Time Series Analysis

Time Series Analysis

A time series is a sequence of observations made: 1) over a continuous time interval, 2) of successive measurements across that interval, 3) using equal spacing between consecutive measurements, 4) with each time unit within the time interval having only one data point. Some examples include: Monthly mean temperature from 2001 2003. Daily Down Jones Industrial average.

Wetland Enhanced Vegetation Index Measurements: 2001-2012 Date EVI Jan-01 1999 Feb-01 421 Mar-01 2258 Apr-01 1897 May-01 1773 Jun-01 1708 Jul-01 1436 Aug-01 1716 Sep-01 1835 Oct-01 2371 Nov-01 1838 Dec-01 2442 Jan-02 1561 Feb-02 1442 Mar-02 2409 Apr-02 2142 May-02 1716 Jun-02 1596 Jul-02 1765 Aug-02 1662 Sep-02 2068 Oct-02 1941 Nov-02 1533 Dec-02 2616 Jan-03 2949 Feb-03 2678 Mar-03 2682 Apr-03 2060 May-03 2442 Jun-03 2108 Jul-03 1763 Aug-03 1768 Sep-03 1772 Oct-03 1730 Nov-03 2201 Dec-03 1413 Date EVI Jan-04 1918 Feb-04 2058 Mar-04 2228 Apr-04 2361 May-04 1661 Jun-04 1838 Jul-04 1660 Aug-04 1521 Sep-04 2205 Oct-04 2361 Nov-04 2331 Dec-04 1859 Jan-05 1222 Feb-05 2756 Mar-05 2382 Apr-05 2210 May-05 1920 Jun-05 1650 Jul-05 1784 Aug-05 1619 Sep-05 1760 Oct-05 2312 Nov-05 2175 Dec-05 2498 Jan-06 1903 Feb-06 914 Mar-06 2068 Apr-06 2483 May-06 1621 Jun-06 1747 Jul-06 1769 Aug-06 1575 Sep-06 1530 Oct-06 1967 Nov-06 2481 Dec-06 2096 Date EVI Jan-07 2508 Feb-07 2648 Mar-07 2342 Apr-07 1962 May-07 1993 Jun-07 1469 Jul-07 1600 Aug-07 1703 Sep-07 2015 Oct-07 2586 Nov-07 2421 Dec-07 2163 Jan-08 2329 Feb-08 2165 Mar-08 2274 Apr-08 2262 May-08 2206 Jun-08 2016 Jul-08 1963 Aug-08 1985 Sep-08 1778 Oct-08 2750 Nov-08 2530 Dec-08 1777 Jan-09 2690 Feb-09 1771 Mar-09 2490 Apr-09 2410 May-09 1888 Jun-09 1910 Jul-09 1709 Aug-09 1965 Sep-09 2167 Oct-09 1950 Nov-09 2689 Dec-09 2595 Date EVI Jan-10 2801 Feb-10 2087 Mar-10 2692 Apr-10 2184 May-10 2148 Jun-10 2020 Jul-10 1724 Aug-10 1581 Sep-10 1684 Oct-10 2673 Nov-10 2577 Dec-10 2193 Jan-11 2329 Feb-11 2525 Mar-11 2754 Apr-11 2381 May-11 2241 Jun-11 1759 Jul-11 1960 Aug-11 2076 Sep-11 2050 Nov-11 2720 Nov-11 2840 Dec-11 1858 Jan-12 1880 Feb-12 3145 Mar-12 2724 Apr-12 2665 May-12 2512 Jun-12 1700 Jul-12 1929 Aug-12 1892 Sep-12 1885 Oct-12 2501 Nov-12 2863 Dec-12 2836

Time Series Analyses allow us to answer the questions: Do these data exhibit seasonal (fixed pattern) variation? Do these data exhibit a trend (increasing or decreasing)? Do these data exhibit a cyclic pattern (non-fixed pattern) variation?

Strong seasonality, strong cyclic behavior, no trend No seasonality, slight cyclic behavior, strong trend Strong seasonality, no cyclic behavior, strong trend No seasonality, cyclic behavior, or trend

These components of a time series data set can be written as: y = S + T + t t t E t where y is the observation at t time, S is the seasonal component, T is the combined trend and cyclic components, and E is the error (remainder) component. This is called an additive model and assumes that the value of the next observation in the sequence is arithmetically associated with the previous observation. For example: In the sequence (2, 4, 2, 4, 2, 4) the next observation in the sequence is arithmetically generated by +2 or 2 of the previous observation.

These components of a time series data set can also be written as: This is called an multiplicative model and assumes that the value of the next observation in the sequence is multiplicative associated with the previous observation. For example: y = S T t t In the sequence (2, 4, -8, -16, 32, 64) the next observation in the sequence is multiplicatively generated by 2 or 2 of the previous observation. t E t

Therefore: If your time series appears to be random (neither increasing or decreasing) over time, use an additive model. If your time series appears to be either increasing or decreasing over time, use a multiplicative model.

An alternative to using a multiplicative model is to first transform the data and then using an additive model: y t = S t T t E t is equivalent to y = log S + logt + t t t log E t However, this makes the interpretation of the results somewhat more difficult.

Time Series Decomposition: a statistical method that deconstructs a time series into notional (i.e. seasonal, trend, and error) components. Moving Average: the method of removing seasonal influences in which each observation is replaced by the average of the x number of observations preceding it. (m centered on y) = T t y m where m is the order (window) of moving average, T is the estimate of the trend-cycle at time t, and y is the observation. The m centered on y term means that the window is centered on the observation. Therefore, if your window is 7, you will average the observation, plus the 3 observation before and after.

Weighted Moving Average: adding a term that will capture the influence each observation has on the whole. For example, if we are interested in plant growth we would want more importance to be given to observations during the growing season and de-emphasize observations during the non-growing season. ( y(m centered on y) T t = m weight)

Moving Average (yearly: m = 12) example Date Data Weights Weighted EVI Jan 01 1999 0.925 1849.08 -- Feb 01 421 0.925 389.43 -- Mar 01 2258 0.975 2201.55 -- Apr 01 1897 0.975 1849.58 -- May 01 1773 1 1773.00 -- Jun 01 1708 1 1708.00 -- Jul 01 1436 1 1436.00 1748.54 Aug 01 1716 1 1716.00 Sep 01 1835 0.975 1789.13 Oct 01 2371 0.975 2311.73 Nov 01 1838 0.925 1700.15 Dec 01 2442 0.925 2258.85 Jan 02 1561 Feb 02 1442 Mar 02 2409 Apr 02 2142 May 02 1716 Jun 02 1596 Jul 02 1765 Aug 02 1662 Sep 02 2068 Oct 02 1941 Nov 02 1533 Dec 02 2616 Yearly Moving Average Trend It is best to have an odd numbered window since it is easier to center.

Bofedal EVI Decomposed Data (2001 only) Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Original Data 1999 421 2258 1897 1773 1708 1436 1716 1835 2371 1838 2442 Seasonal 91.83 98.29 357.07 181.39-74.98-310.8-321.86-328.27-186.45 217.69 235.16 40.94 Trend NA NA NA NA NA NA 1748.54 1813.87 1862.7 1879.2 1887.04 1880 Error NA NA NA NA NA NA 9.32 230.4 158.74 274.1-284.2 521.05 Since we have monthly data on a 12 month repeating cycle, we will use a 12 month moving average, which results in loss of the first and last 6 observations. Most computer programs use a weighted average, with greater weight placed on closer observations. Note: 1748.54 321.86 + 9.32 = 1436 (see box above)

Two Techniques for Estimating the Time Series Components 1. Seasonal averaging: The seasonal figure is computed by averaging, for each time unit, over all periods. All Januaries are averaged, then all Februaries are averaged, etc This results in constant components (e.g. is the same for each year). 2. Seasonal smoothing: The seasonal component is found by smoothing the seasonal sub-series (the series of all January values,...). This results non-constant components.

Decomposition using Seasonal Averaging

Decomposition using Seasonal Smoothing

Seasonal averaging trend Seasonal smoothing trend

Steps in Additive Time Series Decomposition 1. Calculate the trend-cycle component (T t ) using moving averages. 2. Calculate a de-trended series by subtracting the trend from the observation (y t T t ). 3. Estimate the seasonal component (S t ) for each period (e.g. month) by averaging the de-trended values for that period. 4. Calculate the remainder (error) component by subtracting both the seasonal and trend-cycle components from the observation (E t = y t S t T t ).

Steps in Multiplicative Time Series Decomposition 1. Calculate the trend-cycle component (T t ) using moving averages. 2. Calculate a de-trended series by dividing the observation by the trend (y t / T t ). 3. Estimate the seasonal component (S t ) for each period (e.g. month) by averaging the de-trended values for that period. 4. Calculate the remainder (error) component by dividing the observation by the product of the seasonal and trend-cycle components: E t = y t / (S t T t ).

Comments on Classical Time Series Decomposition Classical decomposition methods assume that the seasonal component repeats from year to year. This is not always the case. For example, electricity demand patterns have changed over time as power used increases. External interruptions in the time series will significantly influence the decomposition results. For example, an employee dispute at an airline may alter passenger traffic. Larger moving average windows will remove lager amounts of data from the analyses. If this is anticipated, data should be collected well before and after the period of interest.

Time Series Analysis Example

Stationarity a property of a time series data set where the mean and sd of the series do not change over time. Stationarity is only important if you are trying to forecast. Stationary Data Non-Stationary Data Mean and sd constant over time. Mean increases over time. We will only be describing the time series data, so stationarity is not critical.

The raw EVI data for bofedal #16 shows periodicity, but the magnitude (mean EVI) and duration (sd EVI) of each period changes over time.

A monthly plot shows considerable variation within each month (mean EVI in red), but a predictable trend over the course of the year. These austral winter months have the least variation.

Autocorrelation when the value of y t is correlated with (or not independent of) the value of y t-1. In other words, the current measurement is a function of past measurements.

Nearly all of the vertical lines cross the horizontal dashed line, indicating a high level of autocorrelation. The EVI values are correlated across time, although this diminishes as time increases.

The decomposed EVI data. Note the strong seasonal peaks

The seasonal component is fairly strong, showing a definite single yearly peak.

The residual plot shows that the error component is symmetrical about the mean (0), suggesting there is no systematic bias.

What happened here?