Notes
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Weak Stationarity

Many time series “feel different” depending on when you look: sales might be higher in December, or a sensor might drift upward as it warms up. Weak stationarity is a way to formalize the idea that, despite random ups and downs, the series’ basic statistical behavior stays stable over time.

What weak stationarity means
A time series is weakly stationary (also called covariance-stationary) if its first two moments don’t change with time:

  • The mean is constant: the expected value is the same at all times.
  • The variance is constant and finite: the typical spread doesn’t grow or shrink over time.
  • The autocovariance depends only on the lag (how far apart two points are), not on the calendar time when you measure it.

Intuitively: the “center,” “wiggliness,” and “pattern of correlation” stay the same as the series moves forward.

Concrete examples

  • Exam scores from equally difficult weekly quizzes might be weakly stationary if the class level is stable and there’s no trend or changing variability.
  • House prices are usually not weakly stationary because they often trend upward and can show changing volatility; price changes (returns/differences) are closer to stationary.
  • Daily temperature is typically non-stationary due to seasonality; removing seasonal patterns can help.

Why it matters in AI/ML
Many forecasting and sequence models assume stable relationships over time. With weak stationarity, tools like ARMA/ARIMA, Kalman filters, and autocorrelation-based feature engineering behave predictably. If the series isn’t stationary, estimated correlations can be misleading, uncertainty can be miscalibrated, and models may “learn” trends that later shift.

Common fixes and checks
Practitioners often use differencing, detrending, and seasonal adjustment, and check stationarity with tests like ADF or KPSS (plus plots of ACF/PACF).

Weak Stationarity (covariance stationarity) means a time series has a constant mean over time, constant variance, and an autocovariance that depends only on the time lag, not the specific time. It matters because many statistical and ML time-series models (e.g., ARMA/ARIMA assumptions) rely on stable second-order structure for valid estimation and forecasting. Example: differencing a trending series can make it weakly stationary.

Imagine tracking the temperature in your city every day. Even though it wiggles up and down, you might feel that the “typical” temperature and how wildly it swings stay pretty similar from month to month. That’s the basic idea behind weak stationarity.

In time-series data (numbers recorded over time), a series is weakly stationary if its average level stays constant over time, its overall spread (how variable it is) stays constant, and the way today relates to yesterday depends only on how far apart they are—not on the specific date. Many forecasting methods work best when this is roughly true.