👨‍🏫 Correlação e Regressão

# 👨‍🏫 Correlação e Regressão
## 🔗 <a href="https://ap.metodosquantitativos.com/">Aula Correlação</a>
### Steven Dutt Ross
### UNIRIO

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## .hand[O Diagrama de dispersão]

.hand[O Diagrama de dispersão é um gráfico onde pontos no espaço cartesiano (X,Y) são usados para representar simultaneamente os valores de duas variáveis quantitativas medidas em cada elemento do conjunto de dados.]

---

---

```r
x <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) 
y <- c(100, 200,300,400,500,600, 700, 800, 900, 1000) 
par(mfrow=c(1,2))
plot(x,y, col = "red", pch=21,lwd = 10) 
y <- c(210, 50,280,400,590,540, 730, 770, 800, 1100)
plot(x,y, col = "blue",pch=21, lwd = 10) 
```

```r
par(mfrow=c(1,1))
```

---

```r
idade = c(56, 30, 40, 32, 39, 23, 17, 20, 28, 16)
qtdmiojo = c(17, 29, 27, 35, 27, 56, 58, 54, 50, 38)
datacor = data.frame(qtdmiojo,idade)
datacor
```

```
##    qtdmiojo idade
## 1        17    56
## 2        29    30
## 3        27    40
## 4        35    32
## 5        27    39
## 6        56    23
## 7        58    17
## 8        54    20
## 9        50    28
## 10       38    16
```

---

Este diagrama de dispersão tem um padrão linear geral (reta), mas a relação é negativa.

![](index_files/figure-html/idade3-1.png)

---

Essa padrão linear é facil de ver.

![](index_files/figure-html/idade4-1.png)

---

```r
#Outro exemplo: pena de morte e aborto
penademorte <- c(7,7,3,0,0,10,5,7,0,10,1,8,1,9,8,8,4,10,10,9)
aborto <- c(1,2,7,7,9, 3,9,5,8,4,10,3,9,2,2,6,8,6,6,8)
plot(penademorte,aborto,col = "darkblue",pch=21)
abline(lsfit(penademorte,aborto),col="darkred")
```

![](index_files/figure-html/aborto-1.png)

---

Quais variáveis têm padrão linear positivo? quais têm padrão linear negativo? Quais não tem um padrão (padrão nulo)?

![](index_files/figure-html/tipo-1.png)

---
class: center, middle

---

## O que é correlação?

A Correlação mede a **direção** (positivas ou negativas) e a **intensidade** (força) da relação linear entre duas variáveis quantitativas (relacionamento entre duas variáveis). Costuma-se representar a correlação pela letra *r*.

---

## Fatos sobre a correlação

**A correlação não faz distinção entre variável explicativa e variável resposta** Não faz diferença alguma qual variável você chama de *x* e qual você chama de *y*, ao calcular a correlação.
 
 **r positivo indica uma associação positiva entre as variáveis**, e r negativo indica uma associação negativa.
 
 **A correlação é sempre um número entre -1 e 1.** Valores próximos de zero indicam uma relação linear muito fraca. A intensidade da relação linear cresce, à medida que *r* se afasta de zero em direção a -1 ou 1. Os valores de *r* próximos de -1 ou 1 indicam que os pontos num diagrama de dispersão caem próximos de uma reta. Os valores extremos *r= -1* e *r= 1* ocorrem apenas no caso de relação linear perfeita , quando os pontos caem exatamente sobre a reta.

---

## Coeficiente de correlação produto-momento de Pearson (r)
 Mede a intensidade e a direção da relação entre duas variáveis contínuas

## Tipos de correlações

**Correlação de Pearson** para variáveis continuas
 **Correlação de Spearman** para variáveis ordinais

[Para saber mais clique aqui](http://rstudio-pubs-static.s3.amazonaws.com/10539_9a0d69971efd414d96bfb4b8cc20e76f.html#/6)

[Fonte](http://www.seer.ufu.br/index.php/cieng/issue/view/108)

---

### Interpretação do r - Valores de referência	
#### Negativo 
 -0,2 < r < 0 baixa ou nenhuma associação	
 -0,7 < r < -0,2 	grau fraco/moderado de associação 
 < -0,7 grau excelente de associação
#### Positivo 
 0 < r < 0,2 baixa ou nenhuma associação	(ou -0,2 < r < 0)
 0,2 < r < 0,7 	grau fraco/moderado de associação (ou -0,7 < r < -0,2)
 r > 0,7 grau excelente de associação (ou <-0,7)

---

---

## Fórmula da correlação de Pearson

$$ \Huge{ r =\frac{COV(x,y)}{S_xS_y}  }$$
Onde:
* COV = covariância
* S = Desvio-padrão
* Covariância é o número que reflete o grau em que duas variáveis variam juntas.

$$ \Huge{COV = \frac{\Sigma(X - \overline{X})(Y - \overline{Y})}{N - 1}} $$

---

## Fórmula alternativa

`$$\Huge{r = \frac{n*\Sigma(X*Y) - \Sigma(X)*\Sigma(Y)}{\sqrt{n*\Sigma(X)^2-(\Sigma(X))^2}\sqrt{n*\Sigma(Y)^2-(\Sigma(Y))^2}}}$$`

## Como aplicar a Fórmula em um conjunto de dados?

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## Banco de dados
#### para o cálculo da correlação

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## Variáveis originais e cálculos intermediários

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---
class: center, middle

---

$$ \Huge{r = \frac{n\cdot\Sigma(X\cdot Y) -\Sigma(X)*\Sigma(Y)}{\sqrt{n\cdot\Sigma(X)^2-(\Sigma(X))^2}\sqrt{n\cdot\Sigma(Y)^2-(\Sigma(Y))^2}}} $$

$$ \Huge{r = \frac{8(447) -40\cdot80}{\sqrt{8\cdot228-(40)^2}\sqrt{8\cdot(882)^2-((80))^2}}} $$

$$ \Huge{r = \frac{3576-3200}{\sqrt{1824-1600}\sqrt{7056-6400)}}}  $$

$$ \Huge{r = \frac{376}{383,33}= 0,981 }$$

```r
dados <-data.frame(x=c(2,3,4,5,5,6,7,8),
 y=c(4,7,9,10,11,11,13,15))
cor(dados$x,dados$y)
```

---

## Correlação entre idade e miojo

```r
cor(datacor$idade, datacor$qtdmiojo)
```

```
## [1] -0.8262782
```

```r
plot(datacor$idade, datacor$qtdmiojo)
abline(lsfit(datacor$idade, datacor$qtdmiojo),col="darkred")
```

![](index_files/figure-html/Coeficientes2-1.png)

---

# Correlação de Pearson
Correlação entre as variáveis Kmporlitro e HP e as variáveis Kmporlitro e Peso

```r
cor(CARROS$Kmporlitro,CARROS$HP)
```

```
## [1] -0.7761684
```

```r
cor(CARROS$Kmporlitro,CARROS$Peso)
```

```
## [1] -0.8676594
```

---

##  Correlação de Spearman

```r
var1 = c(10, 9, 5, 6, 7)
var2 = c(3, 6, 10, 5, 4)
cor(var1,  var2, method="spearman")
```

```
## [1] -0.7
```

---

### Na prática, fazemos uma matriz com todas as correlações.

```r
animais = c(10, 13, 14, 11, 10, 17, 10, 7, 12, 13)
frutas = c(11, 11, 14, 9, 7, 14, 9, 4, 13, 12)
fas = c(3, 20, 27, 26, 16, 41, 34, 13, 31, 38)
dados.fv = data.frame(animais, frutas, fas)
#cor(dados.fv)  
pairs(dados.fv)
```

![](index_files/figure-html/Coeficientes4-1.png)

---

## Na prática, fazemos uma matriz com todas as correlações.

```r
cor(CARROS[,c("Preco","RPM","HP","Kmporlitro","Amperagem_circ_eletrico","Peso")])
```

```
##                              Preco         RPM         HP Kmporlitro
## Preco                    1.0000000 -0.43369788  0.7909486 -0.8475514
## RPM                     -0.4336979  1.00000000 -0.7082234  0.4186840
## HP                       0.7909486 -0.70822339  1.0000000 -0.7761684
## Kmporlitro              -0.8475514  0.41868403 -0.7761684  1.0000000
## Amperagem_circ_eletrico -0.7102139  0.09120476 -0.4487591  0.6811719
## Peso                     0.8879799 -0.17471588  0.6587479 -0.8676594
##                         Amperagem_circ_eletrico       Peso
## Preco                               -0.71021393  0.8879799
## RPM                                  0.09120476 -0.1747159
## HP                                  -0.44875912  0.6587479
## Kmporlitro                           0.68117191 -0.8676594
## Amperagem_circ_eletrico              1.00000000 -0.7124406
## Peso                                -0.71244065  1.0000000
```

---

# No entanto, hoje em dia podemos construir uma visualização de dados dessa matriz.

---

## Visualização da Matriz de Correlação
![](index_files/figure-html/Coeficientes7-1.png)

---