Kaplan-Meier
S(t) = prod over event times t_i <= t of (1 - d_i / n_i), with the Greenwood variance
Var = S^2 sum d_i / (n_i (n_i - d_i)). Confidence intervals use the log(-log) transform (lifelines' and R's
default), which keeps them inside [0, 1].
from lifelines import KaplanMeierFitter
km = KaplanMeierFitter().fit(df.duration_days, df.event, label="all")
km.survival_function_at_times([90, 180, 365]); km.median_survival_time_
km.plot_survival_function(at_risk_counts=True)Cox proportional hazards
h(t | x) = h0(t) exp(x' beta). A hazard ratio exp(beta) multiplies the hazard at every time point.
from lifelines import CoxPHFitter
cph = CoxPHFitter(penalizer=0.1).fit(train, "duration_days", "event") # Efron ties
cph.print_summary(); cph.check_assumptions(train, p_value_threshold=0.05)
cph.predict_survival_function(test, times=[90, 180, 365])- Ties: Efron (default in lifelines and R) is better than Breslow when many events share times.
- Stratification (
strata=["region"]) gives each stratum its own baseline hazard, for covariates that violate PH but are not of interest. - Non-linear effects: splines or binning. Interactions: explicit product columns.
Parametric AFT models
log T = x' beta + sigma * epsilon. The coefficients are time ratios: exp(beta) = 1.5 means 50% longer
expected time.
| Distribution | Hazard shape |
|---|---|
| Exponential | constant |
| Weibull | monotone increasing or decreasing |
| Log-normal, log-logistic | rises then falls |
| Generalised gamma | flexible, nests the others |
AFT models extrapolate beyond the follow-up (e.g. lifetime value), but only as well as the distribution fits.
Compare by AIC and with KM curves. Lifelines: WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter.
Machine learning
| Model | Library | Notes |
|---|---|---|
| Random survival forest | sksurv.ensemble.RandomSurvivalForest | non-PH, interactions; min_samples_leaf 5-20 |
| Gradient-boosted Cox | sksurv.ensemble.GradientBoostingSurvivalAnalysis | strong on tabular data |
| XGBoost survival | objective="survival:cox" or "survival:aft" | scale; AFT needs interval labels |
| Penalised Cox (elastic net) | sksurv.linear_model.CoxnetSurvivalAnalysis | many covariates |
| Deep models | DeepSurv, DeepHit (pycox) | large data, unstructured inputs |
Competing risks
When other events prevent the event of interest (a customer who downgrades can no longer churn from premium; death from other causes), 1 - KM overestimates the cumulative incidence.
| Question | Method |
|---|---|
| Cumulative incidence of each event | Aalen-Johansen (lifelines.AalenJohansenFitter) |
| Effect on the cause-specific hazard (aetiology) | cause-specific Cox: treat other events as censored |
| Effect on the cumulative incidence (prediction) | Fine-Gray subdistribution hazard (R cmprsk, scikit-survival in recent versions) |
Discrete time
With monthly data and many rows, expand to person-period rows (one row per subject per month at risk) and fit a logistic regression or gradient boosting on "event this month". This handles time-varying covariates naturally and scales well. Survival is the product of (1 - hazard) over months.
Churn and lifetime value
- Expected lifetime from survival:
E[T] = integral of S(t) dt(restricted to the horizon: RMST). - Customer lifetime value = sum over periods of S(t) x margin(t) x discount(t).
- Parametric or discrete-time models are needed to extrapolate beyond the observed follow-up. State the assumption.
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.