A time series is any sequence of observations ordered in time. The defining feature is that the ordering matters; shuffle the values and you destroy information. Yesterday’s observation is part of what determines today’s, and that dependency is what time series analysis tries to exploit.
Most introductions start with retail sales or temperature and stay there. However, the same data structure appears in very different domains, each with its own distinct characteristics. It’s worth seeing that variety before narrowing down to methods.
Four Structural Types
(a) Trend and seasonality is the most familiar pattern; a long-term direction with a repeating calendar-based rhythm on top. CO₂ levels, airline passengers, quarterly earnings all fall here. This is the primary focus of forecasting work.
(b) Volatility clustering looks different. The series fluctuates around zero with no trend, but the size of the fluctuations is not constant; turbulent periods cluster together, as do calm ones. Financial log returns are the classic example. A large swing today makes a large swing tomorrow more likely, even if the direction is unpredictable.
(c) Quasi-periodic oscillations appear in physical signals like speech. Vocal cords vibrate at a roughly consistent frequency, but the period drifts slightly from cycle to cycle. There’s no trend and no fixed seasonal period; the oscillation itself is the signal.
(d) Multivariate time series arises when multiple variables are recorded simultaneously. In brain imaging, signals from different regions are measured at once, and the correlations between them, which ones move together, which lag behind, are the object of study.
Two Approaches
Time domain asks: how does the past relate to the future? It investigates lagged relationships, does yesterday predict today, and how far back does the dependency reach? Most forecasting tools: ARIMA, exponential smoothing, ACF analysis live here.
Frequency domain asks: what periodic patterns are present, and at what timescales? Instead of tracking observations over time, it decomposes the signal into oscillating components at different frequencies. For speech, seismic data, or any signal driven by physical oscillations, this framing is more natural than asking about lagged values.
For business and economic forecasting, the time domain approach is primary. But knowing the frequency domain exists, and what it’s suited for is useful context.
Additive vs Multiplicative Seasonality
One distinction that matters in practice: whether seasonal swings are constant in size, or grow with the level of the series.
In an additive model, the seasonal effect adds a fixed amount regardless of the current level. In a multiplicative model, it scales with the level, if the series doubles, so do the seasonal swings. The J&J series is a clear example: the quarterly spikes are barely visible in the early years and large by 1980.
The practical fix for multiplicative seasonality is a log transformation, which compresses large values relative to small ones and converts the multiplicative structure into an additive one.
Looking at whether seasonal swings stay constant or grow with the level is usually enough to tell the two apart.
The next post goes into decomposition: how to formally separate a series into trend, seasonal, and remainder components, and what each part tells you.
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