library(zenplots)

# All pairs

A zenplot can show the same information as a pairs plot but with two important display differences.

First, the matrix organization of the pairs layout is replaced by the “zig-zag” layout of zenplot. Second, the number of plots produced is about half that of a pairs plot allowing each plot in a zenplot to be given more visual space.

## Producing all pairs with PairViz

A convenient function to produce all pairs can be found in the PairViz package found on cran and installed in R via install.packages("PairViz").

library(PairViz)

We will illustrate this functionality and the difference between a pairs plot and a zenplot by first considering a small dataset on earthquakes having only a few variates. The difference between the two plots becomes much more important for data having larger numbers of variates – we illustrate the difference again using German data on voting patterns in two elections.

### Example: Ground acceleration of earthquakes

The built-in R data set called attenu contains measurements to estimate the attenuating effect of distance on the ground acceleration of earthquakes in California.

There are 5 different variates used to describe the peak acceleration of 23 California earthquakes measured at different observation stations. The data set contains 182 different peak acceleration measurements and has some missing data. The first few cases of the data set look like

head(attenu)
##   event mag station dist accel
## 1     1 7.0     117   12 0.359
## 2     2 7.4    1083  148 0.014
## 3     2 7.4    1095   42 0.196
## 4     2 7.4     283   85 0.135
## 5     2 7.4     135  107 0.062
## 6     2 7.4     475  109 0.054

Its variates are

## [1] "event"   "mag"     "station" "dist"    "accel"

and we are interested in all pairs of these variates.

To get these, first imagine a graph having as its nodes the variates of the data. An edge of this graph connects two nodes and hence represents a pair of variates. If interest lies in all pairs of varates, then the graph is a complete graph – it will have an edge between every pair of nodes. An ordering of variate pairs corresponds to any path on the graph. To have an ordering of all pairs of variates, the path must visit all edges and is called an Eulerian, or Euler path. Such a path always exists for complete graphs on an odd number of nodes; when the number of nodes is even, extra edges must be added to the graph before an Eulerian can exist.

For a complete graph with n nodes, the function eseq (for Euler sequence) function from the PairViz package returns an order in which the nodes (numbered 1 to n) can be visited to produce an Euler path. It works as follows.

## Since attenu has 5 variates, the complete graph has n=5 nodes
## and an Euler sequence is given as
eseq(5)
##  [1] 1 2 3 1 4 2 5 3 4 5 1

In terms of the variate names of attenu, this is:

names(attenu)[eseq(5)]
##  [1] "event"   "mag"     "station" "event"   "dist"    "mag"     "accel"
##  [8] "station" "dist"    "accel"   "event"

As can be seen in the corresponding complete graph below, this sequence traces an Eulerian path on the complete graph and so presents every variate next to every other variate somewhere in the order.

#### Euler sequences via zenpath

This functionality (and more) from PairViz has been bundled together in the zenplots package as a single function zenpath. For example,

zenpath(5)
##  [1] 5 1 2 3 1 4 2 5 3 4 5

This sequence, while still Eulerian, is slightly different than that returned by eseq(5). The sequence is chosen so that all pairs involving the first index appear earliest in the sequence, then all pairs involving the second index, and so on. We call this a “front loaded” sequence and identify it with the zenpath argument method = "front.loaded". Other possibilities are method = "back.loaded" and method = "balanced" giving the following sequences:

## Back loading ensures all pairs appear latest (back) for
## high values of the indices.
zenpath(5, method = "back.loaded")
##  [1] 1 2 3 1 4 2 5 3 4 5 1
## Frot loading ensures all pairs appear earliest (front) for
## low values of the indices.
zenpath(5, method = "front.loaded")
##  [1] 5 1 2 3 1 4 2 5 3 4 5
## Balanced loading ensures all pairs appear in groups of all
## indices (Hamiltonian paths -> a Hamiltonian decomposition of the Eulerian)
zenpath(5, method = "balanced")
##  [1] 1 2 3 5 4 1 3 4 2 5 1

The differences are easier to see when there are more nodes. Below, we show the index ordering (top to bottom) for each of these three methods when the graph has 15 nodes, here labelled“a” to “o” (to make plotting easier).

Starting from the bottom (the back of the sequence), “back loading” has the last index, “o”, complete its pairing with every other index before “n” completes all of its pairings. All of “n”’s pairings complete before those of “m”, all of “m”’s before “l”, and so on until the last pairing of “a” and “b” are completed. Note that the last indices still appear at the end of the sequence (since the sequence begins at the top of the display and moves down). The term “back loading” is used here in a double sense - the later (back) indices have their pairings appear as closely together as possible towards the back of the returned sequence. A simple reversal, that is rev(zenpath(15, method = "back.loaded")), would have them appear at the beginning of the sequence. In this case the “back loading” would only be in one sense, namely that the later indexed (back) nodes appear first in the reversed sequence.

Analogously, “front loading” has the first (front) indices appear at the front of the sequence with their pairings appear as closely together as possible.

The “balanced” case ensures that all indices appear in each block of pairings. In the figure there are 7 blocks.

All three sequences are Eulerian, meaning all pairs appear somewhere in each sequence.

#### Pairs plots versus zenplots

Eulerian sequences can now be used to compare a pairs plot with a zenplot when all pairs of variates are to be displayed.

First a pairs plot:

## We remove the space between plots and suppress the axes
## so as to give maximal space to the individual scatterplots.
## We also choose a different plotting character and reduce
## its size to better distinguish points.
pairs(attenu, oma=rep(0,4), gap=0, xaxt="n", yaxt="n")

We now effect a display of all pairs using zenplot.

## Plotting character and size are chosen to match that
## of the pairs plot.
## zenpath ensures that all pairs of variates appear
## in the zenplot.
## The last argument, n2dcol, is chosen so that the zenplot
## has the same number of plots across the page as does the
## pairs plot.
zenplot(attenu[, zenpath(ncol(attenu))], n2dcol=4)

Each display shows scatterplot of all choose(5,2) = 10 pairs of variates for this data. Each display occupies the same total area.

With pairs each plot is displayed twice and arranged in a symmetric matrix layout with the variate labels appearing along the diagonal. This makes for easy look-up but uses a lot of space.

With zenplot, each plot appears only once with its coordinate defining variates appearing as labels on horizontal (top or bottom) and vertical (left or right) axis positions. The layout follows the order of the variates in which the variates appear in the call to zenplot beginning in the top left corner of the display and then zig-zagging from top left to bottom right; when the rightmost boundary or the display is reached, the direction is reversed horizontally and the zigzag moves from top right to bottom left. The following display illustrates the pattern (had by simply calling zenplot):

## Call zenplot exactly as before, except that each scatterplot is replaced
## by an arrow that shows the direction of the layout.
zenplot(attenu[, zenpath(ncol(attenu))], plot2d="arrow", n2dcol=4)

The zig zag pattern of plots appears as follows.

• The top left plot (of either zenplot display) has horizontal variate event and vertical variate mag.
• To its right is a plot sharing the same vertical variate mag but now with horizontal variate station. Note that the variate station has some missing values and this is recorded on its label as station (some NA).
• Below this a plot appears with the same horizontal variate station but now with vertical variate event. Since this is the first repeat appearance of event it appears with a suffix as event.1.
• To its right is a plot with shared vertical variate event and new horizontal variate dist.
• Below this is a plot having shared horizontal variate dist and as vertical variate the first repeat of the variate mag.
• To its right is a plot having shared vertical variate mag and new horizontal variate accel.
• The right edge of the display is reached and the zigzag changes horizontal direction repeating the pattern until either the left edge is reached (whereupon the horizontal direction is reversed) or the variates are exhausted.

Like the pairs plot, the zenplot lays its plots out on a two dimensional grid – the argument n2dcol=4 specifies the number of columns for the 2d plots (e.g. scatterplots). As shown, this can lead to a lot of unused space in the display.

The zenplot layout can be made more compact by different choices of the argument n2dcol (odd values provide a more compact layout). For example,

## The default n2dcol is used
zenplot(attenu[, zenpath(ncol(attenu))], n2dcol=5)

By default, zenplot tries to determine a value for n2dcol that minimizes the space unused by its zigzag layout.

## The default n2dcol is used
zenplot(attenu[, zenpath(ncol(attenu))])

with layout directions as

## The directions
zenplot(attenu[, zenpath(ncol(attenu))], plot2d="arrow")

As the direction arrows show, the default layout is to zigzag horizontally first as much as possible.

This is clearly a much more compact display. Again, axes are shared wherever a label appears between plots.

Unless explicitly specified, the value of n2dcol is determined by the aregument scaling which can either be a numerical value specifying the ratio of the height to the width of the zenplot layout or be a string describing a page whose ratio of height to width will be used. The possible strings are letter'' (the default),square’‘, A4'',golden’‘(for the golden ratio), or legal’’.

### Example: German election data

In the zenplots package the data set de_elect contains the district results of two German federal elections (2002 and 2005) as well as a number of socio-economic variates as well.

## Access the German election data from zenplots package
data(de_elect)

There are 299 districts and 68 variates yielding a possible choose(68,2) = 2278 different scatterplots.

This many scatterplots will overwhelm a pairs plot. In its most compact form, the pairs plot for the first 34 variates already occupies a fair bit of space:

## pairs(de_elect[,1:34], oma=rep(0,4), gap=0, pch=".", xaxt="n", yaxt="n")

(N.B. We do not execute any of these large plots simply to keep the storage needs of this vignette to a minimum. We do encourage the reader to execute the code however on their own.)

If you execute the above code you will see interesting point configurations including: some very strong positive correlations, some positive and negative correlations, non-linear relations, the existence of some outlying points, clustering, striation, etc.

Because this scatterplot matrix is for only half of the variates it shows choose(34,2) = 561 different scatterplots, each one twice. For a display of 1122 plots, only about one quarter of all 2278 pairwise variate scatterplots available in the data set appear in this display.

A second scatterplot matrix on the remaining 34 variates would also show only a quarter of the plots. The remaining half, $$34 \times 34 = 1156$$ plots, are missing from both plots.

In contrast, the zenplot shows all 2278 plots at once. In fact, because an Eulerian sequence requires a graph to be even (i.e. each node has an even number of edges), whenever the number of variates, $$p$$, is even zenpath(...) will repeat exactly $$p/2$$ pairs somewhere in the sequence it returns.

To produce the zenplot of all pairs of variates on the German election data we call zenplot(de_elect[,zenpath(68)], pch="."). (Again, we don’t produce it here so as to minimize the storage footprint of this vignette.)

## Try invoking the plot with the following
## zenplot(de_elect[,zenpath(68)], pch=".", n2dcol="square",col=adjustcolor("black",0.5))

In approximately the same visual space as the scatterplot matrix (showing only 561 unique plots), the zenplot has efficiently and compactly laid out all 2278 different plots plus r ncol(de_elect) duplicate plots. This efficient layout means that zenplot can facilitate visual search for interesting point configurations over much larger collections of variate pairs – in the case of the German election data, this all possible pairs of variates are presented simultaneously.

In contrast, all pairs loses most of the detail

## pairs(de_elect, oma=rep(0,4), gap=0, pch=".", xaxt="n", yaxt="n",col=adjustcolor("black",0.5))

# Groups of pairwise plots

Zenplots also accomodate a list of data sets whose pairwise contents are to be displayed. The need for this can arise quite naturally in many applications.

The German election data, for instance, contains socio-economic data whose variates naturally group together. For example, we might gather variates related to education into one group and those related to employment into another.

Education <- c("School.finishers",
"School.wo.2nd", "School.2nd",
"School.Real",  "School.UED")
Employment <- c("Employed", "FFF", "Industry",
"CTT", "OS" )

We could plot all pairs for these two groups in a single zenplot.

EducationData <- de_elect[, Education]
EmploymentData <- de_elect[, Employment]

## Plot all pairs within each group
zenplot(list(Educ= EducationData[, zenpath(ncol(EducationData))],
Empl= EmploymentData[, zenpath(ncol(EmploymentData))]))

All pairs of education variates are plotted first in zigzag order followed by a blank plot then continuing in the same zigzag pattern by plots all pairs of employment variates.

## All pairs by group

In addition to the Education and Employment groups above, a number of different groupings of variates having a shared context. For example, these might include the following:

## Grouping variates in the German election data
Regions <- c("District", "State", "Density")
PopDist <- c("Men", "Citizens", "Pop.18.25", "Pop.25.35",
"Pop.35.60", "Pop.g.60")
PopChange <- c("Births", "Deaths", "Move.in", "Move.out", "Increase")
Agriculture <- c("Farms", "Agriculture")
Mining <- c("Mining", "Mining.employees")
Apt <- c("Apt.new", "Apt")
Motorized <- c("Motorized")
Education <- c("School.finishers",
"School.wo.2nd", "School.2nd",
"School.Real",  "School.UED")
Unemployment <- c("Unemployment.03", "Unemployment.04")
Employment <- c("Employed", "FFF", "Industry", "CTT", "OS" )
Voting.05 <- c("Voters.05", "Votes.05", "Invalid.05", "Valid.05")
Voting.02 <- c("Voters.02", "Votes.02", "Invalid.02", "Valid.02")
Voting <- c(Voting.02, Voting.05)
PercentByParty.02 <- c("SPD.02", "CDU.CSU.02", "Gruene.02",
PercentByParty.05 <- c("SPD.05", "CDU.CSU.05", "Gruene.05",
PercentByParty <- c(PercentByParty.02, PercentByParty.05)

The groups can now be used to explore internal group relations for many different groups in the same plot. Here the following helper function comes in handy.

groups <- list(Regions=Regions, Pop=PopDist,
Change = PopChange, Agric=Agriculture,
Mining=Mining, Apt=Apt,  Cars=Motorized,
Educ=Education, Unemployed=Unemployment, Employed=Employment#,
# Vote02=Voting.02, Vote05=Voting.05,
# Perc02=PercentByParty.02, Perc05=PercentByParty.05
)

group_paths <- lapply(groups, FUN= function(g) g[zenpath(length(g), method = "front.loaded")] )
x <- groupData(de_elect, indices=group_paths)

zenplot(x, pch = ".", cex=0.7, col = "grey10")

All pairs within each group are presented following the zigzag pattern; each group is separated by an empty plot. The zenplot provides a quick overview of the pairwise relationships between variates within all groups.

The plot can be improved some by using shorter names for the variates. With a little work we can replace these within each group of x.

#
## Grouping variates in the German election data
RegionsShort <- c("ED", "State", "density")
PopDistShort <- c("men", "citizen", "18-25", "25-35", "35-60", "> 60")
PopChangeShort <- c("births", "deaths", "in", "out", "up")
AgricultureShort <- c("farms", "hectares")
MiningShort <- c("firms", "employees")
AptShort <- c("new", "all")
TransportationShort <- c("cars")
EducationShort <- c("finishers", "no.2nd", "2nd", "Real", "UED")
UnemploymentShort<- c("03", "04")
EmploymentShort <- c("employed", "FFF", "Industry", "CTT", "OS" )
Voting.05Short <- c("eligible", "votes", "invalid", "valid")
Voting.02Short <- c("eligible", "votes", "invalid", "valid")
PercentByParty.02Short <- c("SPD", "CDU.CSU", "Gruene", "FDP", "Linke", "rest")
PercentByParty.05Short <-  c("SPD", "CDU.CSU", "Gruene", "FDP", "Linke", "rest")

shortNames <- list(RegionsShort, PopDistShort, PopChangeShort, AgricultureShort,
MiningShort, AptShort, TransportationShort, EducationShort,
UnemploymentShort, EmploymentShort, Voting.05Short, Voting.02Short,
PercentByParty.05Short)
# Now replace the names in x by these.

nGroups <- length(x)

for (i in 1:nGroups) {
longNames <- colnames(x[[i]])
newNames <- shortNames[[i]]
oldNames <- groups[[i]]
#print(longNames)
#print(newNames)
for (j in 1:length(longNames)) {
for (k in 1:length(newNames)) {
if (grepl(oldNames[k], longNames[j])) {
longNames[longNames == longNames[j]] <- newNames[k]
}
}
}
colnames(x[[i]]) <- longNames
}

zenplot(x, pch = ".", cex=0.75)

## Crossing pairs between groups

It can also be of interest to compare variates between groups. For example, to compare the various levels of education with the employment categories.

crossedGroups <- c(Employment, Education)
crossedPaths <- zenpath(c(length(Employment), length(Education)), method="eulerian.cross")
zenplot(de_elect[,crossedGroups][crossedPaths])

# Other plots

A zenplot can be thought of as taking a data set whose variates are to be plotted in the order given. A sequence of one dimensional plots, as determined by the argument plot1d, are constructed in the order of the variates. Between each pair of these 1d plots, a two dimensional plot is constructed from the variates of the 1d plots. One variate provides the vertical y values and the other the horizontal x values. If the orientation of the preceding one-dimensional plot is horizontal, then that variate gives the x values; if it’s vertical then the vertical y coordinates.

## Built in 1d and 2d plots

The actual displays depend on the arguments plot1d and plot2d. There are numerous built-in choices provided.

For plot1d any of the following strings may be selected to produce a one-dimensional plot: "label", "rug", "points", "jitter", "density", "boxplot", "hist", "arrow", "rect", "lines". The first in the list is the default.

For plot2d any of the following strings can be given: "points", "density", "axes", "label", "arrow", "rect". Again, the first of these is the default value.

## Arbitrary layout using turns

For example, we could produce a boxplot for the measured variates of the earthquake data as follows:

earthquakes <- attenu[, c(1,2,4,5)]  # ignore the station id
zenplot(earthquakes,
plot1d="boxplot", plot2d=NULL,
width1d=5, width2d=1,
turns=c("r","r","r","r","r","r","r"))

There are a few things to note here. First that every variate in the data set has a boxplot of its values presented and that the extent of boxplot display is that variate’s range. Second, the argument specification plot2d=NULL causes a null plot to be produced for each of the variate pairs. Third, the arguments width1d and width2d determine the relative widths of the two displays.

Finally, the argument turns determines the layout of the plots by specifying where the next display (1d or 2d) is to appear in relation to the current one. Here every display appears to the right of the existing display.

An alternative layout of the same boxplots can be had by adjusting the turns.

zenplot(earthquakes,
plot1d="boxplot", plot2d=NULL,
width1d=1, width2d=1,  # now widths must be the same
turns=c("r","d","d","l","l","u","u"))

To better see how the turns work, arrows could be used instead to show the directions of the turns.

zenplot(earthquakes,
plot1d= "arrow", plot2d="arrow",
width1d=1, width2d=2,
turns=c("r","d","d","l","l","u","u"))

Adding a rectangle to outline the drawing space will make the layout a little clearer and illustrate the arguments that are passed on to the pne and two dimensional plot functions.

zenplot(earthquakes,
plot1d = function(zargs, ...) {
rect_1d_graphics(zargs,  ...)
},
plot2d = function(zargs, ...) {
rect_2d_graphics(zargs,  ...)
arrow_2d_graphics(zargs, col="steelblue", lwd=3, lty=2, add=TRUE, ...)
},
width1d = 1, width2d = 2,
turns=c("r","d","d","l","l","u","u"))

The red arrows are the turns for the 1d plots, the blue dashed arrows for the 2d plots. The turns are interlaced and begin at the topmost red arrow for the 1d plot. It points right, matching the first turn "r" in the list of turns. The remaining arrows follow each other in clockwise order. As can be seen, there is an arrow for each plot: red for the 1d plots, black for the 2d plots.

## Arbitrary plots

The above example also introduces some other important features of zenplot.

The value "arrow" of plot2d argument caused an arrow to be drawn wherever a 2d plot was to appear in the direction of the turn associated with that plot. As the value of plot1d here suggests, the argument could also have been a function.

In fact, when given a string value for plot2d, zenplot calls a function whose name is constructed with this string. In the case of plot2d = "arrow", zenplot calls the function arrow_2d_graphics to produce a 2d plot whenever one is required. The naming convention constructs the function name from the string supplied, here “arrow”, the dimensionality of the plot (here 2d) and the R graphics package that is being used (here the base graphics package). The function of that name is called on arguments appropriate to draw the plot.

The same construction is also used when a string is given as the value of the plot1d argument. For example plot1d = "arrow" causes the function named arrow_1d_graphics to be called to draw the 1d plots.

Should, for example, the user wish to extend the base functionality of zenplot to include say "myplot", they need only write functions myplot_1d_graphics and/or myplot_2d_graphics to allow the value "myplot" to be used for plot1d and/or plot2d in the base graphics package in R. Note that two other R plotting packages besides the base graphics are supported by zenplot, namely either the highly customizable grid package or the highly interactive loon package. The default package is graphics but either of the other two may be specified via the pkg argument to zenplot as in pkg = "grid" or pkg = "loon".

When zenplot is called with plot1d="arrow", say, then one of the functions arrow_1d_graphics, arrow_1d_grid, or arrow_1d_loon will be called upon depending on the value of the argument pkg. This means that a true extension of zenplot to include, say, plot1d = "myplot" would require writing three functions, namely myplot_1d_graphics, myplot_1d_grid, and myplot_1d_loon, to complete the functionality.

More often, as shown in the boxplot example, it will be a one-off functionality that might be required for either plot2d or plot1d.

In this case, any function passed as the argument value will be used by zenplot to construct the corresponding plots. In the boxplot example, the default functions boxplot_1d_graphics and arrow_1d_graphics were both called so that one could be plotted on top of the other in each 1d plot of the zenplot.

### Example: mixing plots to assess distributions

zenplot can be used to layout graphics produced by any of the three pkg (i.e. graphics, grid, or loon) provided all graphics used are from the same package.

For example, suppose we were interested in the marginal distributions of the earthquake data. We might craft the following plot to investigate all marginal distributions and to compare each marginal to every other (i.e. all pairs).

library(qqtest)
zenplot(earthquakes[,zenpath(ncol(earthquakes))],
width1d = 1, width2d = 2, n2dcol=5,
plot1d = function(zargs, ...) {
r <- extract_1d(zargs) # extract arguments for 1d
col <- adjustcolor(if (r$horizontal) "firebrick" else "steelblue", alpha.f = 0.7) hist_1d_graphics(zargs, col=col, ...) }, plot2d=function(zargs, ...) { r <- extract_2d(zargs) # extract arguments for 2d x <- as.matrix(r$x)
xlim <- r$xlim y <- as.matrix(r$y)
ylim <- r\$ylim
qqtest(y, dataTest=x,
xlim=xlim, ylim=ylim,
cex=0.3, col="black",  pch=19,
legend=FALSE, main="", axes=FALSE, ...)

})

Here we are using the function qqtest from the package of that name and whose display capability is built using only the base graphics package. For 2d data qqtest compares the two empirical distributions by drawing an empirical quantile-quantile plot which should be near a straight line if the marginal distributions are of the same shape. The empirical quantile-quantile plot is supplemented by simulated values of empirical quantiles from the empirical distribution of the horizontal variate. The results of 1,000 draws from this distribution are shown in shades of grey on the qqplot. The 1d plots are shown as histograms coloured "grey" when that histogram was used to generate the simulated values and in "steelblue" when the histogram was not.

As can be seen from the qqplots, since numerous points are outside the gray envelopes of each plot, no two marginal distributions would appear to be alike.

## Arguments to plot1d functions

Every plot1d function must take arbitrarily many arguments, accepting at least the following set:

• x, a vector of values for a single variate
• horizontal, which is TRUE if the 1d plot is to be horizontal,
• plotAsp, the aspect ratio of the plot (i.e. the smaller/larger side ration in [0,1])
• turn, the single character turn out from the current plot
• plotID, a list containing information on the identification of the plot.

If the plot1d function is from the package grid, then it should also expect to receive a vp or viewport argument; if it is from the package loon, it might also receive a parent argument. These arguments should be familiar to users of either package.

For a plot1d function, the plotID consists of

• group, the number of the group in which this 1d plot is placed,
• number.within.group, the within group index of this variate,
• index , the index of the variate among all variates in the data set(s)
• label, the variate label, and
• plotNo, the number of the 1d plot being displayed (i.e its position in order among only those which are 1d).

All other functions in the ellipsis, ..., are passed on to the drawing functions.

## Arguments to plot2d functions

Every plot2d function must take arbitrarily many arguments, accepting at least the following set:

• x, a vector of values for the horizontal variate
• y, a vector of values for the vertical variate
• turn, the single character turn out from the current plot
• plotID, a list containing information on the identification of the plot.

If the plot2d function is from the package grid, then it should also expect to receive a vp or viewport argument; if it is from the package loon, it might also receive a parent argument. These arguments should be familiar to users of either package.

For a plot2d function, the plotID consists of

• group, the number of the group in which this 2d plot is placed,
• number.within.group, the within group index for these variates (same index),
• index , the indices of the variates among all variates in the data set(s)
• label, the labels of the variates, and
• plotNo, the number of the 2d the plot being displayed (i.e its position in order among only those which are 2d).

All other functions in the ellipsis, ..., are passed on to the drawing functions.