|
Task
|
MATLAB/Octave
|
Python
NumPy
|
R
|
Julia
|
Task
|
|
CREATING
MATRICES
|
|
Creating
Matrices
(here: 3x3 matrix)
|
M>
A = [1 2 3; 4 5 6; 7 8 9]
A =
1 2
3
4 5 6
7 8
9
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
A
array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
|
R>
A = matrix(c(1,2,3,4,5,6,7,8,9),nrow=3,byrow=T)
#
equivalent to
# A = matrix(1:9,nrow=3,byrow=T)
R>
A
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
[3,] 7 8 9
|
J>
A=[1 2 3; 4 5 6; 7 8 9]
3x3 Array{Int64,2}:
1 2 3
4 5 6
7
8 9
|
Creating
Matrices
(here: 3x3 matrix)
|
|
Creating
an 1D column vector
|
M>
a = [1; 2; 3]
a =
1
2
3
|
P>
a
=
np.array([1,2,3]).reshape(1,3)
P>
b.shape
(1,
3)
|
R>
a = matrix(c(1,2,3), nrow=3, byrow=T)
R>
a
[,1]
[1,] 1
[2,] 2
[3,] 3
|
J>
a=[1; 2; 3]
3-element Array{Int64,1}:
1
2
3
|
Creating
an 1D column vector
|
|
Creating
an
1D row vector
|
M>
b = [1 2 3]
b =
1 2 3
|
P>
b = np.array([1,2,3])
P>
b
array([1, 2, 3])
#
note that numpy doesn't have
# explicit “row-vectors”, but
1-D
# arrays
P>
b.shape
(3,)
|
R>
b = matrix(c(1,2,3), ncol=3)
R>
b
[,1] [,2] [,3]
[1,] 1 2 3
|
J>
b=[1 2 3]
1x3 Array{Int64,2}:
1 2 3
# note that this
is a 2D array.
# vectors in Julia are columns
|
Creating
an
1D row vector
|
|
Creating
a
random m x n matrix
|
M>
rand(3,2)
ans =
0.21977 0.10220
0.38959 0.69911
0.15624 0.65637
|
P>
np.random.rand(3,2)
array([[ 0.29347865, 0.17920462],
[ 0.51615758, 0.64593471],
[ 0.01067605, 0.09692771]])
|
R>
matrix(runif(3*2), ncol=2)
[,1] [,2]
[1,] 0.5675127
0.7751204
[2,] 0.3439412 0.5261893
[3,] 0.2273177 0.223438
|
J>
rand(3,2)
3x2 Array{Float64,2}:
0.36882 0.267725
0.571856
0.601524
0.848084 0.858935
|
Creating
a
random m x n matrix
|
|
Creating
a
zero m x n matrix
|
M>
zeros(3,2)
ans =
0 0
0
0
0 0
|
P>
np.zeros((3,2))
array([[ 0., 0.],
[ 0., 0.],
[ 0.,
0.]])
|
R>
mat.or.vec(3, 2)
[,1] [,2]
[1,] 0 0
[2,] 0 0
[3,] 0 0
|
J>
zeros(3,2)
3x2 Array{Float64,2}:
0.0 0.0
0.0 0.0
0.0
0.0
|
Creating
a
zero m x n matrix
|
|
Creating
an
m x n matrix of ones
|
M>
ones(3,2)
ans =
1 1
1
1
1 1
|
P>
np.ones((3,2))
array([[ 1., 1.],
[ 1., 1.],
[ 1., 1.]])
|
R>
mat.or.vec(3, 2) + 1
[,1] [,2]
[1,] 1 1
[2,] 1 1
[3,]
1 1
|
J>
ones(3,2)
3x2 Array{Float64,2}:
1.0 1.0
1.0 1.0
1.0
1.0
|
Creating
an
m x n matrix of ones
|
|
Creating
an
identity matrix
|
M>
eye(3)
ans =
Diagonal Matrix
1 0
0
0 1 0
0 0
1
|
P>
np.eye(3)
array([[ 1., 0., 0.],
[ 0., 1., 0.],
[
0., 0., 1.]])
|
R>
diag(3)
[,1] [,2] [,3]
[1,] 1 0 0
[2,] 0 1 0
[3,] 0 0
1
|
J>
eye(3)
3x3 Array{Float64,2}:
1.0 0.0 0.0
0.0 1.0 0.0
0.0
0.0 1.0
|
Creating
an
identity matrix
|
|
Creating
a
diagonal matrix
|
M>
a = [1 2 3]
M>
diag(a)
ans =
Diagonal Matrix
1 0
0
0 2 0
0 0
3
|
P>
a = np.array([1,2,3])
P>
np.diag(a)
array([[1, 0, 0],
[0,
2, 0],
[0, 0, 3]])
|
R>
diag(1:3)
[,1] [,2] [,3]
[1,] 1 0 0
[2,] 0 2 0
[3,] 0
0 3
|
J>
a=[1, 2, 3]
# added commas because julia
# vectors are
columnar
J>
diagm(a)
3x3 Array{Int64,2}:
1 0 0
0 2 0
0 0 3
|
Creating
a
diagonal matrix
|
|
ACCESSING
MATRIX ELEMENTS
|
|
Getting
the dimension
of a matrix
(here: 2D, rows x cols)
|
M>
A = [1 2 3; 4 5 6]
A =
1 2 3
4 5 6
M>
size(A)
ans =
2 3
|
P>
A = np.array([ [1,2,3], [4,5,6] ])
P>
A
array([[1, 2, 3],
[4, 5,
6]])
P>
A.shape
(2, 3)
|
R>
A = matrix(1:6,nrow=2,byrow=T)
R>
A
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
R>
dim(A)
[1] 2 3
|
J>
A=[1 2 3; 4 5 6]
2x3 Array{Int64,2}:
1 2 3
4 5 6
J>
size(A)
(2,3)
|
Getting
the dimension
of a matrix
(here: 2D, rows x cols)
|
|
Selecting
rows
|
M>
A = [1 2 3; 4 5 6; 7 8 9]
% 1st row
M>
A(1,:)
ans =
1 2 3
% 1st 2
rows
M>
A(1:2,:)
ans =
1 2 3
4 5 6
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
# 1st row
P>
A[0,:]
array([1, 2, 3])
# 1st 2 rows
P>
A[0:2,:]
array([[1, 2, 3], [4, 5, 6]])
|
R>
A = matrix(1:9,nrow=3,byrow=T)
# 1st row
R>
A[1,]
[1] 1 2 3
# 1st 2 rows
R>
A[1:2,]
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
#semicolon suppresses output
#1st
row
J>
A[1,:]
1x3 Array{Int64,2}:
1 2 3
#1st 2 rows
J>
A[1:2,:]
2x3 Array{Int64,2}:
1 2 3
4 5 6
|
Selecting
rows
|
|
Selecting
columns
|
M>
A = [1 2 3; 4 5 6; 7 8 9]
% 1st column
M>
A(:,1)
ans =
1
4
7
% 1st 2 columns
M>
A(:,1:2)
ans =
1 2
4
5
7 8
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
# 1st column
(as row vector)
P>
A[:,0]
array([1, 4, 7])
# 1st column (as column
vector)
P>
A[:,[0]]
array([[1],
[4],
[7]])
# 1st 2 columns
P>
A[:,0:2]
array([[1, 2],
[4,
5],
[7, 8]])
|
R>
A = matrix(1:9,nrow=3,byrow=T)
# 1st column as row
vector
R>
t(A[,1])
[,1] [,2] [,3]
[1,] 1 4 7
# 1st column
as column vector
R>
A[,1]
[1] 1 4 7
# 1st 2 columns
R>
A[,1:2]
[,1] [,2]
[1,] 1 2
[2,] 4 5
[3,] 7 8
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
#1st column
J>
A[:,1]
3-element Array{Int64,1}:
1
4
7
#1st 2
columns
J>
A[:,1:2]
3x2 Array{Int64,2}:
1 2
4 5
7 8
|
Selecting
columns
|
|
Extracting
rows and columns by criteria
(here: get rows that have
value 9 in column 3)
|
M>
A = [1 2 3; 4 5 9; 7 8 9]
A =
1 2
3
4 5 9
7 8
9
M>
A(A(:,3) == 9,:)
ans =
4 5 9
7 8 9
|
P>
A = np.array([ [1,2,3], [4,5,9], [7,8,9]])
P>
A
array([[1, 2, 3],
[4, 5, 9],
[7, 8, 9]])
P>
A[A[:,2] == 9]
array([[4, 5, 9],
[7, 8, 9]])
|
R>
A = matrix(1:9,nrow=3,byrow=T)
R>
A
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 9
[3,] 7 8
9
R>
matrix(A[A[,3]==9], ncol=3)
[,1] [,2] [,3]
[1,] 4 5 9
[2,]
7 8 9
|
J>
A=[1 2 3; 4 5 9; 7 8 9]
3x3 Array{Int64,2}:
1 2 3
4 5 9
7
8 9
# use '.==' for
# element-wise check
J>
A[ A[:,3] .==9, :]
2x3 Array{Int64,2}:
4 5 9
7 8 9
|
Extracting
rows and columns by criteria
(here: get rows that have
value 9 in column 3)
|
|
Accessing
elements
(here: 1st element)
|
M>
A = [1 2 3; 4 5 6; 7 8 9]
M>
A(1,1)
ans = 1
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
A[0,0]
1
|
R>
A = matrix(c(1,2,3,4,5,9,7,8,9),nrow=3,byrow=T)
R>
A[1,1]
[1] 1
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
J>
A[1,1]
1
|
Accessing
elements
(here: 1st element)
|
|
MANIPULATING
SHAPE AND DIMENSIONS
|
|
Converting
row
to column vectors
|
M>
b = [1 2 3]
M>
b = b'
b =
1
2
3
|
P>
b = np.array([1, 2, 3])
P>
b = b[np.newaxis].T
# alternatively
# b =
b[:,np.newaxis]
P>
b
array([[1],
[2],
[3]])
|
R>
b = matrix(c(1,2,3), ncol=3)
R>
t(b)
[,1]
[1,] 1
[2,] 2
[3,] 3
|
J>
b=vec([1 2 3])
3-element Array{Int64,1}:
1
2
3
|
Converting
row
to column vectors
|
|
Reshaping
Matrices
(here: 3x3 matrix to row vector)
|
M>
A = [1 2 3; 4 5 6; 7 8 9]
A =
1 2
3
4 5 6
7 8
9
M>
total_elements = numel(A)
M>
B = reshape(A,1,total_elements)
% or reshape(A,1,9)
B
=
1 4 7 2 5 8
3 6 9
|
P>
A = np.array([[1,2,3],[4,5,6],[7,8,9]])
P>
A
array([[1, 2, 3],
[4, 5, 9],
[7, 8, 9]])
P>
total_elements = np.prod(A.shape)
P>
B = A.reshape(1, total_elements)
# alternative
shortcut:
# A.reshape(1,-1)
P>
B
array([[1, 2, 3, 4, 5, 6, 7, 8, 9]])
|
R>
A = matrix(1:9,nrow=3,byrow=T)
R>
A
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
[3,] 7 8 9
R>
total_elements = dim(A)[1] * dim(A)[2]
R>
B = matrix(A, ncol=total_elements)
R>
B
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] 1 4 7 2
5 8 3 6 9
|
J>
A=[1 2 3; 4 5 6; 7 8 9]
3x3 Array{Int64,2}:
1 2 3
4 5 6
7
8 9
J>
total_elements=length(A)
9
J>B=reshape(A,1,total_elements)
1x9
Array{Int64,2}:
1 4 7 2 5 8 3 6 9
|
Reshaping
Matrices
(here: 3x3 matrix to row vector)
|
|
Concatenating
matrices
|
M>
A = [1 2 3; 4 5 6]
M>
B = [7 8 9; 10 11 12]
M>
C = [A; B]
1 2 3
4 5 6
7
8 9
10 11 12
|
P>
A = np.array([[1, 2, 3], [4, 5, 6]])
P>
B = np.array([[7, 8, 9],[10,11,12]])
P>
C = np.concatenate((A, B), axis=0)
P>
C
array([[ 1, 2, 3],
[ 4,
5, 6],
[ 7, 8, 9],
[10, 11, 12]])
|
R>
A = matrix(1:6,nrow=2,byrow=T)
R>
B = matrix(7:12,nrow=2,byrow=T)
R>
C = rbind(A,B)
R>
C
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
[3,] 7 8 9
[4,]
10 11 12
|
J>
A=[1 2 3; 4 5 6];
J>
B=[7 8 9; 10 11 12];
J>
C=[A; B]
4x3 Array{Int64,2}:
1 2 3
4 5 6
7 8 9
10
11 12
|
Concatenating
matrices
|
|
Stacking
vectors
and matrices
|
M>
a = [1 2 3]
M>
b = [4 5 6]
M>
c = [a' b']
c =
1 4
2
5
3 6
M>
c = [a; b]
c =
1 2 3
4 5 6
|
P>
a = np.array([1,2,3])
P>
b = np.array([4,5,6])
P>
np.c_[a,b]
array([[1, 4],
[2,
5],
[3, 6]])
P>
np.r_[a,b]
array([[1, 2, 3],
[4,
5, 6]])
|
R>
a = matrix(1:3, ncol=3)
R>
b = matrix(4:6, ncol=3)
R>
matrix(rbind(A, B), ncol=2)
[,1] [,2]
[1,] 1 5
[2,] 4
3
R>
rbind(A,B)
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
|
J>
a=[1 2 3];
J>
b=[4 5 6];
J>
c=[a' b']
3x2 Array{Int64,2}:
1 4
2 5
3 6
J>
c=[a; b]
2x3 Array{Int64,2}:
1 2 3
4 5 6
|
Stacking
vectors
and matrices
|
|
BASIC
MATRIX OPERATIONS
|
|
Matrix-scalar
operations
|
M> A
= [1 2 3; 4 5 6; 7 8 9]
M>
A * 2
ans =
2 4 6
8 10 12
14 16
18
M>
A + 2
M>
A - 2
M>
A / 2
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
A * 2
array([[ 2, 4, 6],
[ 8, 10, 12],
[14, 16, 18]])
P>
A + 2
P>
A - 2
P>
A / 2
#
Note that NumPy was optimized for
# in-place assignments
#
e.g., A += A instead of
# A = A + A
|
R>
A = matrix(1:9, nrow=3, byrow=T)
R>
A * 2
[,1] [,2] [,3]
[1,] 2 4 6
[2,] 8 10 12
[3,] 14
16 18
R>
A + 2
R>
A - 2
R>
A / 2
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
# elementwise operator
J>
A .* 2
3x3 Array{Int64,2}:
2 4 6
8 10 12
14 16 18
J>
A .+ 2;
J>
A .- 2;
J>
A ./ 2;
|
Matrix-scalar
operations
|
|
Matrix-matrix
multiplication
|
M> A
= [1 2 3; 4 5 6; 7 8 9]
M>
A * A
ans =
30 36
42
66 81 96
102 126 150
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
np.dot(A,A) # or A.dot(A)
array([[ 30, 36, 42],
[ 66, 81, 96],
[102, 126, 150]])
|
R>
A = matrix(1:9, nrow=3, byrow=T)
R>
A %*% A
[,1] [,2] [,3]
[1,] 30 36 42
[2,] 66 81 96
[3,]
102 126 150
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
J>
A * A
3x3 Array{Int64,2}:
30 36 42
66 81 96
102 126
150
|
Matrix-matrix
multiplication
|
|
Matrix-vector
multiplication
|
M>
A = [1 2 3; 4 5 6; 7 8 9]
M>
b = [ 1; 2; 3 ]
M>
A * b
ans =
14
32
50
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
b = np.array([ [1], [2], [3] ])
P>
np.dot(A,b) # or A.dot(b)
array([[14], [32], [50]])
|
R>
A = matrix(1:9, ncol=3)
R>
b = matrix(1:3, nrow=3)
R>
t(b %*% A)
[,1]
[1,] 14
[2,] 32
[3,] 50
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
J>
b=[1; 2; 3];
J>
A*b
3-element Array{Int64,1}:
14
32
50
|
Matrix-vector
multiplication
|
|
Element-wise
matrix-matrix operations
|
M> A
= [1 2 3; 4 5 6; 7 8 9]
M>
A .* A
ans =
1 4
9
16 25 36
49
64 81
M>
A .+ A
M>
A .- A
M>
A ./ A
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
A * A
array([[ 1, 4, 9],
[16, 25, 36],
[49, 64, 81]])
P>
A + A
P>
A - A
P>
A / A
#
Note that NumPy was optimized for
# in-place assignments
#
e.g., A += A instead of
# A = A + A
|
R>
A = matrix(1:9, nrow=3, byrow=T)
R>
A * A
[,1] [,2] [,3]
[1,] 1 4 9
[2,] 16 25 36
[3,] 49
64 81
R>
A + A
R>
A - A
R>
A / A
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
J>
A .* A
3x3 Array{Int64,2}:
1 4 9
16 25 36
49 64 81
J>
A .+ A;
J>
A .- A;
J>
A ./ A;
|
Element-wise
matrix-matrix operations
|
|
Matrix
elements to power n
(here: individual elements
squared)
|
M> A
= [1 2 3; 4 5 6; 7 8 9]
M>
A.^2
ans =
1 4 9
16 25 36
49 64 81
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
np.power(A,2)
array([[ 1, 4, 9],
[16, 25, 36],
[49, 64, 81]])
|
R>
A = matrix(1:9, nrow=3, byrow=T)
R>
A ^ 2
[,1] [,2] [,3]
[1,] 1 4 9
[2,] 16 25 36
[3,] 49
64 81
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
J>
A .^ 2
3x3 Array{Int64,2}:
1 4 9
16 25 36
49 64 81
|
Matrix
elements to power n
(here: individual elements
squared)
|
|
Matrix
to power n
(here: matrix-matrix multiplication with
itself)
|
M> A
= [1 2 3; 4 5 6; 7 8 9]
M>
A ^ 2
ans =
30 36
42
66 81 96
102 126 150
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
np.linalg.matrix_power(A,2)
array([[ 30, 36,
42],
[ 66, 81, 96],
[102, 126, 150]])
|
R>
A = matrix(1:9, ncol=3)
# requires the ‘expm’
package
R>
install.packages('expm')
R>
library(expm)
R>
A %^% 2
[,1] [,2] [,3]
[1,] 30 66 102
[2,] 36 81 126
[3,]
42 96 150
|
J>
A=[1 2 3; 4 5 6; 7 8 9];
J>
A ^ 2
3x3 Array{Int64,2}:
30 36 42
66 81 96
102 126
150
|
Matrix
to power n
(here: matrix-matrix multiplication with
itself)
|
|
Matrix
transpose
|
M> A
= [1 2 3; 4 5 6; 7 8 9]
M>
A'
ans =
1 4 7
2
5 8
3 6 9
|
P>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])
P>
A.T
array([[1, 4, 7],
[2, 5,
8],
[3, 6, 9]])
|
R>
A = matrix(1:9, nrow=3, byrow=T)
R>
t(A)
[,1] [,2] [,3]
[1,] 1 4 7
[2,] 2 5 8
[3,] 3 6 9
|
J>
A=[1 2 3; 4 5 6; 7 8 9]
3x3 Array{Int64,2}:
1 2 3
4 5 6
7
8 9
J>
A'
3x3 Array{Int64,2}:
1 4 7
2 5 8
3 6 9
|
Matrix
transpose
|
|
Determinant
of a matrix:
A -> |A|
|
M>
A = [6 1 1; 4 -2 5; 2 8 7]
A =
6 1
1
4 -2 5
2 8
7
M>
det(A)
ans = -306
|
P> A
= np.array([[6,1,1],[4,-2,5],[2,8,7]])
P>
A
array([[ 6, 1, 1],
[ 4, -2, 5],
[ 2, 8,
7]])
P>
np.linalg.det(A)
-306.0
|
R>
A = matrix(c(6,1,1,4,-2,5,2,8,7), nrow=3, byrow=T)
R>
A
[,1] [,2] [,3]
[1,] 6 1 1
[2,] 4 -2 5
[3,] 2 8 7
R>
det(A)
[1] -306
|
J>
A=[6 1 1; 4 -2 5; 2 8 7]
3x3 Array{Int64,2}:
6 1 1
4 -2
5
2 8 7
J>
det(A)
-306.0
|
Determinant
of a matrix:
A -> |A|
|
|
Inverse
of a matrix
|
M>
A = [4 7; 2 6]
A =
4 7
2
6
M>
A_inv = inv(A)
A_inv =
0.60000 -0.70000
-0.20000 0.40000
|
P>
A = np.array([[4, 7], [2, 6]])
P>
A
array([[4, 7],
[2,
6]])
P>
A_inverse = np.linalg.inv(A)
P>
A_inverse
array([[ 0.6, -0.7],
[-0.2, 0.4]])
|
R>
A = matrix(c(4,7,2,6), nrow=2, byrow=T)
R>
A
[,1] [,2]
[1,] 4 7
[2,] 2 6
R>
solve(A)
[,1] [,2]
[1,] 0.6 -0.7
[2,] -0.2 0.4
|
J>
A=[4 7; 2 6]
2x2 Array{Int64,2}:
4 7
2 6
J>
A_inv=inv(A)
2x2 Array{Float64,2}:
0.6 -0.7
-0.2 0.4
|
Inverse
of a matrix
|
|
ADVANCED
MATRIX OPERATIONS
|
|
Calculating
the covariance matrix
of 3 random variables
(here:
covariances of the means
of x1, x2, and x3)
|
M>
x1 = [4.0000 4.2000 3.9000 4.3000 4.1000]’
M>
x2 = [2.0000 2.1000 2.0000 2.1000 2.2000]'
M>
x3 = [0.60000 0.59000 0.58000 0.62000 0.63000]’
M>
cov( [x1,x2,x3] )
ans =
2.5000e-02
7.5000e-03 1.7500e-03
7.5000e-03
7.0000e-03 1.3500e-03
1.7500e-03
1.3500e-03 4.3000e-04
|
P>
x1 = np.array([ 4, 4.2, 3.9, 4.3, 4.1])
P>
x2 = np.array([ 2, 2.1, 2, 2.1, 2.2])
P>
x3 = np.array([ 0.6, 0.59, 0.58, 0.62, 0.63])
P>
np.cov([x1, x2, x3])
Array([[ 0.025 , 0.0075 ,
0.00175],
[ 0.0075 , 0.007
, 0.00135],
[ 0.00175,
0.00135, 0.00043]])
|
R>
x1 = matrix(c(4, 4.2, 3.9, 4.3, 4.1), ncol=5)
R>
x2 = matrix(c(2, 2.1, 2, 2.1, 2.2), ncol=5)
R>
x3 = matrix(c(0.6, 0.59, 0.58, 0.62, 0.63), ncol=5)
R>
cov(matrix(c(x1, x2, x3), ncol=3))
[,1] [,2] [,3]
[1,]
0.02500 0.00750 0.00175
[2,] 0.00750 0.00700 0.00135
[3,]
0.00175 0.00135 0.00043
|
J>
x1=[4.0 4.2 3.9 4.3 4.1]';
J>
x2=[2. 2.1 2. 2.1 2.2]';
J>
x3=[0.6 .59 .58 .62 .63]';
J>
cov([x1 x2 x3])
3x3 Array{Float64,2}:
0.025 0.0075
0.00175
0.0075 0.007 0.00135
0.00175 0.00135 0.00043
|
Calculating
the covariance matrix
of 3 random variables
(here:
covariances of the means
of x1, x2, and x3)
|
|
Calculating
eigenvectors
and eigenvalues
|
M>
A = [3 1; 1 3]
A =
3 1
1
3
M>
[eig_vec,eig_val] = eig(A)
eig_vec =
-0.70711
0.70711
0.70711 0.70711
eig_val
=
Diagonal Matrix
2 0
0
4
|
P>
A = np.array([[3, 1], [1, 3]])
P>
A
array([[3, 1],
[1, 3]])
P>
eig_val, eig_vec = np.linalg.eig(A)
P>
eig_val
array([ 4., 2.])
P>
eig_vec
Array([[ 0.70710678, -0.70710678],
[ 0.70710678, 0.70710678]])
|
R>
A = matrix(c(3,1,1,3), ncol=2)
R>
A
[,1] [,2]
[1,] 3 1
[2,] 1 3
R>
eigen(A)
$values
[1] 4 2
$vectors
[,1] [,2]
[1,]
0.7071068 -0.7071068
[2,] 0.7071068 0.7071068
|
J>
A=[3 1; 1 3]
2x2 Array{Int64,2}:
3 1
1 3
J>
(eig_vec,eig_val)=eig(a)
([2.0,4.0],
2x2
Array{Float64,2}:
-0.707107 0.707107
0.707107 0.707107)
|
Calculating
eigenvectors
and eigenvalues
|
|
Generating
a Gaussian dataset:
creating random vectors from the
multivariate normal
distribution given mean and covariance
matrix
(here: 5 random vectors with
mean 0, covariance
= 0, variance = 2)
|
%
requires statistics toolbox package
% how to install and load
it in Octave:
% download the package from:
%
http://octave.sourceforge.net/packages.php
% pkg install
%
~/Desktop/io-2.0.2.tar.gz
% pkg install
%
~/Desktop/statistics-1.2.3.tar.gz
M>
pkg load statistics
M>
mean = [0 0]
M>
cov = [2 0; 0 2]
cov =
2 0
0
2
M>
mvnrnd(mean,cov,5)
2.480150 -0.559906
-2.933047 0.560212
0.098206
3.055316
-0.985215 -0.990936
1.122528
0.686977
|
P>
mean = np.array([0,0])
P>
cov = np.array([[2,0],[0,2]])
P>
np.random.multivariate_normal(mean, cov, 5)
Array([[
1.55432624, -1.17972629],
[-2.01185294, 1.96081908],
[-2.11810813, 1.45784216],
[-2.93207591, -0.07369322],
[-1.37031244, -1.18408792]])
|
#
requires the ‘mass’ package
R>
install.packages('MASS')
R>
library(MASS)
R>
mvrnorm(n=10, mean, cov)
[,1] [,2]
[1,] -0.8407830
-0.1882706
[2,] 0.8496822 -0.7889329
[3,] -0.1564171
0.8422177
[4,] -0.6288779 1.0618688
[5,] -0.5103879
0.1303697
[6,] 0.8413189 -0.1623758
[7,] -1.0495466
-0.4161082
[8,] -1.3236339 0.7755572
[9,] 0.2771013
1.4900494
[10,] -1.3536268 0.2338913
|
#
requires the Distributions package from
https://github.com/JuliaStats/Distributions.jl
J>
using Distributions
J>
mean=[0., 0.]
2-element Array{Float64,1}:
0.0
0.0
J>
cov=[2. 0.; 0. 2.]
2x2 Array{Float64,2}:
2.0 0.0
0.0
2.0
J>
rand( MvNormal(mean, cov), 5)
2x5 Array{Float64,2}:
-0.527634
0.370725 -0.761928 -3.91747 1.47516
-0.448821 2.21904 2.24561
0.692063 0.390495
|
Generating
a Gaussian dataset:
creating random vectors from the
multivariate normal
distribution given mean and covariance
matrix
(here: 5 random vectors with
mean 0, covariance
= 0, variance = 2)
|