Elbow (inertia) and kneedle
Inertia W_k = Σ ||x − μ_c(x)||² always decreases with k. The elbow is where adding clusters stops paying.
Kneedle (Satopää et al. 2011): normalise k and W_k to [0, 1], take the point with the largest distance below the
chord from the first to the last point. kneed.KneeLocator(ks, W, curve="convex", direction="decreasing") and
yellowbrick's KElbowVisualizer implement it.
Silhouette
For point i: a = mean distance to its own cluster, b = mean distance to the nearest other cluster, s = (b − a) / max(a, b). Mean silhouette ∈ [−1, 1].
from sklearn.metrics import silhouette_score, silhouette_samples
silhouette_score(X, labels, sample_size=5000, random_state=0) # sample for large n (O(n²))Rules of thumb: > 0.5 reasonable structure, 0.25-0.5 weak, < 0.25 no substantial structure. Look at the per-cluster silhouette plot: a cluster with many negative values is poorly separated.
Calinski-Harabasz and Davies-Bouldin
- CH = [B / (k − 1)] / [W / (n − k)] (between vs within dispersion), higher is better.
- DB = mean over clusters of max_j (s_i + s_j) / d(c_i, c_j), lower is better.
calinski_harabasz_score,davies_bouldin_score. Both are cheap and favour convex clusters.
Gap statistic (Tibshirani, Walther & Hastie 2001)
Gap(k) = E*[log W_k] − log W_k (E* over B reference datasets drawn uniformly in the bounding box)
s_k = sd(log W_k*) · sqrt(1 + 1/B)
choose the smallest k with Gap(k) >= Gap(k+1) − s_{k+1}Gap(k) ≈ 0 for all k means the data is not more clustered than uniform noise, the main thing other indices cannot tell you. Variants draw references in the principal-component box (better for correlated data).
Stability
Fit on two (overlapping) subsamples or with different seeds; compare labels on shared rows with the adjusted
Rand index (adjusted_rand_score). Plot mean ARI vs k: the supported k values are stable (ARI > 0.8). Stability
alone is not enough: k-means splits uniform data stably.
BIC / AIC for Gaussian mixtures
from sklearn.mixture import GaussianMixture
bic = [GaussianMixture(k, covariance_type="full", n_init=3, random_state=0).fit(X).bic(X) for k in range(1, 11)]Lower is better; outliers inflate the chosen k (see clustering-algorithm-selector's recipes note).
Cost
| Method | Cost per k |
|---|---|
| Inertia, CH, DB | one fit |
| Silhouette | O(n²) distances (sample) |
| Gap | (B + 1) fits |
| Stability | 2+ fits per repeat |
Putting it together
- Run all cheap indices and the gap statistic over k = 2..10.
- Discard k values that produce clusters below the size floor.
- Among the rest, prefer k values where several criteria agree and stability is high.
- Decide with profiles and the business: the clearer, actionable segmentation wins ties.
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.