Time-series is a sequence of measurements of a variable, approximately equally spaced in time. It typically arises in economic and financial data.
Characteristics of time-series data: trend (drift in moving average) and seasonality (spectral density estimation); autocorrelation (e.g. random walk).
Uncertainty in time-series models are often presented as 95% CI band, as in {Hanna2017}.
Standard single time-series model, as implemented in R forecast
package {Hyndman2008}.
Multiple time-series with black box model, as deployed in Uber {Laptev2017}.
Additive vs multiplicative decomposition:
X = T + C + S + I X = T * C * S * I
Components:
Calendar effects are any effect related to changes in the calendar: trading day (TD) effects; moving holiday effects, holidays whose dates vary from year to year, such as Easter, Passover, Ramadan, Chinese New Year and U.S. Labor Day; working day effects; length of month/quarter effects;
Outliers: Data from an abrupt, untypical movement in the time series, e.g. hurricane or strike, that are likely to distort the estimates of seasonal, trading day or holiday effects. For seasonal adjustment, the software's generic outlier regressors are used to estimate and temporarily (approximately) remove the outliers, in order to prevent distortion of the desired estimates. These protected estimates are removed from the original series to obtain the adjusted series. Consequently, the seasonal and perhaps trading day and holiday effect adjusted series still shows the expected outlier movements. Extreme Value Adjustment refers to the X-11 procedure's more limited form of temporary adjustment, not requiring regression estimation, to protect seasonal effect estimates from distortion by outliers.
No de-trending for non-count series: Intensive quantities e.g. probability density do not need de-trending (normalization).
serial correlation
AutoRegressive Integrated Moving Average (ARIMA) is a versatile family of models for modeling and forecasting time series data. Seasonal ARIMA models have a special form for efficiently modeling many kinds of seasonal time series and are heavily used in seasonal adjustment.
Seasonal adjustment is the removal of the seasonal component from a time-series.
X-13ARIMA-SEATS is a seasonal adjustment software combining X-12-ARIMA (developed by the United States Census Bureau) and TRAMO-SEATS (developed by the Band of Spain). This is the seasonal adjustment software currently used by the US Census Bureau.
The X-11 method decomposes a time-series into trend-cycle, seasonal, and irregular components by iteratively applying linear filters (moving averages).
Time Series Regression with ARIMA noise (TRAMO); Signal Extraction in ARIMA Time Series (SEATS);
Regression+ARIMA models use linear regression to estimate moving holiday, trading day and outlier effects, and then use a seasonal ARIMA model to estimate trend, cycle and seasonal components from the regression residuals.