Applied Time Series Analysis Outcomes

MATH 494R

Time series analysis¹

Time series²

Sampling Interval³

Serial Dependence or Autocorrelation⁴

time series trend⁵

Seasonal Variation⁶

Cycle⁷

Stochastic Trend⁸

Deterministic Trend⁹

smoothing or smoothed¹⁰

centred (center or centered) moving average¹¹

Additive decomposition model¹²

Multiplicative decomposition model¹³

tsibble¹⁴

monthly additive effect¹⁵

ergodic¹⁶

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Footnotes

1. A time series analysis quantifies the main features in data and the random variation. These reasons, combined with improved computing power, have made time series methods widely applicable in government, industry, and commerce. (1.2)

2. Time series are analysed to understand the past and to predict the future, enabling managers or policy makers to make properly informed decisions.

3. When a variable is measured sequentially in time over or at a fixed interval, known as the sampling interval, the resulting data form a time series.

4. A correlation of a variable with itself at different times is known as autocorrelation or serial correlation. (1.2, 2.2.5)

5. In general, a systematic change in a time series that does not appear to be periodic is known as a trend. The simplest model for a trend is a linear increase or decrease, and this is often an adequate approximation. (1.2 1.4.1)

6. Repeated pattern within each year (or any other fixed time period). (1.2)

7. Repeated pattern that does not correspond to some fixed natural period.

8. Random trend that does not follow a discernible or predictable pattern. (1.2)

9. Can be modeled with mathematical functions, facilitating the long-term prediction of the behavior

10. The centred moving average is an example of a smoothing procedure that is applied retrospectively to a time series with the objective of identifying an underlying signal or trend. (1.3 1.5.4)

11. A “centered moving average” is a statistical method used to smooth out short-term fluctuations in time series data by calculating the average of a set of observations, but placing the average value directly in the middle of the data points used, effectively “centering” it on the midpoint of the timeframe, which helps to reduce lag and provide a more accurate representation of the underlying trend compared to a standard moving average. (1.3 1.5.3 1.5.4)

12. \(x_t = m_t + s_t + z_t\) or after taking log \(\log(x_t) = m_t + s_t + z_t\). (1.3 1.5.2)

13. \(x_t = m_t \cdot s_t + z_t\)

14. A tsibble (short for “time series tibble”) is sorted by its key first and index. The index (e.g., Date, POSIXct, yearmonth, yearweek) must be sequential and capable of being ordered. Key: (e.g., a sensor ID or region) One or more variables that uniquely identify each time point. Values: One or more measured variables that correspond to observations at each time point.

15. The centered moving average, , is then used to compute the monthly additive effect

16. A time series model that is stationary in the mean is ergodic in the mean if the time average for a single time series tends to the ensemble mean as the length of the time series increases (2.2 2.2.3).