๐Ÿ—๏ธ Emoji Key โ€” Industry Sectors

๐Ÿ“Š Outsourcing General Commercial & IT

  • ๐Ÿ“Š Stats โ€” descriptive, CI, A/B, precision/recall, robust outliers
  • ๐Ÿค– ML โ€” regression, logistic, regularisation, PCA, Bayes
  • โ˜Ž๏ธ Ops/Queues โ€” Poisson, Erlang-C, smoothing, linear programming
  • ๐ŸŒ Networks & Reliability โ€” availability, PageRank

โšก๏ธ Electricity Retail & Wholesale

  • ๐Ÿ“ˆ Forecasting โ€” demand, ARIMA, Holt-Winters, weather-normalisation
  • ๐Ÿค– ML โ€” GBT/LSTM load & RES forecasts, spike classifiers, anomaly detection, RL battery dispatch
  • ๐Ÿงฐ Operations & Dispatch โ€” economic dispatch, unit commitment, storage, demand response
  • ๐Ÿ’น Risk & Hedging โ€” hedge cost, VaR/CVaR, GARCH, elasticity
  • ๐ŸŒ Network & Settlement โ€” DC-OPF/LMPs, congestion rent, loss factors

โ˜Ž๏ธ Telecommunications Retail & Wholesale

  • ๐Ÿ“ˆ Demand & Forecasting โ€” ARIMAX, Holt-Winters, fraud sanity checks
  • ๐Ÿค– ML โ€” churn + survival, CDR fraud (IsoForest + supervised), traffic GBT/LSTM, QoE regression, graph ML, RL for SON
  • โ˜Ž๏ธ Traffic & Capacity โ€” Erlang-B/C, Engset, Littleโ€™s Law, latency
  • ๐Ÿ›ฐ๏ธ Radio & Throughput โ€” path loss, SINR, Shannon, TCP throughput
  • ๐Ÿ’ท Commercial โ€” ARPU, CLV, churn, LCR, interconnect billing
  • ๐ŸŒ Network & Reliability โ€” availability, PageRank, MOS/QoE

Outsourcing Model Click2Expand

๐Ÿ“Š Stats โ€” Descriptive, CI, A/B, Precision/Recall, Robust Outliers โ–พ
NameFormulaNon-technical descriptionTechnical description ScenarioProsPitfallsFormulas to address pitfalls Python librariesPython codeFormula Steps in Plain EnglishStep-by-step evaluation
๐Ÿ“ Descriptive stats (mean/variance/std/z) xฬ„ = (1/n) ฮฃ xi; sยฒ = (1/(n-1)) ฮฃ (xiโˆ’xฬ„)ยฒ; z = (xโˆ’xฬ„)/s Find average, spread, and how unusual a value is. Estimators for central tendency & dispersion; z standardises units. Average handling time (AHT) in a council helpline. Simple baseline; quick QC. Skew/outliers distort mean & std. median, MAD; robust z: 0.6745(xโˆ’med)/MAD numpy, pandas 
z = (aht - aht.mean())/aht.std(ddof=1)
 1) Add values โ†’ mean. 2) Deviations โ†’ square/average โ†’ variance. 3) โˆšvariance โ†’ std. 4) (xโˆ’mean)/std โ†’ z. Collect AHT โ†’ compute mean/std โ†’ flag |z|>3 โ†’ review calls.
๐ŸŽฏ Confidence interval (mean) xฬ„ ยฑ zฮฑ/2ยท s/โˆšn A safety margin around the average. Normal/t approximation to mean CI. Estimate ticket resolve time ยฑ5 min @95%. Quantifies uncertainty. Autocorrelation breaks iid assumptions. Block bootstrap; Neweyโ€“West SEs. scipy.stats, statsmodels 
stats.t.interval(0.95, n-1, loc=x.mean(), scale=x.std(ddof=1)/n**0.5)
 1) Mean. 2) Std error = std/โˆšn. 3) Margin = zร—SE. 4) Mean ยฑ margin. Size target MOE โ†’ compute n โ†’ sample โ†’ report CI on dashboards.
๐Ÿงช A/B test (means & proportions) t = (xฬ„โ‚โˆ’xฬ„โ‚‚)/(s_pโˆš(1/nโ‚+1/nโ‚‚)); z = (pโ‚โˆ’pโ‚‚)/โˆš(p(1โˆ’p)(1/nโ‚+1/nโ‚‚)) Check if a change really helped. NHST for differences in means/props. Old vs new IVR transfer rate. Clear decision-gate. Multiple tests inflate false positives. Holm/Bonferroni; power analysis (1โˆ’ฮฒ). scipy.stats, statsmodels 
from scipy import stats
stats.ttest_ind(a, b, equal_var=False)
 1) Compare group means/props. 2) Compute test statistic. 3) Get p-value from dist. 4) Decide vs ฮฑ. Define KPI โ†’ randomise โ†’ run test โ†’ correct for multiplicity โ†’ decide rollout.
๐Ÿงฎ Precision / Recall / F1 P=TP/(TP+FP), R=TP/(TP+FN), F1=2PR/(P+R) Quality of yes/no predictions. Confusion-matrix derived metrics. Security alert triage (high precision on P1). Aligns to SLA priorities. Threshold choice changes trade-off. PR curve; optimise Fฮฒ for business cost. scikit-learn 
from sklearn.metrics import f1_score
f1 = f1_score(y_true, (proba>0.42))
 1) Count TP/FP/FN. 2) Compute P, R. 3) Combine into F1. Estimate costs โ†’ pick threshold on PR curve โ†’ monitor drift.
๐Ÿงฏ Robust outlier detection (MAD) MAD = median(|xโˆ’med|); zแตฃ=0.6745(xโˆ’med)/MAD Outlier score that resists skew. Median-based; 50% breakdown point. Flag rogue CPU spikes or ticket times. Works on messy ops data. MAD=0 if flat segments. IQR; STL detrend + MAD on residuals. numpy, pandas 
med = np.median(x)
mad = np.median(np.abs(x-med))
zr  = 0.6745*(x-med)/mad
 1) Median. 2) Absolute deviations. 3) Median of those โ†’ MAD. 4) Scale to zแตฃ. Compute zแตฃ โ†’ set |zแตฃ|>k (e.g., 3.5) โ†’ alert & suppress repeats.
๐Ÿค– ML โ€” Regression, Classification, Regularisation, Optimisation, NLP, PCA, Bayes โ–พ
NameFormulaNon-technical descriptionTechnical description ScenarioProsPitfallsFormulas to address pitfalls Python librariesPython codeFormula Steps in Plain EnglishStep-by-step evaluation
๐Ÿ“ˆ Linear regression ฮฒฬ‚=(Xแต€X)โปยนXแต€y; minimize MSE Best-fit line through points. OLS; BLUE under Gaussโ€“Markov. Forecast backlog from arrivals & staffing. Interpretable, fast. Multicollinearity inflates variance. VIF; Ridge +ฮปโ€–ฮฒโ€–ยฒ. scikit-learn, numpy 
from sklearn.linear_model import LinearRegression
m = LinearRegression().fit(X, y)
 1) Arrange X (inputs), y (target). 2) Compute ฮฒฬ‚. 3) Use Xฮฒฬ‚ to predict. Split train/test โ†’ fit โ†’ residual checks โ†’ deploy & monitor.
๐Ÿ” Logistic regression (binary) P(y=1|x)=1/(1+e^{-(wแต€x+b)}); log-loss Predict yes/no probability. Bernoulli GLM with logit link; MLE. Predict SLA breach risk in 24h. Calibrated probabilities. Class imbalance; miscalibration. Class weights; Platt/Isotonic calibration. scikit-learn 
from sklearn.linear_model import LogisticRegression
m = LogisticRegression(class_weight='balanced').fit(X,y)
 1) Weighted sum โ†’ sigmoid. 2) Output 0โ€“1 probability. Pick threshold for business cost โ†’ calibrate โ†’ track drift.
๐Ÿงฒ Regularisation (L1/L2/ElasticNet) Minimise โ€–yโˆ’Xฮฒโ€–ยฒ + ฮปโ€–ฮฒโ€–ยฒ (L2) or + ฮปโ€–ฮฒโ€–โ‚ (L1) Prevent overfitting; simplify model. Penalised ERM adds bias, reduces variance. Many features, few rows. Stabilises coefficients. Too much ฮป underfits. CV to choose ฮป; 1-SE rule. scikit-learn 
from sklearn.linear_model import ElasticNetCV
m = ElasticNetCV(cv=5).fit(X,y)
 1) Fit with penalty. 2) Tune ฮป via CV. 3) Refit on full data. Grid search ฮป/ฮฑ โ†’ check bias/var โ†’ lock and version.
๐Ÿƒ Gradient descent (SGD) ฮธ_{t+1}=ฮธ_t โˆ’ ฮฑ โˆ‡J(ฮธ_t) Repeated nudges downhill. First-order iterative optimiser. Train classifiers on high-dim data. Scales to big data. Bad learning rate diverges. Line search; Adam; schedules. sklearn, PyTorch, TF 
from sklearn.linear_model import SGDClassifier
m = SGDClassifier().fit(X,y)
 1) Start guess. 2) Compute slope. 3) Step opposite direction ร— ฮฑ. 4) Repeat. Monitor loss vs epochs โ†’ early stopping โ†’ save best weights.
๐Ÿงพ TF-IDF + Cosine similarity tfidf=tfยทlog(N/df); cosฮธ=(uยทv)/(โ€–uโ€–โ€–vโ€–) Turn text into weighted numbers; compare angle. Sparse vectorisation; angular similarity. Auto-route tickets (โ€œpassword resetโ€). Strong baseline for NLP routing. OOV & vocabulary drift. n-grams; hashing; sublinear tf. scikit-learn 
from sklearn.feature_extraction.text import TfidfVectorizer
X = TfidfVectorizer().fit_transform(texts)
 1) Count words. 2) Down-weight common terms. 3) Compare vectors by angle. Build training corpus โ†’ vectorise โ†’ nearest-neighbour route โ†’ retrain monthly.
๐ŸงŠ PCA (dimensionality reduction) ฮฃv=ฮปv; project Z=XV_k Compress features but keep variance. Orthogonal projection onto top eigenvectors. Reduce 300 KPIs to 10 drivers. Speeds training; denoises. Loss of interpretability. Inspect loadings; varimax rotation. scikit-learn, numpy 
from sklearn.decomposition import PCA
Z = PCA(10).fit_transform(X)
 1) Standardise. 2) Covariance & eigenvectors. 3) Keep top k. 4) Project. Choose k by explained variance โ†’ sanity-check loadings โ†’ use Z downstream.
๐Ÿง  Bayesโ€™ theorem P(A|B)=P(B|A)P(A)/P(B) Update belief with new evidence. Posterior โˆ likelihood ร— prior. Probability a ticket is security-related given keywords. Transparent updating. Bad priors bias results. Empirical Bayes; hierarchical models. numpy, scipy 
post=(like*prior)/(like*prior + like0*(1-prior))
 1) Set prior. 2) Compute likelihoods. 3) Apply formula โ†’ posterior. Estimate priors from historicals โ†’ update online โ†’ calibrate with reliability curves.
โ˜Ž๏ธ Ops/Queues โ€” Poisson & Littleโ€™s Law, Erlang-C, Exponential Smoothing, Linear Programming โ–พ
NameFormulaNon-technical descriptionTechnical description ScenarioProsPitfallsFormulas to address pitfalls Python librariesPython codeFormula Steps in Plain EnglishStep-by-step evaluation
โ˜Ž๏ธ Poisson arrivals & Littleโ€™s Law P(N=k)=e^{-ฮปt}(ฮปt)^k/k!; L=ฮปW Calls arrive randomly but at a rate. Poisson process; steady-state relationship L=ฮปW. Inbound calls per 5-min; infer avg queue. Simple, insight-rich. Overdispersion (burstiness). Negative Binomial (Var=ฮผ+ฮบฮผยฒ); ฮป(t) by time-of-day. scipy.stats, statsmodels 
from scipy.stats import poisson
p = poisson.pmf(k, lam*t)
 1) Estimate ฮป. 2) Choose t. 3) Use formula for k arrivals. 4) L=ฮปW links rate & wait. Fit ฮป(t) per interval โ†’ validate dispersion โ†’ apply Littleโ€™s Law for WIP & SLAs.
๐Ÿ‘ฅ Erlang-C (M/M/c) + ASA ฯ=ฮป/(cฮผ); PW= (a^c/(c!(1โˆ’ฯ)))/[ฮฃโ‚€^{cโˆ’1} a^n/n! + a^c/(c!(1โˆ’ฯ))]; a=ฮป/ฮผ; ASA=PW/(cฮผโˆ’ฮป) Predict wait probability & agents needed. Steady-state queueing for c parallel servers. Size a 24/7 helpline to meet 80/20 SLA. Industry standard. Non-Poisson arrivals; skill mismatch. Skills routing; simulation; use ฮป(t). numpy 
def erlang_c(lam, mu, c):
    import numpy as np, math
    a = lam/mu; rho = lam/(c*mu)
    num = a**c/(math.factorial(c)*(1-rho))
    den = sum(a**n/math.factorial(n) for n in range(c)) + num
    return num/den
 1) Get ฮป, ฮผ. 2) Try c. 3) Compute PW, ASA. 4) Iterate c to hit SLA. Forecast interval loads โ†’ compute c by interval โ†’ add shrinkage & occupancy caps.
๐Ÿ“… Exponential smoothing (level) ลทt+1=ฮฑ y_t + (1โˆ’ฮฑ) ลท_t Fast, adaptive short-term forecast. EW squared-error minimiser; 0<ฮฑ<1. 15-min inbound chat forecast. Tiny & robust. No seasonality/trend. Holt-Winters (add trend/seasonal). statsmodels 
from statsmodels.tsa.holtwinters import ExponentialSmoothing
fit = ExponentialSmoothing(y).fit()
fc  = fit.forecast(12)
 1) Pick ฮฑ. 2) New forecast = ฮฑร—latest + (1โˆ’ฮฑ)ร—old forecast. CV ฮฑ per queue โ†’ roll forecasts โ†’ compare MAPE โ†’ feed into staffing LP.
๐Ÿงฎ๐Ÿ“ฆ Linear programming (rostering) min cแต€x s.t. Ax โ‰ฅ b, x โ‰ฅ 0 Optimise resources under limits. Convex optimisation with feasibility regions. Minimise staffing cost while meeting 80/20 & occupancy. Global optimum with proofs. Infeasible if constraints clash. Add slack; sensitivity โˆ‚z/โˆ‚b. PuLP, OR-Tools 
import pulp as pl
x = pl.LpVariable.dicts('shift', S, 0)
prob = pl.LpProblem('staff', pl.LpMinimize)
prob += pl.lpSum(cost[s]*x[s] for s in S)
 1) Define variables & costs. 2) Add constraints. 3) Solve โ†’ schedule. Encode SLAs as constraints โ†’ add shrinkage โ†’ solve & export rota.
๐ŸŒ Networks/Reliability โ€” Availability, PageRank/Centrality โ–พ
NameFormulaNon-technical descriptionTechnical description ScenarioProsPitfallsFormulas to address pitfalls Python librariesPython codeFormula Steps in Plain EnglishStep-by-step evaluation
๐Ÿ›ก๏ธ Availability & redundancy A=MTBF/(MTBF+MTTR); Series: โˆAแตข; Parallel: 1โˆ’โˆ(1โˆ’Aแตข) Uptime of components and combinations. Steady-state availability; assumes independence. Data-centre uptime; redundant links justification. Clear link to SLAs. Common-mode failures break independence. Dependency factoring; fault-tree analysis (FTA). numpy, pandas 
A = mtbf/(mtbf+mttr)
A_par = 1 - np.prod(1-np.array(Ai))
 1) Find MTBF/MTTR. 2) Compute A. 3) Combine in series/parallel. Map dependencies โ†’ compute path availability โ†’ identify single points of failure.
๐ŸŒ PageRank / eigenvector centrality PR(i)=(1โˆ’d)/N + d ฮฃ_{jโˆˆIn(i)} PR(j)/out(j) Importance ranking in a network. Stationary distribution of a random walk with teleport. Rank routers/AD servers for patch priority. Surfaces critical assets. Dangling nodes; directionality. Damping 0<d<1; personalise vector; handle dangling mass. networkx 
import networkx as nx
pr = nx.pagerank(G, alpha=0.85)
 1) Start equal scores. 2) Distribute along edges. 3) Add teleport. 4) Iterate to convergence. Build graph from CMDB โ†’ compute PR โ†’ harden top k assets first.

Electricity Market Model Click2Expand

๐Ÿ“ˆ Forecasting โ€” Holt-Winters, ARIMA/ARIMAX, Weather Normalisation โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
โšก๏ธ Holt-Winters (additive) ลท_{t+h}=(โ„“_t+h b_t) s_{t+h-mk}, updates with ฮฑ,ฮฒ,ฮณ; seasonal period m=48 (HH) Forecast half-hourly demand using level, trend, seasonality. Triple exponential smoothing (ETS AAA) minimises exponentially weighted SSE. Supplier day-ahead demand forecast for hedging. Fast; strong baseline. Breaks on regime shifts (price caps, storms). Refit rolling; add exogenous vars (temp, price): ETSX. statsmodels 
from statsmodels.tsa.holtwinters import ExponentialSmoothing
fit = ExponentialSmoothing(y,trend='add',seasonal='add',seasonal_periods=48).fit()
fc  = fit.forecast(96)
 1) Track current level, trend, seasonality. 2) Update each period. 3) Project forward h steps. Train/validation split โ†’ tune ฮฑฮฒฮณ โ†’ backtest by season โ†’ nightly refit โ†’ push to hedge calc.
๐Ÿ“ˆ ARIMA / ARIMAX ฯ•(B)(1-B)^d y_t = ฮธ(B)ฮต_t + ฮฒX_t Time-series that learns autocorrelation/shocks; can include weather/price. Box-Jenkins with exogenous regressors; choose (p,d,q) by AIC/BIC. Forecast IDNO/DUoS area load with temperature and price. Flexible; interpretable. Non-stationarity; regime changes. Differencing; holiday dummies; rolling refit. pmdarima, statsmodels 
import pmdarima as pm
m = pm.auto_arima(y, X=exog, seasonal=False).fit()
fc = m.predict(n_periods=48, X=exog_fc)
 1) Make series stationary. 2) Fit AR & MA parts. 3) Add exogenous drivers. 4) Forecast & invert differencing. Diagnose residuals โ†’ select order โ†’ backtest multiple windows โ†’ deploy & monitor drift.
๐ŸŒก๏ธ Weather normalisation (HDD/CDD) HDD=max(0,T_b-T), CDD=max(0,T-T_b); regression y=ฮฒโ‚€+ฮฒโ‚HDD+ฮฒโ‚‚CDD+ฮต Adjust demand to โ€œtypical weatherโ€ for fair KPI comparisons. Linear model on degree-days; choose base T_b (e.g., 15.5 ยฐC UK). Compare this winterโ€™s usage vs typical. Fair baselines; portfolio comparability. Wrong base temp; microclimates. Grid-search T_b; add site fixed-effects. pandas, scikit-learn 
X = df[['HDD','CDD']]
from sklearn.linear_model import LinearRegression
m = LinearRegression().fit(X, y)
 1) Compute HDD/CDD from temps. 2) Regress demand on them. 3) Predict at โ€œtypicalโ€ HDD/CDD. Select base โ†’ fit โ†’ report normalised demand/KPIs โ†’ update monthly.
๐Ÿงฐ Operations & Dispatch โ€” Economic Dispatch, Unit Commitment, Storage, DR, Load/Capacity & Diversity โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿงฐ Economic dispatch (no network) Minimise โˆ‘ C_i(P_i) s.t. โˆ‘P_i=D, 0โ‰คP_iโ‰คP_i^{max} Choose cheapest generator mix to meet demand now. Convex cost with KKT multiplier (system lambda). Half-hour balancing by marginal cost stack. Efficient given costs. Ignores start-up/min-up-down. Add UC constraints (MILP). cvxpy 
import cvxpy as cp
P=cp.Variable(n)
prob=cp.Problem(cp.Minimize(c@P),[cp.sum(P)==D,P>=0,P<=Pmax])
prob.solve()
 1) Sort by cost. 2) Fill to meet D. 3) Respect limits. Validate vs actual stack โ†’ quantify savings โ†’ iterate costs.
๐Ÿญ Unit Commitment (UC) Binary on/off u_{i,t}โˆˆ{0,1}, power P_{i,t}, min-up/down, ramps, start-ups. Which plants to have on, considering start/stop costs. MILP over time horizon; operational constraints. DA schedule for CCGT/OCGT fleet. Operationally realistic. Large MILPs can be slow. Lagrangian relax; rolling horizon; heuristics. PuLP, OR-Tools 
# u[i,t] on/off, P[i,t] output; add minup/mindown & ramps
 1) Decide on/off. 2) Constrain time-coupling. 3) Minimise total cost. Solve horizon โ†’ roll forward each HH โ†’ compare to market schedule.
๐Ÿ”‹ Storage arbitrage Maximise โˆ‘ ฯ€_t e_t s.t. SOC dynamics, charge/discharge limits, round-trip ฮท When to charge/discharge to earn from price spreads. LP with state-of-charge constraints. 2-hour battery on GB day-ahead. Clear & solvable. Degradation, cycle limits. Add cycle constraints; aging models. cvxpy 
# soc[t+1]=soc[t]+ฮท_c*ch[t]-dis[t]/ฮท_d
 1) Set prices & limits. 2) Constrain SOC. 3) Optimise schedule. Backtest โ†’ add degradation โ†’ redeploy daily.
๐Ÿงฒ Demand Response baseline & impact Baseline BL_t=avg_{dโˆˆD}(y_{d,t}), Impact ฮ”=BLโˆ’y Estimate savings vs โ€œwhat wouldโ€™ve happenedโ€. Matched-day/10-of-10 + adj; or ML counterfactual. Large C&I reduces load on price signal. Simple accounting. Baseline inflation/erosion; rebound. ML baselines; penalty terms for rebound. pandas, sklearn 
bl = y.shift(7*48).rolling(10).mean()
impact = bl - y
 1) Build baseline. 2) Compare actual. 3) Sum verified kWh. QC with weather/occupancy โ†’ settle payments on verified savings.
๐Ÿงฎ Load factor (LF) LF = ศฒ / P_{max} How โ€œflatโ€ your demand is (near 1 = smoother). Average power divided by peak power over period. Retail portfolio shaping KPI. Intuitive, quick. Misses sub-interval spikes. Add 95th percentile peak, ramp rate. pandas 
lf = y.mean()/y.max()
 1) Mean. 2) Max. 3) Divide. Track LF + 95th pct & ramps for ops decisions.
โš™๏ธ Capacity factor (generation) CF = (โˆ‘ P_t)/(P_{rated}ยทT) How hard an asset ran vs its capability. Time-weighted utilisation metric. Wind farm performance vs P50. Clear health check. Curtailment/outages confound. Decompose by cause codes; weather-adjust. pandas 
cf = gen.sum()/(nameplate*hours)
 1) Sum output. 2) Divide by ratedร—hours. Attribute gaps to wind vs outages vs curtailment.
๐Ÿค Coincidence / Diversity factor CF_agg = P_{max,agg} / โˆ‘ P_{max,i}, Diversity = 1โˆ’CF How individual peaks combine across many customers. Portfolio peak relative to sum of individual peaks. Size capacity/hedge at portfolio (not customer) level. Shows natural smoothing. Herding creates high coincidence. Segment cohorts; stress at peak HH. pandas 
cf = agg.max() / indiv_max.sum()
 1) Portfolio peak. 2) Sum indiv. peaks. 3) Divide. Use CF in capacity/hedge planning; monitor by segment.
๐Ÿ’น Risk & Hedging โ€” Elasticity, Hedge Cost-to-Serve, Mean-Variance, VaR/CVaR, GARCH, Imbalance โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿงฒ Price elasticity of demand ฮต โ‰ˆ (ฮ”Q/Q)/(ฮ”P/P); log-log: ln Q = ฮฑ + ฮต ln P + โ€ฆ How usage changes when price changes. Elasticity from log regression; beware endogeneity. TOU tariff response estimation (EV load). Quantifies DR potential. Price โ†” demand feedback. IV regression; diff-in-diff. statsmodels 
import statsmodels.api as sm
m = sm.OLS(np.log(Q), sm.add_constant(np.c_[np.log(P), Z])).fit()
 1) Take logs. 2) Regress. 3) Coef on log P = elasticity. Check sign/CI โ†’ simulate tariff scenarios โ†’ validate vs trial.
๐Ÿ“ฆ Hedge cost-to-serve C=โˆ‘ (q_h f_h) + โˆ‘ (d_h - q_h) s_h Blended cost from forwards + spot top-up. Portfolio cost under hedge ratio path. Choose 80% day-ahead + 20% intraday. Quantifies hedge value. Over/under-hedge risk. Optimise q_h via mean-CVaR. numpy, cvxpy 
C = (q*f + (d-q)*s).sum()
 1) Pick hedge volumes. 2) Multiply by forward. 3) Residual at spot. 4) Sum. Stress with price scenarios โ†’ pick policy โ†’ monitor tracking error.
๐Ÿ“‰ Mean-Variance (hedge portfolio) Minimise ฮป wแต€ฮฃw โˆ’ ฮผแต€w s.t. โˆ‘w=1, wโ‰ฅ0 Choose mix of forwards to balance risk/return. Markowitz: return = โˆ’cost; variance = price risk. Blend month/quarter/season strips. Simple & fast. Fat-tails break Gaussian assumptions. Use CVaR (coherent) optimisation. numpy, cvxpy 
w=cp.Variable(n)
obj=cp.Minimize(cp.quad_form(w,Sigma) - mu@w)
 1) Estimate ฮผ, ฮฃ. 2) Optimise weights under constraints. Backtest โ†’ set limits โ†’ monitor exposures & VaR.
๐Ÿ“Š VaR / CVaR (energy P&L) VaR: quantile; CVaR: tail mean E[L | L โ‰ฅ VaRฮฑ] Loss at confidence level; average of worst tail. Quantile & tail expectation on P&L distribution. 95% daily VaR/CVaR on hedge book. Focuses on downside. VaR non-coherent; model risk. Prefer CVaR; historical/MC scenarios. numpy, scipy 
var  = np.quantile(L, 0.95)
cvar = L[L>=var].mean()
 1) Simulate P&L. 2) Take quantile. 3) Average the tail. Scenario set incl. spikes โ†’ set limits & capital.
๐ŸŒช๏ธ GARCH (price volatility) ฯƒ_t^2 = ฯ‰ + ฮฑ ฮต_{t-1}^2 + ฮฒ ฯƒ_{t-1}^2 Models time-varying volatility clustering. Conditional heteroskedasticity for returns. Size intraday risk limits. Captures clustering. Regime changes, jumps. EGARCH/GJR; jump components. arch 
from arch import arch_model
am = arch_model(r, vol='Garch', p=1, q=1).fit()
 1) Fit to returns. 2) Get conditional ฯƒ. 3) Forecast risk. Combine with VaR/CVaR sizing; recalibrate periodically.
๐Ÿ”„ Imbalance cash-out Charge = Volume ร— ImbalancePrice What you pay if short/long vs position. Settlement using system price (pay-as-imbalance). Supplier under-forecast in tight evening peak. Clear incentive to balance. Price spikes โ†’ large P&L hits. Better nowcasts; intraday re-trading. pandas 
charge = vol * sysprice
 1) Compute delta vs notified position. 2) Multiply by imbalance price. Attribute P&L by HH โ†’ improve forecast & hedge policy.
๐ŸŒ Network & Settlement โ€” DC-OPF & LMPs, Congestion Rent, Loss Factors โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿงญ DC-OPF & LMPs Minimise cost s.t. balance; line flows F=Bฮธ, |F|โ‰คF^{max}; LMP=ฮป_energy+ฮป_cong(+ฮป_loss) Price at each node reflects energy + congestion (and losses). Linearised (DC) OPF; dual variables give LMPs. Compute nodal prices & congestion rents. Captures grid constraints. Ignores reactive/voltage (AC effects). AC-OPF or PTDF + loss factors. PYPOWER, pandapower 
import pypower.api as pp
r = pp.runopf(case)
LMP = r['bus'][:,13]
 1) Build network model. 2) Optimise dispatch. 3) Read duals โ†’ nodal prices. Analyse hub-node spreads โ†’ hedge congestion exposure.
๐Ÿšฆ Congestion rent Rent = โˆ‘_โ„“ (ฮ”ฯ€_โ„“) ยท F_โ„“ Value collected due to constrained lines. Price difference times flow, summed over lines. Interconnector constraint revenue estimate. Quantifies bottlenecks. Volatile; counterflows. Scenario DC-OPF expected flows. numpy 
rent = np.sum((pi_from - pi_to) * flow)
 1) Compute price diff on each line. 2) Multiply by flow. 3) Sum. Rank lines by rent โ†’ target upgrades/hedges.
๐Ÿ”Œ Loss factors (settlement) E_delivered = E_metered ร— LLF (or 1โˆ’โ„“_dist) Adjust energy for distribution losses so settlement is fair. Apply LLF by profile class/region/time to meter reads. Supplier settlement adjustments (e.g., Elexon LLFs). Simple application. Wrong MPANโ†’LLF mapping. Validate joins; reconciliation vs statements. pandas 
E_del = E_mtr * llf_series
 1) Map MPAN to LLF. 2) Multiply reads by LLF. Reconcile with settlement statements; investigate deltas.
๐Ÿค– ML in Electricity โ€” Forecasting, Risk, Networks โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐ŸŒณ Gradient Boosted Trees (STLF) Stagewise additive model: F_m(x)=F_{m-1}(x)+ฮฝยทh_m(x) minimising loss Many small trees, each fixes last oneโ€™s mistakes. Boosting over CARTs on features (lags, weather, calendar). Half-hourly load forecast with temp & holiday effects. Strong accuracy; little feature scaling needed. Overfit if too deep/too many rounds. Early stopping; shrinkage ฮฝ; regularisation. xgboost, lightgbm, sklearn 
import xgboost as xgb
m = xgb.XGBRegressor(n_estimators=500, max_depth=6, learning_rate=0.05)
m.fit(X_train, y_train)
 1) Build lag & weather features. 2) Train boosted trees. 3) Stop when val error rises. Walk-forward backtest โ†’ SHAP to sanity-check drivers โ†’ deploy nightly.
๐Ÿง  LSTM (sequence-to-one) Recurrent cells: h_t,f_t,i_t,o_t gating long/short memory Neural net that โ€œremembersโ€ recent patterns. Many-to-one RNN for short-term load/RES output. PV/wind intra-day forecast with ramps. Catches non-linear temporal effects. Needs data & tuning; can drift. Dropout; early stopping; recalibration schedule. TensorFlow, PyTorch 
# Pytorch sketch
model = LSTM(input_size=X.shape[-1], hidden=64, layers=2)
 1) Windowise time series. 2) Train on past windows. 3) Predict next horizon. Backtest rolling origin โ†’ compare vs tree baseline โ†’ ensemble.
๐ŸŒฌ๏ธ Random Forest (RES output) Bagging: average of many trees to reduce variance Many decorrelated trees vote an answer. RF regression on wind/pv with NWP features. Day-ahead wind farm output P50/P90. Stable; handles interactions. Less sharp peaks vs boosting. Quantile RF for P90; feature selection. sklearn, skgarden 
from sklearn.ensemble import RandomForestRegressor
m = RandomForestRegressor(n_estimators=400).fit(X,y)
 1) Build met/NWP features. 2) Fit many trees. 3) Average predictions. Calibrate quantiles โ†’ generate P50/P90 bands.
โš ๏ธ XGBoost Classifier (price spike) Boosted trees minimising log-loss Predicts probability of a spike event. Binary classification with class weights. Imbalance price spike nowcast. Probabilities for risk sizing. Class imbalance โ†’ false alarms. Focal loss; threshold from cost curve. xgboost, sklearn 
clf = xgb.XGBClassifier(scale_pos_weight=20).fit(X,y)
 1) Label spikes. 2) Train classifier. 3) Pick threshold to balance cost. Track precision/recall over time โ†’ retrain monthly.
๐Ÿ•ต๏ธ Isolation Forest (anomalies) Isolation via random splits; anomaly score by path length โ€œOutsidersโ€ get isolated quickly. Unsupervised anomaly detection on SCADA/load. Detect metering or telemetry glitches. No labels required. Flags change-points as outliers. Add STL detrend; use rolling training windows. sklearn 
from sklearn.ensemble import IsolationForest
iso = IsolationForest().fit(X); score = iso.decision_function(X)
 1) Build feature set. 2) Fit unsupervised. 3) Score & alert top-k. Whitelist known events; tune contamination.
๐ŸŽฎ RL for Battery Dispatch Q-learning: Q(s,a)โ†(1โˆ’ฮฑ)Q+ฮฑ[r+ฮณ max_a' Q(s',a')] Agent learns charge/discharge to maximise ยฃ. MDP over price & SoC; discrete actions. Day-ahead + intraday arbitrage. Adapts to patterns. Exploration risk; constraint handling. Reward shaping; safety layer on SoC/DoD. stable-baselines3, numpy 
# define env(state: SoC, price); train DQN/PPO agent
 1) Define states/actions. 2) Train with reward=profit. 3) Constrain SoC/cycles. Backtest vs LP benchmark โ†’ deploy with guardrails.

Telecoms Wholesale and retail Click2Expand

๐Ÿ“ˆ Demand & Forecasting โ€” โ€œWhatโ€™s coming down the pipe?โ€ โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿ“… Holtโ€“Winters (additive) ลท_{t+h}=(โ„“_t+h b_t)s_{t+h-mk} with ฮฑ,ฮฒ,ฮณ; seasonal m=24 (hourly) Forecast calls/data sessions with daily/weekly rhythm. Triple exponential smoothing (ETS AAA) minimising EW-SSE. Call centre volume by hour; store staffing. Fast, solid baseline. Breaks on outages, promos, price shocks. Refit rolling; ETSX with exogenous (price, promo, outage dummies). statsmodels 
from statsmodels.tsa.holtwinters import ExponentialSmoothing
fit=ExponentialSmoothing(y,trend='add',seasonal='add',seasonal_periods=24).fit()
fc=fit.forecast(48)
 Track level/trend/seasonal โ†’ update each period โ†’ project h steps ahead. Train/validate โ†’ tune ฮฑฮฒฮณ โ†’ backtest โ†’ publish to WFM & routing.
๐Ÿ“ˆ ARIMAX ฯ•(B)(1โˆ’B)^d y_t = ฮธ(B)ฮต_t + ฮฒX_t Lets history and drivers (price, weather, promos) steer the forecast. Boxโ€“Jenkins with exogenous regressors. Predict broadband tickets after a price change. Flexible; interpretable. Non-stationary, regime switches. Differencing; holiday dummies; rolling refits. pmdarima, statsmodels 
import pmdarima as pm
m=pm.auto_arima(y, X=exog, seasonal=False).fit()
 Make series steady โ†’ fit AR & MA โ†’ plug in drivers โ†’ forecast, undifference. Residual checks โ†’ walk-forward backtest โ†’ deploy & watch drift.
๐Ÿงฎ Benford for CDR/fraud sanity P(d)=log10(1+1/d), dโˆˆ{1..9} First-digit test to sniff odd billing patterns. Benford distribution on aggregated measures. Wholesale CDR audit: spot fabricated volumes. Quick anomaly screen. Not proof of fraud; small samples flaky. Complement with MAD outliers & supervised models. numpy, pandas 
import numpy as np
digits = (np.log10(1+1/np.arange(1,10)))
 Count first digits โ†’ compare to Benford โ†’ flag deviations. Escalate big gaps โ†’ deeper checks on rated events.
โ˜Ž๏ธ Traffic & Capacity โ€” โ€œHow many trunks/agents do we actually need?โ€ โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿ“ฆ Erlang-B (blocking, no queue) B(A,c)=\frac{A^c/c!}{\sum_{k=0}^{c} A^k/k!}, traffic A=ฮป/ฮผ % of calls that get โ€œbusy toneโ€ if all trunks busy. Loss system (M/M/c/c) steady-state blocking probability. SIP trunks for a retail contact centre. Gold standard for trunks. Ignores retrials/queues. Use Erlang-C if you queue; simulate retrials. numpy, math 
def erlang_b(A,c):
    import math
    num=A**c/math.factorial(c)
    den=sum(A**k/math.factorial(k) for k in range(c+1))
    return num/den
 Compute traffic A โ†’ pick trunks c โ†’ get blocking โ†’ tweak until GoS met. Choose GoS (e.g., 1%) โ†’ iterate c โ†’ add safety for peaks.
โณ Erlang-C (wait, with queue) ฯ=ฮป/(cฮผ), P_W via standard Erlang-C; ASA=P_W/(cฮผโˆ’ฮป) Chance a caller waits + expected wait time. M/M/c with infinite buffer; steady-state. Retail care queue meeting 80/20. Battle-tested staffing. Arrival burstiness, skill-mismatch. Skills-based routing; simulate; ฮป(t) per interval. numpy 
# reuse earlier erlang_c; compute ASA from P_W
 Estimate ฮป, ฮผ โ†’ trial c โ†’ compute wait prob & ASA โ†’ adjust. Do per 15-min โ†’ add shrinkage/occupancy caps in WFM.
๐Ÿ‘ฅ Engset (finite sources) Blocking with finite N callers (formula omitted here for brevity) When your caller pool is small (e.g., internal helpdesk). Loss model with finite source population. IT service desk for 300 staff site. More realistic than Erlang-B for small N. Needs N estimate. Cross-check with sim / B as bound. custom 
# use an Engset implementation or simulate
 Estimate sources N โ†’ calls per head โ†’ compute blocking. Sanity-check with discrete-event simulation.
๐Ÿ“œ Littleโ€™s Law (packets, calls) L=ฮปW Average in-system = rate ร— time. Simple and beautiful. Applies across queues in steady-state. Estimate average active calls given arrival & wait. One-liner insight. Not for transients. Use windowed ฮป(t); exclude incident periods. pandas 
W = L/lam
 Pick a stable window โ†’ apply the identity. Cross-validate against measured occupancy.
โŒ› M/M/1 latency (packet queue) W = 1/(ฮผโˆ’ฮป), ฯ=ฮป/ฮผ Delay skyrockets as utilisation nears 1. Ooft. Single-server queue with Poisson/Exp. Edge SBC or NAT box sizing. Back-of-envelope capacity. Traffic not Poisson; service not Exp. M/G/1 (Pollaczekโ€“Khinchine), simulations. numpy 
W = 1/(mu - lam)
 Estimate ฮผ, ฮป โ†’ compute W โ†’ keep ฯ well below 1. Target ฯโ‰ค0.7 for headroom; monitor p95 latency.
๐Ÿ›ฐ๏ธ Radio & Throughput โ€” โ€œWill it actually shift the bits?โ€ โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿ“ก Friis / Path loss (free space) PL(dB)=32.44+20log10(f_MHz)+20log10(d_km) โˆ’ G_t โˆ’ G_r How loudly the signal fades with distance/frequency. Link budget basic; add margins & non-LOS losses in reality. Microwave backhaul feasibility check. Simple first pass. Ignores obstacles & rain fade. ITU-R models; add rain/urban clutter. numpy 
PL=32.44+20*np.log10(f)+20*np.log10(d)-Gt-Gr
 Pick f,d โ†’ compute loss โ†’ check RX level vs sensitivity. Add fades & margins โ†’ confirm availability target.
๐Ÿ“ถ SINR & spectral efficiency SINR = S/(I+N); ฮทโ‰ˆf(SINR) via MCS curve Signal vs noise+interference โ†’ how many bits we can cram in. Maps to modulation/coding โ†’ bits/Hz. 5G small cell planning on busy street. Directly tied to user throughput. Fast fading & scheduling fairness. Use distributions (percentiles), not point SINR. numpy 
sinr = S/(I+N)
 Measure S,I,N โ†’ compute SINR โ†’ lookup MCS โ†’ get ฮท. Plan for p5/p50/p95 SINR โ†’ verify drive-tests.
๐Ÿ“ก Shannonโ€“Hartley (upper bound) C = B log2(1+SNR) Ceiling on throughput for a clean channel. AWGN capacity bound; reality below due to overheads. Rough max throughput for fixed wireless. Gives hard upper bound. Not achievable with real protocols. Subtract overheads; use MCS curves. numpy 
C = B*np.log2(1+snr)
 Bandwidth ร— log(1+SNR) โ†’ bits per second. Apply protocol overhead โ†’ compare to SLA.
๐ŸŒ TCP throughput (Mathis) T โ‰ˆ 1.22 ยท MSS / (RTT ยท โˆšp) Why long RTT + a sniff of loss kills your speed tests. TCP Reno steady-state approximation. International transit troubleshooting. Explains user complaints fast. Modern CCAs differ (Cubic/BBR). Use measured CCA; still a good sanity check. numpy 
T=1.22*MSS/(RTT*np.sqrt(p))
 Measure MSS/RTT/loss โ†’ compute T โ†’ set expectations. Recommend CDN/BBR/MTU tuning if constrained.
๐Ÿ’ท Commercial & Risk โ€” โ€œARPU pays the bills, churn steals your lunch.โ€ โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿ“Š ARPU & CLV (simple) ARPU = Rev/Subs; CLV โ‰ˆ ARPU ยท \frac{m}{r} with margin m, retention r Average revenue per user and lifetime ยฃ value. Steady-state approximation for planning. Retail mobile bundle business case. Quick portfolio health. Ignores cohort & discounting. Discounted CLV: sum of cashflows / (1+r)^t. pandas, numpy 
clv = arpu*margin/retention
 Compute ARPU โ†’ apply margin โ†’ divide by retention. Refine with cohort curves & discount rate.
๐Ÿšช Churn (logistic) + survival P(churn)=ฯƒ(wแต€x); Kaplanโ€“Meier S(t) Whoโ€™s likely to leave, and when. Binary GLM + survival for tenure timing. Flag risky SIMs; save offers. Actionable probabilities. Imbalance; leakage of future info. Class weights; time-aware CV; survival features. sklearn, lifelines 
from sklearn.linear_model import LogisticRegression
m=LogisticRegression(class_weight='balanced').fit(X,y)
 Engineer features โ†’ fit โ†’ calibrate โ†’ threshold on cost curve. Champion-challenger โ†’ monitor drift โ†’ retrain monthly.
๐Ÿ”— Least-Cost Routing (LCR) Minimise โˆ‘ c_{i,d} x_{i,d} s.t. demand & quality constraints Pick the cheapest wholesaler per destination, within QoS. LP/MIP with capacity, A-Z breaks, quality gates. Wholesale voice A-Z routing plan. Direct OPEX impact. Route flapping; QoS drift. Add hysteresis; QoS constraints; penalties. cvxpy, pulp 
# x[i,d] fraction via carrier i to dest d; minimise cost
 Build cost/QoS matrix โ†’ solve โ†’ generate route tables. Monitor CDR KPIs โ†’ auto-reroute on breaches.
๐Ÿ’ฑ Interconnect settlement Charge = โˆ‘ minutes_d ร— rate_d (+ surcharges) Bill each destination at the agreed rate card. Rating by A-Z prefix with time bands, surcharges. Monthly invoice to/ from carriers. Transparent and auditable. Prefix drift; rate card mismatches. Versioned rate tables; prefix normalisation; diffs. pandas 
bill = (mins*rate).groupby('dest').sum()
 Join CDRs to rate card โ†’ multiply โ†’ sum per counterparty. Reconcile vs partner; chase deltas; adjust prefixes.
๐Ÿงฒ Price elasticity (bundles) ฮต โ‰ˆ (ฮ”Q/Q)/(ฮ”P/P); log-log regression How subs change when you tweak price. Elasticity from ln-ln model; beware endogeneity. Broadband + mobile bundle test. Guides promo depth. Confounded by offers/seasonality. Instruments; diff-in-diff; fixed effects. statsmodels 
import statsmodels.api as sm
m=sm.OLS(np.log(Q), sm.add_constant(np.c_[np.log(P), Z])).fit()
 Take logs โ†’ regress โ†’ elasticity is coef on log P. Simulate scenarios โ†’ guardrails for pricing.
๐ŸŒ Network, Routing & Reliability โ€” โ€œFind the choke points before they find you.โ€ โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿ›ก๏ธ Availability (nodes/paths) A=MTBF/(MTBF+MTTR); series: โˆAแตข; parallel: 1โˆ’โˆ(1โˆ’Aแตข) Uptime of gear and combined paths. Steady-state availability; assumes independence. Core + access redundancy score. Clear SLA link. Common-mode failures spoil party. Fault-tree analysis; dependency factors. numpy 
A = mtbf/(mtbf+mttr)
 Get MTBF/MTTR โ†’ compute A โ†’ combine by topology. Identify SPoFs โ†’ prioritise hardening.
๐Ÿงญ PageRank / centrality PR(i)=(1โˆ’d)/N + d ฮฃ PR(j)/out(j) Find the โ€œkingโ€ routers/edges by influence. Eigenvector of random-walk on graph. Prioritise patching/monitoring targets. Surfaces critical hubs. Dangling nodes; directionality. Handle dangling mass; personalise by traffic. networkx 
import networkx as nx
pr = nx.pagerank(G, alpha=0.85)
 Build graph โ†’ run PR โ†’ sort nodes by score. Cross-check with flow; harden top-k.
๐Ÿช™ MOS / E-model (voice QoE) R=Rโ‚€โˆ’I_sโˆ’I_dโˆ’I_e+โ€ฆ; MOSโ‰ˆ1+0.035R+7eโˆ’6R(Rโˆ’60)(100โˆ’R) Maps impairments to a โ€œhow it soundsโ€ score. E-model per ITU-T G.107; codec, delay, loss factors. Wholesale voice route QoE benchmarking. Single MOS for execs. Model assumptions; non-speech effects. Use PESQ/POLQA for lab, E-model for fleet. numpy 
# compute R then MOS from formula
 Measure loss/jitter/latency โ†’ compute R โ†’ map to MOS. Alert on MOS dips โ†’ reroute carriers.
๐Ÿค– ML in Telecoms โ€” Churn, Fraud, Traffic, QoE, Routing โ–พ
NameFormula (๐Ÿ“)Non-technical descriptionTechnical description Scenario ๐ŸงชPros โœ…Pitfalls โš ๏ธFormulas to address pitfalls Python libs ๐Ÿ› ๏ธPython codeFormula Steps in Plain English ๐Ÿ“Step-by-step evaluation ๐Ÿ‘ฃ
๐Ÿšช Churn ML (logistic + survival) P(churn)=ฯƒ(wแต€x); survival S(t) via Cox/KM Who leaves, and how soon. Binary GLM + time-to-event model. Flag risky broadband/mobile subs. Actionable; clear uplift targeting. Leakage; class imbalance. Time-aware CV; class weights; calibration. sklearn, lifelines 
from sklearn.linear_model import LogisticRegression
m=LogisticRegression(class_weight='balanced').fit(X,y)
 1) Engineer features. 2) Fit & calibrate. 3) Score & intervene. Championโ€“challenger offers โ†’ monitor retention lift.
๐Ÿ•ต๏ธ CDR Fraud (IsoForest + supervised) Isolation paths + classifier stack Spot weird call patterns & confirm with labels. Unsupervised pre-filter โ†’ supervised confirmer. Wholesale Aโ€“Z fraud, IRSF, SIM box. High recall on oddities. Alert fatigue. Two-stage: anomaly score โ†’ threshold โ†’ classifier. sklearn, xgboost 
iso=IsolationForest().fit(X); cand=iso.decision_function(X)
 1) Score anomalies. 2) Train classifier on confirmed fraud. Auto-block high score + human-in-loop review.
๐Ÿ“ˆ Traffic Forecast (GBTs/LSTM) Boosting or RNN as above Predict sessions/calls with promos/outages. Tree or seq2seq with exogenous drivers. Contact-centre/Wi-Fi offload planning. Great short-term accuracy. Promo/incident drift. Event features; rapid refits; ensembles. xgboost, PyTorch 
# choose tree or LSTM path per KPI
 1) Build features/windows. 2) Train. 3) Validate & ensemble. Deploy with drift monitors; auto-refresh models.
๐Ÿ›ฐ๏ธ QoE Estimation (MOS regressor) Map KPIs โ†’ MOS via regression Predict โ€œhow it feelsโ€ from KPIs. Nonlinear regressor on latency/jitter/loss/codec. Wholesale route MOS for alerts. Fast, continuous MOS. MOS is a proxy; lab โ‰  field. Retrain per codec/region; add quantile models. sklearn 
from sklearn.ensemble import GradientBoostingRegressor
 1) Collect KPIs. 2) Fit MOS model. 3) Alert when predicted MOS dips. Compare to POLQA samples; recalibrate.
๐Ÿ—บ๏ธ Graph ML (routing anomalies) Node2Vec/GraphSAGE embeddings โ†’ classifier Learn network โ€œshapeโ€ to catch odd flows. Graph embeddings + anomaly/cls. Weird BGP/route changes, choke points. Captures topology context. Needs graph pipeline. Automate graph ETL; rolling windows. networkx, stellargraph, pyg 
# build G โ†’ node2vec โ†’ clf on embeddings
 1) Build graph. 2) Learn embeddings. 3) Detect anomalies. Alert & correlate with NetFlow/SNMP.
๐ŸŽฎ RL for SON / Routing Policy gradient / Q-learning rewards on QoE Auto-tune network knobs for experience. MDP with KPIs as reward; safe constraints. eNodeB power/tilt, handover margins. Adapts to local conditions. Exploration risk on live users. Shadow mode; constrained RL; guardrails. stable-baselines3 
# define env; train PPO with safety constraints
 1) Sim/Shadow learn. 2) Validate. 3) Gradual live rollout. Compare QoE uplift vs control cells; rollback if worse.

๐Ÿ“ Formulas without business context are the Billy-no-mates of algorithms. Full of potential, waiting for a dataset to play with ๐Ÿ“Š.

โž• Mean โ€” baseline average (needs context)
โž• Mean (Average)
โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿ“Š Quick โ€œmiddle groundโ€ check Add โ€™em up โž• and divide โž— Arithmetic mean 1๏ธโƒฃ Add
2๏ธโƒฃ Count
3๏ธโƒฃ Divide		 (2,4,6) โ†’ 12 รท 3 = 4 โšก Average monthly electricity bill from weekly readings
๐Ÿ”ฒ Variance โ€” how spread out values are
๐Ÿ”ฒ Variance (Spread)
โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿ“‰ See how jumpy values are Spread of values Mean squared deviation Subtract โ†’ Square โ†’ Sum โ†’ รทn (2,4,6) โ†’ Varโ‰ˆ2.67 ๐Ÿ“ˆ Month-to-month revenue volatility for a product line
๐Ÿ”— Covariance โ€” do two variables move together?
๐Ÿ”— Covariance (Joint Movement)
โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿค Check if two things rise & fall together Do 2 vars move in sync? Expected joint deviation Subtract means โ†’ Multiply โ†’ Sum โ†’ รทn X=(1,2,3), Y=(2,3,5) โ†’ Cov=1 ๐Ÿ›’ Ad spend moving with weekly sales in a campaign
๐Ÿ“ Correlation โ€” strength & direction (โˆ’1 to +1)
๐Ÿ“ Correlation (Strength of Link)
โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿ“ Ask: how strong is the link? Togetherness measure Cov รท (ฯƒXฯƒY) Cov / product of SDs rโ‰ˆ0.97 ๐Ÿ“ž Call-centre volume vs website outage duration
๐Ÿ“ˆ Linear Regression โ€” best-fit line through points
๐Ÿ“ˆ Linear Regression (Best Fit Line)
โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿ“ Want a line through dots Best-fit straight line Least squares fit ฮฒโ‚=Cov/Var, ฮฒโ‚€=ศณโˆ’ฮฒโ‚xฬ„ ลท=0.33+1.5x โš™๏ธ Predict maintenance time from machine age/usage
๐Ÿงฉ OLS Matrix โ€” matrix solution to regression
๐Ÿงฉ Ordinary Least Squares (Matrix Form)
โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿงฎ Need exact solution for line/plane Matrix shortcut (Xแต€X)โปยนXแต€y Build X โ†’ invert โ†’ multiply Slope=1.5, Int=0.33 ๐Ÿ“Š Fitting BI trendlines across many features fast
๐ŸŽฏ MSE โ€” average squared prediction error
๐ŸŽฏ Mean Squared Error (Loss)
โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐ŸŽฏ Compare modelsโ€™ accuracy Average squared miss Quadratic loss Square โ†’ Sum โ†’ รทn MSEโ‰ˆ0.056 ๐Ÿงช A/B test: pick the model with the lower error on holdout
๐Ÿ”ป Gradient Descent โ€” iterative downhill optimiser
๐Ÿ”ป Gradient Descent (Optimiser)
โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿฅพ When algebra too hard, walk it Iterative optimiser ฮธโ†ฮธโˆ’ฮทโˆ‡L Guess โ†’ Gradient โ†’ Step โ†’ Repeat w=0โ†’0.8โ†’1.44 ๐Ÿค– Train a forecasting model on large datasets
๐Ÿšฟ SGD โ€” online learner (one sample at a time)

๐Ÿšฟ Stochastic Gradient Descent (Online Learning)

โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿบ Stream data one by one Noisy but efficient Update on each sample w=0โ†’0.6โ†’1.56 Clickstream learning Online ads adapting live
๐Ÿ“ˆ Logistic Regression โ€” S-curve for probabilities

๐Ÿ“ˆ Logistic Regression (Classification)

โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿ”ฎ Want yes/no with probabilities Sigmoid transform ลท=1/(1+e^โˆ’(ฮฒโ‚€+ฮฒโ‚x)) ฮฒโ‚€=โˆ’3, ฮฒโ‚=2, x=2 โ†’ p=0.73 Churn yes/no Customer retention analysis
๐ŸŒ€ Cross-Entropy โ€” penalises confident wrong answers

๐ŸŒ€ Cross-Entropy (Log Loss)

โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
โš–๏ธ Penalise confident errors Log-loss โˆ’[ylogp+(1โˆ’y)log(1โˆ’p)] y=1, p=0.73 โ†’ 0.314 Classifier training Email spam filters
๐Ÿงญ Bayes โ€” update beliefs with evidence

๐Ÿงญ Bayesโ€™ Theorem (Update Rule)

โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿง  Update beliefs with evidence Posterior โˆ Priorร—Likelihood P(A|B)=P(B|A)P(A)/P(B) P(spam|WIN)=95% Spam filter Medical diagnosis updates
๐Ÿงฎ Matrix Inversion โ€” algebra behind OLS

๐Ÿงฎ Matrix Inversion (OLS Step)

โฐ When to Apply๐Ÿ—ฃ๏ธ Plain Speak๐Ÿ“ Technical๐Ÿชœ Stepsโœ๏ธ Example๐ŸŒ Scenario
๐Ÿงฎ When you must invert Xแต€X Feature covariance inverse (Xแต€X)โปยน [[14,โˆ’6],[โˆ’6,3]]/6 OLS engine Running regression in BI tools

๐Ÿฆถ note: Machine Learning put into context

The maths are already used; ML just remembers stuff and evolves understanding. After all the world was flat until Aristotle (384 - 322 BCE) figured it out by observing that ships disappear hull-first over the horizon!