Assignment

Overview

This assignment covers the material from Time series modeling in R 1, 2, and 3. You will fit the same family of models we fit in the tutorials – white noise, autoregressive, seasonal ARIMA, time-series linear models, and dynamic regression – to a new ecological time series, and evaluate each one the way we did in class.

Work through the parts in order, which will expand the modeling approach gradually fixing problems you diagnose as you go.

Turn in a single R script (or Quarto document) on Canvas that contains your code, generates your plots, and includes written answers as comments (R script) or text block (Quarto document).

Note

Learning Objectives:

  • Fit white noise, AR, ARIMA, TSLM, and dynamic regression models to a new dataset
  • Use residual to check if a model has captured the structure in a time series
  • Interpret model coefficients

Data

We are exploring dissolved oxygen from Lake Barco, the National Ecological Observatory Network (NEON) aquatic site at the Ordway-Swisher Biological Station in Florida.

Dissolved oxygen is controlled by a combination of temperature and algae and serves as a good indicator of lake condition.

The data is monthly and includes three variables:

  • oxygen – dissolved oxygen concentration (mg/L), our response variable
  • temperature – water temperature (°C)
  • chla – chlorophyll-a (µg/L), a measure of how much algae is in the water

Where the data come from

The data are NEON sensor measurements, obtained through the NEON Ecological Forecasting Challenge, which is run by the Ecological Forecasting Initiative Research Coordination Network and repackages the raw NEON products into tidy daily “target” files.

Two NEON data products underlie the three columns:

Both are collected at the Lake Barco (BARC) field site.

If you want to see how the daily target file was reduced to the monthly series – or to rebuild it with more recent data – the script is make_barco_timeseries.R.

Thomas RQ, Boettiger C, Carey CC, et al. (2023) The NEON Ecological Forecasting Challenge. Frontiers in Ecology and the Environment 21(3): 112–113. doi:10.1002/fee.2616

Part 1: Look at the data

  1. Download the Lake Barco limnology data
  2. Plot the oxygen time series along with its ACF and PACF using gg_tsdisplay().
  3. Briefly describe what you see. E.g., Is there a trend? Is there a seasonal signal? At what lags is the autocorrelation strongest?

Part 2: White noise

  1. Fit a white noise model to oxygen using MEAN().
  2. Report the model with report().
  3. Plot the data with the fitted values on top using augment() and autoplot().
  4. Plot the residuals with gg_tsresiduals().

Question: What structure in the data is this model failing to capture and how do you now?

Part 3: Autoregressive models

  1. Based on the PACF from Part 1, pick the number of autoregressive terms and fit an AR() model to oxygen.
  2. Report the model.
  3. Plot the fitted values over the data, and plot the residuals.

Question: Describe how you chose the AR order. Describe how this month’s oxygen relates to last month’s. Compared to the white noise model, what did the AR model improve. What structure is still left in the residuals?

Part 4: Seasonal ARIMA

  1. Fit an ARIMA() model to oxygen, letting fable select the structure.
  2. Report the model.
  3. Plot the fitted values over the data, and plot the residuals.

Question: What model structure was selected (How many AR terms, how many MA terms, how much differencing, and what seasonal terms). How did the the residual ACF change compared to the AR model.

Part 5: Adding covariates with TSLM

  1. Fit a TSLM() model predicting oxygen from temperature.
  2. Report the model.
  3. Add chla as a second predictor and refit.
  4. Plot the fitted values compared to the data, and plot the residuals of the model with both temperature andchla` as drivers.

Question: Describe the model coefficients including sign and significance. The TSLM() model doesn’t explicitly model seasonality, butit tracks the seasonal ups and downs well. How is it doing that? Are there any issues with the residuals?

Part 6: Dynamic regression

  1. Fit an ARIMA() model with temperature as an external covariate.
  2. Report the model.
  3. Fit a second version that also includes chla.
  4. Compare the two models using the AICc values (lower is better) in their reports and plot the fitted values and residuals for your preferred model.

Question: What error structure did fable choose for the two models? Is the temperature coefficient similar to the one from the TSLM() model? Compare the residuals of this model to the residuals from the TSLM() model in Part 5. What did adding the ARIMA error structure change, and why does it matter for interpreting the coefficients?

Part 7: Wrap-up

Include a short paragraph the answers:

  • Which model would you choose to describe how dissolved oxygen behaves in Lake Barco, and why?
  • One model suggests that oxygen is seasonal; another suggests why it is seasonal. Which is which, and when might you prefer each one?