Explained variance
Eigenvalue λ_i of the covariance (or correlation) matrix = variance along component i. Explained ratio λ_i / Σλ. Cumulative share: keep the smallest k reaching a target (80% exploratory, 90-95% compression).
from sklearn.decomposition import PCA
p = PCA().fit(Z) # Z standardised
cum = p.explained_variance_ratio_.cumsum()
k95 = int((cum >= 0.95).argmax()) + 1Scree plot / elbow
Plot λ_i against i; keep components before the curve levels off. Subjective; useful as a picture.
Kaiser criterion
On the correlation matrix keep λ > 1 (a component explains more than one original standardised variable). Tends to keep too many with many variables and too few with few.
Parallel analysis (Horn 1965)
Compare each eigenvalue with the distribution of the same-rank eigenvalue from data with no correlation
structure: shuffle each column independently (keeps the marginals) or simulate normal data of the same shape.
Keep components while λ_i exceeds the 95th percentile of the random λ_i. The most reliable simple rule for
"how many real dimensions" (implemented in pca_report.py).
Velicer's MAP
Minimum average partial correlation: partial out the first m components and find the m minimising the average
squared partial correlation. Available in R (psych::vss); conservative.
Broken-stick
Keep component i if its share exceeds the expected share of the i-th longest piece of a randomly broken stick:
b_i = (1/p) Σ_{k=i..p} 1/k.
Downstream cross-validation
When PCA is a preprocessing step, treat k as a hyperparameter:
from sklearn.model_selection import GridSearchCV
search = GridSearchCV(make_pipeline(StandardScaler(), PCA(), Ridge()),
{"pca__n_components": [2, 4, 8, 16, 32]}, cv=5, scoring="neg_mean_absolute_error")Probabilistic PCA likelihood
PCA(n_components="mle") (Minka's MLE) or PCA().score(X) (average log-likelihood under probabilistic PCA) on
held-out data can choose k.
Which rule for which purpose
| Purpose | Rule |
|---|---|
| How many underlying factors? | parallel analysis (and MAP) |
| Compress with little information loss | cumulative variance 90-99% |
| Visualise | 2-3 components (check their cumulative share) |
| Model input | CV over k |
anomaly-detection-model-builder
Build and evaluate anomaly detection for tabular, sensor and transaction data with few or no labels: define the anomaly type (point, contextual, collective), choose detectors (robust z-score/MAD and IQR rules, Mahalanobis with robust covariance, k-NN distance, LOF, Isolation Forest, one-class SVM, PCA reconstruction / autoencoders, forecasting residuals for time series), combine them by rank, explain each alert by its deviating features, set thresholds from an alert budget or the score distribution, and evaluate with precision@k, PR-AUC and recall on known incidents, or detector agreement when no labels exist. Includes pure-Python detectors (robust z, Mahalanobis, kNN, Isolation Forest, ensemble) and an evaluation script.
catboost-model-builder
Build CatBoost models that use its strengths correctly: raw categorical columns as cat_features (ordered target statistics, no manual encoding), text_features, Pool construction with NaN-safe categories, ordered boosting and has_time for temporal data, symmetric (oblivious) trees, overfitting detector with eval_set and use_best_model, losses for classification, regression (Quantile, Poisson, Tweedie) and ranking (YetiRank), class weights, monotone constraints, feature importance (PredictionValuesChange, LossFunctionChange) and SHAP values, GPU training and model export. Plans cat/text/numeric columns from a CSV and generates a runnable training script, and checks parameter sets for the conflicts CatBoost rejects (iterations vs n_estimators, Bayesian bootstrap with subsample, multiple class-weight options, depth > 16, use_best_model without eval_set).
categorical-encoding-selector
Choose and implement the right encoding for every categorical column given the model family and the data: one-hot (with infrequent-level grouping and handle_unknown), ordinal with an explicit order, cross-fitted target encoding (scikit-learn TargetEncoder, CatBoost ordered statistics, category_encoders inside CV), frequency/count encoding, feature hashing, native categorical support in LightGBM, HistGradientBoosting, XGBoost and CatBoost, prefix levels for codes (zip3, NAICS-2), and embeddings. Measures cardinality, rare-level share, unseen-level rate on a time-ordered holdout and target signal per column, emits the ColumnTransformer, and quantifies target-encoding leakage (in-sample vs out-of-fold vs holdout AUC).
classification-model-selector
Choose the right classification algorithm for tabular, text or mixed data and prove the choice with a fair bake-off: logistic regression (L2, L1, elastic net), linear SVM, naive Bayes, k-NN, RBF SVM, decision trees, random forest, extra trees, HistGradientBoosting, LightGBM, XGBoost, CatBoost, explainable boosting machines (EBM/GAM), points scorecards and MLPs, matched to data size, width, categorical and high-cardinality share, missingness, class imbalance, text, and constraints such as interpretability, monotonic constraints, calibrated probabilities, latency budgets and tiny deployable models. Ranks candidates with starting hyperparameters and cautions, then generates a bake-off script running every candidate on identical CV folds with per-fold metrics, fit time and prediction latency.
clustering-algorithm-selector
Choose the right clustering algorithm and check that clustering makes sense at all: cluster tendency (Hopkins statistic with a correlated-features caveat, nearest-neighbour isolation), then k-means / MiniBatchKMeans, Gaussian mixtures with BIC, agglomerative (Ward, average, complete, single) with dendrograms, DBSCAN, HDBSCAN, OPTICS, spectral clustering, BIRCH, mean shift, k-prototypes and k-modes for mixed or categorical data, and Gower distance, matched to data size, expected shapes, noise, varying density, known k, soft memberships, hierarchy and streaming. Covers preprocessing (scaling, encoding, dimensionality, distance metrics), validation (internal indices, bootstrap stability, external labels, business profiling) and assigning new points.
collaborative-filtering-implementer
Implement collaborative filtering recommenders that beat popularity: item-based kNN with cosine, Jaccard or asymmetric similarity, shrinkage, neighbourhood size, positive-only or mean-centred rating signals and 'because you liked X' explanations; implicit ALS with confidence weighting, BPR pairwise ranking and the closed-form EASE model in plain NumPy, tuned on a temporal validation split by recall@k and NDCG@k and refit on all data; plus user-kNN, explicit matrix factorisation, the implicit/LightFM libraries, hyperparameter ranges, cold-start fallbacks, scaling with sparse matrices and approximate nearest neighbours. Produces top-k lists in the format of the recommendation evaluator.