Time Series Analysis

Transcription

1 Time Series Analysis

2 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 Daily Down Jones Industrial average.

3 Wetland Enhanced Vegetation Index Measurements: Date EVI Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Date EVI Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Date EVI Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Date EVI Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Nov Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

4 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?

5 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

6 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.

7 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

8 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.

9 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.

10 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.

11 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)

12 Moving Average (yearly: m = 12) example Date Data Weights Weighted EVI Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Yearly Moving Average Trend It is best to have an odd numbered window since it is easier to center.

13 Bofedal EVI Decomposed Data (2001 only) Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Original Data Seasonal Trend NA NA NA NA NA NA Error NA NA NA NA NA NA 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: = 1436 (see box above)

14 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.

15 Decomposition using Seasonal Averaging

16 Decomposition using Seasonal Smoothing

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18 Seasonal averaging trend Seasonal smoothing trend

19 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 ).

20 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 ).

21 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.

22 Time Series Analysis Example

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25 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.

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

27 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.

28 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.

29 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.

30 The decomposed EVI data. Note the strong seasonal peaks

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

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

33 What happened here?

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