Introdução

Este relatório apresenta uma análise aprofundada do dataset Groceries, que contém transações de compras em um supermercado. O objetivo é demonstrar técnicas de manipulação de dados, análise exploratória e mineração de regras de associação utilizando R.

O dataset Groceries é amplamente utilizado em estudos de Market Basket Analysis (Análise de Cesta de Compras), permitindo identificar padrões de consumo e relações entre produtos.

1 Carregamento e Manipulação de Dados

1.1 Carregando o Dataset

# Carregar pacotes necessários
library(arules)      # Para análise de regras de associação
library(arulesViz)   # Para visualização de regras
library(DT)          # Para tabelas interativas
library(dplyr)       # Para manipulação de dados
library(ggplot2)     # Para gráficos avançados
library(tidyr)       # Para organização de dados
library(knitr)       # Para formatação
library(kableExtra)  # Para tabelas elegantes

# Configurações globais
opts_chunk$set(
  fig.align = 'center',
  message = FALSE,
  warning = FALSE,
  fig.width = 10,
  fig.height = 6
)

# Carregar o dataset groceries
data("Groceries")

# Informações básicas
cat("=== INFORMAÇÕES DO DATASET ===\n")

## === INFORMAÇÕES DO DATASET ===

cat("Número de transações:", length(Groceries), "\n")

## Número de transações: 9835

cat("Número de itens únicos:", length(itemLabels(Groceries)), "\n")

## Número de itens únicos: 169

# Calcular densidade corretamente
densidade <- round(size(Groceries) / (length(Groceries) * length(itemLabels(Groceries))), 4)
cat("Densidade da matriz:", densidade, "\n")

## Densidade da matriz: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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1.1.1 Estrutura dos Dados

# Visualizar as primeiras transações
inspect(head(Groceries, 10))

##      items                     
## [1]  {citrus fruit,            
##       semi-finished bread,     
##       margarine,               
##       ready soups}             
## [2]  {tropical fruit,          
##       yogurt,                  
##       coffee}                  
## [3]  {whole milk}              
## [4]  {pip fruit,               
##       yogurt,                  
##       cream cheese ,           
##       meat spreads}            
## [5]  {other vegetables,        
##       whole milk,              
##       condensed milk,          
##       long life bakery product}
## [6]  {whole milk,              
##       butter,                  
##       yogurt,                  
##       rice,                    
##       abrasive cleaner}        
## [7]  {rolls/buns}              
## [8]  {other vegetables,        
##       UHT-milk,                
##       rolls/buns,              
##       bottled beer,            
##       liquor (appetizer)}      
## [9]  {pot plants}              
## [10] {whole milk,              
##       cereals}

# Converter para dataframe para análises adicionais
groceries_df <- as(Groceries, "data.frame")

# Exibir amostra do dataframe
kable(head(groceries_df, 15), 
      caption = "Primeiras 15 linhas do Dataset Groceries") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE)

Primeiras 15 linhas do Dataset Groceries
items
{citrus fruit,semi-finished bread,margarine,ready soups}
{tropical fruit,yogurt,coffee}
{whole milk}
{pip fruit,yogurt,cream cheese ,meat spreads}
{other vegetables,whole milk,condensed milk,long life bakery product}
{whole milk,butter,yogurt,rice,abrasive cleaner}
{rolls/buns}
{other vegetables,UHT-milk,rolls/buns,bottled beer,liquor (appetizer)}
{pot plants}
{whole milk,cereals}
{tropical fruit,other vegetables,white bread,bottled water,chocolate}
{citrus fruit,tropical fruit,whole milk,butter,curd,yogurt,flour,bottled water,dishes}
{beef}
{frankfurter,rolls/buns,soda}
{chicken,tropical fruit}

1.2 Análise Exploratória Detalhada

1.2.1 Frequência de Itens

# Calcular frequências absolutas e relativas
item_freq_abs <- itemFrequency(Groceries, type = "absolute")
item_freq_rel <- itemFrequency(Groceries, type = "relative")

# Criar dataframe completo
item_freq_df <- data.frame(
  Item = names(item_freq_abs),
  Frequencia_Absoluta = as.numeric(item_freq_abs),
  Frequencia_Relativa = round(as.numeric(item_freq_rel) * 100, 2),
  Porcentagem = paste0(round(as.numeric(item_freq_rel) * 100, 2), "%")
) %>%
  arrange(desc(Frequencia_Absoluta))

# Exibir top 20
kable(head(item_freq_df, 20), 
      caption = "Top 20 Produtos Mais Comprados",
      col.names = c("Produto", "Freq. Absoluta", "Freq. Relativa (%)", "Porcentagem")) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE) %>%
  row_spec(1:3, bold = TRUE, color = "white", background = "#3498db")

Top 20 Produtos Mais Comprados
Produto	Freq. Absoluta	Freq. Relativa (%)	Porcentagem
whole milk	2513	25.55	25.55%
other vegetables	1903	19.35	19.35%
rolls/buns	1809	18.39	18.39%
soda	1715	17.44	17.44%
yogurt	1372	13.95	13.95%
bottled water	1087	11.05	11.05%
root vegetables	1072	10.90	10.9%
tropical fruit	1032	10.49	10.49%
shopping bags	969	9.85	9.85%
sausage	924	9.40	9.4%
pastry	875	8.90	8.9%
citrus fruit	814	8.28	8.28%
bottled beer	792	8.05	8.05%
newspapers	785	7.98	7.98%
canned beer	764	7.77	7.77%
pip fruit	744	7.56	7.56%
fruit/vegetable juice	711	7.23	7.23%
whipped/sour cream	705	7.17	7.17%
brown bread	638	6.49	6.49%
domestic eggs	624	6.34	6.34%

1.2.2 Manipulações Avançadas

# 1. FILTRAR itens com frequência > 1000
top_items <- item_freq_df %>%
  filter(Frequencia_Absoluta > 1000) %>%
  arrange(desc(Frequencia_Absoluta))

# 2. CRIAR categorias de popularidade
top_items <- top_items %>%
  mutate(
    Categoria = case_when(
      Frequencia_Absoluta > 2000 ~ "Muito Popular",
      Frequencia_Absoluta > 1500 ~ "Popular",
      Frequencia_Absoluta > 1000 ~ "Moderado",
      TRUE ~ "Baixo"
    ),
    Quartil = ntile(Frequencia_Absoluta, 4),
    Score_Normalizado = round((Frequencia_Absoluta - min(Frequencia_Absoluta)) / 
                               (max(Frequencia_Absoluta) - min(Frequencia_Absoluta)), 3)
  )

# 3. ORDENAR por múltiplos critérios
top_items <- top_items %>%
  arrange(desc(Categoria), desc(Frequencia_Absoluta))

# Exibir resultado
cat("\n=== RESUMO DAS MANIPULAÇÕES ===\n")

## 
## === RESUMO DAS MANIPULAÇÕES ===

cat("Itens filtrados (freq > 1000):", nrow(top_items), "\n")

## Itens filtrados (freq > 1000): 8

cat("Categorias criadas:", length(unique(top_items$Categoria)), "\n\n")

## Categorias criadas: 3

print(top_items)

##               Item Frequencia_Absoluta Frequencia_Relativa Porcentagem
## 1 other vegetables                1903               19.35      19.35%
## 2       rolls/buns                1809               18.39      18.39%
## 3             soda                1715               17.44      17.44%
## 4       whole milk                2513               25.55      25.55%
## 5           yogurt                1372               13.95      13.95%
## 6    bottled water                1087               11.05      11.05%
## 7  root vegetables                1072               10.90       10.9%
## 8   tropical fruit                1032               10.49      10.49%
##       Categoria Quartil Score_Normalizado
## 1       Popular       4             0.588
## 2       Popular       3             0.525
## 3       Popular       3             0.461
## 4 Muito Popular       4             1.000
## 5      Moderado       2             0.230
## 6      Moderado       2             0.037
## 7      Moderado       1             0.027
## 8      Moderado       1             0.000

1.2.3 Estatísticas Descritivas

# Estatísticas por categoria
stats_categoria <- top_items %>%
  group_by(Categoria) %>%
  summarise(
    N_Produtos = n(),
    Freq_Media = round(mean(Frequencia_Absoluta), 2),
    Freq_Mediana = median(Frequencia_Absoluta),
    Freq_Min = min(Frequencia_Absoluta),
    Freq_Max = max(Frequencia_Absoluta),
    Desvio_Padrao = round(sd(Frequencia_Absoluta), 2)
  ) %>%
  arrange(desc(Freq_Media))

kable(stats_categoria, 
      caption = "Estatísticas Descritivas por Categoria de Popularidade") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE)

Estatísticas Descritivas por Categoria de Popularidade
Categoria	N_Produtos	Freq_Media	Freq_Mediana	Freq_Min	Freq_Max	Desvio_Padrao
Muito Popular	1	2513.00	2513.0	2513	2513	NA
Popular	3	1809.00	1809.0	1715	1903	94.0
Moderado	4	1140.75	1079.5	1032	1372	155.9

# Box plot das frequências
boxplot(Frequencia_Absoluta ~ Categoria, 
        data = top_items,
        main = "Distribuição de Frequências por Categoria",
        xlab = "Categoria",
        ylab = "Frequência Absoluta",
        col = c("#e74c3c", "#f39c12", "#3498db"),
        border = "darkblue",
        notch = TRUE)

1.3 Visualizações Avançadas

1.3.1 Gráfico de Barras Horizontal

# Top 20 produtos
top_20 <- head(top_items, 20)

par(mar = c(5, 10, 4, 2))
barplot(
  top_20$Frequencia_Absoluta,
  names.arg = top_20$Item,
  horiz = TRUE,
  las = 1,
  col = colorRampPalette(c("#3498db", "#e74c3c"))(20),
  main = "Top 20 Produtos Mais Comprados",
  xlab = "Frequência Absoluta",
  cex.names = 0.8,
  border = NA
)
grid(nx = 10, ny = 0, col = "gray", lty = "dotted")

1.3.2 Distribuição de Frequências

hist(item_freq_df$Frequencia_Absoluta,
     breaks = 50,
     col = "#3498db",
     border = "white",
     main = "Distribuição de Frequências dos Produtos",
     xlab = "Frequência Absoluta",
     ylab = "Número de Produtos",
     xlim = c(0, 3000))

# Adicionar linha da média
abline(v = mean(item_freq_df$Frequencia_Absoluta), 
       col = "red", 
       lwd = 2, 
       lty = 2)
legend("topright", 
       legend = paste("Média =", round(mean(item_freq_df$Frequencia_Absoluta), 2)),
       col = "red", 
       lty = 2, 
       lwd = 2)

1.3.3 Análise de Pareto (80/20)

# Calcular frequência acumulada
item_freq_df <- item_freq_df %>%
  mutate(
    Freq_Acumulada = cumsum(Frequencia_Absoluta),
    Perc_Acumulada = (Freq_Acumulada / sum(Frequencia_Absoluta)) * 100
  )

# Encontrar ponto onde atinge 80%
ponto_80 <- min(which(item_freq_df$Perc_Acumulada >= 80))

cat("=== ANÁLISE DE PARETO ===\n")

## === ANÁLISE DE PARETO ===

cat("80% das vendas concentram-se em", ponto_80, "produtos\n")

## 80% das vendas concentram-se em 54 produtos

cat("Isso representa", round((ponto_80 / nrow(item_freq_df)) * 100, 2), 
    "% do total de produtos\n")

## Isso representa 31.95 % do total de produtos

1.4 Regras de Associação

1.4.1 Mineração de Regras

# Gerar regras de associação
regras <- apriori(Groceries,
                  parameter = list(
                    support = 0.001,     # Suporte mínimo
                    confidence = 0.5,    # Confiança mínima
                    minlen = 2,          # Tamanho mínimo da regra
                    maxlen = 5           # Tamanho máximo da regra
                  ))

## Apriori
## 
## Parameter specification:
##  confidence minval smax arem  aval originalSupport maxtime support minlen
##         0.5    0.1    1 none FALSE            TRUE       5   0.001      2
##  maxlen target  ext
##       5  rules TRUE
## 
## Algorithmic control:
##  filter tree heap memopt load sort verbose
##     0.1 TRUE TRUE  FALSE TRUE    2    TRUE
## 
## Absolute minimum support count: 9 
## 
## set item appearances ...[0 item(s)] done [0.00s].
## set transactions ...[169 item(s), 9835 transaction(s)] done [0.00s].
## sorting and recoding items ... [157 item(s)] done [0.00s].
## creating transaction tree ... done [0.00s].
## checking subsets of size 1 2 3 4 5

##  done [0.01s].
## writing ... [5622 rule(s)] done [0.00s].
## creating S4 object  ... done [0.00s].

cat("=== REGRAS DE ASSOCIAÇÃO GERADAS ===\n")

## === REGRAS DE ASSOCIAÇÃO GERADAS ===

cat("Total de regras encontradas:", length(regras), "\n\n")

## Total de regras encontradas: 5622

# Ordenar por lift
regras_ordenadas <- sort(regras, by = "lift", decreasing = TRUE)

# Top 10 regras com maior lift
cat("Top 10 Regras com Maior Lift:\n")

## Top 10 Regras com Maior Lift:

inspect(head(regras_ordenadas, 10))

##      lhs                         rhs                  support confidence    coverage     lift count
## [1]  {Instant food products,                                                                       
##       soda}                   => {hamburger meat} 0.001220132  0.6315789 0.001931876 18.99565    12
## [2]  {soda,                                                                                        
##       popcorn}                => {salty snack}    0.001220132  0.6315789 0.001931876 16.69779    12
## [3]  {flour,                                                                                       
##       baking powder}          => {sugar}          0.001016777  0.5555556 0.001830198 16.40807    10
## [4]  {ham,                                                                                         
##       processed cheese}       => {white bread}    0.001931876  0.6333333 0.003050330 15.04549    19
## [5]  {whole milk,                                                                                  
##       Instant food products}  => {hamburger meat} 0.001525165  0.5000000 0.003050330 15.03823    15
## [6]  {other vegetables,                                                                            
##       curd,                                                                                        
##       yogurt,                                                                                      
##       whipped/sour cream}     => {cream cheese }  0.001016777  0.5882353 0.001728521 14.83409    10
## [7]  {processed cheese,                                                                            
##       domestic eggs}          => {white bread}    0.001118454  0.5238095 0.002135231 12.44364    11
## [8]  {tropical fruit,                                                                              
##       other vegetables,                                                                            
##       yogurt,                                                                                      
##       white bread}            => {butter}         0.001016777  0.6666667 0.001525165 12.03058    10
## [9]  {hamburger meat,                                                                              
##       yogurt,                                                                                      
##       whipped/sour cream}     => {butter}         0.001016777  0.6250000 0.001626843 11.27867    10
## [10] {liquor,                                                                                      
##       red/blush wine}         => {bottled beer}   0.001931876  0.9047619 0.002135231 11.23527    19

1.4.2 Análise das Melhores Regras

# Converter regras para dataframe
regras_df <- as(regras_ordenadas, "data.frame")

# Selecionar top 15
top_regras <- head(regras_df, 15) %>%
  mutate(
    support = round(support, 4),
    confidence = round(confidence, 4),
    coverage = round(coverage, 4),
    lift = round(lift, 2),
    count = round(count, 0)
  )

# Criar descrição mais legível
top_regras$Descricao <- paste0(
  "Se compra ", 
  gsub("[{}]", "", top_regras$rules),
  " (Lift: ", top_regras$lift, ")"
)

2 Tabela Interativa com DT

2.1 Tabela de Produtos

Tabela interativa completa com todos os produtos e suas estatísticas:

datatable(
  top_items %>% select(Item, Frequencia_Absoluta, Frequencia_Relativa, 
                       Categoria, Quartil, Score_Normalizado),
  caption = "Tabela Interativa: Análise Completa dos Produtos",
  filter = "top",
  extensions = c('Buttons', 'FixedHeader'),
  options = list(
    pageLength = 15,
    autoWidth = TRUE,
    searchHighlight = TRUE,
    dom = 'Bfrtip',
    buttons = c('copy', 'csv', 'excel', 'pdf', 'print'),
    fixedHeader = TRUE,
    scrollX = TRUE,
    columnDefs = list(
      list(className = 'dt-center', targets = 1:5)
    )
  ),
  rownames = FALSE,
  class = 'cell-border stripe hover'
) %>%
  formatStyle(
    'Categoria',
    backgroundColor = styleEqual(
      c('Muito Popular', 'Popular', 'Moderado'),
      c('#e74c3c', '#f39c12', '#3498db')
    ),
    color = 'white',
    fontWeight = 'bold'
  ) %>%
  formatPercentage('Score_Normalizado', 1) %>%
  formatRound(c('Frequencia_Absoluta', 'Frequencia_Relativa'), 0)

Funcionalidades disponíveis:

🔍 Busca global e por coluna
📊 Ordenação clicável em qualquer coluna
📄 Paginação com controle de registros por página
📥 Export para CSV, Excel, PDF e impressão
🎨 Destaque de categorias por cor
📌 Cabeçalho fixo ao rolar

2.2 Tabela de Regras

Regras de associação mais relevantes:

datatable(
  top_regras %>% select(rules, support, confidence, lift, count),
  caption = "Top 15 Regras de Associação (ordenadas por Lift)",
  filter = "top",
  extensions = 'Buttons',
  options = list(
    pageLength = 10,
    dom = 'Bfrtip',
    buttons = c('copy', 'csv', 'excel'),
    scrollX = TRUE,
    columnDefs = list(
      list(width = '400px', targets = 0)
    )
  ),
  rownames = FALSE,
  class = 'cell-border stripe'
) %>%
  formatRound(c('support', 'confidence'), 4) %>%
  formatRound('lift', 2) %>%
  formatStyle(
    'lift',
    background = styleColorBar(range(top_regras$lift), '#3498db'),
    backgroundSize = '100% 90%',
    backgroundRepeat = 'no-repeat',
    backgroundPosition = 'center'
  )

3 Equações Matemáticas em LaTeX

3.1 Equação 1: Suporte (Support)

3.1.1 Formulação Matemática

\[ \text{supp}(X) = \frac{|\{t \in T \mid X \subseteq t\}|}{|T|} \]

Onde: - \(T\) = conjunto de todas as transações - \(X\) = conjunto de itens (itemset) - \(t\) = uma transação individual - \(|\cdot|\) = cardinalidade do conjunto

3.1.2 Significado e Interpretação

O suporte mede a frequência relativa de um conjunto de itens \(X\) no dataset. Representa a proporção de transações que contêm todos os itens de \(X\).

Interpretação prática: - Suporte alto (\(\geq\) 0.3): item muito comum - Suporte médio (0.1 - 0.3): item moderadamente comum - Suporte baixo (\(<\) 0.1): item raro

Exemplo: Se {leite} aparece em 2.513 de 9.835 transações: \[\text{supp}(\text{leite}) = \frac{2513}{9835} = 0.2555 = 25.55\%\]

3.2 Equação 2: Confiança (Confidence)

3.2.1 Formulação Matemática

\[ \text{conf}(X \Rightarrow Y) = \frac{\text{supp}(X \cup Y)}{\text{supp}(X)} = P(Y|X) \]

Forma alternativa: \[ \text{conf}(X \Rightarrow Y) = \frac{|\{t \in T \mid X \cup Y \subseteq t\}|}{|\{t \in T \mid X \subseteq t\}|} \]

3.2.2 Significado e Interpretação

A confiança mede a probabilidade condicional de encontrar \(Y\) dado que \(X\) está presente na transação. É fundamental para avaliar a força de regras de associação.

Interpretação prática: - Confiança alta (\(\geq\) 0.7): regra forte e confiável - Confiança média (0.5 - 0.7): regra moderada - Confiança baixa (\(<\) 0.5): regra fraca

Exemplo: Se 80% das pessoas que compram pão também compram leite: \[\text{conf}(\text{pão} \Rightarrow \text{leite}) = 0.80 = 80\%\]

3.3 Equação 3: Lift

3.3.1 Formulação Matemática

\[ \text{lift}(X \Rightarrow Y) = \frac{\text{supp}(X \cup Y)}{\text{supp}(X) \times \text{supp}(Y)} = \frac{\text{conf}(X \Rightarrow Y)}{\text{supp}(Y)} \]

Forma equivalente usando probabilidades: \[ \text{lift}(X \Rightarrow Y) = \frac{P(X \cap Y)}{P(X) \cdot P(Y)} \]

3.3.2 Significado e Interpretação

O lift indica a força da associação entre \(X\) e \(Y\), comparando a frequência observada com a esperada sob independência.

Interpretação prática: - Lift \(> 1\): correlação positiva (compram juntos mais que o esperado) - Lift \(= 1\): independência (sem associação) - Lift \(< 1\): correlação negativa (compram juntos menos que o esperado)

Exemplo: Se lift = 2.5, significa que comprar \(X\) torna 2.5 vezes mais provável comprar \(Y\).

3.4 Equação 4: Entropia de Shannon

3.4.1 Formulação Matemática

\[ H(X) = -\sum_{i=1}^{n} P(x_i) \log_2 P(x_i) = \mathbb{E}[-\log_2 P(X)] \]

Para distribuições contínuas: \[ H(X) = -\int_{-\infty}^{\infty} f(x) \log_2 f(x) \, dx \]

3.4.2 Significado e Interpretação

A entropia mede a quantidade de informação ou incerteza em um conjunto de dados. Quanto maior a entropia, maior a imprevisibilidade dos dados.

Interpretação prática: - Entropia máxima: distribuição uniforme (máxima incerteza) - Entropia mínima (0): distribuição determinística (certeza total) - Unidade: bits (quando usa log₂)

Aplicações em Data Science: - Seleção de atributos em árvores de decisão (ID3, C4.5) - Avaliação de modelos de linguagem - Compressão de dados - Teoria da informação

Exemplo: Para uma moeda justa: \[H = -\left(\frac{1}{2}\log_2\frac{1}{2} + \frac{1}{2}\log_2\frac{1}{2}\right) = 1 \text{ bit}\]

3.5 Equação 5: Índice de Gini

3.5.1 Formulação Matemática

\[ \text{Gini} = 1 - \sum_{i=1}^{n} p_i^2 \]

Forma alternativa (impureza): \[ \text{Gini}(D) = 1 - \sum_{i=1}^{C} \left(\frac{|D_i|}{|D|}\right)^2 \]

Onde \(C\) é o número de classes e \(D\) é o conjunto de dados.

3.5.2 Significado e Interpretação

O índice de Gini mede a impureza ou heterogeneidade de um conjunto. É amplamente usado em árvores de decisão (CART) para determinar a melhor divisão dos dados.

Interpretação prática: - Gini = 0: pureza máxima (todos os elementos da mesma classe) - Gini = 0.5: máxima impureza (distribuição uniforme entre 2 classes) - Gini \(\in [0, 1-\frac{1}{n}]\) para \(n\) classes

Comparação com Entropia:

Métrica	Fórmula	Uso Principal
Gini	\(1 - \sum p_i^2\)	CART, Random Forest
Entropia	\(-\sum p_i \log p_i\)	ID3, C4.5

Aplicações: - Árvores de decisão (critério de divisão) - Random Forests - Análise de desigualdade econômica - Classificação de imagens

Exemplo: Para um nó com 60% classe A e 40% classe B: \[\text{Gini} = 1 - (0.6^2 + 0.4^2) = 1 - 0.52 = 0.48\]

4 Figuras e Diagramas

4.1 Figura 1: CRISP-DM

4.1.1 Metodologia CRISP-DM

Ciclo de Vida CRISP-DM para Ciência de Dados

4.1.2 Descrição Detalhada

CRISP-DM (Cross-Industry Standard Process for Data Mining) é a metodologia mais utilizada em projetos de ciência de dados. O processo é iterativo e cíclico, composto por 6 fases:

4.1.2.1 1. Business Understanding (Entendimento do Negócio)

Definir objetivos do projeto
Avaliar situação atual
Determinar metas de data mining
Produzir plano de projeto

4.1.2.2 2. Data Understanding (Entendimento dos Dados)

Coletar dados iniciais
Descrever os dados
Explorar os dados
Verificar qualidade dos dados

4.1.2.3 3. Data Preparation (Preparação dos Dados)

Selecionar dados
Limpar dados
Construir dados
Integrar dados
Formatar dados

4.1.2.4 4. Modeling (Modelagem)

Selecionar técnicas de modelagem
Gerar design de teste
Construir modelo
Avaliar modelo

4.1.2.5 5. Evaluation (Avaliação)

Avaliar resultados
Revisar processo
Determinar próximos passos

4.1.2.6 6. Deployment (Implantação)

Planejar implantação
Planejar monitoramento e manutenção
Produzir relatório final
Revisar projeto

Vantagem: O CRISP-DM enfatiza que ciência de dados é um processo iterativo, onde insights de fases posteriores frequentemente levam a revisões de fases anteriores.

4.2 Figura 2: Machine Learning

4.2.1 Mapa Mental de Machine Learning

Algoritmos e Técnicas de Machine Learning

4.2.2 Taxonomia de Machine Learning

4.2.2.1 1. Aprendizado Supervisionado (Supervised Learning)

Objetivo: Aprender mapeamento de entrada → saída usando dados rotulados

Classificação: - Regressão Logística - SVM (Support Vector Machines) - Árvores de Decisão - Random Forest - Gradient Boosting (XGBoost, LightGBM) - Redes Neurais

Regressão: - Regressão Linear - Regressão Polinomial - Ridge/Lasso - SVR (Support Vector Regression)

4.2.2.2 2. Aprendizado Não Supervisionado (Unsupervised Learning)

Objetivo: Encontrar padrões em dados não rotulados

Clustering: - K-Means - DBSCAN - Hierarchical Clustering - Gaussian Mixture Models

Redução de Dimensionalidade: - PCA (Principal Component Analysis) - t-SNE - UMAP - Autoencoders

Regras de Associação: - Apriori (usado neste trabalho!) - FP-Growth - Eclat

4.2.2.3 3. Aprendizado por Reforço (Reinforcement Learning)

Objetivo: Aprender através de interação com ambiente

Q-Learning
Deep Q-Networks (DQN)
Policy Gradients
Actor-Critic

4.2.2.4 4. Ensemble Methods

Objetivo: Combinar múltiplos modelos para melhor performance

Bagging
Boosting
Stacking
Voting

Critérios de Seleção:

Critério	Método Recomendado
Dados rotulados + Predição	Supervisionado
Dados não rotulados + Padrões	Não Supervisionado
Interação sequencial	Reforço
Alta dimensionalidade	Redução de Dimensionalidade
Grupos naturais	Clustering

5 Referências Bibliográficas

5.1 Referências Acadêmicas

Hahsler, M., Grün, B., & Hornik, K. (2005). arules – A Computational Environment for Mining Association Rules and Frequent Item Sets. Journal of Statistical Software, 14(15), 1-25.
DOI: 10.18637/jss.v014.i15
Relevância: Documentação fundamental do pacote arules utilizado neste trabalho.
Wickham, H., & Grolemund, G. (2017). R for Data Science: Import, Tidy, Transform, Visualize, and Model Data. O’Reilly Media.
ISBN: 978-1491910399
Disponível em: r4ds.had.co.nz
Relevância: Referência essencial para manipulação de dados com tidyverse.
Han, J., Kamber, M., & Pei, J. (2011). Data Mining: Concepts and Techniques (3rd ed.). Morgan Kaufmann.
ISBN: 978-0123814791
Relevância: Livro clássico sobre mineração de dados, incluindo regras de associação e algoritmo Apriori.
James, G., Witten, D., Hastie, T., & Tibshirani, R. (2021). An Introduction to Statistical Learning with Applications in R (2nd ed.). Springer.
ISBN: 978-1071614174
Disponível em: statlearning.com
Relevância: Fundamentos estatísticos de machine learning aplicados em R.
Provost, F., & Fawcett, T. (2013). Data Science for Business: What You Need to Know about Data Mining and Data-Analytic Thinking. O’Reilly Media.
ISBN: 978-1449361327
Relevância: Perspectiva de negócios aplicada à ciência de dados.

Conclusão

5.2 Síntese do Trabalho

Esse relatório demonstrou um fluxo completo de análise de dados utilizando R, abrangendo:

Carregamento e manipulação de dados transacionais
Análise exploratória com estatísticas descritivas e visualizações
Mineração de regras de associação com algoritmo Apriori
Tabelas interativas com funcionalidades avançadas
Fundamentação matemática com equações LaTeX

Análise Avançada do Dataset Groceries

Mineração de Dados e Regras de Associação

João Guilherme Miranda de Sousa Bispo

2025-11-03