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#
# Examples from the forthcoming Monograph:
# Rmetrics - Financial Engineering and Computational Finance
# written by Diethelm Wuertz
# ISBN to be published
#
# Details:
# Chapter 4.6
# Matrix and Linear Algebra Addon:
#
# List of Examples, Exercises and Code Snippets:
#
# Example: 1 Overview of Acvailable Functions
# Example: 2 Generation of Matrices
# Example: 3 Matrices and Linear Algebra
# Example: 4 Basic Statistics
# Example: 5 Column and Row Vectors
# Example: 6 Stack Operations and Matrix Products
# Example: 7 Missing Values in a Data Matrix
#
# *** This list is not yet complete ***
#
#
# Author:
# (C) 2002-2005, Diethelm Wuertz, GPL
# www.rmetrics.org
# www.itp.phys.ethz.ch
# www.finance.ch
#
################################################################################
### Example: 1 Overview of Available Functions
# GENERATION: DESCRIPTION:
# matrix R creates a matrix from the given set of values
# diag R creates a diagonal matrix or extracts diagonals
# triang S extracs the lower tridiagonal part from a matrix
# Triang S extracs the upper tridiagonal part from a matrix
# pascal S creates a pascal matrix
# colVec S creates a column vector from a vector
# rowVec S creates a row vector from a vector
# as.matrix R attempts to turn its argument into a matrix
# is.matrix R tests if its argument is a (strict) matrix
# dimnames R retrieves or sets the dimnames of an object
# colnames|rownames R retrieves or sets the row or column names
# colIds|rowIds S ... use alternatively
###
# SUBSETS: DESCRIPTION:
# dim R returns the dimension of a matrix object
# ncol|nrow R counts columns|rows of a matrix object
# colNums|rowNums S ... use alternatively
# length R counts elements of a matrix object
# "["|"[[" R subsets a matrix object
# (Arith) R Elementwise Arithmetic: + - * /
# (Lops) R Elementwise logical Ops: > < >= <= == !=
# cbind|rbind R augments a matrix object by columns|rows
# na.omit R removes NA from a matrix object
###
# BASIC STATISTICS: DESCRIPTION:
# var returns the variance matrix
# cov returns the covariance matrix
# col|rowStats * calculates column|row statistics
# col|rowMeans calculates column|row means
# col|rowAvgs * calculates column|row averages
# col|rowVars * calculates column|row variances
# col|rowStdevs * calculates column|row standard deviations
# col|rowSkewness * calculates column|row skewness
# col|rowKurtosis * calculates column|row kurtosis
# col|rowCumsums * calculates column|row cumulated sums
###
# LINEAR ALGEBRA: DESCRIPTION:
# det returns the determinante of a matrix
# norm returns the norm of a matrix
# rk * returns the rank of a matrix
# tr * returns trace of a matrix
# %*% R returns the product of two matrices
# %x%|kron R|S returns the Kronecker product
# t R returns the transposed matrix
# inv|chol2inv * returns the inverse of a matrix
###
# MORE LINEAR ALGEBRA: DESCRIPTION:
# chol R returns the Cholesky factor matrix
# eigen R returns eigenvalues and eigenvectors
# svd R returns the singular value decomposition
# qr R returns the QR decomposition of a matrix
# ginv R returns Moore-Penrose generalized inverse
# solve R solves a system of linear equations
# backsolve R ... use when the matrix is upper triangular
# forwardsolve R ... use when the matrix is lower triangular
###
# ------------------------------------------------------------------------------
### Example: 2 Generation of Matrices
# matrix -
# Create Matrixes from Scratch:
Q = matrix(1:16, 4); Q
G = matrix(1:16, 8, 2); G
R = matrix(rnorm(20), byrow = TRUE, ncol = 2); R
###
# Dimension Names:
R
colnames(R) = c("C1", "C2"); R
rownames(R) = as.character(1:nrow(R)); R
dimnames(R) = NULL; R
dimnames(R) = list(as.character(1:nrow(R)), c("C1", "C2")); R
colIds(R)
###
# Create Lower/Upper Triangle Matrices:
Triang(Q) # returns upper triangle as matrix
triang(Q) # returns lower triangle as matrix
Q[upper.tri(Q)] # returns upper triangle as vector
Q[lower.tri(Q)] # returns lower triangle as vector
###
# Create Diagonal Matrixes:
diag(3) # 3 x 3 Identity matrix
diag(1, 3) # ... another way to create it
diag(1:3, 3) # A diagonal matrix with elements 1,2,3
D = diag(c(1,2,6), 3); D # ... another diagonal atrix
###
# Create Pascal Matrixes:
P = pascal(3); P
###
# ------------------------------------------------------------------------------
### Example: 3 Matrices and Linear Algebra
# Add/Subtract/Multiply/Divide a Constant or Matrix:
X + 3 # adds a constant to a matrix
3 - X # subtracts from a constant
X * 0.5 # Multiplies with a constant
X / 6 # devides by constant
6 / X # devides a constant by a matrix
X + D # adds two matrices element by element
D - X # subtracts two matrices element by element
X * D # subtracts two matrices element by element
D / X # devides two matrices element by element
X %*% diag(1:3) # multiplies rows of a matrix by a vector
diag(v) %*% X # multiplies columns of a matrix by a vector
###
# Operate on Subsets of a Matrix:
n = 3; i = 2; j = 3
dim(X) # returns the dimension of a matrix
ncol(X) # counts columns of a matrix
nrow(X) # counts rows of a matrix
length(X) # counts elements of a matrix
X[, ncol(X)] # the last colum of a matrix
X[nrow(X), ] # the last row of a matrix
X[, -i] # deletes a column of a matrix
X[-i, ] # deletes a row of a matrix
X[, i] + X[, j] # adds two columns of a matrix
X[i, ] - X[j, ] # subtracts two rows of a matrix
X[c(3, 1, 2), ] # permutes the rows
X[, c(3, 1, 2)] # permutes the coliumns
cbind(X, 1:3) # augments horizontally
rbind(X, D) # augments vertically
X[, c(1,1,1)] # repeats first column 3 times
X[rep(1:3, rep(2, 3)), ] # replicates every other line
diag(X) # extracs diagonal as vector
###
# Apply a function to all Elements of a Matrix:
sqrt(X); X^1/2 # returns square root for each element
abs(-X); sign(R) # returns abs|sign root for each element
min(X); max(X) # returns min|max root for each element
cos(X); sin(X); tan(X) # returns trigonometric function elementwise
acos(X); asin(X); atan(X) # returns inverse trigonometric functions
exp(X); 10^X # returns the exponential|power functions
log(X); log10(X) # returns logarithmic function elementwise
apply(x, 2, "min") # minimum value of each column
apply(x, 1, "max") # maximum value of each row
all( R > 0 ) # tests on all elements of a matrix
any( R > 0 ) # tests on any element in a matrix
###
# More Matrix Operations:
X %*% D # returns the product of two matrices
X %x% D # returns the Kronecker Product
t(X) # returns the transposed matrix
inv(X) # returns the inverse of a matrix
norm(X) # returns the norm of a matrix
det(X) # returns the determinante of a matrix
rk(X) # returns the rank of a matrix
tr(X) # returns trace of a matrix
var(X) # returns the variance matrix
cov(X) # returns the covariance matrix
###
# More Linear Algebra:
b = c(1, 2, 3)
chol(X) # returnsns the Cholesky factor matrix
eigen(X) # returns eigenvalues and eigenvectors
svd(X) # returns the singular value decomposition
kappa(X) # returns the condition number
qr(X) # returns the QR decomposition of a matrix
ginv(X) # returns Moore-Penrose generalized inverse
solve(X, b) # solves a system of linear equations
backsolve(R) # ... use when the matrix is upper triangular
forwardsolve(L) # ... use when the matrix is upper triangular
###
# ------------------------------------------------------------------------------
### Example: 4 Basic Statistics
# Create matrix:
ts = matrix(rnorm(100), 50)
var(ts)
cov(ts)
###
# col* -
# Column Operations:
colMeans(ts)
###
# ------------------------------------------------------------------------------
### Example: 5 Column and Row Vectors
# Define and use vectors:
u = c(3, 1, 4)
v = c(2, 0, -1)
u * v; v * u
u %*% v; v %*% u
# Convert to column and row vectors:
U = colVec(u); U
V = rowVec(v); V
# Dot Product or Inner Product:
# Row x Col == Scalar
V %*% U
# Outer Product:
# Col x Row == Matrix
U %*% V
# Matrix * Column Vector and Row Vector * Matrix
X = matrix(rep(1, 9), 3)
X %*% U
V %*% X
###
# ------------------------------------------------------------------------------
### Example: 6 Stack Operations and Matrix Products
# Show, that the following relation holds:
# vec ( A %*% B %*% C ) = ( t(C) %x% A ) %*% vec(B)
###
# Three random Matrixes:
A = matrix(rnorm(9), 3)
B = matrix(rnorm(9), 3)
C = matrix(rnorm(9), 3)
###
# Print vech:
A
vech(A)
###
# Print vec:
B
vec(B)
###
# Vector Product:
A %*% B
###
# Kronecker Product:
A %x% B
# Left Hand Side:
LHS = vec ( A %*% B %*% C )
# Right Hand Side:
# %x# denotes the Kronecker Product
RHS = ( t(C) %x% A ) %*% vec(B)
# Test the Difference:
data.frame(LHS, RHS, LHS-RHS)
###
# ------------------------------------------------------------------------------
### Example: 7 Missing Values in a Data Matrix
# Write R functions which remove, substitute, interpolate and
# impute missing values in a matrix object:
# 1. removeNA removes NAs from a matrix
# object
# 2. subtituteNA substitutes NAs by zeroes, the column
# mean or column median
# 3. interpNA interpolate NAs using R's "approx"
# function
# 4. knnNA imputes NAs by the knn-Algorithm using R's
# contributed function "knn" from the "EMV" Package
# Notes:
# We didn't take care, that NAs at the border of a matrix are
# properly considered. That has still to be done!
# The R functions can be found in "funSeries.R"
###
# Generate a matrix with missing values:
M = matrix(round(rnorm(40), 2), ncol = 5)
M[2:3, 1] = M[4, 3:4] = c(NA, NA)
M[7, 1] = NA
colnames(M) = c("a", "b", "c", "d", "e")
rownames(M) = paste("R", as.character(1:8), sep = "")
M
###
# Remove rows with missing values from M:
M
removeNA(M)
# Remove Columns with NAs from M:
t(removeNA(t(M)))
###
# Substitute missing values with Zeros, the Median and Mean:
M
substituteNA(M)
substituteNA(M, "median")
round(substituteNA(M, "mean"), 2)
###
# Interpolate by Columns:
M
interpNA(M, "before")
interpNA(M, "after")
interpNA(M, "linear")
# By Rows:
M
t(interpNA(t(M), "before"))
t(interpNA(t(M), "after"))
t(interpNA(t(M), "linear"))
###
# Use "knn" Algorithm:
knnNA(M, k=2)
round(knnNA(M, k = 2, correlation = TRUE), 2)
# Apply it to the transposed matrix:
t(round(knnNA(t(M), k = 2), 2))
t(round(knnNA(t(M), k = 2, correlation = TRUE), 2))
###
################################################################################
|
