [1,]  4.0  3.0  6.0 # [2,]  0.0  1.5  0.5 # [3,]  0.0  0.0  1.0 Summary Vectors: Addition, subtraction, scalar multiplication Dot product, vector norm Matrices: Creation, addition, subtraction, scalar multiplication Matrix multiplication, transposition Determinant, inverse Eigenvalues and Eigenvectors: Calculation of eigenvalues and eigenvectors Solving Linear Systems: Solving equations of the form Ax=bAx = bAx=b Singular Value Decomposition (SVD): Performing SVD on matrices Matrix Decomposition: LU Decomposition

Vector Sorting in R Sorting vectors in R is a common operation that allows you to rearrange the elements of a vector in ascending or descending order. Here are the main functions and techniques for sorting vectors in R. sort() Function The sort() function is used to sort a vector in ascending or descending order. Ascending Order Example:  # Vector to sort vec <- c(5, 2, 9, 1, 7) # Ascending order sorted_vec <- sort(vec) print(sorted_vec) # [1] 1 2 5 7 9  Descending Order Example:  # Descending order sorted_vec_desc <- sort(vec, decreasing = TRUE) print(sorted_vec_desc) # [1] 9 7 5 2 1 order() Function The order() function returns the indices that would sort a vector, which can be used to sort other vectors consistently. Sorting a Vector Using order() Example:  # Vector to sort vec <- c(5, 2, 9, 1, 7) # Get sorting indices indices <- order(vec) print(indices) # [1] 4 2 5 1 3 # Sort the vector using indices sorted_vec <- vec[indices] print(sorted_vec) # [1] 1 2 5 7 9  Sorting Multiple Vectors Simultaneously Example:  # Vectors to sort vec1 <- c(5, 2, 9, 1, 7) vec2 <- c(‘E’, ‘B’, ‘D’, ‘A’, ‘C’) # Get sorting indices for vec1 indices <- order(vec1) # Sort vec1 and vec2 using the indices sorted_vec1 <- vec1[indices] sorted_vec2 <- vec2[indices] print(sorted_vec1) # [1] 1 2 5 7 9 print(sorted_vec2) # [1] “A” “B” “C” “E” “D” rank() Function The rank() function assigns ranks to each element of the vector, showing the order of elements without actually sorting them. Example:  # Vector to rank vec <- c(5, 2, 9, 1, 7) # Get ranks ranks <- rank(vec) print(ranks) # [1] 4 2 5 1 3 Sorting Factor Vectors For factors, sorting is generally done based on the levels of the factors. Example:  # Factor vector fact <- factor(c(‘red’, ‘blue’, ‘green’, ‘blue’, ‘red’)) # Sorting factor levels sorted_fact <- sort(fact) print(sorted_fact) # [1] blue blue green red red # Levels: blue green red Advanced Sorting with order() Sorting by Multiple Criteria You can sort by multiple vectors or criteria. Example:  # Vectors to sort vec1 <- c(5, 2, 9, 1, 7) vec2 <- c(10, 20, 10, 30, 20) # Get indices sorted by vec1, then by vec2 indices <- order(vec1, vec2) # Sort the vectors sorted_vec1 <- vec1[indices] sorted_vec2 <- vec2[indices] print(sorted_vec1) # [1] 1 2 5 7 9 print(sorted_vec2) # [1] 30 20 10 10 20 Sorting in data.frame For sorting columns in a data.frame, you can use order() or arrange() from the dplyr package. Using order() with data.frame Example:  # Create a data.frame df <- data.frame(Name = c(‘Alice’, ‘Bob’, ‘Charlie’, ‘David’),                  Score = c(85, 92, 78, 90)) # Sort the data.frame by Score df_sorted <- df[order(df$Score), ] print(df_sorted) #     Name Score # 3 Charlie    78 # 1   Alice    85 # 4   David    90 # 2     Bob    92 Using arrange() from the dplyr Package Installation and Loading:  install.packages(“dplyr”) library(dplyr) Example:  # Sort the data.frame by Score with dplyr df_sorted <- df %>% arrange(Score) print(df_sorted) #     Name Score # 3 Charlie    78 # 1   Alice    85 # 4   David    90 # 2     Bob    92 Summary sort(): Sorts vectors in ascending or descending order. order(): Returns indices to sort vectors and allows simultaneous sorting of multiple vectors. rank(): Assigns ranks to elements of a vector. Factor Sorting: Sorts levels of factor vectors. Advanced Sorting with order(): Sorts based on multiple criteria. Sorting data.frame: Uses order() or arrange() from dplyr for sorting columns.

Functions for Statistical Distributions in R R provides a comprehensive set of functions for working with statistical distributions. These functions allow you to compute densities, probabilities, quantiles, and generate random numbers from various distributions. Normal Distributions dnorm() Function Returns the density of the normal distribution for a given value. Example:  # Density of the normal distribution at x = 0 density <- dnorm(0) print(density) # [1] 0.3989423  pnorm() Function Returns the cumulative distribution function (CDF) value for a given point. Example:  # CDF value for x = 1 cdf <- pnorm(1) print(cdf) # [1] 0.8413447  qnorm() Function Returns the quantile corresponding to a given probability. Example:  # Quantile for a probability of 0.95 quantile <- qnorm(0.95) print(quantile) # [1] 1.644854  rnorm() Function Generates random values from a normal distribution. Example:  # Generate 5 random values with mean = 0 and standard deviation = 1 random_values <- rnorm(5) print(random_values) # [1] -0.9785523  1.0217856  0.0638721 -0.2428278 -0.5025177 Binomial and Bernoulli Distributions dbinom() Function Returns the probability of a specific number of successes in a binomial distribution. Example:  # Probability of exactly 3 successes in 10 trials with a success probability of 0.5 prob <- dbinom(3, size = 10, prob = 0.5) print(prob) # [1] 0.1171875  pbinom() Function Returns the cumulative distribution function (CDF) value for a binomial distribution. Example:  # CDF value for up to 3 successes in 10 trials with a success probability of 0.5 cdf <- pbinom(3, size = 10, prob = 0.5) print(cdf) # [1] 0.2167969 qbinom() Function Returns the quantile corresponding to a given probability in a binomial distribution. Example:  # Quantile for a probability of 0.95 in a binomial distribution with 10 trials and success probability 0.5 quantile <- qbinom(0.95, size = 10, prob = 0.5) print(quantile) # [1] 8  rbinom() Function Generates random values from a binomial distribution. Example:  # Generate 5 random values from a binomial distribution with 10 trials and a success probability of 0.5 random_values <- rbinom(5, size = 10, prob = 0.5) print(random_values) # [1] 7 5 6 4 6 Poisson Distributions dpois() Function Returns the probability of a specific number of events in a Poisson distribution. Example:  # Probability of exactly 3 events with a mean rate of 2 prob <- dpois(3, lambda = 2) print(prob) # [1] 0.180447  ppois() Function Returns the cumulative distribution function (CDF) value for a Poisson distribution. Example:  # CDF value for up to 3 events with a mean rate of 2 cdf <- ppois(3, lambda = 2) print(cdf) # [1] 0.815263  qpois() Function Returns the quantile corresponding to a given probability in a Poisson distribution. Example:  # Quantile for a probability of 0.95 with a mean rate of 2 quantile <- qpois(0.95, lambda = 2) print(quantile) # [1] 4  rpois() Function Generates random values from a Poisson distribution. Example:  # Generate 5 random values with a mean rate of 2 random_values <- rpois(5, lambda = 2) print(random_values) # [1] 1 2 4 1 3 Exponential Distributions dexp() Function Returns the density of the exponential distribution for a given value. Example:  # Density of the exponential distribution at x = 1 with rate = 1 density <- dexp(1, rate = 1) print(density) # [1] 0.3678794  pexp() Function Returns the cumulative distribution function (CDF) value for an exponential distribution. Example:  # CDF value for x = 1 with rate = 1 cdf <- pexp(1, rate = 1) print(cdf) # [1] 0.6321206  qexp() Function Returns the quantile corresponding to a given probability in an exponential distribution. Example:  # Quantile for a probability of 0.95 with rate = 1 quantile <- qexp(0.95, rate = 1) print(quantile) # [1] 2.995732  rexp() Function Generates random values from an exponential distribution. Example:  # Generate 5 random values with rate = 1 random_values <- rexp(5, rate = 1) print(random_values) # [1] 0.4161032 0.2125973 0.5747574 1.2425372 0.3794684 Chi-Square Distributions dchisq() Function Returns the density of the chi-square distribution for a given value. Example:  # Density of the chi-square distribution at x = 5 with 2 degrees of freedom density <- dchisq(5, df = 2) print(density) # [1] 0.2650259  pchisq() Function Returns the cumulative distribution function (CDF) value for a chi-square distribution. Example:  # CDF value for x = 5 with 2 degrees of freedom cdf <- pchisq(5, df = 2) print(cdf) # [1] 0.878813  qchisq() Function Returns the quantile corresponding to a given probability in a chi-square distribution. Example:  # Quantile for a probability of 0.95 with 2 degrees of freedom quantile <- qchisq(0.95, df = 2) print(quantile) # [1] 7.378699 rchisq() Function Generates random values from a chi-square distribution. Exemple :  # Generate 5 random values with 2 degrees of freedom random_values <- rchisq(5, df = 2) print(random_values) # [1] 1.326664 3.046862 1.685939 3.128086 2.093674 Summary Normal Distributions: dnorm(), pnorm(), qnorm(), rnorm() Binomial Distributions: dbinom(), pbinom(), qbinom(), rbinom() Poisson Distributions: dpois(), ppois(), qpois(), rpois() Exponential Distributions: dexp(), pexp(), qexp(), rexp() Chi-Square Distributions: dchisq(), pchisq(), qchisq(), rchisq()

Numerical Integration and Differentiation in R Numerical Differentiation Numerical differentiation estimates the derivative of a function at a given point. This is useful when you can’t solve the derivative analytically. Using the pracma Package The pracma package provides functions for numerical differentiation. Installation and Loading:  install.packages(“pracma”) library(pracma) Example: Calculate the derivative of the function f(x)=x2−4x+4f(x) = x^2 – 4x + 4f(x)=x2−4x+4 at x=2x = 2x=2.  # Define the function f <- function(x) x^2 – 4*x + 4 # Calculate the derivative at x = 2 df <- grad(f, 2) print(df) # [1] 0 In this example, the gradient (or derivative) of the function f(x)=x2−4x+4f(x) = x^2 – 4x + 4f(x)=x2−4x+4 at x=2x = 2x=2 is 0, which corresponds to the minimum point of the quadratic function. Using Finite Differences If you prefer not to use additional packages, you can manually compute derivatives using finite differences. Example: Calculate the derivative using a small increment hhh.  # Define the function f <- function(x) x^2 – 4*x + 4 # Finite difference derivative function finite_diff <- function(f, x, h = 1e-5) {   (f(x + h) – f(x – h)) / (2 * h) } # Calculate the derivative at x = 2 df <- finite_diff(f, 2) print(df) # [1] 0 Numerical Integration Numerical integration approximates the integral of a function. This is useful for functions where an analytical integral is difficult or impossible to find. Using the integrate() Function R’s base integrate() function is used for numerical integration. Example: Integrate the function f(x)=x2f(x) = x^2f(x)=x2 from 0 to 1.  # Define the function f <- function(x) x^2 # Perform the integration result <- integrate(f, lower = 0, upper = 1) print(result$value) # [1] 0.3333333 The result here is 13\frac{1}{3}31​, which is the exact integral of x2x^2×2 from 0 to 1. Using the pracma Package The pracma package also provides numerical integration methods. Example: Using integral() from pracma:  # Define the function f <- function(x) x^2 # Perform the integration result <- integral(f, 0, 1) print(result) # [1] 0.3333333 Advanced Integration and Differentiation Techniques Symbolic Integration and Differentiation For symbolic integration and differentiation, use the SymPy package, which interfaces with Python’s SymPy library. Installation and Loading:  install.packages(“reticulate”) reticulate::py_install(“sympy”) library(reticulate) sympy <- import(“sympy”) Example: Perform symbolic differentiation and integration.  # Define the symbol x <- sympy$Symbol(‘x’) # Define the function f <- x^2 – 4*x + 4 # Differentiate df <- sympy$diff(f, x) print(df) # Integrate integral <- sympy$integrate(f, x) print(integral)  Practical Examples Example: Finding the Area under a Curve Integrate f(x)=e−x2f(x) = e^{-x^2}f(x)=e−x2 from 0 to 2.  # Define the function f <- function(x) exp(-x^2) # Perform the integration result <- integrate(f, lower = 0, upper = 2) print(result$value) # [1] 0.6826895 Example: Derivative of a Trigonometric Function Calculate the derivative of f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x) at x=π/4x = \pi/4x=π/4.  # Define the function f <- function(x) sin(x) # Calculate the derivative at x = pi/4 df <- finite_diff(f, pi/4) print(df) # [1] 0.7071068 Summary Numerical Differentiation: Use grad() from pracma or finite differences for approximating derivatives. Numerical Integration: Use integrate() for basic integration, or integral() from pracma for additional methods. Advanced Techniques: Use SymPy for symbolic differentiation and integration.

Minima and Maxima in R Introduction to Concepts In mathematics and programming, minima and maxima refer to the lowest and highest values in a dataset, respectively. These can be categorized into: Local Minima: Points where a function takes a value lower than its immediate neighbors. Local Maxima: Points where a function takes a value higher than its immediate neighbors. Global Minima: The lowest value over the entire domain of the function. Global Maxima: The highest value over the entire domain of the function. Basic Functions for Finding Minima and Maxima min() Function Returns the minimum value of a vector or matrix. Example:  # Example vector vec <- c(7, 2, 5, 8, 1, 9) # Find the minimum value min_val <- min(vec) print(min_val) # [1] 1 max() Function Returns the maximum value of a vector or matrix. Example:  # Example vector vec <- c(7, 2, 5, 8, 1, 9) # Find the maximum value max_val <- max(vec) print(max_val) # [1] 9 Finding the Indices of Minima and Maxima which.min() Function Returns the index of the first minimum in a vector. Example:  # Example vector vec <- c(7, 2, 5, 8, 1, 9) # Find the index of the minimum min_index <- which.min(vec) print(min_index) # [1] 5  which.max() Function Returns the index of the first maximum in a vector. Example:  # Example vector vec <- c(7, 2, 5, 8, 1, 9) # Find the index of the maximum max_index <- which.max(vec) print(max_index) # [1] 6 Finding Local Minima and Maxima For more complex functions or time series, you may need to find local minima and maxima, which basic functions min() and max() cannot directly handle. Using the pracma Package The pracma package provides tools for finding local extrema. Installation and Loading the Package:  install.packages(“pracma”) library(pracma) Example:  # Example function f <- function(x) x^3 – 6*x^2 + 9*x # Find local extrema in the interval [0, 4] x <- seq(0, 4, length.out = 100) y <- f(x) # Find local minima and maxima extrema <- findpeaks(y) print(extrema) Using the stats Package For continuous functions, you can use optimization methods. Example:  # Example function f <- function(x) x^2 – 4*x + 4 # Find the global minimum using `optimize` opt <- optimize(f, interval = c(0, 4)) print(opt$minimum) print(opt$objective) Minima and Maxima in Matrices Finding Minima and Maxima in a Matrix For matrices, min() and max() applied directly return the minimum and maximum values over the entire matrix. Example:  # Example matrix mat <- matrix(c(1, 2, 3, 4, 5, 6, 7, 8, 9), nrow = 3) # Find minima and maxima in the matrix min_mat <- min(mat) max_mat <- max(mat) print(min_mat) # [1] 1 print(max_mat) # [1] 9  Finding Minima and Maxima by Column or Row Example:  # Find the minimum by column min_col <- apply(mat, 2, min) print(min_col) # [1] 1 2 3 # Find the maximum by row max_row <- apply(mat, 1, max) print(max_row) # [1] 3 6 9 Practical Example: Finding Minima and Maxima of a Function Consider a quadratic function and find its minima and maxima. Example:  # Define the function f <- function(x) (x – 2)^2 + 1 # Find the global minimum opt <- optimize(f, interval = c(-10, 10)) print(paste(“Minima:”, opt$minimum)) print(paste(“Value at minima:”, opt$objective)) Summary Minima and Maxima: Use min() and max() for global values in vectors and matrices. Indices of Minima and Maxima: Use which.min() and which.max() for their indices. Local Minima and Maxima: Use packages like pracma or optimization techniques for complex functions. Matrices: Use apply() to find minima and maxima by column or row.

Cumulative Sums and Products Cumulative sums and products are powerful tools in data analysis that help in performing progressive calculations on vectors of data. In R, you can use the functions cumsum() and cumprod() for these operations. Cumulative Sum (cumsum()) The cumsum() function computes the cumulative sum of elements in a vector. For each element in the vector, it returns the sum of all previous elements up to that element. Syntax:  cumsum(x) Example:  # Example vector vec <- c(2, 3, 5, 7, 11) # Calculate cumulative sum cumulative_sum <- cumsum(vec) print(cumulative_sum) # Result: 2, 5, 10, 17, 28 Explanation: For the first element (2), the cumulative sum is 2. For the second element (3), the cumulative sum is 2 + 3 = 5. For the third element (5), the cumulative sum is 5 + 5 = 10, and so on. Cumulative Product (cumprod()) The cumprod() function computes the cumulative product of elements in a vector. For each element in the vector, it returns the product of all previous elements up to that element. Syntax:  cumprod(x) Example:  # Example vector vec <- c(2, 3, 5, 7, 11) # Calculate cumulative product cumulative_product <- cumprod(vec) print(cumulative_product) # Result: 2, 6, 30, 210, 2310 Explanation: For the first element (2), the cumulative product is 2. For the second element (3), the cumulative product is 2 * 3 = 6. For the third element (5), the cumulative product is 6 * 5 = 30, and so on. Practical Applications Calculating Total Monthly Sales Suppose you have monthly sales data and want to know the cumulative total up to each month.  # Monthly sales sales <- c(100, 150, 200, 250, 300) # Calculate cumulative sales total_sales <- cumsum(sales) print(total_sales) # Result: 100, 250, 450, 700, 1000 Compound Interest Calculation Suppose you are calculating compound interest with varying monthly rates.  # Monthly interest rates interest_rates <- c(1.01, 1.02, 1.015, 1.01, 1.03) # Calculate cumulative product for interest cumulative_interest <- cumprod(interest_rates) print(cumulative_interest) # Result: 1.01, 1.0302, 1.0456303, 1.0560798, 1.0890712 Financial Data Analysis You can use these functions to analyze financial data, such as cumulative revenue or expenses over time. Visualization You can also visualize cumulative sums and products using plots. Example Visualization:  library(ggplot2) # Data x <- 1:5 sales <- c(100, 150, 200, 250, 300) cumulative_sales <- cumsum(sales) # Create a dataframe for ggplot data <- data.frame(Month=x, Sales=sales, Cumulative=cumulative_sales) # Plot ggplot(data, aes(x=Month)) + geom_line(aes(y=Cumulative, color=”Cumulative”), size=1) + geom_point(aes(y=Cumulative, color=”Cumulative”)) + labs(title=”Cumulative Sales”, x=”Month”, y=”Cumulative Total”) + theme_minimal() Explanation: This code generates a plot showing the progression of cumulative sales over the months. Using cumsum() and cumprod(), you can easily perform progressive calculations and cumulative analysis for various types of data, providing valuable insights into trends and accumulations in your datasets.

Math Functions in R R provides a comprehensive set of mathematical functions to perform basic and advanced calculations. Here’s a detailed look at some common functions. Basic Arithmetic Functions Addition, Subtraction, Multiplication, and Division Addition (+)  5 + 3  # Result: 8 Subtraction (–)  10 – 4  # Result: 6 Multiplication (*)  6 * 7  # Result: 42 Division (/)  15 / 3  # Result: 5 Exponentiation (^ or **) Exponentiation  2^3  # Result: 8 4**0.5  # Result: 2 (square root of 4) Mathematical Functions Square Root (sqrt()) Description: Calculates the square root of a number. Syntax:  sqrt(x) Example:  sqrt(16)  # Result: 4 sqrt(25)  # Result: 5  Logarithms (log(), log10(), log2()) Description: Computes logarithms, natural base (log()), base 10 (log10()), base 2 (log2()). Syntax:  log(x, base) log10(x) log2(x) Examples:  log(10)  # Natural logarithm of 10 log10(100)  # Base 10 logarithm of 100 log2(32)  # Base 2 logarithm of 32 Exponential (exp()) Description: Computes the exponential of a number. Syntax:  exp(x) Example:  exp(1)  # Result: 2.7182818 (e) exp(2)  # Result: 7.389056  Trigonometric Functions (sin(), cos(), tan()) Description: Calculates trigonometric functions for angles in radians. Syntax:  sin(x) cos(x) tan(x) Examples:  sin(pi/2)  # Result: 1 cos(pi)  # Result: -1 tan(pi/4)  # Result: 1 Hyperbolic Functions (sinh(), cosh(), tanh()) Description: Computes hyperbolic functions. Syntax:  sinh(x) cosh(x) tanh(x) Examples:  sinh(1)  # Result: 1.175201 cosh(1)  # Result: 1.543081 tanh(1)  # Result: 0.761594 Absolute Value (abs()) Description: Computes the absolute value of a number. Syntax:  abs(x) Example:  abs(-5)  # Result: 5 abs(3)   # Result: 3 Ceiling (ceiling()) and Floor (floor()) Description: Rounds up to the nearest integer (ceiling()) or down to the nearest integer (floor()). Syntax:  ceiling(x) floor(x) Examples:  ceiling(2.3)  # Result: 3 floor(2.7)    # Result: 2 Rounding (round(), trunc(), signif()) Description: Rounds numbers to a specified number of significant digits. Syntax:  round(x, digits) trunc(x) signif(x, digits) Examples:  round(3.14159, 2)  # Result: 3.14 trunc(3.14159)    # Result: 3 signif(3.14159, 3)  # Result: 3.14 Complex Numbers (Re(), Im(), Mod(), Arg()) Description: Manipulates complex numbers. Syntax:  Re(z) Im(z) Mod(z) Arg(z) Example:  z <- 3 + 4i Re(z)  # Result: 3 Im(z)  # Result: 4 Mod(z)  # Result: 5 Arg(z)  # Result: 0.9272952 (radians) Practical Examples and Exercises Solve a Quadratic Equation Example: Solve the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.  a <- 1 b <- -3 c <- 2 # Calculate roots discriminant <- b^2 – 4*a*c root1 <- (-b + sqrt(discriminant)) / (2*a) root2 <- (-b – sqrt(discriminant)) / (2*a) root1  # Result: 2 root2  # Result: 1 Calculate the Area of a Circle Example: Use the pi function to calculate the area of a circle with radius 5.  radius <- 5 area <- pi * radius^2 area  # Result: 78.53982 Convert Between Degrees and Radians Example: Convert 90 degrees to radians.  degrees <- 90 radians <- degrees * pi / 180 radians  # Result: 1.570796 Calculate the Compound Interest Example: Calculate the compound interest for a principal of $1000 with an annual interest rate of 5% over 3 years.  principal <- 1000 rate <- 0.05 time <- 3 compound_interest <- principal * (1 + rate)^time – principal compound_interest  # Result: 157.625