footer: 정보시각화 고전 독서 모임 - The Grammar of Ghapcis 5장 © [nacyot](https://twitter.com/nacyo_t) 2015 -
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# [fit] The Grammar of Graphics
## [fit] Algebra

---

## 정보시각화 고전 독서 모임 5번째

### 2015. 8. 18.

---

![fit](http://i.imgur.com/BVz3cIx.png)

---

# 2.1.8 Variable

a variable $$X$$ is $$X = [O, V, f]$$

* the domain $$O$$ is a set of objects,
* the codomain $$V$$ is a set of values,
* the function $$f$$ assigns to each element of $$O$$ an element in $$V$$

---

# 2.1.8 변수(Variable)

변수 $$X$$는 $$X = [O, V, f]$$

* 정의역 $$O$$은 객체들의 집합
* 공역 $$V$$은 값들의 집합
* 함수 $$f$$는 $$O$$의 각 값들을 $$V$$의 요소에 대응시킨다.

---

# $$O$$

## a set of objects

## $$\{ 1, 2, 3, 4, 5 \}$$

---

# $$V$$

## a set of values

## $$\{ ant, bee, drone, moth, locust \}$$

---

# $$f$$

## $$f(O_{i}) = V_{i}$$

---

# Variable $$X$$



$$
[
\{1,2,3,4\},
\{ant, bee, drone, moth\},
f
]
$$

* 테이블 표현과 비슷
  * 1 -> ant
  * 2 -> bee
  * 3 -> drone
  * 4 -> moth

---

# 2.1.9 Varsets

a varset $$\mathsf{X}$$ is $$\mathsf{X} = [V, \tilde{O}, f]$$

* the domain $$V$$ is a set of values,
* the codomain $$\tilde{O}$$ is a set of all possible bags of objects,
* the function $$f$$ assigns to each element of $$V$$ an element in $$\tilde{O}$$

---

# 2.1.9 Varsets

varset $$\mathsf{X}$$는 $$\mathsf{X} = [V, \tilde{O}, f]$$

* 도메인 $$V$$는 값들의 집합
* 공역 $$\tilde{O}$$는 모든 가능한 백(bags)들의 집합
* 함수 $$f$$는 $$V$$의 각 값들을 $$\tilde{O}$$에 대응시킨다

---

# $$V$$

## a set of Values

## $$\{ red, blue \}$$

---

# $$\tilde{O}$$

## [fit] set of all possible bags of objects,

## $$\{ <\cdot>, <\cdot, \cdot>,\ ... \}$$

---

# $$f$$

## $$red \to <1, 2>$$
## $$blue \to <3>$$

---

# 다른 Variable Y 예제

$$
[
\{1,2,3\},
\{red, blue\},
f
]
$$

* 1 -> red
* 2 -> red
* 3 -> blue

---

# Varset Y

$$
[
\{red, blue\},
\{<1>, <2>, <3>, <1,2>, <2,3>, <3,1>, <1,2,3>\},
f
]
$$

* red -> <1,2>
* blue -> <3>

---

# Varset은
# [fit] Variable의 값들로 인덱스된 형태

---

# Table

$$
\begin{array}{}
  KEY & VALUE \\ \hline
  1 \T & red \\ 
  2 \T & red \\ 
  3 \T & blue \\ \hline
\end{array}
$$

---

# [fit] $$Variable\ \approx \ Varset\ \approx \ Table(Column)$$

---

$$
Y = [
\{1,2,3\},
\{red, blue\},
f_{1}
]
$$

$$
\mathsf{Y} = [
\{red, blue\},
\{<\cdot>, <\cdot, \cdot>,\ ... \},
f_{2}
]
$$

$$
Table\ Y = 
\begin{array}{}
  KEY & VALUE \\ \hline
  1 \T & red \\ 
  2 \T & red \\ 
  3 \T & blue \\ \hline
\end{array}
$$

---

# [fit] 2. How To Make a Pie

## 2.2 Recipe

---

# Variables(32p)

![inline 100%](http://i.imgur.com/rob2S91.png)

---

# Variables 재정의

---

# [fit] Data Definition Statements

```
SOURCE: <source name> = <fn>(<args>)
DATA: <variable name> = <fn>(<args>)
TRANS: <variable name> = <fn>(<args>)
```

---

# SOURCE

```
SOURCE: acls_data = csvSource(path("C:\\acls.csv")) 
```

csv file(이 존재한다고 가정)

```
CaseID, Gender, Response, ...
1, Male, Frequently, ...
2, Female, Not Sure, ...
3, Male, Frequently, ...
...
3834, Male, Rarely, ...
3835, Female, Infrequently, ...
````

---

# Source Table

$$ 
\begin{array}{}
  CaseID & Gender & Response & ... \\ \hline
  1 \T & Male & Frequently & ... \\ 
  2 \T & Female & Not Sure & ... \\ 
  3 \T & Male & Frequently & ... \\
  ... \T  & ... & ... & ... \\ 
  3834 \T & Male & Rarely & ... \\ 
  3835 \T & Female & Infrequently & ... \\ \hline
\end{array}
$$


---

# DATA

```
SOURCE: acls_data = csvSource(path("C:\\acls.csv"))
DATA: gender = col(source(acls_data), name("Gender"))
DATA: response = col(source(acls_data), name("Response"))
```

col 함수를 통해서 gender, responce 변수 할당

---

# gender (1)

$$
Variable\ gender = [ \{1, 2, ..., 3835\}, \{Male, Female\}, f_{1}]
$$

$$
Varset\ \mathsf{gender} = [ \{Male, Female\}, \{...\}, f_{2}]
$$

---

# gender (2)

```
select CaseID, Gender from acls_data
```

$$
Table\ gender = 
\begin{array}{}
  CaseID & Gender \\ \hline
  1 \T & Male \\ 
  2 \T & Female \\ 
  3 \T & Male \\
  ... \T  & ... \\ 
  3834 \T & Male \\ 
  3835 \T & Female \\ \hline
\end{array}
$$

---

# response (1)

$$
Variable\ responese = [ \{1, 2, ..., 3835\}, \{R, I, O, F, N\}, f_{1}]
$$

$$
Varset\ \mathsf{response} = [ \{R, I, O, F, N\}, \{...\}, f_{2}]
$$

---

# response (2)

```
select CaseID, Response from acls_data
```

$$
Table\ response = 
\begin{array}{}
  CaseID & Response \\ \hline
  1 \T & Frequently \\ 
  2 \T & Not Sure \\ 
  3 \T & Frequently \\
  ... \T  & ... \\ 
  3834 \T & Rarely \\ 
  3835 \T & Infrequently \\ \hline
  \end{array}
$$

---

# Algebra

---

# Varset Algebra

---

# Algebra와 Frames (1)

> Graphics algebra provides a method for specifying  F(which we call a frame) when we wish to construct agraphic based on some function of a set of data.
-- [nVIZn](http://www.cs.uic.edu/~wilkinson/Publications/ibm.pdf) 3p

---

# Algebra와 Frames (2)

> This chapter deals with restoring and balancing sets of variables in order to create the specification for the frames in which graphs are embedded.
-- GoG Ch5. Algebra 63p

---

# Frame이란?

> A frame is a set of tuple $$(x_1, ..., x_p)$$ ranging over all possible values in the domain of a p-dimensional varset.
-- GoG 30p

그래프가 그려지는 공간을 정의. (2개 varset이 있을 때는 일반적으로 xy평면과 같은 2차원 평면을 정의)

---

# Frame Specification

* 1판에서는 frame spcification이 있었음(40p)

```
FRAME: x * y
GRAPH: point()
```

* 2판의 표현법(39p)

```
ELEMENT: point(position(x * y))
```

---

# Symbol

A symbol is used to represent an entity operated on by an algebra. The symbols in varset algebra are varsets.

심볼은 대수식에서 계산되어지는 항목을 나타낸다. varset 대수식에서 심볼은 varset이다.

---

# Operators

An operator is a method for relating symbols in an algebra.

연산자는 대수식에서 심볼들을 관계짓는 메서드이다.

* Cross(*), Nest(/), Blend(+)
* Varsets($$\mathsf{A}$$ and $$\mathsf{B}$$)
* Value Domain of Varsets($$V_{\mathsf{A}}$$ and $$V_{\mathsf{B}}$$)

---

# Cross: $$*$$

---

# Cross($$*$$) 수학적 정의

## Varset 정의

$$\mathsf{X} = [V, \tilde{O}, f]$$

## Cross 정의

$$
A*B: (s, t) \to A(s) \cap B(t)
$$

$$
V_{A*B} = \{ (s, t),  \forall s \in V_A, \forall t \in V_B \}
$$

---

# [fit] 테이블(Variable) 표현으로 이해하기

$$
color = 
\begin{array}{}
  key & color \\ \hline
  1 \T & red \\ 
  2 \T & blue \\ 
  3 \T & blue \\
  4 \T & red \\ \hline
\end{array}
,\ \ 
car = 
\begin{array}{}
  key & response \\ \hline
  1 \T & Supercar \\ 
  2 \T & Sportcar \\ 
  3 \T & Minicar \\
  4 \T & Supercar \\ \hline
\end{array}
$$

---

# $$V_{ \mathsf{color * car} } \approx Frame$$ 

```
SELECT DISTINCT color.color, car.car
FROM color CROSS JOIN car;
```

```
color       car
----------  ----------
red         supercar
red         sportcar
red         minicar
blue        supercar
blue        sportcar
blue        minicar
```

---

# $$Frame\ of\ ( \mathsf{color * car} )$$

![Imgur inline](http://i.imgur.com/Gnc73Lz.png)

---

# $$f:V \to \tilde{O}$$ (1) sql

```
SELECT
  ('(' || color.color || ', ' || car.car || ')')
    AS "Value Domain",
  GROUP_CONCAT(
    CASE WHEN color.key = car.key THEN color.key ELSE NULL END
  ) AS "Set of Cases"
FROM color CROSS JOIN car
GROUP BY "Value Domain";
```

---

# $$f:V \to \tilde{O}$$ (2) result

```
Value Domain      Set of Cases
----------------  ------------
(blue, minicar)   3
(blue, sportcar)  2
(blue, supercar)
(red, minicar)
(red, sportcar)
(red, supercar)   1,4
```

---

# [fit] Cross의 테이블(Variable) 표현

SELECT color.key, color.color, car.car
FROM color, car WHERE color.key = car.key;

$$
Table\ color * key =
\begin{array}{}
  key & color & car \\ \hline
  1 \T & red & Supercar\\ 
  2 \T & blue & Sportcar\\ 
  3 \T & blue & Minicar\\
  4 \T & red  & Supercar \\ \hline
\end{array}
$$

단, 정의역($$V_{A*B}$$) 전체가 표현된다는 보장은 없음.

---

# Cross의 그래프 표현

```
ELEMENT: point(position(color*car), text=key)
```

![inline 100%](http://i.imgur.com/rrYJZrV.png)

---

# Nest: $$/$$

---

# Cross와 Nest의 수학적 정의

Cross($$*$$)

$$
A*B: (s, t) \to A(s) \cap B(t)
$$

$$
V_{A*B} = \{ (s, t),  \forall s \in V_A, \forall t \in V_B \}
$$

Nest($$/$$)

$$
A/B: (s, t) \to A(s) \cap B(t)
$$

$$
V_{A/B} = \{ (s, t),  \forall s \in V_A, \forall t \in V_B : A(s) \cap B(t) \neq \varnothing \}
$$

---

# [fit] 앞선 예제의 테이블(Variable) 표현

$$
color = 
\begin{array}{}
  key & color \\ \hline
  1 \T & red \\ 
  2 \T & blue \\ 
  3 \T & blue \\
  4 \T & red \\ \hline
\end{array}
,\ \ 
car = 
\begin{array}{}
  key & response \\ \hline
  1 \T & Supercar \\ 
  2 \T & Sportcar \\ 
  3 \T & Minicar \\
  4 \T & Supercar \\ \hline
\end{array}
$$

---

# $$V_{ \mathsf{color / car} } \neq Frame$$ 

```
SELECT DISTINCT color.color, car.car
FROM color INNER JOIN car
WHERE color.key = car.key;
```

```
color       car
----------  ----------
red         supercar
blue        sportcar
blue        minicar
```

---

# $$V_{ \mathsf{color / car} } \neq V_{ \mathsf{color * car} }$$

```
# cross                     # nest
color       car             color       car
----------  ----------      ----------  ----------
red         supercar        red         supercar
red         sportcar        blue        sportcar
red         minicar         blue        minicar
blue        supercar
blue        sportcar
blue        minicar
```

---

## $$V_{ \mathsf{color / car} } \neq V_{ \mathsf{color * car} } \to$$
## $$\mathsf{color / car} \neq \mathsf{color * car} $$

---

# $$f:V \to \tilde{O}\ of\ \mathsf{ color * car }$$

```
Value Domain      Set of Cases
----------------  ------------
(blue, minicar)   3
(blue, sportcar)  2
(blue, supercar)
(red, minicar)
(red, sportcar)
(red, supercar)   1,4
```

---

# $$f:V \to \tilde{O}\ of\ \mathsf{ color / car }$$

```
Value Domain      Set of Cases
----------------  ------------
(blue, minicar)   3
(blue, sportcar)  2
(red, supercar)   1,4
```

```
SELECT
  ('('||color.color||','||car.car||')') AS "Value Domain",
  GROUP_CONCAT(car.key) as "Set of Cases"
FROM color INNER JOIN car WHERE color.key = car.key
GROUP BY "Value Domain";
```

---

# Nest연산은
#[fit] Frame을 만들지 않는다

---

# Nest 연산의 역할 (1)

> Wilkinson presents three algebraic operators called cross ($$*$$), nest($$/$$), and blend ($$+$$), together with the rules for their use. Theyare derived from the set operators product ($$*$$), discreteunion ($$\sqcup$$), and union ($$\cup$$), respectively.
-- [nVIZn](http://www.cs.uic.edu/~wilkinson/Publications/ibm.pdf) 3p

---

# Nest 연산의 역할 (2)

> The disjoint union of two sets A and B, denoted by A $$\sqcup$$ B, produces a set whose members are tagged elements. A tagged element is one of the form x:$, where x $$\in$$X is the element and the symbol $ is the tag. A tag is some- times called an identifier or a color; it may be a string, a numerical value, or another piece of information.
-- GoG 26p

---

# Nest 연산의 역할 (3)

> The nesting variable tags the nested variable such that (s, t) implies (s : t). We will not use this notation below, but we will assume an algebra system can identify the nesting elements through a tagging index.
-- GoG 65p

---

# Nest 연산의 용도

* Tagging
  * Data values
  * Metadata
  * Data organization

---

# 테이블로 직관적으로 이해하기

$$
Table\ color / key =
\begin{array}{}
  key & color/car \\ \hline
  1 \T & red:Supercar\\ 
  2 \T & blue:Sportcar\\ 
  3 \T & blue:Minicar\\
  4 \T & red:Supercar \\ \hline
\end{array}
$$

---

# [fit] d3.js의 nest 함수로 이해하기 (1)

```javascript
data = [
  {color: 'red', car: 'supercar'},
  {color: 'blue', car: 'sportcar'},
  {color: 'blue', car: 'minicar'},
  {color: 'red', car: 'supercar'}
]
```

---

# [fit] `d3.js`의 `nest` 함수로 이해하기 (2)

```javascript
var nestedData = d3.nest()
  .key(function(d) { return d.car; })
  .map(dataSet);
```

```json
{
'supercar':[
    {color: 'red', car: 'supercar'},
    {color: 'red', car: 'supercar'},
  ],
  'sportcar':[
    {color: 'blue', car: 'sportcar'},
  ],
  'minicar':[
    {color: 'blue', car: minicar'},
  ]
}
```

---

# [fit] $$/\ \approx\ d3.nest\ \approx\ group\ by$$

---

# Blend: $$+$$

---

# Blend의 수학적 정의

$$
\mathsf{A + B} : s \to \mathsf{A}(s) \cup \mathsf{B}(s)
$$

$$
V_{\mathsf{A+B}} = V_{\mathsf{A}} \cup V_{\mathsf{B}}
$$

---

# 항등원(Identity emelemnt) 

* +의 항등원: 공집합
* cross와 nest의 항등원 varset

$$
1: unity \to \Omega
$$

$$
V_1 = unity
$$

* 차수(order)를 맞추기 위해서도 사용

---

# cross 항등원(같은 표현)

```
ELEMENT: point(position(pop1980))
```

```
ELEMENT: point(position(pop1980*1))
```

![inline 150%](http://i.imgur.com/7uyoOzT.png)

---

# Expression

* term(expression with no + operator)
* factor(expression with no * operator)
* monomial(one term)
* polynomial(more than one term)
* order(number of factors in term)
* k(largest order among the monomials)

---

# Algebraic Form

An algebraic form is a monomial or a polynomial whose terms all have the same number of factors.

* $$\mathsf{G * 1 + A * C / D + B * C / D}$$
* $$factors = { \mathsf{G, 1, A, C / D, B} }$$
* $$terms = { \mathsf{G * 1, A * C / D, B * C / D} }$$
* $$k = 2$$

---

# 왜 차수를 맞춰야할까?

* 차수는 그래프가 그려지는 공간의 차원을 결정
* 차수가 같아야 Blend 연산을 할 때 의미를 가짐

---

# Operator Precedence

$$/ \to * \to +$$

---

# 그래프의 세계로

---

# 1차원 그래프

---

# pop1980

```
COORD: rect(dim(1))
ELEMENT: point(position(pop1980))
```

![inline 150%](http://i.imgur.com/7uyoOzT.png)

---

# 2차원 그래프

---

# pop1980 * 1

```
COORD: rect(dim(1, 2))
ELEMENT: point(position(pop1980 * 1))
# FRAME = [0, 30] x unity
```

![inline 150%](http://i.imgur.com/7uyoOzT.png)

---

# pop1980 * pop2000

```
COORD: rect(dim(1,2))
ELEMENT: point(position(pop1980*pop2000))
# FRAME = [0, 30] x [0, 30]
```

![inline](http://i.imgur.com/Zl2sZnB.png)

---

# 3차원 그래프?

---

# 책의 예제 (1)

```
ELEMENT: point(position(city*pop1980*group))
# FRAME = (Tokyo, ...) x [0, 30] x (World, USA)
```

![inline](http://i.imgur.com/ZQR16Re.png)

---

# 책의 예제(2) - 좀 더 정확한 표현

```
COORD: rect(dim(3), rect(dim(1,2)))
ELEMENT: point(position(city*pop1980*group))
```

![inline](http://i.imgur.com/ZQR16Re.png)

---

# 3차원 그래프!

![inline](http://i.imgur.com/O49Mkrl.png)

---

# 앞의 예제를 3차원으로

```
COORD: rect(dim(1,2,3))
ELEMENT: point(position(city*pop1980*group))
# FRAME = (Tokyo, ...) x [0, 30] x (World, USA)
```

![inline](http://i.imgur.com/Mqr4xLw.jpg)


---

# Step by Step

## [fit] <br>`rect(dim(1,2,3)) to rect(dim(3), rect(dim(1,2)))`

---

# 2차원

![inline](http://i.imgur.com/LgrWS1k.jpg)

---

# 3차원

![inline](http://i.imgur.com/Mqr4xLw.jpg)

---

# World 평면으로 잘라내기

![inline](http://i.imgur.com/yxwxZGJ.jpg)

---

# USA 평면으로 잘라내기

![inline](http://i.imgur.com/Ud1vcMe.jpg)

---

# [fit] World, USA 평면을 따로 펼쳐놓기

![inline](http://i.imgur.com/KaHr7dE.jpg)

---

# `city * pop1980 * group`

* 같은 2차원 Frame(`city*pop1980`)의 평면들

```
COORD: rect(dim(3), rect(dim(1,2)))
ELEMENT: point(position(city*pop1980*group))
```

![inline](http://i.imgur.com/ZQR16Re.png)

---

# [fit] Nest 연산 $$/$$ 이해하기

---

# 책의 예제


```
ELEMENT: point(position(city/group*pop2000))
```

![inline](http://i.imgur.com/aMnhvlc.png)

---

# city/group

civy/group의 도메인

$$
V_{city/group} = \{ Tokyo:world, Mumbai:world, New York:usa, ...\}
$$

각 태그(그룹)으로 나눈 도메인

$$
V_{city_{usa}} = \{ New York:usa, Los Angeles:usa, ... \}
$$

$$
V_{city_{world}} = \{ Tokyo:world, Mumbai:world, ... \}
$$

---

# 서로 다른 2개의 평면!

## $$city_{usa} * pop2000$$
## $$city_{world} * pop2000$$

---

# $$city_{usa} * pop2000$$

![inline](http://i.imgur.com/7DhGKKG.png)

---

# $$city_{world} * pop2000$$

![inline](http://i.imgur.com/A0zg4of.png)

---

# Figure 5.6

```
DATA: p1980 = "1980"
DATA: p2000 = "2000"
ELEMENT: point(position(city*(pop1980+pop2000)), color(p1980 + p2000))
```

![inline](http://i.imgur.com/E1x6gST.png)

---

# Frame of Figure 5.6

* city*(pop1980+pop2000)

```
element: point(position(city*(pop1980+pop2000)))
# FRAME = {Tokyo, ...} x [0, 30]
```

* city*pop2000

```
element: point(position(city*pop2000))
# FRAME = {Tokyo, ...} x [0, 30]
```

---

# Color Blend Operator

![inline 100%](http://i.imgur.com/99X1MF1.png)

GoG 310p

---

# Figure 5.6, again

```
DATA: p1980 = "1980"
DATA: p2000 = "2000"
ELEMENT: point(position(city*(pop1980+pop2000)), color(p1980 + p2000))
```

![inline](http://i.imgur.com/E1x6gST.png)

---

# Figure 5.8

```
DATA: p1980 = "1980"
DATA: p2000 = "2000"
ELEMENT: point(position((city/group)*(pop1980+pop2000), color(p1980 + p2000))
```

![inline](http://i.imgur.com/b1NLx9D.png)

---

# Figure 5.9

```
DATA: p1980 = "Pop 1980 (millions)"
DATA: p2000 = "Pop 2000 (millions)"
ELEMENT: point(position(city*(pop1980/p1980+pop2000/p2000)))
```

![inline](http://i.imgur.com/mAXxVsW.png)

---

# [fit] ggplot2로 ch5 예제 재구현

## [fit] [GOG ch5 Varget Algebra Example - Gist](https://gist.github.com/nacyot/ba7eb605b71ae140690c)

---

# ggplot2와 GoG

* ggplot2
  * Varset Algebra 개념을 지원하지 않음
  * 시각화하는 data.frame을 미리 조작해야함

`scales = 'free'` 옵션으로 nest와 비슷하게 구현

```
ggplot(pop, aes(x = city2, y = pop2000_d)) + 
  geom_point(colour = I('darkblue')) +
  # scales = 'free'를 통해서 평면에서 공집합이 되는 (a, b) 카테고리를 제거 
  facet_grid(. ~ group, scales = 'free') +
  scale_y_continuous(limits=c(0, 30)) 
```

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# Thank you!

## [@nacyo_t](https://twitter.com/nacyo_t)