Data Mining

1. Overview

• Data mining extracts meaningful patterns and knowledge from massive datasets.
• Pipeline: data cleaning → feature construction → modelling/pattern discovery → evaluation & visualisation.
• Algorithms must account for the “4 V’s” of big data: volume, variety, velocity, veracity.

2. Frequent Itemset Mining

2.1 Concepts

• Identify sets of items that co-occur in transaction data.
• Support: fraction of transactions containing itemset $X$
  $$\mathrm{support}(X) = \frac{#\text{transactions containing }X}{#\text{all transactions}}$$
• Confidence: probability that $Y$ appears when $X$ does in rule $X \rightarrow Y$.
• Lift: $ \frac{P(X,Y)}{P(X)P(Y)}$ measures how much more often $X$ and $Y$ appear together than by independence.

2.2 Apriori Algorithm

• Top-down search that grows larger frequent itemsets from smaller ones.
• Apriori principle: if an itemset is not frequent, none of its supersets can be frequent.
• Steps:
  1. Generate candidate 1-itemsets and compute support.
  2. Keep itemsets above the minimum support threshold.
  3. Combine survivors to form $(k+1)$-item candidates.
  4. Repeat until no more frequent sets remain.
• Limitation: repeated data scans and huge candidate sets.

2.3 FP-Growth

• Compress transactions into an FP-tree without candidate generation.
• Process: order items by support → insert transactions into the tree sharing prefixes → build conditional FP-trees recursively.
• Requires less I/O than Apriori and scales well to large datasets.

2.4 Association Rules

• Split each frequent itemset into antecedent/consequent pairs to form rule candidates.
• Filter by support, confidence, lift, etc.
• Applications: recommendation, market-basket analysis, intrusion detection.

3. Similar Item Search: Locality-Sensitive Hashing (LSH)

3.1 Problem

• Quickly find similar objects in very large datasets (duplicate documents, image search, record linkage).

3.2 Core Idea

• Design hash families so that similar items collide with high probability.
• Choose hash functions according to similarity metrics:
  • Jaccard → MinHash
  • Cosine → Random hyperplane
  • $L_p$ distance → $p$-stable hashes

3.3 MinHash with Banding

• Represent documents as sets and build MinHash signatures (minimum values of multiple hash functions).
• Split the signature matrix into $b$ bands of $r$ rows; hash each band to buckets.
• If two signatures match in any band, treat them as a candidate pair and compute exact similarity.
• Adjust $(r, b)$ to balance false negatives (more bands) vs. false positives (more rows).

3.4 LSH Pipeline

1. Preprocess data into sets or vectors.
2. Generate LSH signatures.
3. Extract candidate pairs.
4. Compute exact distances and return top-$k$ or threshold matches.

4. Clustering

4.1 Essentials

• Discover natural groupings in unlabeled data.
• Objectives: maximise intra-cluster similarity and minimise inter-cluster similarity.
• Distance metrics: Euclidean, cosine, Hamming, DTW, etc.

4.2 K-means

• Classic algorithm for roughly spherical clusters.
• Procedure: choose $K$ centres (random or k-means++), assign points to nearest centre, recompute centres, repeat until convergence.
• Watch out for $K$ selection (elbow, silhouette), sensitivity to outliers, and initial seeds.

4.3 Hierarchical Clustering

• Builds a dendrogram based on pairwise distances.
• Agglomerative (bottom-up): start with singletons and merge clusters using linkage criteria (single, complete, average, Ward).
• Divisive (top-down): start with all points and split recursively.
• Cut the tree at a chosen level to obtain the desired number of clusters.

4.4 Density-Based Clustering

• DBSCAN/HDBSCAN treat dense regions as clusters and sparse areas as noise.
• Parameters: radius $\varepsilon$ and minimum points MinPts.
• Captures arbitrarily shaped clusters and automatically flags outliers.

5. Community Detection in Graphs

5.1 Definition

• Partition a graph into subgraphs with dense internal connections and sparse external ones.
• Use cases: social networks, recommendation engines, biological networks.

5.2 Modularity

• Measures the difference between intra-cluster and inter-cluster edge density.
• Louvain/Leiden algorithms greedily maximise modularity for large graphs.

5.3 Spectral Clustering

• Uses eigenvectors of the graph Laplacian to embed nodes into a low-dimensional space.
• Workflow: build adjacency matrix $W$, degree matrix $D$, compute normalised Laplacian $L = D^{-1/2}(D-W)D^{-1/2}$, take the smallest $k$ eigenvectors, and run k-means.
• Handles nonlinear structures and non-spherical clusters.

5.4 Comparison

• Modularity optimisation is fast but subject to resolution limits.
• Spectral methods are more costly (eigen decomposition) yet yield high-quality partitions.
• Choose based on graph size, sparsity, and runtime constraints.

6. Dimensionality Reduction

6.1 Motivation

• High-dimensional data worsens computation, visualisation, and the curse of dimensionality.
• Dimensionality reduction keeps as much information as possible while compressing representation.

6.2 Principal Component Analysis (PCA)

• Eigen-decompose the covariance matrix to find directions of maximal variance.
• Steps: standardise data → compute covariance → eigenvalues/vectors → project onto top-$k$ components.
• Linear, orthogonal transform that preserves maximum variance; sensitive to nonlinearity/outliers.

6.3 Linear Discriminant Analysis (LDA)

• Maximises between-class variance while minimising within-class variance.
• Supervised dimensionality reduction; boosts classification performance.

6.4 Nonlinear Techniques

• t-SNE: excellent for local structure visualisation, but distorts global layout.
• UMAP: faster than t-SNE with better global preservation.
• Autoencoders: neural networks that learn nonlinear embeddings.

6.5 Working with LSH/Clustering

• Reduce dimensionality first to improve LSH efficiency and denoise features.
• Applying PCA before clustering often yields higher silhouette scores.

7. Project Tips

• Prioritise data quality: handle missing values, detect outliers, and scale appropriately.
• Evaluation metrics:
  • Frequent patterns: support, confidence, lift, conviction, Kulczynski, etc.
  • Clustering/community detection: silhouette, Davies–Bouldin, modularity.
  • Dimensionality reduction: reconstruction error, downstream model accuracy.
• Visualise results with heatmaps, correlation matrices, network graphs, and 2D embeddings (PCA, t-SNE).
• In practice, LSH is just a candidate generator—always follow up with precise distance checks.
• Document your pipeline and tuning experiments to ensure reproducibility.