Introdução aos Modelos Lineares

.title[
# Introdução aos Modelos Lineares
]
.subtitle[
## com códigos em R
]
.author[
### <img src = 'https://d33wubrfki0l68.cloudfront.net/9b0699f18268059bdd2e5c21538a29eade7cbd2b/67e5c/img/logo/cursor1-5.png' width = '40%'>
]
.date[
### março de 2024
]

---

# Agenda

.pull-left[
- O que é e quando usar
- Parâmetro vs estimador
- Teste de Hipóteses e valor-p
- Interpretação dos parâmetros
- Regressão Linear Múltipla
- Preditores Categóricos
- Transformações Não Lineares dos Preditores

]

---

## Referências

- [Aprendizagem de Máquinas: Uma Abordagem Estatística (Rafael Izbicki e Thiago Mendonça, 2020)](http://www.rizbicki.ufscar.br/AME.pdf)

- [Introduction to Statistical Learning (Hastie, et al)](https://hastie.su.domains/ISLR2/ISLRv2_website.pdf)

- [Ciência de Dados: Fundamentos e Aplicações](https://curso-r.github.io/main-regressao-linear/referencias/Ci%C3%AAncia%20de%20Dados.%20Fundamentos%20e%20Aplica%C3%A7%C3%B5es.%20Vers%C3%A3o%20parcial%20preliminar.%20maio%20Pedro%20A.%20Morettin%20Julio%20M.%20Singer.pdf)

---

# Introdução

---

# Motivação

Somos consultores e fomos contratados para dar conselhos para uma empresa aumentar as suas vendas.

Obtivemos o seguinte banco de dados

* PERGUNTA: Jornal tem mais retorno do que as demais mídias? Quantas vendas terão se eu investir X em jornais?

---

# Motivação

Somos consultores e fomos contratados para dar conselhos para uma empresa aumentar as suas vendas.

Obtivemos o seguinte banco de dados

* PERGUNTA: Jornal tem mais retorno do que as demais mídias? Quantas vendas terão se eu investir X em jornais?

---

# Modo - Regressão e Classificação

Existem dois principais tipos de problemas em modelagem estatística:

## Regressão

__Y__ é uma variável quantitativa contínua

- Volume de vendas
- Peso
- Temperatura
- Valor de Ações

]

## Classificação

__Y__ é uma variável qualitativa discreta.

- Fraude/Não Fraude
- Pegou em dia/Não pagou
- Cancelou assinatura/Não cancelou
- Gato/Cachorro/Cavalo/Outro

]

---

# Exemplo de reta de regressão

---

# Definições e Nomenclaturas

### A tabela por trás (do excel, do sql, etc.)

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> midia </th>
   <th style="text-align:right;"> investimento </th>
   <th style="text-align:right;"> vendas </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> TV </td>
   <td style="text-align:right;"> 220.3 </td>
   <td style="text-align:right;"> 24.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> newspaper </td>
   <td style="text-align:right;"> 25.6 </td>
   <td style="text-align:right;"> 5.3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> newspaper </td>
   <td style="text-align:right;"> 38.7 </td>
   <td style="text-align:right;"> 18.3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> radio </td>
   <td style="text-align:right;"> 42.3 </td>
   <td style="text-align:right;"> 25.4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> radio </td>
   <td style="text-align:right;"> 43.9 </td>
   <td style="text-align:right;"> 22.3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> TV </td>
   <td style="text-align:right;"> 139.5 </td>
   <td style="text-align:right;"> 10.3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> radio </td>
   <td style="text-align:right;"> 11.0 </td>
   <td style="text-align:right;"> 7.2 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> radio </td>
   <td style="text-align:right;"> 1.6 </td>
   <td style="text-align:right;"> 6.9 </td>
  </tr>
</tbody>
</table>

---

# Definições e Nomenclaturas

* `$X_1$`, `$X_2$`, ..., `$X_p$`: variáveis explicativas (ou variáveis independentes ou *features* ou preditores).

- `$\boldsymbol{X} = {X_1, X_2, \dots, X_p}$`: conjunto de todas as *features*.

* __Y__: variável resposta (ou variável dependente ou *target*). 
* __Ŷ__: valor **esperado** (ou predição ou estimado ou *fitted*). 
* `$f(X)$` também é conhecida também como "Modelo" ou "Hipótese".

## No exemplo:

- `$X_1$`: `midia` - indicadador de se a propaganda é para jornal, rádio, ou TV.
- `$X_2$`: `investimento` - valor do orçamento

* __Y__: `vendas` - qtd vendida

---

# Definições e Nomenclaturas

### **Observado** *versus* **Esperado**

- __Y__ é um valor **observado** (ou verdade ou *truth*)
- __Ŷ__ é um valor **esperado** (ou predição ou estimado ou *fitted*). 
- __Y__ - __Ŷ__ é o resíduo (ou erro)

Por definição, `$\hat{Y} = f(x)$` que é o valor que a função `$f$` retorna.

---

# Definições e Nomenclaturas

### A tabela por trás depois das predições

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> midia </th>
   <th style="text-align:right;"> investimento </th>
   <th style="text-align:right;"> vendas </th>
   <th style="text-align:right;"> regressao_linear </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> TV </td>
   <td style="text-align:right;"> 220.3 </td>
   <td style="text-align:right;"> 24.7 </td>
   <td style="text-align:right;font-weight: bold;color: purple !important;"> 17.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> newspaper </td>
   <td style="text-align:right;"> 25.6 </td>
   <td style="text-align:right;"> 5.3 </td>
   <td style="text-align:right;font-weight: bold;color: purple !important;"> 13.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> newspaper </td>
   <td style="text-align:right;"> 38.7 </td>
   <td style="text-align:right;"> 18.3 </td>
   <td style="text-align:right;font-weight: bold;color: purple !important;"> 14.4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> radio </td>
   <td style="text-align:right;"> 42.3 </td>
   <td style="text-align:right;"> 25.4 </td>
   <td style="text-align:right;font-weight: bold;color: purple !important;"> 15.0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> radio </td>
   <td style="text-align:right;"> 43.9 </td>
   <td style="text-align:right;"> 22.3 </td>
   <td style="text-align:right;font-weight: bold;color: purple !important;"> 15.1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> TV </td>
   <td style="text-align:right;"> 139.5 </td>
   <td style="text-align:right;"> 10.3 </td>
   <td style="text-align:right;font-weight: bold;color: purple !important;"> 13.6 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> radio </td>
   <td style="text-align:right;"> 11.0 </td>
   <td style="text-align:right;"> 7.2 </td>
   <td style="text-align:right;font-weight: bold;color: purple !important;"> 13.4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> radio </td>
   <td style="text-align:right;"> 1.6 </td>
   <td style="text-align:right;"> 6.9 </td>
   <td style="text-align:right;font-weight: bold;color: purple !important;"> 12.9 </td>
  </tr>
</tbody>
</table>

---

# Por que ajustar uma f?

* Predição
* Inferência

## Predição

Em muitas situações X está disponível facilmente mas, Y não é fácil de descobrir. (Ou mesmo não é possível descobrí-lo). Queremos que `$\hat{Y} = \hat{f}(X)$` seja uma boa estimativa (preveja bem o futuro).
Neste caso não estamos interessados em como é a estrutura `$\hat{f}$` desde que ela apresente predições boas para `$Y$`.

Por exemplo:

* Meu cliente vai atrasar a fatura no mês que vem?

---

# Por que ajustar uma f?

* Predição
* Inferência

## Inferência

Em inferência estamos mais interessados em entender a relação entre as variáveis explicativas `$X$` e a variável resposta `$Y$`.

Por exemplo:

* A droga é eficaz para o tratamento da doença X? 
* **Quanto que é** o impacto nas vendas para cada real investido em TV?

Neste material focaremos em **inferência**.

---

# Por que ajustar uma f?

---

## O que é e quando usar

### Regressão Linear Simples

$$
y = \beta_0 + \beta_1x
$$

### Exemplo:

$$
dist = \beta_0 + \beta_1speed
$$

]

]

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 61 (Simple Linear Regression).
]

---

## O que é e quando usar

### Regressão Linear Simples

$$
y = \beta_0 + \beta_1x
$$

### Exemplo:

$$
dist = \beta_0 + \beta_1speed
$$

]

]

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 61 (Simple Linear Regression).
]

---

## O que é e quando usar

### Regressão Linear Simples

$$
y = \beta_0 + \beta_1x
$$

### Exemplo:

$$
dist = \beta_0 + \beta_1speed
$$

]

]

---

## O que é e quando usar

### Regressão Linear Simples

$$
y = \beta_0 + \beta_1x
$$

]

]

```r
# ajuste de uma regressão linear simples no R
*melhor_reta <- lm(dist ~ speed, data = cars)
melhor_reta
```

```
## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## Coefficients:
## (Intercept)        speed  
##     -17.579        3.932
```

---

## Métricas - "Melhor reta" segundo o quê?

Queremos a reta que **erre menos**.

Exemplo de medida de erro: **R**oot **M**ean **S**quared **E**rror.

$$
RMSE = \sqrt{\frac{1}{N}\sum(y_i - \hat{y_i})^2}
$$

---

## Métricas - "Melhor reta" segundo o quê?

Queremos a reta que **erre menos**.

Exemplo de medida de erro: **R**oot **M**ean **S**quared **E**rror.

$$
RMSE = \sqrt{\frac{1}{N}\sum(y_i - (\hat{\beta_0} + \hat{\beta_1}x))^2}
$$

Ou seja, nosso **objetivo** é

## Encontrar `$\hat{\beta}_0$` e `$\hat{\beta}_1$` que nos retorne o ~menor~ RMSE.

---

## Qual o valor ótimo para `$\beta_0$` e `$\beta_1$`?

No nosso exemplo, a nossa **HIPÓTESE** é de que

$$
dist = \beta_0 + \beta_1speed
$$

Então podemos escrever o Erro Quadrático Médio como

$$
EQM = \frac{1}{N}\sum(y_i - \hat{y_i})^2 = \frac{1}{N}\sum(y_i -  \color{red}{(\hat{\beta}_0 + \hat{\beta}_1speed)})^2 
$$

Com ajuda do Cálculo é possível mostrar que os valores ótimos para `$\beta_0$` e `$\beta_1$` são

`$\hat{\beta}_1 = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2}$`

`$\hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x}$`

```r
# lembrete: exercício 2 do script!
```
]

---

## Depois de estimar...

$$
\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x
$$

### Exemplo:

$$
\hat{dist} = \hat{\beta}_0 + \hat{\beta}_1speed
$$

Colocamos um `$\hat{}$` em cima dos termos para representar "estimativas". Ou seja, `$\hat{y}_i$` é uma estimativa de `$y_i$`.

No nosso exemplo,

- `$\hat{\beta}_0$` é uma estimativa de `$\beta_0$` e vale `-17.579`.
- `$\hat{\beta}_1$` é uma estimativa de `$\beta_1$` e vale `3.932`.
- `$\hat{dist}$` é uma estimativa de `$dist$` e vale `-17.579 + 3.932 x speed`.

```r
# Exercício: se speed for 15 m/h, quanto que 
# seria a distância dist esperada?
```

]

---

## Curiosidade - Método numérico

Podemos encontrar a reta que **erre menos** por meio de algoritmos numéricos.

Exemplo: Modelo de regressão linear `$f(x) = \beta_0 + \beta_1 x$`.

Fonte: [https://alykhantejani.github.io/images/gradient_descent_line_graph.gif](https://alykhantejani.github.io/images/gradient_descent_line_graph.gif)

]

---

## Teste de Hipóteses e valor-p

Exemplo: relação entre População Urbana e Assassinatos.

Modelo proposto:

`$$y = \beta_0 + \beta_1 x$$`

]

Hipótese do pesquisador: 
> "Assassinatos não estão relacionados com a proporção de população urbana de uma cidade."

Tradução da hipótese em termos matemáticos:

$$
H_0: \beta_1 = 0 \space\space\space\space\space vs \space\space\space\space H_a: \beta_1 \neq 0
$$
Se a hipótese for verdade, então o `$\beta_1$` deveria ser zero. Porém, os dados disseram que `$\hat{\beta}_1 = 0.02$`.

#### 0.02 é diferente de 0.00?

]

---

## 0.02 é diferente de 0.00?

Saída do R

```
## 
## Call:
## lm(formula = Murder ~ UrbanPop, data = USArrests)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -6.537 -3.736 -0.779  3.332  9.728 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  6.41594    2.90669   2.207   0.0321 *
*## UrbanPop     0.02093    0.04333   0.483   0.6312  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.39 on 48 degrees of freedom
## Multiple R-squared:  0.00484,	Adjusted R-squared:  -0.01589 
## F-statistic: 0.2335 on 1 and 48 DF,  p-value: 0.6312
```

]

---

## 0.02 é diferente de 0.00?

Conceito importante: Os estimadores ( `$\hat{\beta}_0$` e `$\hat{\beta}_1$` no nosso caso) têm distribuições de probabilidade.

### Simulação de 1000 retas (ajustadas com dados diferentes).

![distrib_params](static/img/combined_gif.gif)

---

## 0.02 é diferente de 0.00?

Conceito importante: Os estimadores ( `$\hat{\beta}_0$` e `$\hat{\beta}_1$` no nosso caso) têm distribuições de probabilidade.

### A Teoria Assintótica nos fornece o seguinte resultado:

`$t = \frac{\hat{\beta_1} - \beta_1}{\hat{\sigma}_{\beta_1}} \overset{\text{a}}{\sim}  t(N - 2)$`

#### Em que

`$\hat{\sigma}_{\beta_1} = \sqrt{\frac{EQM}{\sum(x_i - \bar{x})^2}}$`

Usamos essas distribuições assintóticas para testar as hipóteses.

]

]

---

## 0.02 é diferente de 0.00?

Conceito importante: Os estimadores ( `$\hat{\beta}_0$` e `$\hat{\beta}_1$` no nosso caso) têm distribuições de probabilidade.

### A Teoria Assintótica nos fornece o seguinte resultado:

`$t = \frac{\hat{\beta_1} - \beta_1}{\hat{\sigma}_{\beta_1}} \overset{\text{a}}{\sim}  t(N - 2)$`

#### Em que

`$\hat{\sigma}_{\beta_1} = \sqrt{\frac{EQM}{\sum(x_i - \bar{x})^2}}$`

Usamos essas distribuições assintóticas para testar as hipóteses.

]

No nosso exemplo, a hipótese é `$H_0: \beta_1 = 0$`, então

`$$t = \frac{0.02 - 0}{0.04} = 0.48$$`

]

---

## 0.02 é diferente de 0.00?

.letrinha[
Então agora podemos tomar decisão! Se a estimativa cair muito distante da distribuição t da hipótese 0, decidimos por **rejeitá-la**. Caso contrário, decidimos por **não rejeitá-la** como verdade.
]

![testes_t_estrelas](static/img/testes_t_estrelas.png)

```
## NO R:
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  6.41594    2.90669   2.207   0.0321 *
## UrbanPop     0.02093    0.04333   0.483   0.6312  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]

---

## Interpretação dos parâmetros

$$
y = \color{darkgblue}{\beta_0} + \color{darkgreen}{\beta_1}x
$$

]

### Interpretações matemáticas

`$\color{darkgblue}{\beta_0}$` é o lugar em que a reta cruza o eixo Y.

`$\color{darkgreen}{\beta_1}$` é a derivada de Y em relação ao X. É quanto Y varia quando X varia em 1 unidade.

### Interpretações estatísticas

`$\color{darkgblue}{\beta_0}$` é a distância percorrida esperada quando o carro está parado (X = 0).

`$\color{darkgreen}{\beta_1}$` é o efeito médio na distância por variar 1 ml/h na velocidade do carro.

]

---

## Teste de Hipóteses e valor-p

### Exercício 3 do script:

No R, use a função `summary(melhor_reta)` (ver slide 9) para decidir se `speed` está associado com `dist`. Descubra o valor-p associado.

lembrete: o banco de dados se chama `cars`.

### Exercício 4 do script:

Interprete o parâmetro `$\beta_1$`.

---

## Intervalo de confiança para `$\beta_1$`

#### Intervalo de 95%
`$$[\hat{\beta} - 1,96 * \hat{\sigma}_{\beta_1}, \hat{\beta} + 1,96 * \hat{\sigma}_{\beta_1}]$$`

#### Intervalo de 90%
`$$[\hat{\beta} - 1,64 * \hat{\sigma}_{\beta_1}, \hat{\beta} + 1,64 * \hat{\sigma}_{\beta_1}]$$`

#### Intervalo de 1 - `$\alpha$`%

`$$[\hat{\beta}_1 - q_{\alpha} * \hat{\sigma}_{\beta_1}, \hat{\beta} + q_{\alpha} * \hat{\sigma}_{\beta_1}]$$`

em que `$q_\alpha$` é o quantil da `$Normal(0, 1)$`.

.letrinha[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 66 (Assessing the Accuracy of the Model).
]

---

## O modelo está bom?

### EQM e EPR

ERP significa *E*rro *P*adrão dos *R*esíduos e é definido como

$$
EPR = \frac{\sum(y_i - \hat{y_i})^2}{N - 2} = \frac{SQR}{N - 2}
$$
O **2** no denominador decorre do fato de termos **2 parâmetros** para estimar no modelo.

- Se `$y_i = \hat{y}_i \space\space\space \rightarrow \color{green}{EPR = 0 \downarrow}$`
- Se `$y_i >> \hat{y}_i \rightarrow \color{red}{EPR = alto \uparrow}$`
- Se `$y_i << \hat{y}_i \rightarrow \color{red}{EPR = alto \uparrow}$`

Problema: Como sabemos se o EPR é grande ou pequeno?

.letrinha[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 68 (Assessing the Accuracy of the Model).

]

---

## O modelo está bom?

### R-quadrado ( `$R^2$` )

$$
R^2 = 1 - \frac{\sum(y_i - \color{salmon}{\hat{y_i}})^2}{\sum(y_i - \color{royalblue}{\bar{y}})^2} = 1 - \frac{\color{salmon}{SQR}}{\color{royalblue}{SQT}}
$$

]

`$R^2 \approx 1 \rightarrow \color{salmon}{SQR} << \color{royalblue}{SQT}$`.
`$R^2 \approx 0 \rightarrow \color{salmon}{reta} \text{ em cima da } \color{royalblue}{reta}$`.

Problema do `$R^2$` é que ele sempre aumenta conforme novos preditores vão sendo incluídos.

]

.letrinha[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 68 (Assessing the Accuracy of the Model).
]

---

## O modelo está bom?

### R-quadrado ajustado

$$
R^2 = 1 - \frac{\color{salmon}{SQR}}{\color{royalblue}{SQT}}\frac{\color{royalblue}{N-1}}{\color{salmon}{N-p}}
$$

Em que `$p$` é o número de parâmetros do modelo (no caso da regressão linear simples, `$p = 2$`).

```r
# lembrete: exercícios 5 e 6 do script!
```

---

# Regressão Linear Múltipla

### Regressão Linear Simples

$$
y = \beta_0 + \beta_1x
$$

### Exemplo:

$$
dist = \beta_0 + \beta_1speed
$$

```r
### No R:
lm(dist ~ speed, data=cars)
```

]

.letrinha[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 61 (Simple Linear Regression).
]

]

---

# Regressão Linear Múltipla

### Regressão Linear Múltipla

$$
y = \beta_0 + \beta_1x_1 + \dots + \beta_px_p
$$

### Exemplo:

$$
mpg = \beta_0 + \beta_1wt + \beta_2disp
$$

```r
### No R:
lm(mpg ~ wt + disp, data=mtcars)
```

]

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]

---

## Regressão Linear Múltipla

### Regressão Linear Simples

$$
y = \beta_0 + \beta_1x
$$

### Regressão Linear Múltipla

$$
y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \dots + \beta_px_p
$$

```r
# ajuste de uma regressão linear múltipla no R
*modelo_boston <- lm(medv ~ lstat + age, data = Boston)
summary(modelo_boston)
#             Estimate Std.Error t value Pr(>|t|)    
# (Intercept) 33.22    0.73      45.4    < 2e-16 ***
# lstat       -1.03    0.04     -21.4    < 2e-16 ***
# age          0.03    0.01       2.8    0.00491 ** 
```

---

## Preditores Categóricos

### Preditor com apenas 2 categorias

Tamanho das nadadeiras de pinguins são diferentes entre os sexos?

```r
summary(lm(flipper_length_mm ~ sex, data = penguins))
# Coefficients:
#             Estimate Std. Error t value Pr(>|t|)    
# (Intercept)  197.364      1.057 186.792  < 2e-16 ***
# sexmale        7.142      1.488   4.801 2.39e-06 ***
```

---

## Preditores Categóricos

### Preditor com apenas 2 categorias

Tamanho das nadadeiras de pinguins são diferentes entre os sexos?

$$
y_i = \beta_0 + \beta_1x_i \space\space\space\space\space\space \text{em que}\space\space\space\space\space\space x_i = \Bigg\\{\begin{array}{ll}1&\text{se a i-ésimo animal for }\texttt{female}\\\\
0&\text{se a i-ésimo animal for } \texttt{male}\end{array}
$$

```r
# lembrete: exercícios 8, 9 e 10 do script!
```

Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 84 (Predictors with Only Two Levels).
]

---

## Preditores Categóricos

### Preditor com 3 ou mais categorias

```r
summary(lm(flipper_length_mm ~ species, data = penguins))
#                  Estimate Std. Error t value Pr(>|t|)    
# (Intercept)      189.9536     0.5405 351.454  < 2e-16 ***
# speciesChinstrap   5.8699     0.9699   6.052 3.79e-09 ***
# speciesGentoo     27.2333     0.8067  33.760  < 2e-16 ***
```

]

Modelo

`$$y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i}$$`

Em que

`$x_{1i} = \Bigg \{ \begin{array}{ll} 1 & \text{se for }\texttt{Chinstrap}\\0&\text{caso contrário}\end{array}$`

`$x_{2i} = \Bigg \{ \begin{array}{ll} 1 & \text{se for }\texttt{Gentoo}\\0&\text{caso contrário}\end{array}$`

]

---

## Preditores Categóricos

### Preditor com 3 ou mais categorias

"One hot enconding" ou "Dummies" ou "Indicadores".

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> species </th>
   <th style="text-align:right;"> (Intercept) </th>
   <th style="text-align:right;"> speciesChinstrap </th>
   <th style="text-align:right;"> speciesGentoo </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Adelie </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Chinstrap </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Gentoo </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Adelie </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Gentoo </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Adelie </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Adelie </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Chinstrap </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

---

## Preditores Categóricos

### Preditor com 3 ou mais categorias

Interpretação dos parâmetros:

`$y_{i} = \left\{ \begin{array}{ll} \beta_0 & \text{se for }\texttt{Adelie}\\ \beta_0 + \beta_1&\text{se for } \texttt{Chinstrap}\\ \beta_0 + \beta_2&\text{se for } \texttt{Gentoo}\end{array}\right.$`

```r
# interprete cada um dos três parâmetros individualmente.
# lembrete: exercício 11 do script!
```

---

## Transformações Não Lineares dos Preditores

### Exemplo: log

Outras transformações comuns: raíz quadrada, polinômios, Box-Cox, ...

```r
# lembrete: exercício 13 do script!
```
]

---

## Transformações Não Lineares dos Preditores

### Exemplo: log

Outras transformações comuns: raíz quadrada, polinômios, Box-Cox, ...

```r
# lembrete: exercício 13 do script!
```

---

## Transformações Não Lineares dos Preditores

### Gráfico de Resíduos

---

## Transformações Não Lineares dos Preditores

### Exemplo: Regressão Polinomial

]

]

```r
# lembrete: exercício 14 do script!
```

]

---

## Transformações Não Lineares dos Preditores

### Exemplo: Regressão Polinomial

]

]

```r
# lembrete: exercício 14 do script!
```
]

---

## Transformações Não Lineares dos Preditores

### Gráfico de Resíduos

---

## Interações

Interação entre duas variáveis explicativas: `Species` e `Sepal.Length`

---

## Interações

Modelo proposto (Matemático): Seja `y = Sepal.Width` e `x = Sepal.Length`,

`$$\small \begin{array}{l} y = \beta_0 + \beta_1x\end{array}$$`

Modelo proposto (em R): `Sepal.Width ~ Sepal.Length`

```r
# lembrete: exercícios 14 ao 17 do script!
```

---

## Interações

Modelo proposto (Matemático): Seja `y = Sepal.Width` e `x = Sepal.Length`,

`$$\small \begin{array}{l} y = \beta_0 + \beta_1x + \beta_2I_{versicolor} + \beta_3I_{virginica}\end{array}$$`

Modelo proposto (em R): `Sepal.Width ~ Sepal.Length + Species`

```r
# lembrete: exercícios 14 ao 17 do script!
```

---

## Interações

Modelo proposto (Matemático): Seja `y = Sepal.Width` e `x = Sepal.Length`,

`$$\small \begin{array}{l} y = \beta_0 + \beta_1x + \beta_2I_{versicolor} + \beta_3I_{virginica} + \beta_4\color{red}{xI_{versicolor}} + \beta_5\color{red}{xI_{virginica}}\end{array}$$`

Modelo proposto (em R): `Sepal.Width ~ Sepal.Length * Species`

```r
# lembrete: exercícios 14 ao 17 do script!
```

---

## Heterocedasticidade

---

## Heterocedasticidade

---

## Heterocedasticidade

---

## Heterocedasticidade

### Problema

- O estimador 
`$\hat{\sigma}_{\beta_1} = \sqrt{\frac{EQM}{\sum(x_i - \bar{x})^2}}$` deixa de ter as melhores propriedades. Poderíamos ter conclusões estranhas para `$\beta_1$`.

### Diagnóstico

- Visualização dos resíduos;
- Testes formais (Breuch-Pagan, White, etc)

### Tratamentos

- Transformações na variável resposta. log(y), sqrt(y), 1/y, etc;
- Mínimos Quadrados Ponderados/Generalizados

---

## Multicolinearidade

]

Modelo 1: sem colineares

<table class="table" style="font-size: 13px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:left;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> -173.41 </td>
   <td style="text-align:left;"> <span style="     color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;text-align: r;">43.83</span> </td>
   <td style="text-align:right;"> -3.96 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Limit </td>
   <td style="text-align:right;"> 0.17 </td>
   <td style="text-align:left;"> <span style="     color: white !important;padding-right: 4px; padding-left: 4px; background-color: darkred !important;text-align: r;">0.01</span> </td>
   <td style="text-align:right;"> 34.50 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Age </td>
   <td style="text-align:right;"> -2.29 </td>
   <td style="text-align:left;"> <span style="     color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;text-align: r;">0.67</span> </td>
   <td style="text-align:right;"> -3.41 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

Modelo 2: com colineares

<table class="table" style="font-size: 13px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:left;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> -377.54 </td>
   <td style="text-align:left;"> <span style="     color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;text-align: r;">45.25</span> </td>
   <td style="text-align:right;"> -8.34 </td>
   <td style="text-align:right;"> 0.00 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Limit </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:left;"> <span style="     color: white !important;padding-right: 4px; padding-left: 4px; background-color: darkred !important;text-align: r;">0.06</span> </td>
   <td style="text-align:right;"> 0.38 </td>
   <td style="text-align:right;"> 0.70 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Rating </td>
   <td style="text-align:right;"> 2.20 </td>
   <td style="text-align:left;"> <span style="     color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;text-align: r;">0.95</span> </td>
   <td style="text-align:right;"> 2.31 </td>
   <td style="text-align:right;"> 0.02 </td>
  </tr>
</tbody>
</table>

]

Problema: Instabilidade numérica, desvios padrão inflados e interpretação comprometida.

Soluções: eliminar uma das variáveis muito correlacionadas ou Consultar o VIF (Variance Inflation Factor)

---

## Multicolinearidade

### VIF (Variance Inflation Factor)

Detecta preditores que são combinações lineares de outros preditores.

**Procedimento:** Para cada preditor `$X_j$`,

1) Ajusta regressão linear com as demais: `lm(X_j ~ X_1 + ... + X_p)`.

2) Calcula-se o R-quadrado dessa regressão e aplica a fórmula abaixo

`$$\small VIF(\hat{\beta}_j) = \frac{1}{1 - R^2_{X_j|X_{-j}}}$$`

3) Remova o preditor se VIF maior que 5 (regra de bolso).

```r
# lembrete: exercícios 18 do script!
```

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 101.
]

---

## Limitações da Regressão Linear

- Variável resposta Não Normal
- Variável resposta Positiva
- Variável resposta com mais de duas categorias
- Relação funcional não linear entre X e Y
- Muitas variáveis disponíveis
- Muitas interações para testar entre as preditoras