Time Series Forecasting

Time series forecasting predicts future values based on historical patterns. Revenue forecasting, capacity planning, demand prediction, and anomaly detection all depend on accurate time series models. The challenge is that time series data has unique properties — trend, seasonality, autocorrelation — that standard ML models do not handle well out of the box.

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Time Series Components

Raw signal = Trend + Seasonality + Residual

Trend:       Long-term direction (growing, declining, flat)
Seasonality: Repeating patterns (daily, weekly, monthly, yearly)
Residual:    Random noise after removing trend and seasonality

Example (e-commerce revenue):
  Trend: 15% YoY growth
  Weekly seasonality: Peak on weekends
  Yearly seasonality: Holiday spikes (Nov-Dec)
  Residual: Day-to-day variation

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Statistical Methods

ARIMA

from statsmodels.tsa.arima.model import ARIMA

# ARIMA(p, d, q)
# p: autoregressive order (how many past values)
# d: differencing order (how many times to difference)
# q: moving average order (how many past errors)

model = ARIMA(train_data, order=(2, 1, 2))
fitted = model.fit()

# Forecast next 30 days
forecast = fitted.forecast(steps=30)

# With confidence intervals
forecast_result = fitted.get_forecast(steps=30)
ci = forecast_result.conf_int(alpha=0.05)  # 95% CI

Prophet (Meta)

from prophet import Prophet

# Prophet handles trend, seasonality, and holidays automatically
model = Prophet(
    yearly_seasonality=True,
    weekly_seasonality=True,
    daily_seasonality=False,
    changepoint_prior_scale=0.05,  # Flexibility of trend
)

# Add custom seasonality
model.add_seasonality(name='monthly', period=30.5, fourier_order=5)

# Add holidays
model.add_country_holidays(country_name='US')

# Fit
model.fit(df[['ds', 'y']])  # ds: date, y: value

# Forecast
future = model.make_future_dataframe(periods=90)
forecast = model.predict(future)

# Components (trend, seasonality breakdown)
model.plot_components(forecast)

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Feature Engineering for Temporal Data

def create_temporal_features(df, date_col='date'):
    """Create features that capture temporal patterns."""
    df = df.copy()
    
    # Calendar features
    df['day_of_week'] = df[date_col].dt.dayofweek
    df['month'] = df[date_col].dt.month
    df['quarter'] = df[date_col].dt.quarter
    df['is_weekend'] = df['day_of_week'].isin([5, 6]).astype(int)
    df['is_month_end'] = df[date_col].dt.is_month_end.astype(int)
    
    # Lag features
    for lag in [1, 7, 14, 28]:
        df[f'lag_{lag}'] = df['value'].shift(lag)
    
    # Rolling statistics
    for window in [7, 14, 28]:
        df[f'rolling_mean_{window}'] = df['value'].rolling(window).mean()
        df[f'rolling_std_{window}'] = df['value'].rolling(window).std()
    
    # Expanding features
    df['expanding_mean'] = df['value'].expanding().mean()
    
    return df

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Evaluation

# Time series cross-validation (walk-forward)
from sklearn.model_selection import TimeSeriesSplit

tscv = TimeSeriesSplit(n_splits=5)
errors = []

for train_idx, test_idx in tscv.split(data):
    train = data.iloc[train_idx]
    test = data.iloc[test_idx]
    
    model.fit(train)
    predictions = model.predict(test)
    
    mape = mean_absolute_percentage_error(test['y'], predictions)
    errors.append(mape)

print(f"Average MAPE across folds: {np.mean(errors):.2%}")

Metric	Formula	When to Use
MAE	mean(abs(actual - predicted))	When all errors matter equally
MAPE	mean(abs(actual - predicted) / actual)	When relative error matters
RMSE	sqrt(mean((actual - predicted)²))	When large errors are costly
sMAPE	Symmetric MAPE	When values cross zero

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Anti-Patterns

Anti-Pattern	Consequence	Fix
Random train/test split	Data leakage, inflated metrics	Walk-forward / time series CV
Ignoring seasonality	Model misses repeating patterns	Decompose and model seasonality
No baseline comparison	Cannot assess model value	Compare vs naive forecast (last value)
Forecasting too far ahead	Accuracy degrades rapidly	Limit horizon, quantify uncertainty
Single point forecast	False precision, no uncertainty	Prediction intervals with confidence levels

Time series forecasting is not about predicting the future perfectly — it is about quantifying uncertainty and providing decision-makers with ranges and probabilities.