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.
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
usesvoronoi
methodology with a special weighting.
Weighting:
- Both
nearest
andvoronoi
use distance power-based weight (d**power
) laplace
uses the ratio of the voronoi-neighbor edge length do distance between neighbors.
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)
The example below uses default interpolation methods on California Housing prices interpolations based solely on longitude, latitude. Not a great test, because we know that the model will not fit well, but it demonstrates 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['corr'] = 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
statdf.loc[['mean', 'std', 'rmse']].round(2)
# Output:
# lat lon y yhat
# mean 35.63 -119.57 2.07 2.09
# std 2.14 2.00 1.15 1.00
# rmse 33.66 121.66 0.00 0.98
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 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.
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.
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 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 (
Note: The GMOS implemented here does not have elevation or land/ocean awareness. Nor does it implement iterative QC as described by Glahn (2009).
[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.