Function for identifying clusters in bivariate polar plots (polarPlot());
identifying clusters in the original data for subsequent processing.
polarCluster(
mydata,
pollutant = "nox",
x = "ws",
wd = "wd",
n.clusters = 6,
after = NA,
cols = "Paired",
angle.scale = 315,
units = x,
auto.text = TRUE,
plot = TRUE,
plot.data = FALSE,
...
)an openair object. The object includes four main
components: call, the command used to generate the plot;
data, by default the original data frame with a new field
cluster identifying the cluster, clust_stats giving the
contributions made by each cluster to number of measurements, their
percentage and the percentage by pollutant; and plot, the plot
itself. Note that any rows where the value of pollutant is NA
are ignored so that the returned data frame may have fewer rows than the original.
If the clustering is carried out considering differences, i.e., an
after data frame is supplied, the output also includes the
after data frame with cluster identified.
A data frame minimally containing wd, another variable
to plot in polar coordinates (the default is a column “ws” --- wind
speed) and a pollutant. Should also contain date if plots by time
period are required.
Mandatory. A pollutant name corresponding to a variable in a
data frame should be supplied e.g. pollutant = "nox". Only one
pollutant can be chosen.
Name of variable to plot against wind direction in polar coordinates, the default is wind speed, “ws”.
Name of wind direction field.
Number of clusters to use. If n.clusters is more
than length 1, then a lattice panel plot will be output showing the
clusters identified for each one of n.clusters.
The function can be applied to differences between polar plot
surfaces (see polarDiff for details). If an after data frame
is supplied, the clustering will be carried out on the differences between
after and mydata in the same way as polarDiff.
Colours to be used for plotting. Options include
“default”, “increment”, “heat”, “jet” and
RColorBrewer colours --- see the openair openColours
function for more details. For user defined the user can supply a list of
colour names recognised by R (type colours() to see the full list).
An example would be cols = c("yellow", "green", "blue"). cols
can also take the values "viridis", "magma",
"inferno", or "plasma" which are the viridis colour maps
ported from Python's Matplotlib library.
Sometimes the placement of the scale may interfere with an
interesting feature. The user can therefore set angle.scale to any
value between 0 and 360 degrees to mitigate such problems. For example
angle.scale = 45 will draw the scale heading in a NE direction.
The units shown on the polar axis scale.
Either TRUE (default) or FALSE. If TRUE
titles and axis labels will automatically try and format pollutant names
and units properly e.g. by subscripting the `2' in NO2.
Should a plot be produced? FALSE can be useful when
analysing data to extract plot components and plotting them in other ways.
By default, the data component of polarCluster()
contains the original data frame appended with a new "cluster" column. When
plot.data = TRUE, the data component instead contains data to reproduce
the clustered polar plot itself (similar to data returned by
polarPlot()). This may be useful for re-plotting the polarCluster()
plot in other ways.
Arguments passed on to polarPlot
typetype determines how the data are split i.e. conditioned,
and then plotted. The default is will produce a single plot using the
entire data. Type can be one of the built-in types as detailed in
cutData e.g. “season”, “year”, “weekday” and so
on. For example, type = "season" will produce four plots --- one for
each season.
It is also possible to choose type as another variable in the data
frame. If that variable is numeric, then the data will be split into four
quantiles (if possible) and labelled accordingly. If type is an existing
character or factor variable, then those categories/levels will be used
directly. This offers great flexibility for understanding the variation of
different variables and how they depend on one another.
Type can be up length two e.g. type = c("season", "weekday") will
produce a 2x2 plot split by season and day of the week. Note, when two
types are provided the first forms the columns and the second the rows.
statisticThe statistic that should be applied to each wind
speed/direction bin. Because of the smoothing involved, the colour scale
for some of these statistics is only to provide an indication of overall
pattern and should not be interpreted in concentration units e.g. for
statistic = "weighted.mean" where the bin mean is multiplied by the
bin frequency and divided by the total frequency. In many cases using
polarFreq will be better. Setting statistic = "weighted.mean"
can be useful because it provides an indication of the concentration *
frequency of occurrence and will highlight the wind speed/direction
conditions that dominate the overall mean.Can be:
“mean” (default), “median”, “max” (maximum), “frequency”. “stdev” (standard deviation), “weighted.mean”.
statistic = "nwr" Implements the Non-parametric Wind
Regression approach of Henry et al. (2009) that uses kernel smoothers. The
openair implementation is not identical because Gaussian kernels are
used for both wind direction and speed. The smoothing is controlled by
ws_spread and wd_spread.
statistic = "cpf" the conditional probability function (CPF)
is plotted and a single (usually high) percentile level is supplied. The
CPF is defined as CPF = my/ny, where my is the number of samples in the y
bin (by default a wind direction, wind speed interval) with mixing ratios
greater than the overall percentile concentration, and ny is the
total number of samples in the same wind sector (see Ashbaugh et al.,
1985). Note that percentile intervals can also be considered; see
percentile for details.
When statistic = "r" or statistic = "Pearson", the
Pearson correlation coefficient is calculated for two pollutants.
The calculation involves a weighted Pearson correlation coefficient, which
is weighted by Gaussian kernels for wind direction an the radial variable
(by default wind speed). More weight is assigned to values close to a wind
speed-direction interval. Kernel weighting is used to ensure that all data
are used rather than relying on the potentially small number of values in a
wind speed-direction interval.
When statistic = "Spearman", the Spearman correlation
coefficient is calculated for two pollutants. The calculation
involves a weighted Spearman correlation coefficient, which is weighted by
Gaussian kernels for wind direction an the radial variable (by default wind
speed). More weight is assigned to values close to a wind speed-direction
interval. Kernel weighting is used to ensure that all data are used rather
than relying on the potentially small number of values in a wind
speed-direction interval.
"robust_slope" is another option for pair-wise statistics and
"quantile.slope", which uses quantile regression to estimate the
slope for a particular quantile level (see also tau for setting the
quantile level).
"york_slope" is another option for pair-wise statistics which
uses the York regression method to estimate the slope. In this
method the uncertainties in x and y are used in the
determination of the slope. The uncertainties are provided by
x_error and y_error --- see below.
limitsThe function does its best to choose sensible limits
automatically. However, there are circumstances when the user will wish to
set different ones. An example would be a series of plots showing each year
of data separately. The limits are set in the form c(lower, upper),
so limits = c(0, 100) would force the plot limits to span 0-100.
exclude.missingSetting this option to TRUE (the default)
removes points from the plot that are too far from the original data. The
smoothing routines will produce predictions at points where no data exist
i.e. they predict. By removing the points too far from the original data
produces a plot where it is clear where the original data lie. If set to
FALSE missing data will be interpolated.
uncertaintyShould the uncertainty in the calculated surface be shown?
If TRUE three plots are produced on the same scale showing the
predicted surface together with the estimated lower and upper uncertainties
at the 95% confidence interval. Calculating the uncertainties is useful to
understand whether features are real or not. For example, at high wind
speeds where there are few data there is greater uncertainty over the
predicted values. The uncertainties are calculated using the GAM and
weighting is done by the frequency of measurements in each wind
speed-direction bin. Note that if uncertainties are calculated then the
type is set to "default".
percentileIf statistic = "percentile" then percentile
is used, expressed from 0 to 100. Note that the percentile value is
calculated in the wind speed, wind direction ‘bins’. For this reason
it can also be useful to set min.bin to ensure there are a
sufficient number of points available to estimate a percentile. See
quantile for more details of how percentiles are calculated.
percentile is also used for the Conditional Probability Function
(CPF) plots. percentile can be of length two, in which case the
percentile interval is considered for use with CPF. For example,
percentile = c(90, 100) will plot the CPF for concentrations between
the 90 and 100th percentiles. Percentile intervals can be useful for
identifying specific sources. In addition, percentile can also be of
length 3. The third value is the ‘trim’ value to be applied. When
calculating percentile intervals many can cover very low values where there
is no useful information. The trim value ensures that values greater than
or equal to the trim * mean value are considered before the
percentile intervals are calculated. The effect is to extract more detail
from many source signatures. See the manual for examples. Finally, if the
trim value is less than zero the percentile range is interpreted as
absolute concentration values and subsetting is carried out directly.
weightsAt the edges of the plot there may only be a few data points
in each wind speed-direction interval, which could in some situations
distort the plot if the concentrations are high. weights applies a
weighting to reduce their influence. For example and by default if only a
single data point exists then the weighting factor is 0.25 and for two
points 0.5. To not apply any weighting and use the data as is, use
weights = c(1, 1, 1).
An alternative to down-weighting these points they can be removed
altogether using min.bin.
min.binThe minimum number of points allowed in a wind speed/wind
direction bin. The default is 1. A value of two requires at least 2 valid
records in each bin an so on; bins with less than 2 valid records are set
to NA. Care should be taken when using a value > 1 because of the risk of
removing real data points. It is recommended to consider your data with
care. Also, the polarFreq function can be of use in such
circumstances.
mis.colWhen min.bin is > 1 it can be useful to show where data
are removed on the plots. This is done by shading the missing data in
mis.col. To not highlight missing data when min.bin > 1
choose mis.col = "transparent".
alphaThe alpha transparency to use for the plotting surface (a value
between 0 and 1 with zero being fully transparent and 1 fully opaque).
Setting a value below 1 can be useful when plotting surfaces on a map using
the package openairmaps.
upperThis sets the upper limit wind speed to be used. Often there are only a relatively few data points at very high wind speeds and plotting all of them can reduce the useful information in the plot.
force.positiveThe default is TRUE. Sometimes if smoothing data
with steep gradients it is possible for predicted values to be negative.
force.positive = TRUE ensures that predictions remain positive. This
is useful for several reasons. First, with lots of missing data more
interpolation is needed and this can result in artefacts because the
predictions are too far from the original data. Second, if it is known
beforehand that the data are all positive, then this option carries that
assumption through to the prediction. The only likely time where setting
force.positive = FALSE would be if background concentrations were
first subtracted resulting in data that is legitimately negative. For the
vast majority of situations it is expected that the user will not need to
alter the default option.
kThis is the smoothing parameter used by the gam function in
package mgcv. Typically, value of around 100 (the default) seems to
be suitable and will resolve important features in the plot. The most
appropriate choice of k is problem-dependent; but extensive testing
of polar plots for many different problems suggests a value of k of
about 100 is suitable. Setting k to higher values will not tend to
affect the surface predictions by much but will add to the computation
time. Lower values of k will increase smoothing. Sometimes with few
data to plot polarPlot will fail. Under these circumstances it can
be worth lowering the value of k.
normaliseIf TRUE concentrations are normalised by dividing by
their mean value. This is done after fitting the smooth surface.
This option is particularly useful if one is interested in the patterns of
concentrations for several pollutants on different scales e.g. NOx and CO.
Often useful if more than one pollutant is chosen.
key.headerAdds additional text/labels to the scale key. For example,
passing the options key.header = "header", key.footer = "footer1"
adds addition text above and below the scale key. These arguments are
passed to drawOpenKey via quickText, applying the
auto.text argument, to handle formatting.
key.footersee key.footer.
key.positionLocation where the scale key is to plotted. Allowed
arguments currently include "top", "right", "bottom"
and "left".
keyFine control of the scale key via drawOpenKey. See
drawOpenKey for further details.
ws_spreadThe value of sigma used for Gaussian kernel weighting of
wind speed when statistic = "nwr" or when correlation and regression
statistics are used such as r. Default is 0.5.
wd_spreadThe value of sigma used for Gaussian kernel weighting of
wind direction when statistic = "nwr" or when correlation and
regression statistics are used such as r. Default is 4.
x_errorThe x error / uncertainty used when statistic =
"york_slope".
y_errorThe y error / uncertainty used when statistic =
"york_slope".
kernelType of kernel used for the weighting procedure for when
correlation or regression techniques are used. Only "gaussian" is
supported but this may be enhanced in the future.
formula.labelWhen pair-wise statistics such as regression slopes are calculated and plotted, should a formula label be displayed?
tauThe quantile to be estimated when statistic is set to
"quantile.slope". Default is 0.5 which is equal to the median
and will be ignored if "quantile.slope" is not used.
David Carslaw
Bivariate polar plots generated using the polarPlot function provide a
very useful graphical technique for identifying and characterising different
air pollution sources. While bivariate polar plots provide a useful graphical
indication of potential sources, their location and wind-speed or other
variable dependence, they do have several limitations. Often, a `feature'
will be detected in a plot but the subsequent analysis of data meeting
particular wind speed/direction criteria will be based only on the judgement
of the investigator concerning the wind speed-direction intervals of
interest. Furthermore, the identification of a feature can depend on the
choice of the colour scale used, making the process somewhat arbitrary.
polarCluster applies Partition Around Medoids (PAM) clustering
techniques to polarPlot() surfaces to help identify potentially interesting
features for further analysis. Details of PAM can be found in the
cluster package (a core R package that will be pre-installed on all R
systems). PAM clustering is similar to k-means but has several advantages
e.g. is more robust to outliers. The clustering is based on the equal
contribution assumed from the u and v wind components and the associated
concentration. The data are standardized before clustering takes place.
The function works best by first trying different numbers of clusters and
plotting them. This is achieved by setting n.clusters to be of length
more than 1. For example, if n.clusters = 2:10 then a plot will be
output showing the 9 cluster levels 2 to 10.
The clustering can also be applied to differences in polar plot surfaces (see
polarDiff()). On this case a second data frame (after) should be
supplied.
Note that clustering is computationally intensive and the function can take a
long time to run --- particularly when the number of clusters is increased.
For this reason it can be a good idea to run a few clusters first to get a
feel for it e.g. n.clusters = 2:5.
Once the number of clusters has been decided, the user can then run
polarCluster to return the original data frame together with a new
column cluster, which gives the cluster number as a character (see
example). Note that any rows where the value of pollutant is NA
are ignored so that the returned data frame may have fewer rows than the
original.
Note that there are no automatic ways in ensuring the most appropriate number of clusters as this is application dependent. However, there is often a-priori information available on what different features in polar plots correspond to. Nevertheless, the appropriateness of different clusters is best determined by post-processing the data. The Carslaw and Beevers (2012) paper discusses these issues in more detail.
Note that unlike most other openair functions only a single
type “default” is allowed.
Carslaw, D.C., Beevers, S.D, Ropkins, K and M.C. Bell (2006). Detecting and quantifying aircraft and other on-airport contributions to ambient nitrogen oxides in the vicinity of a large international airport. Atmospheric Environment. 40/28 pp 5424-5434.
Carslaw, D.C., & Beevers, S.D. (2013). Characterising and understanding emission sources using bivariate polar plots and k-means clustering. Environmental Modelling & Software, 40, 325-329. doi:10.1016/j.envsoft.2012.09.005
Other polar directional analysis functions:
percentileRose(),
polarAnnulus(),
polarDiff(),
polarFreq(),
polarPlot(),
pollutionRose(),
windRose()
Other cluster analysis functions:
timeProp(),
trajCluster()
if (FALSE) {
## plot 2-8 clusters. Warning! This can take several minutes...
polarCluster(mydata, pollutant = "nox", n.clusters = 2:8)
# basic plot with 6 clusters
results <- polarCluster(mydata, pollutant = "nox", n.clusters = 6)
## get results, could read into a new data frame to make it easier to refer to
## e.g. results <- results$data...
head(results$data)
## how many points are there in each cluster?
table(results$data$cluster)
## plot clusters 3 and 4 as a timeVariation plot using SAME colours as in
## cluster plot
timeVariation(subset(results$data, cluster %in% c("3", "4")),
pollutant = "nox",
group = "cluster", col = openColours("Paired", 6)[c(3, 4)]
)
}
Run the code above in your browser using DataLab