I'd like to ggplot 3 pairs (factor x0) of 2 conditions (factor cond0), using boxplots with specific quantile limits.
The 2 problems are:
There are 3 groups in condition A (red), but only 2 groups in condition B (blue). Since group 1B is missing, the boxplot of group 1A occupies its space on the graph (twice as large). I would like its width to be as narrow as the others, and the space of the missing group 1B to be maintained even if it is empty.
Since the group 3B has only one value (and therefore no outlier), the outliers of group 3A are located in the middle of the pair instead of being aligned with boxplot 3A.
Would there be a solution to these problems?
Thanks for help
library(dplyr)
library(ggplot2)
# dataframe
x1 <- rep(1:3, each=60)
y1 <- rnorm(180, rep(c(20,35,50), each=60), 10)
cond1 <- rep("A", each=180)
dat1 <- data.frame(x1, y1, cond1)
dat1$x1 <- as.factor(dat1$x1)
dat1$cond1 <- as.factor(dat1$cond1)
dat1 <- dat1 %>% rename(x0 = x1, y0 = y1, cond0 = cond1)
x2 <- rep(2:3, each = 179, len = 180) ; y2
y2 <- rnorm(180, rep(c(30,60), each=90), 7) ; x2
cond2 <- rep("B", each=180)
dat2 <- data.frame(x2, y2, cond2)
dat2$x2 <- as.factor(dat2$x2)
dat2$cond2 <- as.factor(dat2$cond2)
dat2 <- dat2 %>% rename(x0 = x2, y0 = y2, cond0 = cond2)
dat <- rbind(dat1,dat2)
# define boxplots limits
dat_boxlim <- function(x) {
r <- quantile(x, probs = c(0.1, 0.4, 0.5, 0.8, 0.9))
names(r) <- c("ymin", "lower", "middle", "upper", "ymax")
r
}
# define outliers limits
dat_boxout <- function(x) {
subset(x, x < quantile(x, 0.1) | x > quantile(x, 0.9))
}
# figure
ggplot(dat, aes(x0, y0, group=interaction(cond0, x0), fill = cond0))+
stat_summary(fun.data = dat_boxlim, geom = "boxplot", position = position_dodge(0.7), width = 0.5, show.legend = TRUE) +
stat_summary(fun = dat_boxout, geom = "point", size=2, position = position_dodge(0.7), show.legend = FALSE)
The first problem is solved by using "position = position_dodge2(preserve = "single")" in stat_summary geom="boxplot".
The second problem is solved using the new formula below.
The whole appropriate code is:
library(dplyr)
library(ggplot2)
# dataframe
x1 <- rep(1:3, each=60)
y1 <- rnorm(180, rep(c(20,35,50), each=60), 10)
cond1 <- rep("A", each=180)
dat1 <- data.frame(x1, y1, cond1)
dat1$x1 <- as.factor(dat1$x1)
dat1$cond1 <- as.factor(dat1$cond1)
dat1 <- dat1 %>% rename(x0 = x1, y0 = y1, cond0 = cond1)
x2 <- rep(2:3, each = 179, len = 180) ; y2
y2 <- rnorm(180, rep(c(30,60), each=90), 7) ; x2
cond2 <- rep("B", each=180)
dat2 <- data.frame(x2, y2, cond2)
dat2$x2 <- as.factor(dat2$x2)
dat2$cond2 <- as.factor(dat2$cond2)
dat2 <- dat2 %>% rename(x0 = x2, y0 = y2, cond0 = cond2)
dat <- rbind(dat1,dat2)
# define boxplots limits
dat_boxlim <- function(x) {
r <- quantile(x, probs = c(0.1, 0.4, 0.5, 0.8, 0.9))
names(r) <- c("ymin", "lower", "middle", "upper", "ymax")
r
}
# define outliers limits
dat_boxout <- function(x) {
if (length(x) > 1) { # or other length if needed (e.g. > 7)
return(subset(x, x < quantile(x, 0.1) | x > quantile(x, 0.9))) # only for low outliers
} else {
return(NA)
}
}
# figure
ggplot(dat, aes(x0, y0, group=interaction(cond0, x0), fill = cond0))+
stat_summary(fun.data = dat_boxlim, geom = "boxplot", position = position_dodge2(preserve = "single", 0.7, padding = 0.1), width = 0.5, show.legend = TRUE) +
stat_summary(fun = dat_boxout, geom = "point", size=2, position = position_dodge(preserve = "total", 0.5), show.legend = FALSE)
Related
I have a set of points such as this, and I am trying to group or cluster them into four groups of equal size based on their distance to their mean.
df <- data.frame(x = rnorm(50, 0, 1),
y = rnorm(50, -0, 0.7))
ggplot(data = df, aes(x = x, y = y)) +
geom_point() +
geom_abline(slope = 0.7, intercept = 0) +
geom_abline(slope = -0.7, intercept = 0)
In this case, I would want each point to be assigned a group based on how far they are with respect to the average (here it would be (0,0)) and whether this distance trends more up, down, left or right. Thanks for the help.
df <- data.frame(x = rnorm(50, 0, 1),
y = rnorm(50, -0, 0.7),
center = rep(0,50)) # not really needed
df$v1 <- -df$x^2 # vector one 0(center) - df$x2
df$v2 <- -df$y^2 # vector two 0(center) - df$y2
df$dist = df$v1 + df$v2
df$len = sqrt(abs(df$dist)) #lenght of vector betwen center and point
df$group = cut(df$len, 10,labels = paste("Level",LETTERS[1:10])) # grouping into 10 bins
df<-df %>% mutate(posit= case_when(x < mean(x) & y < mean(y) ~ "A",
x > mean(x) & y > mean(y) ~ "B",
x <mean(x) & y >mean(y) ~ "C",
x >mean(x) & y < mean(y) ~ "D"))
ggplot(data = df ) +
geom_point(aes(x = x, y = y,col= group, shape= posit, size=2)) +
geom_vline(xintercept =mean(df$x))+
geom_abline(slope = -0.0, intercept =mean(df$y))
I frequently use kernel density plots to illustrate distributions. These are easy and fast to create in R like so:
set.seed(1)
draws <- rnorm(100)^2
dens <- density(draws)
plot(dens)
#or in one line like this: plot(density(rnorm(100)^2))
Which gives me this nice little PDF:
I'd like to shade the area under the PDF from the 75th to 95th percentiles. It's easy to calculate the points using the quantile function:
q75 <- quantile(draws, .75)
q95 <- quantile(draws, .95)
But how do I shade the the area between q75 and q95?
With the polygon() function, see its help page and I believe we had similar questions here too.
You need to find the index of the quantile values to get the actual (x,y) pairs.
Edit: Here you go:
x1 <- min(which(dens$x >= q75))
x2 <- max(which(dens$x < q95))
with(dens, polygon(x=c(x[c(x1,x1:x2,x2)]), y= c(0, y[x1:x2], 0), col="gray"))
Output (added by JDL)
Another solution:
dd <- with(dens,data.frame(x,y))
library(ggplot2)
qplot(x,y,data=dd,geom="line")+
geom_ribbon(data=subset(dd,x>q75 & x<q95),aes(ymax=y),ymin=0,
fill="red",colour=NA,alpha=0.5)
Result:
An expanded solution:
If you wanted to shade both tails (copy & paste of Dirk's code) and use known x values:
set.seed(1)
draws <- rnorm(100)^2
dens <- density(draws)
plot(dens)
q2 <- 2
q65 <- 6.5
qn08 <- -0.8
qn02 <- -0.2
x1 <- min(which(dens$x >= q2))
x2 <- max(which(dens$x < q65))
x3 <- min(which(dens$x >= qn08))
x4 <- max(which(dens$x < qn02))
with(dens, polygon(x=c(x[c(x1,x1:x2,x2)]), y= c(0, y[x1:x2], 0), col="gray"))
with(dens, polygon(x=c(x[c(x3,x3:x4,x4)]), y= c(0, y[x3:x4], 0), col="gray"))
Result:
This question needs a lattice answer. Here's a very basic one, simply adapting the method employed by Dirk and others:
#Set up the data
set.seed(1)
draws <- rnorm(100)^2
dens <- density(draws)
#Put in a simple data frame
d <- data.frame(x = dens$x, y = dens$y)
#Define a custom panel function;
# Options like color don't need to be hard coded
shadePanel <- function(x,y,shadeLims){
panel.lines(x,y)
m1 <- min(which(x >= shadeLims[1]))
m2 <- max(which(x <= shadeLims[2]))
tmp <- data.frame(x1 = x[c(m1,m1:m2,m2)], y1 = c(0,y[m1:m2],0))
panel.polygon(tmp$x1,tmp$y1,col = "blue")
}
#Plot
xyplot(y~x,data = d, panel = shadePanel, shadeLims = c(1,3))
Here's another ggplot2 variant based on a function that approximates the kernel density at the original data values:
approxdens <- function(x) {
dens <- density(x)
f <- with(dens, approxfun(x, y))
f(x)
}
Using the original data (rather than producing a new data frame with the density estimate's x and y values) has the benefit of also working in faceted plots where the quantile values depend on the variable by which the data is being grouped:
Code used
library(tidyverse)
library(RColorBrewer)
# dummy data
set.seed(1)
n <- 1e2
dt <- tibble(value = rnorm(n)^2)
# function that approximates the density at the provided values
approxdens <- function(x) {
dens <- density(x)
f <- with(dens, approxfun(x, y))
f(x)
}
probs <- c(0.75, 0.95)
dt <- dt %>%
mutate(dy = approxdens(value), # calculate density
p = percent_rank(value), # percentile rank
pcat = as.factor(cut(p, breaks = probs, # percentile category based on probs
include.lowest = TRUE)))
ggplot(dt, aes(value, dy)) +
geom_ribbon(aes(ymin = 0, ymax = dy, fill = pcat)) +
geom_line() +
scale_fill_brewer(guide = "none") +
theme_bw()
# dummy data with 2 groups
dt2 <- tibble(category = c(rep("A", n), rep("B", n)),
value = c(rnorm(n)^2, rnorm(n, mean = 2)))
dt2 <- dt2 %>%
group_by(category) %>%
mutate(dy = approxdens(value),
p = percent_rank(value),
pcat = as.factor(cut(p, breaks = probs,
include.lowest = TRUE)))
# faceted plot
ggplot(dt2, aes(value, dy)) +
geom_ribbon(aes(ymin = 0, ymax = dy, fill = pcat)) +
geom_line() +
facet_wrap(~ category, nrow = 2, scales = "fixed") +
scale_fill_brewer(guide = "none") +
theme_bw()
Created on 2018-07-13 by the reprex package (v0.2.0).
How can I get ggplot to produce something similar like
library(ggplot2)
library(reshape2)
library(ecp)
synthetic_control.data <- read.table("/path/synthetic_control.data.txt", quote="\"", comment.char="")
n <- 2
s <- sample(1:100, n)
idx <- c(s, 100+s, 200+s, 300+s, 400+s, 500+s)
sample2 <- synthetic_control.data[idx,]
df = as.data.frame(t(as.matrix(sample2)))
#calculate the change points
changeP <- e.divisive(as.matrix(df[1]), k=8, R = 400, alpha = 2, min.size = 3)
changeP = changeP$estimates
changeP = changeP[-c(1,length(changeP))]
changePoints = data.frame(changeP,variable=colnames(df)[1])
for(series in 2:ncol(df)){
changeP <- e.divisive(as.matrix(df[series]), k=8, R = 400, alpha = 2, min.size = 3)
changeP = changeP$estimates
changeP = changeP[-c(1,length(changeP))]
changePoints = rbind(changePoints, data.frame(changeP,variable=colnames(df)[2]))
}
this is the interesting part about the plot:
df$id = 1:nrow(df)
dfMelt <- reshape2::melt(df, id.vars = "id")
p = ggplot(dfMelt,aes(x=id,y=value))+geom_line(color = "steelblue")+ facet_grid(variable ~ ., scales = 'free_y')
p + geom_vline(aes(xintercept=changeP), data=changePoints, linetype='dashed')
So far my result is: https://www.dropbox.com/s/mysadkruo946oox/changePoint.pdf which means that there is something wrong with my array passed to the geom_vlines.
Could you point me in the right direction why I only get vlines in the first 2 plots?
This is the solution:
library(ggplot2)
library(reshape2)
library(ecp)
synthetic_control.data <- read.table("/Users/geoHeil/Dropbox/6.Semester/BachelorThesis/rResearch/data/synthetic_control.data.txt", quote="\"", comment.char="")
n <- 2
s <- sample(1:100, n)
idx <- c(s, 100+s, 200+s, 300+s, 400+s, 500+s)
sample2 <- synthetic_control.data[idx,]
df = as.data.frame(t(as.matrix(sample2)))
#calculate the change points
changeP <- e.divisive(as.matrix(df[1]), k=8, R = 400, alpha = 2, min.size = 3)
changeP = changeP$estimates
changeP = changeP[-c(1,length(changeP))]
changePoints = data.frame(changeP,variable=colnames(df)[1])
for(series in 2:ncol(df)){
changeP <- e.divisive(as.matrix(df[series]), k=8, R = 400, alpha = 2, min.size = 3)
changeP = changeP$estimates
changeP = changeP[-c(1,length(changeP))]
changePoints = rbind(changePoints, data.frame(changeP,variable=colnames(df)[series]))
}
# plot
df$id = 1:nrow(df)
dfMelt <- reshape2::melt(df, id.vars = "id")
p = ggplot(dfMelt,aes(x=id,y=value))+geom_line(color = "steelblue")+ facet_grid(variable ~ ., scales = 'free_y')
p + geom_vline(aes(xintercept=changeP), data=changePoints, linetype='dashed', colour='darkgreen')
I'm trying to use ggplot2 / geom_boxplot to produce a boxplot where the whiskers are defined as the 5 and 95th percentile instead of 0.25 - 1.5 IQR / 0.75 + IQR and outliers from those new whiskers are plotted as usual. I can see that the geom_boxplot aesthetics include ymax / ymin, but it's not clear to me how I put values in here. It seems like:
stat_quantile(quantiles = c(0.05, 0.25, 0.5, 0.75, 0.95))
should be able to help, but I don't know how to relate the results of this stat to set the appropriate geom_boxplot() aesthetics:
geom_boxplot(aes(ymin, lower, middle, upper, ymax))
I've seen other posts where people mention essentially building a boxplot-like object manually, but I'd rather keep the whole boxplot gestalt intact, just revising the meaning of two of the variables being drawn.
geom_boxplot with stat_summary can do it:
# define the summary function
f <- function(x) {
r <- quantile(x, probs = c(0.05, 0.25, 0.5, 0.75, 0.95))
names(r) <- c("ymin", "lower", "middle", "upper", "ymax")
r
}
# sample data
d <- data.frame(x=gl(2,50), y=rnorm(100))
# do it
ggplot(d, aes(x, y)) + stat_summary(fun.data = f, geom="boxplot")
# example with outliers
# define outlier as you want
o <- function(x) {
subset(x, x < quantile(x)[2] | quantile(x)[4] < x)
}
# do it
ggplot(d, aes(x, y)) +
stat_summary(fun.data=f, geom="boxplot") +
stat_summary(fun.y = o, geom="point")
It is now possible to specify the whiskers endpoints in ggplot2_2.1.0. Copying from the examples in ?geom_boxplot:
# It's possible to draw a boxplot with your own computations if you
# use stat = "identity":
y <- rnorm(100)
df <- data.frame(
x = 1,
y0 = min(y),
y25 = quantile(y, 0.25),
y50 = median(y),
y75 = quantile(y, 0.75),
y100 = max(y)
)
ggplot(df, aes(x)) +
geom_boxplot(
aes(ymin = y0, lower = y25, middle = y50, upper = y75, ymax = y100),
stat = "identity"
)
Building on #konvas's answer, beginning in ggplot2.0.x, you can extend ggplot using the ggproto system and define your own stat.
By copying the ggplot2 stat_boxplot code and making a few edits, you can quickly define a new stat (stat_boxplot_custom) that takes the percentiles you want to use as an argument (qs) instead of the coef argument that stat_boxplot uses. The new stat is defined here:
# modified from https://github.com/tidyverse/ggplot2/blob/master/R/stat-boxplot.r
library(ggplot2)
stat_boxplot_custom <- function(mapping = NULL, data = NULL,
geom = "boxplot", position = "dodge",
...,
qs = c(.05, .25, 0.5, 0.75, 0.95),
na.rm = FALSE,
show.legend = NA,
inherit.aes = TRUE) {
layer(
data = data,
mapping = mapping,
stat = StatBoxplotCustom,
geom = geom,
position = position,
show.legend = show.legend,
inherit.aes = inherit.aes,
params = list(
na.rm = na.rm,
qs = qs,
...
)
)
}
Then, the layer function is defined. Note that b/c I copied directly from stat_boxplot, you have to access a few internal ggplot2 functions using :::. This includes a lot of stuff copied directly over from StatBoxplot, but the key area is in computing the stats directly from the qs argument: stats <- as.numeric(stats::quantile(data$y, qs)) inside of the compute_group function.
StatBoxplotCustom <- ggproto("StatBoxplotCustom", Stat,
required_aes = c("x", "y"),
non_missing_aes = "weight",
setup_params = function(data, params) {
params$width <- ggplot2:::"%||%"(
params$width, (resolution(data$x) * 0.75)
)
if (is.double(data$x) && !ggplot2:::has_groups(data) && any(data$x != data$x[1L])) {
warning(
"Continuous x aesthetic -- did you forget aes(group=...)?",
call. = FALSE
)
}
params
},
compute_group = function(data, scales, width = NULL, na.rm = FALSE, qs = c(.05, .25, 0.5, 0.75, 0.95)) {
if (!is.null(data$weight)) {
mod <- quantreg::rq(y ~ 1, weights = weight, data = data, tau = qs)
stats <- as.numeric(stats::coef(mod))
} else {
stats <- as.numeric(stats::quantile(data$y, qs))
}
names(stats) <- c("ymin", "lower", "middle", "upper", "ymax")
iqr <- diff(stats[c(2, 4)])
outliers <- (data$y < stats[1]) | (data$y > stats[5])
if (length(unique(data$x)) > 1)
width <- diff(range(data$x)) * 0.9
df <- as.data.frame(as.list(stats))
df$outliers <- list(data$y[outliers])
if (is.null(data$weight)) {
n <- sum(!is.na(data$y))
} else {
# Sum up weights for non-NA positions of y and weight
n <- sum(data$weight[!is.na(data$y) & !is.na(data$weight)])
}
df$notchupper <- df$middle + 1.58 * iqr / sqrt(n)
df$notchlower <- df$middle - 1.58 * iqr / sqrt(n)
df$x <- if (is.factor(data$x)) data$x[1] else mean(range(data$x))
df$width <- width
df$relvarwidth <- sqrt(n)
df
}
)
There is also a gist here, containing this code.
Then, stat_boxplot_custom can be called just like stat_boxplot:
library(ggplot2)
y <- rnorm(100)
df <- data.frame(x = 1, y = y)
# whiskers extend to 5/95th percentiles by default
ggplot(df, aes(x = x, y = y)) +
stat_boxplot_custom()
# or extend the whiskers to min/max
ggplot(df, aes(x = x, y = y)) +
stat_boxplot_custom(qs = c(0, 0.25, 0.5, 0.75, 1))
I am trying to output multiple density plot from a function, by dividing the dataframe into pieces such that separate density for each level of a factor for corresponding yvar.
set.seed(1234)
Aa = c(rnorm(40000, 50, 10))
Bb = c(rnorm(4000, 70, 10))
Cc = c(rnorm(400, 75, 10))
Dd = c(rnorm(40, 80, 10))
yvar = c(Aa, Bb, Cc, Dd)
gen <- c(rep("Aa", length(Aa)),rep("Bb", length(Bb)), rep("Cc", length(Cc)),
rep("Dd", length(Dd)))
mydf <- data.frame(gen, yvar)
minyvar <- min(yvar)
maxyvar <- max(yvar)
par(mfrow = c(length(levels(mydf$gen)),1))
plotdensity <- function (xf, minyvar, maxyvar){
plot(density(xf), xlim=c(minyvar, maxyvar), main = paste (names(xf),
"distribution", sep = ""))
dens <- density(xf)
x1 <- min(which(dens$x >= quantile(xf, .80)))
x2 <- max(which(dens$x < max(dens$x)))
with(dens, polygon(x=c(x[c(x1,x1:x2,x2)]), y= c(0, y[x1:x2], 0), col="blu4"))
abline(v= mean(xf), col = "black", lty = 1, lwd =2)
}
require(plyr)
ddply(mydf, .(mydf$gen), plotdensity, yvar, minyvar, maxyvar)
Error in .fun(piece, ...) : unused argument(s) (111.544494112914)
My specific expectation are each plot is named by name of level for example Aa, Bb, Cc, Dd
Arrangement of the graphs see the parameter set, so that we compare density changes and means. compact - Low space between the graphs.
Help appreciated.
Edits:
The following graphs are individually produced, although I want to develop a function that can be applicable to x level for a factor.
I see that #Andrie just beat me to most of this. I'm still going to post my answer, since filling only certain quantiles of the distribution requires a slightly different approach.
set.seed(1234)
Aa = c(rnorm(40000, 50, 10))
Bb = c(rnorm(4000, 70, 10))
Cc = c(rnorm(400, 75, 10))
Dd = c(rnorm(40, 80, 10))
yvar = c(Aa, Bb, Cc, Dd)
gen <- c(rep("Aa", length(Aa)),rep("Bb", length(Bb)), rep("Cc", length(Cc)),
rep("Dd", length(Dd)))
mydf <- data.frame(grp = gen,x = c(Aa,Bb,Cc,Dd))
#Calculate the densities and an indicator for the desire quantile
# for later use in subsetting
mydf <- ddply(mydf,.(grp),.fun = function(x){
tmp <- density(x$x)
x1 <- tmp$x
y1 <- tmp$y
q80 <- x1 >= quantile(x$x,0.8)
data.frame(x=x1,y=y1,q80=q80)
})
#Separate data frame for the means
mydfMean <- ddply(mydf,.(grp),summarise,mn = mean(x))
ggplot(mydf,aes(x = x)) +
facet_wrap(~grp) +
geom_line(aes(y = y)) +
geom_ribbon(data = subset(mydf,q80),aes(ymax = y),ymin = 0, fill = "black") +
geom_vline(data = mydfMean,aes(xintercept = mn),colour = "black")
Here is a way of doing it in ggplot:
set.seed(1234)
mydf <- rbind(
data.frame(gen="Aa", yvar= rnorm(40000, 50, 10)),
data.frame(gen="Bb", yvar=rnorm(4000, 70, 10)),
data.frame(gen="Cc", yvar=rnorm(400, 75, 10)),
data.frame(gen="Dd", yvar=rnorm(40, 80, 10))
)
labels <- ddply(mydf, .(gen), nrow)
means <- ddply(mydf, .(gen), summarize, mean=mean(yvar))
ggplot(mydf, aes(x=yvar)) +
stat_density(fill="blue") +
facet_grid(gen~.) +
theme_bw() +
geom_vline(data=means, aes(xintercept=mean), colour="red") +
geom_text(data=labels, aes(label=paste("n =", V1)), x=5, y=0,
hjust=0, vjust=0) +
opts(title="Distribution")
With sincere thanks to joran and Andrie, the following is just compilation of my favorite from above two posts, just some of readers might want to see.
require(ggplot2)
set.seed(1234)
Aa = c(rnorm(40000, 50, 10))
Bb = c(rnorm(4000, 70, 10))
Cc = c(rnorm(400, 75, 10))
Dd = c(rnorm(40, 80, 10))
yvar = c(Aa, Bb, Cc, Dd)
gen <- c(rep("Aa", length(Aa)),rep("Bb", length(Bb)), rep("Cc", length(Cc)),
rep("Dd", length(Dd)))
mydf <- data.frame(grp = gen,x = c(Aa,Bb,Cc,Dd))
mydf1 <- mydf
#Calculate the densities and an indicator for the desire quantile
# for later use in subsetting
mydf <- ddply(mydf,.(grp),.fun = function(x){
tmp <- density(x$x)
x1 <- tmp$x
y1 <- tmp$y
q80 <- x1 >= quantile(x$x,0.8)
data.frame(x=x1,y=y1,q80=q80)
})
#Separate data frame for the means
mydfMean <- ddply(mydf,.(grp),summarise,mn = mean(x))
labels <- ddply(mydf1, .(grp), nrow)
ggplot(mydf,aes(x = x)) +
facet_grid(grp~.) +
geom_line(aes(y = y)) +
geom_ribbon(data = subset(mydf,q80),aes(ymax = y),ymin = 0,
fill = "black") +
geom_vline(data = mydfMean,aes(xintercept = mn),
colour = "black") + geom_text(data=labels,
aes(label=paste("n =", labels$V1)), x=5, y=0,
hjust=0, vjust=0) +
opts(title="Distribution") + theme_bw()