Time Series Analysis & Forecasting

Time series data is ordered by time — stock prices, weather, sales. Forecasting predicts future values.

Time Series Components

Trend: Long-term increase/decrease
Seasonality: Repeating patterns (daily, weekly, yearly)
Cyclical: Irregular patterns (not fixed period)
Noise: Random variation

Decomposition:
y(t) = Trend(t) + Seasonal(t) + Residual(t)

Or multiplicative:
y(t) = Trend(t) × Seasonal(t) × Residual(t)

Stationarity

Stationary: Statistical properties don't change over time
├─ Constant mean
├─ Constant variance
└─ Constant autocorrelation

Why it matters: Most models assume stationarity

Tests: Augmented Dickey-Fuller (ADF) test
p-value < 0.05 → stationary

Make non-stationary:
├─ Differencing: y'(t) = y(t) - y(t-1)
├─ Log transform
└─ Detrending

ARIMA

ARIMA(p, d, q):
├─ AR (p): Autoregressive — uses past values
├─ I (d): Integrated — differencing order
└─ MA (q): Moving average — uses past errors

AR(1): y(t) = c + φ₁y(t-1) + ε(t)
MA(1): y(t) = c + ε(t) + θ₁ε(t-1)

from statsmodels.tsa.arima.model import ARIMA

model = ARIMA(train, order=(1, 1, 1))
fitted = model.fit()
forecast = fitted.forecast(steps=30)

Facebook Prophet

from prophet import Prophet

# Data must have 'ds' and 'y' columns
df = pd.DataFrame({'ds': dates, 'y': values})

model = Prophet(
    yearly_seasonality=True,
    weekly_seasonality=True,
    daily_seasonality=False
)
model.fit(df)

future = model.make_future_dataframe(periods=365)
forecast = model.predict(future)
model.plot(forecast)

Key Takeaways

1. Stationarity is assumed by most time series models
2. ARIMA is the classical approach
3. Prophet handles seasonality and holidays automatically
4. LSTM neural networks capture complex patterns
5. Always split by time (not random) for validation
6. Feature engineering (lags, rolling stats) helps ML models
7. Exponential smoothing is simple but effective
8. Ensemble multiple forecasting methods for best results