Time series Analysis with SARIMA Model
For Key West, Florida Maximum Monthly Sea Water Level Elevation
Time series refer to datasets that are indexed by times. In other words, a time series is a sequence of numerical data points in successive order. Time series analysis is useful to see how given variables change over time. Followings are some explanations on time series characteristics that are important to understand before choosing the model to use in future predictions.
- Autoregressive (AR)
An autoregressive (AR) is a characteristic of time series when there is some correlation between the values under consideration and the values that precede and succeed them. For this type of time series, we only use past data to model the behavior of the series.
- Moving Average (MA)
Moving Average (MA) time series are those that can be modeled using their moving average. For example, the moving average of 12 month air temperature may be computed by taking the monthly average from January to December, and then from February to January, and so on. It progresses by dropping the earliest value and adding the latest value.
- Autoregressive Moving Average (ARMA)
From its name, we can clearly see that ARMA time series are those with autoregressive and moving average characteristics. In ARMA, it is assumed that the series is stationary and fluctuates around a particular time.
- Autoregressive Integrated Moving Average (ARIMA)
This is a generalization of ARMA model. We can apply either AR or MA or both models into ARIMA model. ARIMA models are applied in some cases where data show evidence of non-stationarity, where a differencing step can be applied to eliminate the non-stationarity.
- Seasonal Autoregressive Integrated Moving Average (SARIMA)
This is an extension to ARIMA model, applied to ARIMA time series that shows seasonal patterns. In earth science, this model is widely used to predict future climate variables.
- SARIMAX
One step further, SARIMAX model applied to time series that have SARIMA characteristics with the additional exogenous prediction variables. For example, we are trying to predict future bus ridership in Seattle, Washington using SARIMA. Based on the strong relationship between the bus ridership and precipitation, we may want to add the precipitation data as our exogenous variable to change SARIMA to SARIMAX model for a better prediction.
Maximum Monthly Sea Water Level Future Prediction for Key West, Florida
- Data Interpretation
National Oceanography and Atmospheric Administration (NOAA) published recorded data of maximum monthly sea water level from 1913 through 1940, and from 1971 through 2019. The following figures show the changes in maximum sea water level trends in four and five year periods (1935 through 1940, 1971 through 1976, 1995 through 2000 and 2014 through 2018).
The important take-aways from the above figures are:
- Change in trends, from declining in the thirties, to significantly inclining in the nineties through 2018.
- Change in the minimum values over 10 decades, from below 1.5 to above 2.0 feet.
We should ask ourselves why is this happening? Please watch this video on perennial Arctic ice melt simulation, published by NASSA. Is there any relationship between the Arctic ice melt and the increasing sea water level in Key West? I will let the readers answer this question.
- Prediction of Future Maximum Monthly Sea Water Level with SARIMA Model
Base on my observation on the trends described above, I am concentrating my prediction on the last 5 years of seasonal pattern. Also, from the figure above, it is clear that the data is non-stationary (changes in trends). In time series future prediction, by all means, the goal is not to replicate the observed data, but to predict the future condition. Therefore, there is no reason why we should capture the data trend from the forties, even the seventies. The last five years of records are more important. With that in mind, I chose my model so it captures the latest trend and the extreme value, in addition to the general statistics of the entire data population from 1971 through 2018.
The result of my prediction is presented in the following figure.
It can be seen that the predicted trend is reasonably good. So far the maximum recorded water level is 4.47 ft, which occurred on September 10, 2017. The maximum predicted water level is 4.76 feet, and the predicted average maximum water level is 2.33 feet. The maximum recorded water level in 2019 is 2. 98 ft.
I will close this blog with a little message: “Sea water levels are rising”.
Sources and References:
- NOAA (https://tidesandcurrents.noaa.gov/waterlevels.html?id=8724580&units=standard&timezone=GMT&datum=MLLW&interval=m&mm=MHHW&bdate=20190110&edate=20191210)
- NASA Scientific Visualization Studio
- http://www.statslab.cam.ac.uk/~rrw1/timeseries/t.pdf
Note: The datum of sea water level data used in this analysis is MLLW (Mean Low Lower Level).
