Time series — trend ও seasonality

এটি classical time series statistics-এর foundation। Stationarity-র necessity বুঝলে — অনেক method-এর behavior intuitive।

Stationarity-র সংজ্ঞা (strict):

• Joint distribution of any subset $(Y_{t_1}, ..., Y_{t_k})$ time-shift invariant।
• $P(Y_{t_1} \le y_1, ..., Y_{t_k} \le y_k) = P(Y_{t_1+\tau} \le y_1, ..., Y_{t_k+\tau} \le y_k)$।
• Practically too strict; weak (covariance) stationarity used।

Weak stationarity:

• Mean constant: $E[Y_t] = \mu$ — সব $t$-তে same।
• Variance constant: $\text{Var}(Y_t) = \sigma^2$ — সব $t$-তে same।
• Covariance lag-only: $\text{Cov}(Y_t, Y_{t+k}) = \gamma_k$ — depends on $k$, not $t$।

কেন ARIMA-এর জন্য stationarity essential:

(১) Statistical estimation valid

• OLS estimation — sample-based mean ও variance population-এর estimate।
• Non-stationary-এ — sample mean time-window-এ change, "true mean" undefined।
• Estimator unbiased property break।

(২) Forecasting validity

• "Past pattern = future pattern" assumption।
• Stationary-তে এটা true।
• Non-stationary-এ পুরোনো data future-এর প্রতিনিধি না।

(৩) Spurious regression

• Granger ও Newbold (১৯৭৪): দু'টি independent random walk-এর regression-এ R² = 0.7+।
• Pure noise correlation — meaningless।
• Bangladesh GDP ও global temperature both rising → "GDP causes warming" — spurious।

(৪) Confidence interval

• Standard error formula stationary assume।
• Non-stationary-এ — under-coverage; nominal 95% actually 60%।

উদাহরণ — non-stationary failure:

• Bangladesh GDP series (rising trend)।
• ARMA(1,1) fit — $Y_t = \phi Y_{t-1} + \theta \epsilon_{t-1}$।
• $\phi$ estimate ১.০ (random walk fit) — but ARMA assumes $|\phi| < 1$।
• Forecast unbounded variance।

Solutions — making stationary:

(ক) Differencing:

• $\Delta Y_t = Y_t - Y_{t-1}$।
• Linear trend remove।
• Random walk → white noise।

(খ) Log + differencing:

• Variance growing — log।
• Then trend → differencing।
• Standard for financial data।

(গ) Seasonal differencing:

• $\Delta_{12} Y_t = Y_t - Y_{t-12}$।
• Yearly seasonality remove (monthly data)।

(ঘ) Detrending (regression):

• Linear/polynomial trend regress।
• Residual model।
• Risky if trend changes dynamically।

ARIMA-এর "I" — Integrated:

• $d$-time differencing — stationary বানায়।
• ARIMA(p, d, q) — d differences, then ARMA।
• Forecast → un-difference → original scale।

Cointegration — special case:

• দু'টি non-stationary series, কিন্তু linear combo stationary।
• "Long-term equilibrium" relationship।
• Engle-Granger, Johansen tests।
• Bangladesh-এর USD-BDT rate vs reserve — cointegrated possible।

Modern alternatives — relaxed assumption:

• State-space models: Kalman filter — time-varying parameter handle।
• Prophet: non-stationarity OK — components separately model।
• Deep learning (LSTM, Transformer): non-stationarity tolerate।
• Bayesian time series: uncertainty propagate।

Practical guidance:

• Always plot first — visual stationarity check।
• ADF + KPSS dual test।
• If borderline — try both differenced ও non-differenced model।
• Out-of-sample validation — final arbiter।

Bangladesh examples:

• Daily Daraz revenue: log + differencing।
• BSEC index: differencing (returns)।
• Monthly inflation: usually stationary as-is।
• Annual GDP: differencing — growth rate stationary।

মূল উপলব্ধি: Stationarity = "predictable patterns persist"। ARIMA assume; modern method (Prophet, DL) relax। বুঝে — সঠিক tool বাছা যায়।

এটি Bangladesh, Indonesia, Pakistan, Egypt-এর সব e-commerce ও fintech-এর crucial challenge। Standard solar-calendar-based seasonality এই situations-এ inadequate।

সমস্যা:

• Eid-ul-Fitr lunar — প্রতিবছর ~১১ দিন আগে আসে।
• ২০২৩-এ April, ২০২৪-এ April-early, ২০২৫-এ March-late।
• Standard yearly seasonality (period=365) এই shift ধরে না।
• April-এর gross average misleading।

Solution approaches:

(১) Holiday regressor (Prophet-style)

• Eid dates explicit list করুন।
• Prophet-এ holidays argument:

holidays = pd.DataFrame({
    'holiday': 'eid_fitr',
    'ds': pd.to_datetime(['2023-04-22','2024-04-10','2025-03-30']),
    'lower_window': -3, 'upper_window': 3
})
m = Prophet(holidays=holidays)

• $\pm 3$ days window — pre-Eid shopping surge।
• Model holiday-specific effect learn।

(২) Hijri (Islamic) calendar features

• Each row-এ Hijri month, day add feature।
• Ramadan = Hijri month 9, Eid = Hijri month 10 day 1।
• Tree model auto-discover এই pattern।
• Library: convertdate Python।

(৩) Days-to-Eid feature

• প্রতিটি row-এ "next Eid কত দিন দূরে" feature।
• $-30$ থেকে $+30$ continuous feature।
• Pre-Eid surge এই feature-এ encoded।

(৪) Two-fold seasonality (Prophet support)

• Standard yearly + custom Eid seasonality।
• Eid Fourier basis Hijri-anchored।

Multiple Eid:

• Eid-ul-Fitr (after Ramadan) — biggest e-commerce surge।
• Eid-ul-Adha (qurbani) — different shopping pattern (livestock)।
• দু'টোকে আলাদা holiday feature।

Other Bangladesh holidays:

• Pohela Boishakh (April 14): solar — fixed। Standard yearly seasonality OK।
• Victory Day (Dec 16): solar — fixed।
• Independence Day (March 26): solar — fixed।
• Durga Puja: lunar (Hindu calendar) — separate handle।

Ramadan-specific patterns:

• Daytime e-commerce slow (fasting)।
• Night shopping surge (after iftar)।
• Last 10 days — major Eid-shopping increase।
• "Day-of-Ramadan" feature (1-30) — non-linear pattern।

Local cultural events:

• Cricket matches — internet usage spike।
• Durga Puja — Hindu-majority area surge।
• Boi Mela (Feb) — book industry।
• School admission season (Dec-Jan)।
• Wedding season (Nov-Feb) — gold, sari market।

Seasonality interaction:

• Eid + weekend — multiplied effect।
• Eid + bonus salary day — compound।
• Tree model interaction capture; linear model explicit।

Practical pipeline:

# Hijri date features
import datetime
from hijri_converter import convert
df['hijri_month'] = df['date'].apply(lambda d: convert.Gregorian(d.year, d.month, d.day).to_hijri().month)

# Days to next Eid
eid_dates = pd.to_datetime(['2023-04-22','2024-04-10','2025-03-30','2026-03-20'])
def days_to_eid(d):
    diffs = (eid_dates - d).days
    return min(d for d in diffs if d > -7)
df['days_to_eid'] = df['date'].apply(days_to_eid)

Validation:

• Out-of-sample include Eid period।
• Walk-forward across multiple Eids।
• Prophet plot: holiday effect visualize।

Industry practice:

• Daraz: holiday calendar in feature store, separate model heads।
• bKash: surge prediction Hijri-aware।
• Pathao: holiday-specific surge multipliers।
• Robi: SMS campaigns adjusted।

Common mistake:

• "Eid is in April" hard-code — fails next year।
• Solar yearly seasonality standalone — Eid effect blurred।
• Single "is_holiday" boolean — all holidays same effect (wrong)।

মূল উপলব্ধি: Bangladesh time series Western-textbook examples-এ direct port না। Lunar calendar-aware feature engineering — local data scientist-এর critical skill। Cultural context = better model।

এটি time-series ML-এর সবচেয়ে fundamental concept। প্রতিটি forecasting practitioner-কে জানতে হয়।

Random split-এর সমস্যা:

• Train ও test mixed timestamps।
• Model train data future থেকে peek করে — leakage।
• Production-এ এই info available না।
• Test accuracy inflated।

Concrete example:

• Daily revenue ২০২৩ Jan-Dec।
• Random 80/20 split।
• Train-এ Dec 1, Dec 5, Dec 9...; test-এ Dec 2, Dec 6, Dec 10...।
• Test-এর Dec 6 predict-এ Dec 5, Dec 7 train-এ → trivial।
• RMSE artificially low।

Production reality:

• আজ থেকে আগামীকাল predict — কাল-পরশু-এর data নেই।
• Random split এই reality reflect করে না।

Time-based split (basic):

• Train: 2023 Jan-Aug।
• Test: 2023 Sep-Dec।
• Realistic: train past, test future।

সমস্যা — single split:

• One test period — variance high।
• Sep-Dec lucky? unlucky?
• Need multiple test windows।

Walk-forward validation (a.k.a. expanding window):

1. Initial train: data up to time $t_0$।
2. Predict $t_0 + h$ (forecast horizon)।
3. Compute error।
4. Add $t_0$-এর actual data to train।
5. Next: train up to $t_0 + 1$, predict $t_0 + h + 1$।
6. Repeat through all data।
7. Average errors।

Visualization:

Step 1: [-train-]|test|
Step 2: [--train--]|test|
Step 3: [---train---]|test|
Step 4: [----train----]|test|
...

Sliding window (alternative):

• Fixed train length, slide forward।
• Recent history-এ focus।
• Old data discard।
• Drift handling।

Sklearn-এ TimeSeriesSplit:

from sklearn.model_selection import TimeSeriesSplit
tscv = TimeSeriesSplit(n_splits=5)
for train_idx, test_idx in tscv.split(X):
    X_train, X_test = X[train_idx], X[test_idx]
    # train, predict, evaluate

Considerations:

(১) Forecast horizon

• Single-step (1-day-ahead) — easier।
• Multi-step (30-day-ahead) — harder; cumulative error।
• Walk-forward প্রতিটি horizon-এ separately validate।

(২) Computational cost

• প্রতিটি step-এ retrain — expensive।
• Compromise: every k-th step retrain।

(৩) Hyperparameter tuning

• Nested walk-forward।
• Inner: tune; outer: validate।
• Time-consuming but principled।

(৪) Seasonality preserve

• Train period must include full seasonal cycle।
• 1 year minimum for yearly seasonality।
• 2 years preferred।

Anti-pattern — feature with future:

• "Rolling 30-day mean" — must use only past 30 days at scoring time।
• Not centered (which uses future)।
• Pandas: .rolling(window=30, min_periods=1).mean() — past-only।

Cross-validation alternatives:

• Blocked CV: contiguous blocks, gap between train/test।
• Purged CV: remove temporal proximity rows (Marcos López de Prado)।
• Combinatorial purged CV: finance-specific।

Bangladesh forecasting examples:

• bKash daily transaction: walk-forward, 1-day horizon।
• Daraz weekly inventory: walk-forward, 7-day।
• Pathao surge: rolling, hour-ahead।
• BSEC index: walk-forward with regime-change handling।

Backtest pitfalls:

• Look-ahead bias: feature uses future।
• Survivorship bias: failed entities excluded।
• Selection bias: cherry-picked period।
• Overfitting: many hyperparameter trials।

Reporting:

• Mean error across walk-forward folds।
• Std deviation — robustness।
• Worst-case error — risk।
• Per-period error (Eid effect-এ separately)।

মূল উপলব্ধি: Time series-এ "test set" temporal future। Walk-forward production-এর simulation। Random split = academic shortcut, real-world deception। Senior practitioner-এর হাতিয়ার।

Real production scenario। Each component-এ thoughtful approach।

Expected components:

(১) Trend

• Bangladesh mobile-banking penetration growing।
• 2017: 30M users; 2024: 70M+।
• Daily transaction count steady upward।
• Saturation eventual — but not yet।
• Linear or exponential growth।

(২) Yearly seasonality

• Eid-ul-Fitr — massive spike (gift money, salary)।
• Eid-ul-Adha — qurbani transactions।
• Pohela Boishakh — slight bump (new year shopping)।
• Ramadan — different daytime/nighttime pattern।
• December — year-end bonus।
• February — Boi Mela, school admission।

(৩) Monthly seasonality

• 1st of month — salary deposit + cash-out।
• 15th — mid-month bills।
• End of month — utility payments।
• Regular monthly cycle।

(৪) Weekly seasonality

• Sunday-Thursday: working days, business transactions।
• Friday-Saturday: weekend, more personal transactions।
• Bangladesh weekend = Friday-Saturday (different from West)।
• Pattern stable।

(৫) Daily/hourly seasonality

• Morning rush 9-11 AM।
• Lunch dip।
• Evening rush 6-9 PM।
• Late night quiet।

(৬) Cyclicity

• Macro-economic cycles — multi-year।
• 2024 inflation surge → transaction shift।
• Reserve crisis effect।
• Hard to predict, longer trends।

(৭) Noise

• Random daily fluctuation।
• Localized events (cricket match, weather)।
• System outages — momentary dips।

(৮) Special events

• Government policy changes (cash limit)।
• Hartal/lockdown।
• COVID-19 — dramatic shift (cash-out drop, online surge)।
• Anomaly handling।

Modeling approach:

(ক) Decomposition first

• STL decomposition (better than seasonal_decompose)।
• Trend, seasonal (multiple periods), residual।
• Visual inspection।

(খ) Prophet — production sweet spot

from prophet import Prophet
m = Prophet(
    yearly_seasonality=True,
    weekly_seasonality=True,
    daily_seasonality=False,  # daily count, not hourly
    holidays=bangladesh_holidays,  # Eid, etc.
    changepoint_prior_scale=0.05  # trend flexibility
)
m.fit(df)

(গ) Multiple seasonality (TBATS)

• Yearly (365.25), weekly (7), monthly (~30)।
• TBATS supports multiple।
• Computationally heavier।

(ঘ) Deep learning (LSTM/Transformer)

• Large training data needed।
• Complex pattern auto-learn।
• Less interpretable।
• Production-এ ensemble component।

(ঙ) ARIMA — baseline

• Daily after differencing + log।
• SARIMA for seasonality।
• Limited multiple-seasonality।

Feature engineering:

• day_of_week, day_of_month, month, year।
• is_holiday, is_eid, is_ramadan।
• days_to_next_eid, days_after_eid।
• weather (rainy, hot)।
• cricket_match flag।
• economic_indicator (USD-BDT, inflation)।

Anomaly handling:

• COVID period exclude from training (or model break)।
• Outlier capping।
• Robust regression।

Validation:

• Walk-forward, weekly retrain।
• Per-component error analysis।
• "Eid forecast accuracy" separately track।

Operationalization:

• Hourly retrain (live transaction data)।
• 1-hour, 1-day, 7-day forecast horizons।
• Capacity planning input।
• Anomaly alerting।

Use cases:

• Server capacity scaling।
• Customer-care staffing।
• Cash management (agent network)।
• Fraud baseline — expected vs actual gap।
• Marketing campaign timing।

Bangladesh context-specific:

• NID linking changes affecting onboarding।
• Bangladesh Bank policy (transaction limit)।
• Cellular network outages (Grameenphone, Robi disruptions)।
• Salary disbursement schedule (govt vs private)।

মূল উপলব্ধি: Real time series multi-component, multi-seasonality, complex। Single model rarely sufficient। Ensemble + domain knowledge + careful validation = production excellence।