Introduction to Julia Programming¶

Collections, arrays and linear algebra¶

by Yueh-Hua Tu

Outline¶

Collections
- Arrays
- Sets
- Dictionaries
Linear algebra

Collections¶

Arrays¶

The fundamental collection and data structure in a language.

Create an array¶

In [1]:

x = []

Out[1]:

0-element Array{Any,1}

Homogeneous: 同質性，Array 中只能放入屬於同一型別的物件

In [2]:

Any[]

Out[2]:

0-element Array{Any,1}

In [3]:

Int64[]

Out[3]:

0-element Array{Int64,1}

Type inference on array¶

In [4]:

x = [1, 2, 3]

Out[4]:

3-element Array{Int64,1}:
 1
 2
 3

In [5]:

x = [1, 1.2]

Out[5]:

2-element Array{Float64,1}:
 1.0
 1.2

Specified array type¶

In [6]:

Int8[1, 2, 3, 4]

Out[6]:

4-element Array{Int8,1}:
 1
 2
 3
 4

In [7]:

Array{Int8, 1}(undef, 5)  # uninitialized

Out[7]:

5-element Array{Int8,1}:
 -112
   -8
   86
   -1
  -27

Indexing¶

Index starts from 1.

☐☐☐

1 2 3

In [8]:

Out[8]:

2-element Array{Float64,1}:
 1.0
 1.2

In [9]:

x[1]

Out[9]:

1.0

In [10]:

x[2]

Out[10]:

1.2

In [11]:

length(x)

Out[11]:

In [12]:

x = [6.0, 3.2, 7.6, 0.9, 2.3];

In [13]:

x[1:2]

Out[13]:

2-element Array{Float64,1}:
 6.0
 3.2

In [14]:

x[3:end]

Out[14]:

3-element Array{Float64,1}:
 7.6
 0.9
 2.3

In [15]:

x[begin:3]

Out[15]:

3-element Array{Float64,1}:
 6.0
 3.2
 7.6

In [16]:

x[1:2:end]

Out[16]:

3-element Array{Float64,1}:
 6.0
 7.6
 2.3

In [17]:

view(x, 1:3)

Out[17]:

3-element view(::Array{Float64,1}, 1:3) with eltype Float64:
 6.0
 3.2
 7.6

Assign value¶

In [18]:

x[2] = 7.5

Out[18]:

7.5

In [19]:

Out[19]:

5-element Array{Float64,1}:
 6.0
 7.5
 7.6
 0.9
 2.3

Useful operations¶

In [20]:

push!(x, 9.0)

Out[20]:

6-element Array{Float64,1}:
 6.0
 7.5
 7.6
 0.9
 2.3
 9.0

In [21]:

y = [10.0, 3.4]
append!(x, y)

Out[21]:

8-element Array{Float64,1}:
  6.0
  7.5
  7.6
  0.9
  2.3
  9.0
 10.0
  3.4

In [22]:

Out[22]:

8-element Array{Float64,1}:
  6.0
  7.5
  7.6
  0.9
  2.3
  9.0
 10.0
  3.4

In [23]:

pop!(x)

Out[23]:

3.4

In [24]:

Out[24]:

7-element Array{Float64,1}:
  6.0
  7.5
  7.6
  0.9
  2.3
  9.0
 10.0

In [25]:

popfirst!(x)

Out[25]:

6.0

In [26]:

Out[26]:

6-element Array{Float64,1}:
  7.5
  7.6
  0.9
  2.3
  9.0
 10.0

In [27]:

pushfirst!(x, 6.0)

Out[27]:

7-element Array{Float64,1}:
  6.0
  7.5
  7.6
  0.9
  2.3
  9.0
 10.0

Random array¶

In [28]:

x = rand(5)

Out[28]:

5-element Array{Float64,1}:
 0.09810929831480641
 0.46112452279406857
 0.9618021721401293
 0.19292860821288849
 0.41253510699062246

In [29]:

sort(x)

Out[29]:

5-element Array{Float64,1}:
 0.09810929831480641
 0.19292860821288849
 0.41253510699062246
 0.46112452279406857
 0.9618021721401293

In [30]:

Out[30]:

5-element Array{Float64,1}:
 0.09810929831480641
 0.46112452279406857
 0.9618021721401293
 0.19292860821288849
 0.41253510699062246

In [31]:

sort!(x)

Out[31]:

5-element Array{Float64,1}:
 0.09810929831480641
 0.19292860821288849
 0.41253510699062246
 0.46112452279406857
 0.9618021721401293

In [32]:

Out[32]:

5-element Array{Float64,1}:
 0.09810929831480641
 0.19292860821288849
 0.41253510699062246
 0.46112452279406857
 0.9618021721401293

Sorting: big to small¶

In [33]:

sort(x, rev=true)

Out[33]:

5-element Array{Float64,1}:
 0.9618021721401293
 0.46112452279406857
 0.41253510699062246
 0.19292860821288849
 0.09810929831480641

Sorting by absolute value¶

In [34]:

x = randn(10)

Out[34]:

10-element Array{Float64,1}:
 -0.04449731782053265
 -0.24067299751804175
  0.42097259514342966
 -1.2941611564169055
  0.9251854732954373
  3.265601659774667
  1.2862744750362891
  0.06376759068522922
 -0.6098980867393012
 -0.08696207979292975

In [35]:

sort(x, by=abs)

Out[35]:

10-element Array{Float64,1}:
 -0.04449731782053265
  0.06376759068522922
 -0.08696207979292975
 -0.24067299751804175
  0.42097259514342966
 -0.6098980867393012
  0.9251854732954373
  1.2862744750362891
 -1.2941611564169055
  3.265601659774667

Iteration¶

In [36]:

for i in x
    println(i)
end

-0.04449731782053265
-0.24067299751804175
0.42097259514342966
-1.2941611564169055
0.9251854732954373
3.265601659774667
1.2862744750362891
0.06376759068522922
-0.6098980867393012
-0.08696207979292975

請造出一個陣列，當中的數值是均勻分佈，從-345到957.6

提示： $\LARGE y = \frac{x - min(x)}{max(x) - min(x)}$

請造出一個陣列，當中的數值是服從常態分佈

請造出一個陣列，當中的數值是服從常態分佈，μ=3.5，σ=2.5

提示： $\LARGE y = \frac{x - \mu}{\sigma}$

Sets¶

Mathematical set.

In [40]:

x = Set([1, 2, 3, 4])

Out[40]:

Set{Int64} with 4 elements:
  4
  2
  3
  1

In [41]:

push!(x, 5)

Out[41]:

Set{Int64} with 5 elements:
  4
  2
  3
  5
  1

In [42]:

pop!(x)

Out[42]:

In [43]:

Out[43]:

Set{Int64} with 4 elements:
  2
  3
  5
  1

Exists¶

In [44]:

3 in x

Out[44]:

true

In [45]:

4 in x

Out[45]:

false

Equivalent¶

In [46]:

x == Set([3, 2, 1, 5])

Out[46]:

true

Iteration¶

In [47]:

for i in x
    println(i)
end

請告訴我以下資料有幾種數值

[8, 4, 1, 2, 9, 4, 5, 4, 5, ...]

Dictionaries¶

A data structure stores key-value pairs.

In [50]:

x = Dict("1" => 1, "2" => 2, "3" => 3)

Out[50]:

Dict{String,Int64} with 3 entries:
  "1" => 1
  "2" => 2
  "3" => 3

In [51]:

x["1"]

Out[51]:

In [52]:

x["A"]

KeyError: key "A" not found

Stacktrace:
 [1] getindex(::Dict{String,Int64}, ::String) at ./dict.jl:477
 [2] top-level scope at In[52]:1

Add new pair¶

In [53]:

x["4"] = 4

Out[53]:

In [54]:

Out[54]:

Dict{String,Int64} with 4 entries:
  "4" => 4
  "1" => 1
  "2" => 2
  "3" => 3

Overwrite¶

In [55]:

x["1"] = 5

Out[55]:

In [56]:

Out[56]:

Dict{String,Int64} with 4 entries:
  "4" => 4
  "1" => 5
  "2" => 2
  "3" => 3

keys and values¶

In [57]:

keys(x)

Out[57]:

Base.KeySet for a Dict{String,Int64} with 4 entries. Keys:
  "4"
  "1"
  "2"
  "3"

In [58]:

values(x)

Out[58]:

Base.ValueIterator for a Dict{String,Int64} with 4 entries. Values:
  4
  5
  2
  3

Iteration¶

In [59]:

for (k, v) in x
    println(k, "->", v)
end

4->4
1->5
2->2
3->3

Linear Algebra¶

Linear systems¶

Linear systems are most common in scientific computing. We have a linear system such as follow:

$$ \begin{equation} \left\{ \begin{array}{lr} w_{11}x + w_{12}y + w_{13}z = a \\ w_{21}x + w_{22}y + w_{23}z = b \\ w_{31}x + w_{32}y + w_{33}z = c \end{array} \right. \end{equation} $$

We usually express in matrix form:

$$ \begin{bmatrix} w_{11}& w_{12}& w_{13} \\ w_{21}& w_{22}& w_{23} \\ w_{31}& w_{32}& w_{33} \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} a \\ b \\ c \end{bmatrix} $$

And we use symbols to represent matrices and vectors.

$$ Ax = b $$

Inner product¶

Inner product is defined as following, and transpose is needed to calculate inner product.

$$ A^TB $$

In [60]:

A = [1 2 3; 4 5 6; 7 8 9]
B = [12 11 10 9; 8 7 6 5; 4 3 2 1]
A' * B

Out[60]:

3×4 Array{Int64,2}:
  72   60  48  36
  96   81  66  51
 120  102  84  66

Determinant¶

Determinant (行列式) is calculated by det. Here we need built-in LinearAlgebra package.

In [61]:

using LinearAlgebra

In [62]:

det(A)

Out[62]:

6.661338147750939e-16

Rank¶

Rank (秩) is calculated by rank. LinearAlgebra package is needed.

In [63]:

rank(A)

Out[63]:

Trace¶

Trace is calculated by tr. LinearAlgebra package is needed.

In [64]:

tr(A)

Out[64]:

Norm¶

Norm is used to calculate the "length" of a vector. Usually, we use Euclidean norm (歐幾里得範數). LinearAlgebra package is needed.

$$ ||v||_2 = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} $$

In [65]:

b = [5, 5, 5]
norm(b)

Out[65]:

8.660254037844387

If a norm is calculated over a matrix, Frobenius norm is calculated. LinearAlgebra package is needed.

In [66]:

norm(A)

Out[66]:

16.881943016134134

p-norm¶

P-norm is the norm which is generalized to the power of p.

$$ ||v||_p = (\sum_i |v_i|^p)^{\frac{1}{p}} $$

In [67]:

norm(b, 5)

Out[67]:

6.228654698077587

A Manhattan Distance (or Taxicab norm, $l_1$ norm) is often used in machine learning models.

$$ ||v||_1 = |v_1| + |v_2| + \cdots + |v_n| $$

In [68]:

norm(b, 1)

Out[68]:

15.0

Infinity norm¶

$$ ||v||_\infty = max(|v_1|, |v_2|, \dots, |v_n|) $$

In [69]:

norm(b, Inf)

Out[69]:

5.0

Identity matrix¶

Identity matrix is really easy to use in Julia, unlike eye in other languages. One can represent 3x3 identity matrix as follow:

In [70]:

Matrix{Float64}(I, 3, 3)

Out[70]:

3×3 Array{Float64,2}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

A more convenient way is to use I to represent identity. The dimension of an identity matrix is determined automatically.

In [71]:

A * I

Out[71]:

3×3 Array{Int64,2}:
 1  2  3
 4  5  6
 7  8  9

Inverse¶

An inverse matrix (反矩陣) is calculated by inv.

In [72]:

inv(A)

Out[72]:

3×3 Array{Float64,2}:
 -4.5036e15   9.0072e15   -4.5036e15
  9.0072e15  -1.80144e16   9.0072e15
 -4.5036e15   9.0072e15   -4.5036e15

If a matrix is not a square and full-rank, Moore-Penrose pseudo-inverse is calculated by pinv.

In [73]:

pinv(B)

Out[73]:

4×3 Array{Float64,2}:
 -0.1125      0.1         0.3125
 -0.0166667   0.0333333   0.0833333
  0.0791667  -0.0333333  -0.145833
  0.175      -0.1        -0.375

Solve linear systems¶

To solve a linear system, we express a linear system as follow.

$$ Ax = b $$

If matrix A is invertible (可逆的), \ operation is used to solve the system.

$$ x = A^{-1}b $$

In [74]:

x = A \ b

Out[74]:

3-element Array{Float64,1}:
 -8.999999999999998
 12.999999999999998
 -4.0

Eigenvalue decomposition¶

Eigenvalue decomposition (特徵值分解) is very important and widely used. eigen returns the eigenvalues (特徵值) and eigenvectors (特徵向量) of a matrix.

In [75]:

vals, vecs = eigen(A)

Out[75]:

Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}
values:
3-element Array{Float64,1}:
 -1.1168439698070427
 -1.3036777264747022e-15
 16.116843969807043
vectors:
3×3 Array{Float64,2}:
 -0.78583     0.408248  -0.231971
 -0.0867513  -0.816497  -0.525322
  0.612328    0.408248  -0.818673

Eigenvalue decomposition¶

eigvals returns eigenvalues only.

In [76]:

eigvals(A)

Out[76]:

3-element Array{Float64,1}:
 -1.1168439698070427
 -1.3036777264747022e-15
 16.116843969807043

eigvecs returns only eigenvectors, which are in column-wise manner.

In [77]:

eigvecs(A)

Out[77]:

3×3 Array{Float64,2}:
 -0.78583     0.408248  -0.231971
 -0.0867513  -0.816497  -0.525322
  0.612328    0.408248  -0.818673

Eigenvalue decomposition¶

eigmax returns maximum of eigenvalues.

In [78]:

eigmax(A)

Out[78]:

16.116843969807043

eigmin returns minimum of eigenvalues.

In [79]:

eigmin(A)

Out[79]:

-1.1168439698070427

Singular value decomposition¶

Singular value decomposition (奇異值分解) is also a widely used matrix decomposition method. svd is used to get the results.

In [80]:

U, Σ, V = svd(A)

Out[80]:

SVD{Float64,Float64,Array{Float64,2}}
U factor:
3×3 Array{Float64,2}:
 -0.214837   0.887231   0.408248
 -0.520587   0.249644  -0.816497
 -0.826338  -0.387943   0.408248
singular values:
3-element Array{Float64,1}:
 16.84810335261421
  1.0683695145547103
  1.472808250397788e-16
Vt factor:
3×3 Array{Float64,2}:
 -0.479671  -0.572368   -0.665064
 -0.776691  -0.0756865   0.625318
  0.408248  -0.816497    0.408248

Singular value decomposition¶

svdvals returns singular values (奇異值) in descending order.

In [81]:

svdvals(A)

Out[81]:

3-element Array{Float64,1}:
 16.84810335261421
  1.0683695145547103
  1.4728082503977878e-16

Pearson correlation coefficient¶

In [82]:

using Statistics
A = [1.2 5.21 5.8;
     1.3 8.6 7.4;
     12. 6. 3.]
b = [5.2, 4.6, 7.2]

Out[82]:

3-element Array{Float64,1}:
 5.2
 4.6
 7.2

In [83]:

cor(b)

Out[83]:

1.0

In [84]:

cor(A)

Out[84]:

3×3 Array{Float64,2}:
  1.0       -0.286874  -0.930333
 -0.286874   1.0        0.618192
 -0.930333   0.618192   1.0

Pearson correlation coefficient¶

In [85]:

cor(A, b)

Out[85]:

3×1 Array{Float64,2}:
  0.973610396858362
 -0.4979278975048893
 -0.9894723165936354

In [86]:

B = [1.3 5.6 5.7 4.1;
     5.4 2.3 4.5 7.7;
     1.2 1.4 8.7 9.5]
cor(A, B)

Out[86]:

3×4 Array{Float64,2}:
 -0.511052  -0.671761   0.958503   0.761178
  0.970029  -0.516922  -0.548068   0.402921
  0.79066    0.353308  -0.996271  -0.470317

Introduction to Julia Programming¶

Collections, arrays and linear algebra¶

Outline¶

Collections¶

Arrays¶

Create an array¶

Type inference on array¶

Specified array type¶

Indexing¶

Assign value¶

Useful operations¶

Random array¶

Sorting: big to small¶

Sorting by absolute value¶

Iteration¶

Sets¶

Exists¶

Equivalent¶

Iteration¶

Dictionaries¶

Add new pair¶

Overwrite¶

keys and values¶

Iteration¶

Linear Algebra¶

Linear systems¶

Inner product¶

Determinant¶

Rank¶

Trace¶

Norm¶

p-norm¶

Infinity norm¶

Identity matrix¶

Inverse¶

Solve linear systems¶

Eigenvalue decomposition¶

Eigenvalue decomposition¶

Eigenvalue decomposition¶

Singular value decomposition¶

Singular value decomposition¶

Pearson correlation coefficient¶

Pearson correlation coefficient¶

Q & A¶