Function for identifying clusters in bivariate polar plots (`polarPlot()`

);
identifying clusters in the original data for subsequent processing.

## Usage

```
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,
...
)
```

## Arguments

- mydata
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.- pollutant
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.- x
Name of variable to plot against wind direction in polar coordinates, the default is wind speed, “ws”.

- wd
Name of wind direction field.

- n.clusters
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`

.- after
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.- cols
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.- angle.scale
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.- units
The units shown on the polar axis scale.

- auto.text
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.- plot
Should a plot be produced?

`FALSE`

can be useful when analysing data to extract plot components and plotting them in other ways.- plot.data
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`

`type`

`type`

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.`statistic`

The 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.

`limits`

The 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.missing`

Setting 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.`uncertainty`

Should 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".`percentile`

If

`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.`weights`

At 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.bin`

The 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.col`

When

`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"`

.`alpha`

The 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`

.`upper`

This 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.positive`

The 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.`k`

This 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`

.`normalise`

If

`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.header`

Adds 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.footer`

see

`key.footer`

.`key.position`

Location where the scale key is to plotted. Allowed arguments currently include

`"top"`

,`"right"`

,`"bottom"`

and`"left"`

.`key`

Fine control of the scale key via

`drawOpenKey`

. See`drawOpenKey`

for further details.`ws_spread`

The 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_spread`

The 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_error`

The

`x`

error / uncertainty used when`statistic = "york_slope"`

.`y_error`

The

`y`

error / uncertainty used when`statistic = "york_slope"`

.`kernel`

Type 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.label`

When pair-wise statistics such as regression slopes are calculated and plotted, should a formula label be displayed?

`formula.label`

will also determine whether concentration information is printed when`statistic = "cpf"`

.`tau`

The 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.

## Value

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.

## Details

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.

## References

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

## See also

Other polar directional analysis functions:
`percentileRose()`

,
`polarAnnulus()`

,
`polarDiff()`

,
`polarFreq()`

,
`polarPlot()`

,
`pollutionRose()`

,
`windRose()`

Other cluster analysis functions:
`timeProp()`

,
`trajCluster()`

## Examples

```
if (FALSE) { # \dontrun{
## 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)]
)
} # }
```