A time series is a sequence of observations in time order — sales per day, CPU load per minute. Order matters: neighbouring values are correlated, the future must never leak into the past, and a simple “same as last week” baseline is surprisingly hard to beat.

Component	Meaning	Example
Trend	Long-term increase or decrease	Growing user base
Seasonality	Repeating pattern with a fixed period	Weekly shopping cycle, yearly peaks
Cycles	Rises and falls without a fixed period	Business cycles
Noise	Irregular variation	Random day-to-day fluctuation

A forecasting example

Daily demand with a trend, a weekly pattern and noise. Forecast the last 28 days using only information available the day before:

Python

import numpy as np
import pandas as pd
from sklearn.linear_model import Ridge
from sklearn.metrics import mean_absolute_error

rng = np.random.default_rng(0)
days = pd.date_range("2024-01-01", periods=364, freq="D")
t = np.arange(364)
weekly = np.array([0, 2, 3, 3, 5, 12, 9])[days.dayofweek]          # busier on weekends
y = pd.Series(50 + 0.05 * t + weekly + rng.normal(0, 2, 364), index=days)

print(round(y.autocorr(lag=1), 2), round(y.autocorr(lag=7), 2))     # 0.68 0.91 ← strong weekly autocorrelation

df = pd.DataFrame({"y": y})
df["lag_1"] = y.shift(1)                                  # yesterday
df["lag_7"] = y.shift(7)                                  # same weekday last week
df["roll_7"] = y.shift(1).rolling(7).mean()               # past week's mean — shift FIRST, then roll
df["dow"] = df.index.dayofweek
df = pd.get_dummies(df, columns=["dow"], dtype=float).dropna()

train, test = df.iloc[:-28], df.iloc[-28:]                 # split by time, never shuffle
X_cols = [c for c in df.columns if c != "y"]
model = Ridge(alpha=1.0).fit(train[X_cols], train["y"])

print(round(mean_absolute_error(test["y"], test["lag_1"]), 2))                 # 5.18 ← naive: tomorrow = today
print(round(mean_absolute_error(test["y"], test["lag_7"]), 2))                 # 2.51 ← seasonal naive: = last week
print(round(mean_absolute_error(test["y"], model.predict(test[X_cols])), 2))   # 1.99 ← model with lag + calendar features

Approaches

Family	Examples	Good for
Baselines	Naive, seasonal naive, moving average	Always — the bar to beat
Classical statistical	Exponential smoothing (ETS, Holt-Winters), ARIMA/SARIMA (`statsmodels`)	Single series, clear trend/seasonality, small data, prediction intervals
ML on engineered features	Lags, rolling stats, calendar and holiday features + gradient boosting / linear models	Many related series, external drivers (price, promotions, weather)
Deep learning	RNN/LSTM, temporal CNNs, transformers (see Deep Learning)	Very many long series, complex patterns

Practical points

Multi-step horizons: recursive (feed predictions back as lags) accumulates errors; direct (one model per horizon) avoids that at more cost.
Metrics: MAE/RMSE per horizon; MASE = MAE ÷ MAE of the seasonal-naive forecast (below 1 beats the baseline); avoid MAPE with values near 0.
Validation: rolling-origin (expanding window) evaluation mirrors how forecasts are produced in production.
Label availability: if yesterday's sales are only final two days later, lag_1 isn't available at prediction time — use a gap.
Report prediction intervals, not only point forecasts — inventory and staffing decisions depend on the uncertainty.

How is validating a time-series model different?

What is stationarity and why does it matter?

What baseline would you use for a forecasting problem?