Introdução ao Machine Learning com R

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# Introdução ao Machine Learning com R
### <img src = 'https://d33wubrfki0l68.cloudfront.net/9b0699f18268059bdd2e5c21538a29eade7cbd2b/67e5c/img/logo/cursor1-5.png' width = '40%'>
### abril de 2021

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# Ciência de dados

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# Referências

.pull-left[
<a href = "http://www-bcf.usc.edu/~gareth/ISL/">
<img src="static/img/islr.png" style=" display: block; margin-left: auto; margin-right: auto;"></img>
</a>
]

.pull-right[
<a href = "https://web.stanford.edu/~hastie/Papers/ESLII.pdf">
<img src="static/img/esl.jpg" width = 58% style=" display: block; margin-left: auto; margin-right: auto;"></img>
</a>
]

---

# Referências

.pull-left[
<a href = "https://r4ds.had.co.nz/">
<img src="static/img/r4ds.png"  style=" display: block; margin-left: auto; margin-right: auto;"></img>
</a>
]

.pull-right[
<a href = "https://www.tidymodels.org/">
<img src="static/img/tidymodels.png" width = 75% style=" display: block; margin-left: auto; margin-right: auto;"></img>
</a>
]

---

class: middle, center, inverse

# Introdução

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# O que é Machine Learning?

<br>

- Termo criado por Arthur Samuel, em 1959

- Machine Learning é o campo de estudo que dá aos computadores a habilidade de aprenderem sem serem explicitamente programados (Arthur Samuel, 1959).

- Modelagem preditiva é um framework de análise de dados que visa gerar a estimativa mais precisa possível para uma quantidade ou fenômeno (Max Kuhn, 2014).

---

# Motivação

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

Obtivemos o seguinte banco de dados

* PERGUNTA 1: Se eu investir `$X_1$` em propaganda numa certa mídia `$X_2$`, quanto vou vender, que vamos representar pela letra$Y$?

---

# Motivação

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

Obtivemos o seguinte banco de dados

* PERGUNTA 1: Se eu investir `$X_1$` em propaganda numa certa mídia `$X_2$`, quanto vou vender, que vamos representar pela letra$Y$?
* PERGUNTA 2: Como um computador pode captar essa relação?

---

# Definições e Nomenclaturas

### A tabela por trás

<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>

---

# Machine Learning

Se eu investir `$X_1$` em propaganda numa certa mídia `$X_2$`, quanto vou vender, que vamos representar pela letra$Y$?

Para responder esse pergunta matematicamente, queremos encontrar uma função `$f()$`, e mais especificamente que o **computador** encontre, uma função tal que:

---

# 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

---

# Exemplos de f(x)

---

# Exemplos de f(x)

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# Por que ajustar uma f?

* Predição
* Inferência

## Inferência

Em inferência estamos mais interessados em entender a relação entre as variáveis que conseguimos alterar (o investimento `$X$`) e a variável que queremos prever (a venda `$Y$`).

Por exemplo:

* Qual tipo de mídia influencia mais o `$Y$`?
* Existe alguma mídia em que o investimento praticamente não faz efeito?

Neste curso, vamos falar principalmente sobre **predição**.

---

# 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). Então nós usamos `$f(X)$` como estimativa.

Neste caso não importa se `$f(X)$` é uma reta ou uma escadinha desde que ela apresente predições boas para `$Y$`.

---

background-image: url(static/img/usos_do_ml.png)
background-position: left 140px
background-size: contain

# Por que ajustar uma f?

---

# Definições e Nomenclaturas

`$f(X) \sim Y$` quer dizer que `$f(X)$` está próximo de `$Y$`. Nós qualificamos essa aproximação comparando o dado observado com o dado 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, `$\hat{Y} = f(x)$` é o valor que a função `$f$` retorna.

---

# Definições e Nomenclaturas

### A tabela por trás

<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;"> arvore </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;"> 18.1 </td>
   <td style="text-align:right;"> 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;"> 12.2 </td>
   <td style="text-align:right;"> 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;"> 14.9 </td>
   <td style="text-align:right;"> 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;"> 21.9 </td>
   <td style="text-align:right;"> 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;"> 16.8 </td>
   <td style="text-align:right;"> 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;"> 14.2 </td>
   <td style="text-align:right;"> 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;"> 12.2 </td>
   <td style="text-align:right;"> 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;"> 12.2 </td>
   <td style="text-align:right;"> 12.9 </td>
  </tr>
</tbody>
</table>

---

# Métricas - quão boa é a aproximação de `$f(x)$`?

Queremos medir o erro cometido por `$f(x)$`.

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

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

Ou seja, nosso **objetivo** pode ser

## Encontrar `$f(x)$` que nos retorne o ~menor~ RMSE.

---

# Métricas - quão boa é a aproximação de `$f(x)$`?

Encontrar a reta que **erra menos** é um problema muito conhecido.

Exemplo: Modelo de regressão linear `$y = b + m x$`, precisamos procurar os `$b$` e `$m$` que nos dão os melhores erros.

![](static/img/0_D7zG46WrdKx54pbU.gif)

.footnote[

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

]

---

# Métricas - "Melhor f(x)" segundo o quê?

Queremos a `$f(x)$` que **erre menos**.

Exemplos de medida de erro: **R**oot **M**ean **S**quared **E**rror não é a única.

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

.pull-left[

MAE: Mean Absolute Error

$$
MAE = \frac{1}{N}\sum|y_i - \hat{y_i}|
$$

]

.pull-right[

R2: R-squared

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

---

## Um pouco sobre R-quadrado ( `$R^2$` )

Muitas medidas de erro, que são gerais, são motivadas por ideias da regressão linear:

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

.pull-left[

]

.pull-right[

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

`$R^2$` pode ser calculado para qualquer tipo de modelo.

]

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

---

# Flexibilidade ou Interpretabilidade de modelos de Machine Learning

---

# Overfitting (sobreajuste)

Procurar a `$f(x)$` que se ajuste perfeitamente aos dados é uma boa estratégia?

Intuição (TREINO: dados observados primeiro, TESTE: dados novos)

![scatter_eqm](static/img/overfiting_scatter_eqm.gif)

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

---

# Overfitting (sobreajuste)

---

# Overfitting (sobreajuste)

---

# Overfitting (sobreajuste)

---

# Overfitting (sobreajuste)

---

# Dados novos vs antigos

- **Base de Treino** (dados antigos): a base de histórico que usamos para ajustar o modelo.

- **Base de Teste** (dados novos): a base que irá simular a chegada de dados novos, "em produção".

.pull-left[

```r
initial_split(dados, prop=3/4)
```

> "Quanto mais complexo for o modelo, menor será o **erro de treino.**"

> "Porém, o que importa é o **erro de teste**."

]

.pull-right[
<img src="static/img/erro_treino_erro_teste.png" width = "500px">

]

---

# Dados novos vs antigos

## Estratégia

### 1) Separar inicialmente a base de dados em duas: treino e teste.

```r
initial_split(dados, prop=3/4) # 3/4 da base será de treino
```

A base de teste que só será tocada quando a modelagem terminar. Ela nunca deverá influenciar as decisões que tomamos no período da modelagem.

---

# Hiperparâmetros

São parâmetros que têm que ser definidos antes de ajustar o modelo. Não há como achar o valor ótimo diretamente nas funções de custo. Precisam ser achados "na força bruta".

Exemplo: `tree_depth` das árvores (profundidade das árvores)

.pull-left[

```
decision_tree(tree_depth = 2)
```

]

.pull-right[

]

---

# Hiperparâmetros

São parâmetros que têm que ser definidos antes de ajustar o modelo. Não há como achar o valor ótimo diretamente nas funções de custo. Precisam ser achados "na força bruta".

Exemplo: `tree_depth` das árvores (profundidade das árvores)

.pull-left[

```
decision_tree(tree_depth = 5)
```

]

.pull-right[

]

---

# Cross-validation (validação cruzada)

**O que Validação cruzada faz:** estima (muito bem) o erro de predição.
**Objetivo da Validação cruzada:** encontrar o melhor conjunto de hiperparâmetros.

## Estratégia

.pull-left[

1) Dividir o banco de dados em K partes. (Por ex, K = 5 como na figura)

2) Ajustar o mesmo modelo K vezes, deixar sempre um pedaço de fora para servir de base de teste.

3) Teremos K valores de erros de teste. Tira-se a média dos erros.

]

.pull-right[
<img src="static/img/k-fold-cv.png">

---------------------------------> linhas
]

---

# Cross-validation (validação cruzada)

```
## #  5-fold cross-validation 
## # A tibble: 5 x 6
##   splits          id    n_treino n_teste regressao rmse_teste
##   <list>          <chr>    <dbl>   <dbl> <list>         <dbl>
## 1 <split [40/10]> Fold1       40      10 <lm>            12.0
## 2 <split [40/10]> Fold2       40      10 <lm>            21.4
## 3 <split [40/10]> Fold3       40      10 <lm>            16.6
## 4 <split [40/10]> Fold4       40      10 <lm>            11.3
## 5 <split [40/10]> Fold5       40      10 <lm>            13.8
```

ERRO DE VALIDAÇÃO CRUZADA: `$$RMSE_{cv} = \sqrt{\frac{1}{5}\sum_{i=1}^{5}RMSE_{Fold_i}} = 3,88$$`

---

# Cross-validation (validação cruzada)

### Esquema das divisões de bases:

.footnote[
Fonte: [bookdown.org/max/FES/resampling.html](https://bookdown.org/max/FES/resampling.html)
]

---

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background-image: url(static/img/ml_101.png)
background-position: left 140px
background-size: contain

# Resumo dos conceitos

---

---

# Regressão Linear

.pull-left[

### Regressão Linear Simples

$$
y = \beta_0 + \beta_1x
$$

### Exemplo:

$$
dist = \beta_0 + \beta_1speed
$$

]

.pull-right[

]

### No R:

```r
linear_reg() %>% 
  fit(dist ~ speed, data=cars)
```

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

---

# Regressão Linear

.pull-left[

### Regressão Linear Múltipla

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

### Exemplo:

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

]

.pull-right[

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]

### No R:

```r
linear_reg() %>% 
  fit(disp ~ wt + mpg, data=mtcars)
```

.footnote[
Fonte: [sthda.com/impressive-package-for-3d](http://www.sthda.com/english/wiki/impressive-package-for-3d-and-4d-graph-r-software-and-data-visualization)
]

---

# Regressão Linear - "Melhor Reta"

Queremos a reta que **erre menos**.

Modelo: `$y = b + m x$`

![](static/img/0_D7zG46WrdKx54pbU.gif)

.footnote[

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

]

---

# Regressão Linear - "Melhor Reta"

Queremos a reta que **erre menos**.

Uma medida de erro: RMSE

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

Ou seja, nosso é **encontrar os `$\hat{\beta}'s$` que nos retorne o ~menor~ RMSE.**

#### IMPORTANTE!

o RMSE é a nossa **Função de Custo** e pode ser diferente da **Métrica** que vimos nos slides anteriores!

- **Função de Custo** - para encontrar os melhores parâmetros.
- **Métrica** - para encontrar os melhores hiperparâmetros.

---

## 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 RMSE

$$
RMSE = \sqrt{\frac{1}{N}\sum(y_i - \hat{y_i})^2} = \sqrt{\frac{1}{N}\sum(y_i -  \color{red}{(\hat{\beta}_0 + \hat{\beta}_1speed)})^2} 
$$
.pull-left[
Método mais utilizado para otimizar modelos com parâmetros: **Gradient Descent**

Ver [Wikipedia do Gradient Descent](https://en.wikipedia.org/wiki/Gradient_descent)

]

.pull-right[
<img src = "static/img/gradient_descent.png" width = 65%>
]

---

## 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 RMSE

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

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

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

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

OBS: Já que vieram do EQM, eles são chamados de **Estimadores de Mínimos Quadrados**.

---

## 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?
```

---

## Interpretação dos parâmetros

.pull-left[

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

]

.pull-right[

### 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.

]

---

## Regularização - LASSO

Relembrando o nossa **função de custo** RMSE.

`$$RMSE = \sqrt{\frac{1}{N}\sum(y_i - \hat{y_i})^2} = \sqrt{\frac{1}{N}\sum(y_i -  \color{red}{(\hat{\beta}_0 + \hat{\beta}_1x_{1i} + \dots + \hat{\beta}_px_{pi})})^2}$$`

Regularizar é "não deixar os `$\beta's$` soltos demais".

`$$RMSE_{regularizado} = RMSE + \color{red}{\lambda}\sum_{j = 1}^{p}|\beta_j|$$`

Ou seja, **penalizamos** a função de custo se os `$\beta's$` forem muito grandes.

**PS1:** O `$\color{red}{\lambda}$` é um hiperparâmetro para a Regressão Linear.

**PS2:** Quanto maior o `$\color{red}{\lambda}$`, mais penalizamos os `$\beta's$` por serem grandes.

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 203 (Linear Model Selection and Regularization).
]

---

## Regularização - LASSO

Existe um `$\color{red}{\lambda}$` que retorna o menor erro de cross-validation.

![scatter_eqm](static/img/lasso_lambda.png)

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 219 (The LASSO).
]

---

## Regularização - LASSO

Conforme aumentamos o `$\color{red}{\lambda}$`, forçamos os `$\beta's$` a serem cada vez menores.

![scatter_eqm](static/img/betas.png)

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 219 (The LASSO).
]

---

## Ir para o R

---

## Preditores Categóricos

### Preditor com apenas 2 categorias

Saldo médio no cartão de crédito é diferente entre homens e mulheres?

```r
summary(lm(Balance ~ Gender, data = Credit))
# Coefficients:
#              Estimate  Std.Error  t value Pr(>|t|)    
# (Intercept)    509.80  33.13      15.389   <2e-16 ***
# GenderFemale    19.73  46.05       0.429    0.669   
```

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

---

## Preditores Categóricos

### Preditor com apenas 2 categorias

Saldo médio no cartão de crédito é diferente entre homens e mulheres?

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

.footnote[
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

.pull-left[

```r
summary(lm(Balance ~ Ethnicity, data = Credit))
#                    Estimate  Std.Error t value Pr(>|t|)    
# (Intercept)          531.00  46.32     11.464   <2e-16 ***
# EthnicityAsian       -18.69  65.02     -0.287    0.774    
# EthnicityCaucasian   -12.50  56.68     -0.221    0.826  
```

]

.pull-right[

Modelo

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

Em que

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

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

]

---

## Preditores Categóricos

### Preditor com 3 ou mais categorias

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

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Ethnicity </th>
   <th style="text-align:right;"> (Intercept) </th>
   <th style="text-align:right;"> EthnicityAsian </th>
   <th style="text-align:right;"> EthnicityCaucasian </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Caucasian </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;"> Asian </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;"> Asian </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;"> Asian </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;"> Caucasian </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;"> Caucasian </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;"> African American </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;"> Asian </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>

steps: `step_dummy()`

---

## Preditores Categóricos

### Preditor com 3 ou mais categorias

Interpretação dos parâmetros:

`$y_{i} = \left\{ \begin{array}{ll} \beta_0 & \text{se for }\texttt{Afro American}\\ \beta_0 + \beta_1&\text{se for } \texttt{Asian}\\ \beta_0 + \beta_2&\text{se for } \texttt{Caucasian}\end{array}\right.$`

```r
# interprete cada um dos três parâmetros individualmente.
```

---

## Principais problemas dos modelos lineares

### Vamos estudar nesse curso

- Escala das variáveis explicativas (apenas LASSO sofre)

- Relação não-linear entre X e Y

- Outliers/Pontos de alavanca

### Não vamos estudar nesse curso

- Heterocedasticidade: Variância dos erros não constantes

- Multicolinearidade (LASSO resolve)

- Correlação entre os erros (séries temporais, por ex.)

---

## Transformações Não Lineares dos Preditores

### Exemplo: log

.pull-left[
Modelo real: `$y = 10 + 0.5log(x)$` 
]

.pull-right[
Modelo proposto: `$\small y = \beta_0 + \beta_1log(x)$` 
]

Outras transformações comuns: raíz quadrada, Box-Cox.

steps: `step_log()`, `step_BoxCox()`, `step_sqrt()`

---

## Transformações Não Lineares dos Preditores

### Exemplo: Regressão Polinomial

.pull-left[
Modelo real: `$y = 500 + 0.4(x-10)^3$` 
]

.pull-right[
Modelo proposto: `$y = \beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3$` 
]

Outras expansões comuns: b-splines, natural splines.

steps: `step_poly()`, `step_bs()`, `step_ns`

---

## Interações

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

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

step: `step_interact(terms = ~ Sepal.Length:Species)`

---

## Outliers

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 96 (Outliers).
]

---

## Outliers

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 96 (Outliers).
]

---

## Outliers

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 96 (Outliers).
]

---

## Outliers

### Diagnóstico

- Visualização univariada (histograma, boxplot);

- Comparação do valor com o desvio padrão;

### Tratamentos

- Transformações log(), etc;

- Categorização;

- Remover os valores extremos (raramente boa ideia);

- Usar Random Forest/XGBoost;

- Usar MAE em vez de RMSE;

---

## Ir para o R

---
class: inverse, center, middle

# CLASSIFICAÇÃO

---

# Regressão Logística

.pull-left[

### Para  `$Y \in \{0, 1\}$` (binário)

$$
log\left$\frac{p}{1-p}\right$ = \beta_0 + \beta_1x
$$

em que `$p = P(Y = 1|x)$`. Ou...

$$
p = \frac{1}{1 + e^{-(\beta_0 + \beta_1x)}}
$$

]

.pull-right[

]

### No R:

```r
logistic_reg() %>% fit(dist ~ speed, data=Default)
```

.footnote[
Ver [ISL](https://www.ime.unicamp.br/~dias/Intoduction%20to%20Statistical%20Learning.pdf) página 131 (Logistic Regression).
]

---

# Regressão Logística

---

# Regressão Logística

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---

# Árvore de Decisão

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---

# Regressão Logística - Custo e Regularização
A **função de custo** da Regressão Logística chama-se *log-loss* (ou  *deviance* ou *Binary Cross-Entropy*):

`$$D = \frac{-1}{N}\sum[y_i \log\hat{y_i} + (1 - y_i )\log(1 - \hat{y_i})]$$`

---

# Regressão Logística - Custo e Regularização
A **função de custo** da Regressão Logística chama-se *log-loss* (ou  *deviance* ou *Binary Cross-Entropy*):

`$$D = \frac{-1}{N}\sum[y_i \log\hat{y_i} + (1 - y_i )\log(1 - \hat{y_i})]$$`

Para cada linha da base de dados seria assim...

`$$D_i = \begin{cases} -\log(\hat{y}_i) & \text{quando} \space y_i = 1 \\\\ -\log(1-\hat{y}_i) & \text{quando} \space y_i = 0 \end{cases}$$`

---

# Regressão Logística - Custo e Regularização

A **função de custo** da Regressão Logística chama-se *log-loss* (ou  *deviance* ou *Binary Cross-Entropy*):

`$$D = \frac{-1}{N}\sum[y_i \log\hat{y_i} + (1 - y_i )\log(1 - \hat{y_i})]$$`

Regularizar é analogo a Regressão Linear.

`$$D_{regularizado} = D + \color{red}{\lambda}\sum_{j = 1}^{p}|\beta_j|$$`

**PS1:** `$\hat{y_i} = \hat{p_i} = \hat{P}(Y = 1|X)$`.

**PS2:** Se `$\log\left(\frac{\hat{p_i}}{1-\hat{p_i}}\right) = \beta_0 + \beta_1x$` então 
`$\hat{p_i} = \frac{1}{1 + e^{-(\beta_0 + \beta_1x)}}$`.

---

# Regressão Logística - Predições

O "produto final" será um vetor de probabilidades estimadas.

.pull-left[

<table>
 <thead>
  <tr>
   <th style="text-align:right;background-color: white !important;text-align: center;"> pts excl </th>
   <th style="text-align:left;background-color: white !important;text-align: center;"> classe observada </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> prob </th>
   <th style="text-align:left;background-color: white !important;text-align: center;"> classe predita </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 167 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 0.79 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 129 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 0.45 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Não Spam </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 299 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 1.00 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 270 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 1.00 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 187 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 0.89 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Spam </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 85 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Não Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;"> 0.12 </td>
   <td style="text-align:left;background-color: white !important;text-align: center;"> Não Spam </td>
  </tr>
</tbody>
</table>

]

.pull-right[

]

---

# Matriz de Confusão

.pull-left[
<table class="table table-bordered" style="font-size: 20px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Observado</div></th>
</tr>
  <tr>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Predito </th>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Neg      </th>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Pos  </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Neg </td>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; "> TN </td>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 2in; "> FN </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Pos </td>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; "> FP </td>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 2in; "> TP </td>
  </tr>
</tbody>
</table>

<br/>

<table class="table table-bordered" style="font-size: 20px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: red !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="1"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">p &gt; 50%</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Observado</div></th>
</tr>
  <tr>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Predito </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Não Spam </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Spam </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Não Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 410 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 73 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 52 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 465 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

$$
\begin{array}{lcc}
\mbox{accuracy}  & = & \frac{TP + TN}{TP + TN + FP + FN}\\\\
&   & \\\\
\mbox{precision} & = & \frac{TP}{TP + FP}\\\\
&   & \\\\
\mbox{recall/TPR}    & = & \frac{TP}{TP + FN} \\\\
&   & \\\\
\mbox{F1 score}       & =& \frac{2}{1/\mbox{precision} + 1/\mbox{recall}}\\\\
&   & \\\\
\mbox{FPR}    & = & \frac{FP}{FP + TN}
\end{array}`
$$

]

---

# Nota de Corte (Threshold)

.pull-left[

<table class="table table-bordered" style="font-size: 16px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: red !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="1"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">p &gt; 10%</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Observado</div></th>
</tr>
  <tr>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Predito </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Não Spam </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Spam </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Não Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 267 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 8 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 195 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 530 </td>
  </tr>
</tbody>
</table>

<table class="table table-bordered" style="font-size: 16px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: red !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="1"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">p &gt; 25%</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Observado</div></th>
</tr>
  <tr>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Predito </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Não Spam </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Spam </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Não Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 332 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 28 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 130 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 510 </td>
  </tr>
</tbody>
</table>

<table class="table table-bordered" style="font-size: 16px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: red !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="1"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">p &gt; 50%</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Observado</div></th>
</tr>
  <tr>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Predito </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Não Spam </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Spam </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Não Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 410 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 73 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 52 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 465 </td>
  </tr>
</tbody>
</table>

]

.pull-right[

<table class="table table-bordered" style="font-size: 16px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: red !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="1"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">p &gt; 75%</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Observado</div></th>
</tr>
  <tr>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Predito </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Não Spam </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Spam </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Não Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 443 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 112 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 19 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 426 </td>
  </tr>
</tbody>
</table>

<table class="table table-bordered" style="font-size: 16px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: red !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="1"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">p &gt; 90%</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; color: black !important;padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Observado</div></th>
</tr>
  <tr>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Predito </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Não Spam </th>
   <th style="text-align:right;background-color: white !important;text-align: center;"> Spam </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Não Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 456 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 171 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Spam </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 3in; "> 6 </td>
   <td style="text-align:right;background-color: white !important;text-align: center;width: 2in; "> 367 </td>
  </tr>
</tbody>
</table>

]

---

# Curva ROC

.pull-left[
<img src="01-intro-ml_files/figure-html/unnamed-chunk-58-1.png" style="display: block; margin: auto;" />

]

.pull-right[

<br/>

<table class="table table-bordered" style="font-size: 20px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; padding-right: 4px; padding-left: 4px; background-color: white !important;" colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Observado</div></th>
</tr>
  <tr>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Predito </th>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Neg      </th>
   <th style="text-align:left;background-color: white !important;text-align: center;"> Pos  </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Neg </td>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; "> TN </td>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 2in; "> FN </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; font-weight: bold;"> Pos </td>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 3in; "> FP </td>
   <td style="text-align:left;background-color: white !important;text-align: center;width: 2in; "> TP </td>
  </tr>
</tbody>
</table>

$$
\begin{array}{lcc}
\mbox{TPR}    & = & \frac{TP}{TP + FN} \\\\
&   & \\\\
\mbox{FPR}    & = & \frac{FP}{FP + TN}
\end{array}`
$$

]

.footnote[
[An introduction to ROC analysis](https://people.inf.elte.hu/kiss/11dwhdm/roc.pdf)
]

---

# Curva ROC - Métrica AUC

.pull-left[

]

.pull-right[

<br/>

$$
\mbox{AUC} = \mbox{Area Under The ROC Curve}
$$
]

**PS:** AUC varia de 0.5 a 1.0. O que significa se AUC for zero?

.footnote[
[An introduction to ROC analysis](https://people.inf.elte.hu/kiss/11dwhdm/roc.pdf)
]

---

# Curva ROC - Playground

---

## Ir para o R