In the 1950s, the geochemist Charles David Keeling observed a seasonal pattern in the amount of carbon dioxide present in air samples collected over the course of several years. He was able to attribute this pattern to the variation in global rates of photosynthesis throughout the year, caused by the difference in land area and vegetation cover between the Earth’s northern and southern hemispheres.
In 1958 Keeling began continuous monitoring of atmospheric carbon dioxide concentrations from the Mauna Loa Observatory in Hawaii and soon observed a trend increase carbon dioxide levels in addition to the seasonal cycle. He was able to attribute this trend increase to growth in global rates of fossil fuel combustion. This trend has continued to the present, and is known as the “Keeling Curve.”
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Your goal in this assignment is to produce a comprehensive analysis of the Mona Loa CO2 data that you will be read by an interested, supervising data scientist. Rather than this being a final report, you might think of this as being a contribution to your laboratory. You and your group have been initially charged with the task of investigating the trends of global CO2, and told that if you find “anything interesting” that the team may invest more resources into assessing the question.
Because this is the scenario that you are responding to:
- Your writing needs to be clear, well-reasoned, and concise. Your peers will be reading this, and you have a reputation to maintain.
- Decisions that you make for your analysis need also be clear and well-reasoned. While the main narrative of your deliverable might only present the modeling choices that you determine are the most appropriate, there might exist supporting materials that examine what the consequences of other choices would be. As a concrete example, if you determine that a series is an AR(1) process your main analysis might provide the results of the critical test that led you to that determination and the results of the rest of the analysis under AR(1) modeling choices. However, in an appendix or separate document that is linked in your main report, you might show what a MA model would have meant for your results instead.
- Your code and repository are a part of the deliverable. If you were to make a clear argument that this is a question worth pursuing, but then when the team turned to continue the work they found a repository that was a jumble of coding idioms, version-ed or outdated files, and skeletons it would be a disappointment.
For the first part of this task, suspend reality for a short period of
time and conduct your analysis from the point of view of a data
scientist doing their work in the early months of 1998. Do this by using
data that is included in every R implementation, the co2 dataset.
This dataset is lazily loaded with every R instance, and is stored in an
object called co2.
Introduce the question to your audience. Suppose that they could be interested in the question, but they don’t have a deep background in the area. What is the question that you are addressing, why is it worth addressing, and what are you going to find at the completion of your analysis. Here are a few resource that you might use to start this motivation.
Conduct a comprehensive Exploratory Data Analysis on the co2 series.
This should include (without being limited to) a description of how,
where and why
the data is generated, a thorough investigation of the trend, seasonal
and irregular elements. Trends both in levels and growth rates should be
discussed (consider expressing longer-run growth rates as annualized
averages).
What you report in the deliverable should not be your own process of discovery, but rather a guided discussion that you have constructed so that your audience can come to an understanding as succinctly and successfully as possible. This means that figures should be thoughtfully constructed and what you learn from them should be discussed in text; to the extent that there is any raw output from your analysis, you should intend for people to read and interpret it, and you should write your own interpretation as well.
Fit a linear time trend model to the co2 series, and examine the
characteristics of the residuals. Compare this to a quadratic time trend
model. Discuss whether a logarithmic transformation of the data would be
appropriate. Fit a polynomial time trend model that incorporates
seasonal dummy variables, and use this model to generate forecasts to
the year 2020.
Following all appropriate steps, choose an ARIMA model to fit to the series. Discuss the characteristics of your model and how you selected between alternative ARIMA specifications. Use your model (or models) to generate forecasts to the year 2022.
Generate predictions for when atmospheric CO2 is expected to be at 420 ppm and 500 ppm levels for the first and final times (consider prediction intervals as well as point estimates in your answer). Generate a prediction for atmospheric CO2 levels in the year 2100. How confident are you that these will be accurate predictions?
One of the very interesting features of Keeling and colleagues’ research is that they were able to evaluate, and re-evaluate the data as new series of measurements were released. This permitted the evaluation of previous models’ performance and a much more difficult question: If their models’ predictions were “off” was this the result of a failure of the model, or a change in the system?
In this introduction, you can assume that your reader will have just read your 1997 report. In this introduction, very briefly pose the question that you are evaluating, and describe what (if anything) has changed in the data generating process between 1997 and the present.
The most current data is provided by the United States’ National Oceanic and Atmospheric Administration, on a data page [here]. Gather the most recent weekly data from this page. (A group that is interested in even more data management might choose to work with the hourly data.)
Create a data pipeline that starts by reading from the appropriate URL,
and ends by saving an object called co2_present that is a suitable
time series object.
Conduct the same EDA on this data. Describe how the Keeling Curve evolved from 1997 to the present, noting where the series seems to be following similar trends to the series that you “evaluated in 1997” and where the series seems to be following different trends. This EDA can use the same, or very similar tools and views as you provided in your 1997 report.
Descriptively compare realized atmospheric CO2 levels to those predicted by your forecast from a linear time model in 1997 (i.e. “Task 2a”). (You do not need to run any formal tests for this task.)
Descriptively compare realized atmospheric CO2 levels to those predicted by your forecast from the ARIMA model that you fitted in 1997 (i.e. “Task 3a”). Describe how the Keeling Curve evolved from 1997 to the present.
In 1997 you made predictions about the first time that CO2 would cross 420 ppm. How close were your models to the truth?
After reflecting on your performance on this threshold-prediction task, continue to use the weekly data to generate a month-average series from 1997 to the present, and compare the overall forecasting performance of your models from Parts 2a and 3b over the entire period. (You should conduct formal tests for this task.)
Seasonally adjust the weekly NOAA data, and split both seasonally-adjusted (SA) and non-seasonally-adjusted (NSA) series into training and test sets, using the last two years of observations as the test sets. For both SA and NSA series, fit ARIMA models using all appropriate steps. Measure and discuss how your models perform in-sample and (psuedo-) out-of-sample, comparing candidate models and explaining your choice. In addition, fit a polynomial time-trend model to the seasonally-adjusted series and compare its performance to that of your ARIMA model.
With the non-seasonally adjusted data series, generate predictions for when atmospheric CO2 is expected to be at 420 ppm and 500 ppm levels for the first and final times (consider prediction intervals as well as point estimates in your answer). Generate a prediction for atmospheric CO2 levels in the year 2122. How confident are you that these will be accurate predictions?