regression

# Visually weighted/ Watercolor Plots, new variants: Please vote!

Update Oct-23: Added a new parameter

to the function. Now multiple groups can be plotted in a single plot (see example in my comment)

As a follow-up on my R implementation of Solomon’s watercolor plots, I made some improvements to the function. I fine-tuned the graphical parameters (the median smoother line now diminishes faster with increasing CIs, and the shaded watercolors look more pretty). Furthermore, the function is faster and has more features:

• You can define any standard regression function for the bootstrap procedure.
• vwReg(y ~ x, df, method=lm)
• vwReg(y ~ x + I(x^2), df, method=lm)
• Provide parameters for the fitting function.
• You can make the smoother’s span larger. Then it takes more points into account when doing the local fitting. Per default, the smoother fits a polynomial of degree two – that means as you increase span you will approach the overall quadratic fit: vwReg(y ~ x, df, span=2)
• You can also make the smoother’s span smaller, then it takes less points for local fitting. If it is too small, it will overfit and approach each single data point. The default span (.75) seemed to be the best choice for me for a variety of data sets: vwReg(y ~ x, df, span=0.5)
• Use a robust M-estimator for the smoother; see ?loess for details: vwReg(y ~ x, df, family=”symmetric”)
• Provide your own color scheme (or, for example, a black-and-white scheme). Examples see pictures below.
• Quantize the color ramp, so that regions for 1, 2, and 3 SD have the same color (an idea proposed by John Mashey).

At the end of this post is the source code for the R function.

## Some picture – please vote!

Here are some variants of the watercolor plots – at the end, you can vote for your favorite (or write something into the comments). I am still fine-tuning the default parameters, and I am interested in your opinions what would be the best default.

Plot 1: The current default

Plot 2: Using an M-estimator for bootstrap smoothers. Usually you get wider confidence intervalls.

Plot 3:Increasing the span of the smoothers

Plot 4: Decreasing the span of the smoothers

Plot 5: Changing the color scheme, using a predefined ColorBrewer palette. You can see all available palettes by using this command: library(RColorBrewer); display.brewer.all()

Plot 6: Using a custom-made palette

Plot 7: Using a custom-made palette; with the parameter bias you can shift the color ramp to the “higher” colors:

Plot 8: A black and white version of the plot

Plot 9: The anti-Tufte-plot: Using as much ink as possible by reversing black and white (a.k.a. “the Milky-Way Plot“)

Plot 10: The Northern Light Plot/ fMRI plot. This plotting technique already has been used by a suspicious company (called IRET – never heard of that). I hurried to publish the R code under a FreeBSD license before they can patent it! Feel free to use, share, or change the code for whatever purpose you need. Isn’t that beautiful?

Plot 11: The 1-2-3-SD plot. You can use your own color schemes as well, e.g.: vwReg(y~x, df, bw=TRUE, quantize=”SD”)

Any comments or ideas? Or just a vote? If you produce some nice plots with your data, you can send it to me, and I will post a gallery of the most impressive “data art”!

Cheers,

Felix

Which water color plot do you like most?

View Results

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# Version history:
# 0.1: original code
# 0.1.1: changed license to FreeBSD; re-established compability to ggplot2 (new version 0.9.2)

## Visually weighted regression / Watercolor plots
## Idea: Solomon Hsiang, with additional ideas from many blog commenters

# B = number bootstrapped smoothers
# shade.alpha: should the CI shading fade out at the edges? (by reducing alpha; 0 = no alpha decrease, 0.1 = medium alpha decrease, 0.5 = strong alpha decrease)
# spag: plot spaghetti lines?
# spag.color: color of spaghetti lines
# mweight: should the median smoother be visually weighted?
# show.lm: should the linear regresison line be plotted?
# show.CI: should the 95% CI limits be plotted?
# show.median: should the median smoother be plotted?
# median.col: color of the median smoother
# shape: shape of points
# method: the fitting function for the spaghettis; default: loess
# bw = TRUE: define a default b&w-palette
# slices: number of slices in x and y direction for the shaded region. Higher numbers make a smoother plot, but takes longer to draw. I wouldn'T go beyond 500
# palette: provide a custom color palette for the watercolors
# ylim: restrict range of the watercoloring
# quantize: either "continuous", or "SD". In the latter case, we get three color regions for 1, 2, and 3 SD (an idea of John Mashey)
# add: if add == FALSE, a new ggplot is returned. If add == TRUE, only the elements are returned, which can be added to an existing ggplot (with the '+' operator)
# ...: further parameters passed to the fitting function, in the case of loess, for example, "span = .9", or "family = 'symmetric'"
vwReg <- function(formula, data, title="", B=1000, shade=TRUE, shade.alpha=.1, spag=FALSE, spag.color="darkblue", mweight=TRUE, show.lm=FALSE, show.median = TRUE, median.col = "white", shape = 21, show.CI=FALSE, method=loess, bw=FALSE, slices=200, palette=colorRampPalette(c("#FFEDA0", "#DD0000"), bias=2)(20), ylim=NULL, quantize = "continuous",  add=FALSE, ...) {
IV <- all.vars(formula)[2]
DV <- all.vars(formula)[1]
data <- na.omit(data[order(data[, IV]), c(IV, DV)])

if (bw == TRUE) {
palette <- colorRampPalette(c("#EEEEEE", "#999999", "#333333"), bias=2)(20)
}

print("Computing boostrapped smoothers ...")
newx <- data.frame(seq(min(data[, IV]), max(data[, IV]), length=slices))
colnames(newx) <- IV
l0.boot <- matrix(NA, nrow=nrow(newx), ncol=B)

l0 <- method(formula, data)
for (i in 1:B) {
data2 <- data[sample(nrow(data), replace=TRUE), ]
data2 <- data2[order(data2[, IV]), ]
if (class(l0)=="loess") {
m1 <- method(formula, data2, control = loess.control(surface = "i", statistics="a", trace.hat="a"), ...)
} else {
m1 <- method(formula, data2, ...)
}
l0.boot[, i] <- predict(m1, newdata=newx)
}

# compute median and CI limits of bootstrap
library(plyr)
library(reshape2)
CI.boot <- adply(l0.boot, 1, function(x) quantile(x, prob=c(.025, .5, .975, pnorm(c(-3, -2, -1, 0, 1, 2, 3))), na.rm=TRUE))[, -1]
colnames(CI.boot)[1:10] <- c("LL", "M", "UL", paste0("SD", 1:7))
CI.boot\$x <- newx[, 1]
CI.boot\$width <- CI.boot\$UL - CI.boot\$LL

# scale the CI width to the range 0 to 1 and flip it (bigger numbers = narrower CI)
CI.boot\$w2 <- (CI.boot\$width - min(CI.boot\$width))
CI.boot\$w3 <- 1-(CI.boot\$w2/max(CI.boot\$w2))

# convert bootstrapped spaghettis to long format
b2 <- melt(l0.boot)
b2\$x <- newx[,1]
colnames(b2) <- c("index", "B", "value", "x")

library(ggplot2)
library(RColorBrewer)

# Construct ggplot
# All plot elements are constructed as a list, so they can be added to an existing ggplot

# if add == FALSE: provide the basic ggplot object
p0 <- ggplot(data, aes_string(x=IV, y=DV)) + theme_bw()

# initialize elements with NULL (if they are defined, they are overwritten with something meaningful)
gg.tiles <- gg.poly <- gg.spag <- gg.median <- gg.CI1 <- gg.CI2 <- gg.lm <- gg.points <- gg.title <- NULL

quantize <- match.arg(quantize, c("continuous", "SD"))
if (quantize == "continuous") {
print("Computing density estimates for each vertical cut ...")
flush.console()

if (is.null(ylim)) {
min_value <- min(min(l0.boot, na.rm=TRUE), min(data[, DV], na.rm=TRUE))
max_value <- max(max(l0.boot, na.rm=TRUE), max(data[, DV], na.rm=TRUE))
ylim <- c(min_value, max_value)
}

# vertical cross-sectional density estimate
d2 <- ddply(b2[, c("x", "value")], .(x), function(df) {
res <- data.frame(density(df\$value, na.rm=TRUE, n=slices, from=ylim[1], to=ylim[2])[c("x", "y")])
#res <- data.frame(density(df\$value, na.rm=TRUE, n=slices)[c("x", "y")])
colnames(res) <- c("y", "dens")
return(res)
}, .progress="text")

maxdens <- max(d2\$dens)
mindens <- min(d2\$dens)
d2\$dens.scaled <- (d2\$dens - mindens)/maxdens

## Tile approach
gg.tiles <-  list(geom_tile(data=d2, aes(x=x, y=y, fill=dens.scaled, alpha=alpha.factor)), scale_fill_gradientn("dens.scaled", colours=palette), scale_alpha_continuous(range=c(0.001, 1)))
}
if (quantize == "SD") {
## Polygon approach

SDs <- melt(CI.boot[, c("x", paste0("SD", 1:7))], id.vars="x")
count <- 0
d3 <- data.frame()
col <- c(1,2,3,3,2,1)
for (i in 1:6) {
seg1 <- SDs[SDs\$variable == paste0("SD", i), ]
seg2 <- SDs[SDs\$variable == paste0("SD", i+1), ]
seg <- rbind(seg1, seg2[nrow(seg2):1, ])
seg\$group <- count
seg\$col <- col[i]
count <- count + 1
d3 <- rbind(d3, seg)
}

gg.poly <-  list(geom_polygon(data=d3, aes(x=x, y=value, color=NULL, fill=col, group=group)), scale_fill_gradientn("dens.scaled", colours=palette, values=seq(-1, 3, 1)))
}
}

print("Build ggplot figure ...")
flush.console()

if (spag==TRUE) {
gg.spag <-  geom_path(data=b2, aes(x=x, y=value, group=B), size=0.7, alpha=10/B, color=spag.color)
}

if (show.median == TRUE) {
if (mweight == TRUE) {
gg.median <-  geom_path(data=CI.boot, aes(x=x, y=M, alpha=w3^3), size=.6, linejoin="mitre", color=median.col)
} else {
gg.median <-  geom_path(data=CI.boot, aes(x=x, y=M), size = 0.6, linejoin="mitre", color=median.col)
}
}

# Confidence limits
if (show.CI == TRUE) {
gg.CI1 <- geom_path(data=CI.boot, aes(x=x, y=UL), size=1, color="red")
gg.CI2 <- geom_path(data=CI.boot, aes(x=x, y=LL), size=1, color="red")
}

# plain linear regression line
if (show.lm==TRUE) {gg.lm <- geom_smooth(method="lm", color="darkgreen", se=FALSE)}

gg.points <- geom_point(data=data, aes_string(x=IV, y=DV), size=1, shape=shape, fill="white", color="black")

if (title != "") {
gg.title <- theme(title=title)
}

gg.elements <- list(gg.tiles, gg.poly, gg.spag, gg.median, gg.CI1, gg.CI2, gg.lm, gg.points, gg.title, theme(legend.position="none"))

return(p0 + gg.elements)
} else {
return(gg.elements)
}
}

# Visually weighted regression in R (à la Solomon Hsiang)

[Update 1: Sep 5, 2012: Explore the Magical Data Enhancer by IRES, using this visualization technique]

[Update 2: Sep 6, 2012: See new improved plots, and new R code!

Solomon Hsiang proposed an appealing method for visually displaying the uncertainty in regressions (see his blog [1][2], and also the discussions on the Statistical Modeling, Causal Inference, and Social Science Blog [1][2]).

I implemented the method in R (using ggplot2), and used an additional method of determining the shading (especially concerning Andrew Gelman’s comment that traditional statistical summaries (such as 95% intervals) give too much weight to the edges. In the following I will show how to produce plots like that:

I used following procedure:

1. Compute smoothers from 1000 bootstrap samples of the original sample (this results in a spaghetti plot)
2. Calculate a density estimate for each vertical cut through the bootstrapped smoothers. The area under the density curve always is 1, so the ink is constant for each y-slice.
3. Shade the figure according to these density estimates.

## Now let’s construct some plots!

The basic scatter plot:

No we show the bootstrapped smoothers (a “spaghetti plot”). Each spaghetti has a low alpha. That means that overlapping spaghettis produce a darker color and already give weight to highly populated regions.

Here is the shading according to the smoother’s density:

Now, we can overplot the median smoother estimate for each x value (the “median smoother”):

Or, a visually weighted smoother:

Finally, we can add the plain linear regression line (which obviously does not refelct the data points very well):

At the end of this post is the function that produces all of these plots. The function returns a ggplot object, so you can modify it afterwards, e.g.:

```vwReg(y~x, df, shade=FALSE, spag=TRUE) + xlab("Implicit power motive") + ylab("Corrugator activity during preparation")[/cc]

Here are two plots with actual data I am working on:
The correlation of both variables is .22 (p = .003).
A) As a heat map (note: the vertical breaks at the left and right end occur due to single data points that get either sampled or not during the bootstrap):

B) As a spaghetti plot:

Finally, here's the code (sometimes the code box is collapsed - click the arrow on the top right of the box to open it). Comments and additions are welcome.
[Update: I removed the code, methods/html/as.html">as an updated version has been published <a title="Visually weighted/ Watercolor Plots, new variants: Please vote!" href="https://www.nicebread.de/visually-weighted-watercolor-plots-new-variants-please-vote/">here</a> (see at the end of the post)]