Nearest Neighbor Methods

Nearest Neighbor methods used for 2-dimensional interpolation. These methods include two methods of neighbor selection and two methods of weighting. To access any of the functionality, use the Nearest Neighbor Averaging (NNA) class.

Methodology

NNA

Neighbor Selection:

• nearest selects a number (k) of nearest neighbors using euclidian distance.
• voronoi selects the Voronoi neighbors from within its k-nearest neighbors.
• laplace uses voronoi methodology with a special weighting.

Weighting:

• Both nearest and voronoi use distance power-based weight (d**power)
• laplace uses the ratio of the voronoi-neighbor edge length do distance between neighbors.

GMOS

An additional option has been implemented to use iterative radius searches and subsequent smoothing. This method is based on work done at NOAA/NWS by Glahn (2009)

Examples

The example below uses default interpolation methods on California Housing prices interpolations based solely on interpolating sparse data. We do not expect the model to fit well and the correlation is around 0.6. The intent is not to make a great model of housing, but to show the ease of use.

import pandas as pd
import numpy as np
from nna_methods import NNA
from sklearn.model_selection import KFold
from sklearn.datasets import fetch_california_housing

# last two X featurs are lat and lon
X, y = fetch_california_housing(return_X_y=True)
df = pd.DataFrame(dict(lat=X[:, -2], lon=X[:, -1], y=y, yhat=y*np.nan))
kf = KFold(n_splits=10)

nn = NNA()
for trainidx, testidx in kf.split(X, y):
  nn.fit(X[trainidx, -2:], y[trainidx])
  df.loc[testidx, 'yhat'] = nn.predict(X[testidx, -2:])

statdf = df.describe()
statdf.loc['r'] = df.corr()['y']
statdf.loc['mb'] = df.subtract(df['y'], axis=0).mean()
statdf.loc['rmse'] = (df.subtract(df['y'], axis=0)**2).mean()**0.5
print(statdf.loc[['mean', 'std', 'rmse', 'r']].round(2).to_markdown())
# Output:
# |      |   lat |     lon |    y |   yhat |
# |:-----|------:|--------:|-----:|-------:|
# | mean | 35.63 | -119.57 | 2.07 |   2.09 |
# | std  |  2.14 |    2    | 1.15 |   1    |
# | rmse | 33.66 |  121.66 | 0    |   0.98 |
# | r    | -0.14 |   -0.05 | 1    |   0.59 |

By default, this exmample uses NNA options method='nearest', k=10, and power=-2. That means it selects the 10 nearest neighbors and uses inverse distance squared weights. You can change that by modifying the line nn = NNA(). For example, nn = NNA(method='voronoi', k=30, power=-3) would select Voronoi neighbors from the nearest 30 points and apply inverse distance cubed weighting.

NNA, eVNA, and aVNA

NNA easily implements the method used by the EPA called extended Voronoi Neighbor Averaging or eVNA. eVNA is used to adjust models to better reflect observations. For reference, nn = NNA(method='voronoi', k=30, power=-2) is equivalent to the standard options for EPA's eVNA method.[1,2] In eVNA, the obs:model ratio at monitor sites is interpolated to grid cell centers and then multiplied by the model. The value of k=30 generally agrees well softwares like SMAT-CE and DFT that use an 8 degree radius with a minimum of 20 neighbors.

NNA can also produce a variant of eVNA that I refer to as additive eVNA or aVNA for short. Instead of interpolating the obs:model ratio, it interpolates the bias. Then, the bias is subtracted from the model to adjust the model to better reflect the observations.

NNA, eIDW, and aIDW

The eVNA and aVNA approaches described above can be implemented with nearest instead of vornoi neighbors in about 2% of the walltime. Although it is fast, IDW is subject to artifacts due to the spatially biased observation network that eVNA was designed to address. In particular, the cluster observations around an urban area can overwhelm observations that are closer, but isolated. This can be overcome increasing the magnitude of the power (e.g., -2 to -5). For some applications, that might be fine.

DelaunayInterp, eDNA, and aDNA

The eVNA and aVNA approaches described above can be implemented with Delaunay triangulation in about 0.25% of the walltime. This is very fast, but uses only the three points of the simplex that each prediction coordinate is within. This has the benefit of now allowing an isolated point to be overwhelmed by a cluster, but has ridges associated with the boundaries of interpolation within tesselation (triangle) borders.

GMOS, eGMOS, and aGMOS

GMOS is an implementation of the gridding of model output statistics (GMOS). GMOS was coined by Glahn et al. (2009) and a clear implementation is available in Glahn, Im and Wagner (2012). This technique is used by Djalalova (2015) and subsequently by NOAA for various "adjustments" to their forecasts. Below is my broad-brush understanding:

i \in AirNow stations
y_i = NAQFC_i
o_{a,i}, y_{a,i} = Analog(y_i, t_i, ws_i, wd_i, sr_i)
k_i = KalmanFilter(o_{a,i})
c_i = y_i - mean(y_{a,i}) 
b_i = y_i - (k_i + c_i)
y'_x = y_x - GMOS(b_i)

If we are interested in applying a known bias, unlike Djalalova (2015), then b_i does not need to be predicted. In that case, bias ($b_i$) can be estimated from the difference between the model ($NAQFC_i$) and the observed value at the station ($o_i$). Then, GMOS can be used in place of VNA for an aVNA-like product. Or, the ratio could be used to create an eVNA-like product. The GMOS method is about 25% faster than the VNA equivalent. The results are more similar to voronoi than IDW. This likely due to the concentric circles reducing in successive iterations. Thus, reducing the influence of non-voronoi neighbors.

Note: The GMOS implemented here does not have elevation or land/ocean awareness. Nor does it implement iterative QC as described by Glahn (2009).

References

[1] Timin, Wesson, Thurman, Chapter 2.12 of Air pollution modeling and its application XX. (Eds. Steyn, D. G., & Rao, S. T.) Dordrecht: Springer Verlag. May 2009.

[2] Abt, MATS User Guide 2007 or 2010

[3] Glahn, Gilbert, Cosgrove, Ruth, and Sheets: The Gridding of MOS, Weather and Forecasting, 24, 520-529, https://doi.org/10.1175/2008WAF2007080.1, 2009.

[4] Glahn, B., Im, J. S., and Wagner, G.: 8.4 Objective Analysis of MOS Forecasts and Observations in Sparse Data Regions, 2012.

[5] Djalalova, I., Delle Monache, L., and Wilczak, J.: PM2.5 analog forecast and Kalman filter post-processing for the Community Multiscale Air Quality (CMAQ) model, Atmospheric Environment, 108, 76-87, https://doi.org/10.1016/j.atmosenv.2015.02.021, 2015.