# Local Polynomial Regression 4: Application to Global Warming

In this final post on local polynomial regression we apply the local polynomial estimator to global warming data from NASA.

## Data

We use time-series data from NASA’s Goddard Institute for Space. The data set takes the form of yearly measurements of global temperature anomaly between 1880 and 2021, giving 142 samples in total. We set the independent variable $x_i$ as the measurement year and the dependent variable $y_i$ as the temperature anomaly, defined as the difference between the average yearly temperature and the average temperature between 1951 and 1980.

## Estimator

To fit our model we use a local linear smoother (polynomial of degree 1). As we saw in the first three posts on local polynomial regression, this estimator has the following attractive properties:

- Flexible estimation of a wide variety of regression functions
- Local estimation (predictions only affected by “nearby” data points)
- Only one hyperparameter to choose (the bandwidth)
- Easy bandwidth selection via leave-one-out cross-validation (LOO-CV)
- Robustness against boundary bias and other first-order bias

The data is also well-suited to such an estimator as:

- The independent variable is low-dimensional (one-dimensional in this case)
- The independent variable is evenly distributed with no large gaps
- It is not clear that the regression function should take any specific parametric form
- There are no significant outliers visible in the data
- The data set is small so estimation is computationally tractable

## Results

In Figure 1 we plot the fit of the estimator, animated over a range of bandwidths. As expected, larger bandwidths give a smoother curve with more bias and less variance.

Selecting the bandwidth with leave-one-out cross-validation as in the second post in this series yields Figure 2.

It is possible that the fit is slightly undersmoothed here
(bandwidth too small), as the line appears to be more
jagged than one might expect.
This may be because LOO-CV relies on the assumption
that the samples are *independent*,
which may well not be true of measurements in
consecutive years.
More sophisticated methods are available for the
time-series setting which address this issue.

Nonetheless the fit seems reasonable and captures the overall trend of the data. We note that the smoothed version presented by NASA indeed seems to use a larger bandwidth than we obtained by LOO-CV, though their methodology may be slightly different.

## References

- Data from NASA’s Goddard Institute for Space, accessed July 2022