Lexis Surfaces
Tim Riffe
7/14/2021
Introduction
This tutorial gives code for some basic diagnostic Lexis surfaces. We’ll use HMD data as an example. In general, we’ll want to start from a tidy data format with columns for year, age, and the variable of interest. I’ll also give a code snippet for transforming a data matrix into such a tidy format.
Example data
This is how I got the data, FYI, but you can load the data straight from this repository in the following code chunk.
library(HMDHFDplus)
library(readr)
MX <- readHMDweb("CZE","Mx_1x1",us,pw)
write_csv(MX, file = "MX.csv")You can read in straight from github using this:
library(tidyverse)## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ──## ✓ ggplot2 3.3.5 ✓ purrr 0.3.4
## ✓ tibble 3.1.2 ✓ dplyr 1.0.7
## ✓ tidyr 1.1.3 ✓ stringr 1.4.0
## ✓ readr 1.4.0 ✓ forcats 0.5.1## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()library(readr)
MX <- read_csv("https://raw.githubusercontent.com/timriffe/DemoTutorials/master/Data/MX.csv")##
## ── Column specification ────────────────────────────────────────────────────────
## cols(
## Year = col_double(),
## Age = col_double(),
## Female = col_double(),
## Male = col_double(),
## Total = col_double(),
## OpenInterval = col_logical()
## )Make a simple surface
Weĺl compose the Lexis plot in steps, taking care of details one at a time. At the end the same plot will be built in a single statement that would otherwise be harder to explain.
Here’s the basic setup that we’ll end up modifying some. We use geom_tile() to map fill color to the value on the surface we want to plot.
MX %>%
ggplot(aes(x = Year, y = Age, fill = Male)) +
geom_tile()
We’ll have to modify the plot quite a bit to get a signal out of it. Namely, these empirical \(m(x)\) values can get well over 1 in the highest ages, so they drown out the variation below. You could do log(Male) and that’d help some, but the breaks wouldn’t be good. One thing at a time. First we’ll be explicit about color choices.
Choose a color palette
Use a sequential palette for absolute or log quantities plotted on the surface. Note the name of the palette you want
library(colorspace)
hcl_palettes(type = "sequential",plot = TRUE)
Use a divergent palette for log rate ratios, or differences, usually it makes sense to choose a neutral color (ratio = 1 or difference = 0) that matches your background…
hcl_palettes(type = "diverging",plot = TRUE)
This palette system plays nicely together with ggplot2. scale_fill_continuous_sequential() applies the selected sequential palette.
MX %>%
ggplot(aes(x = Year, y = Age, fill = Male)) +
geom_tile() +
scale_fill_continuous_sequential(palette = "BurgYl")
And get a good color signal but bad breaks by logging:
MX %>%
ggplot(aes(x = Year, y = Age, fill = log(Male))) +
geom_tile() +
scale_fill_continuous_sequential(palette = "BurgYl")
Here the breaks apply to the logged values, and we don’t really have an intuitive sense of what \(e^{-2.5}\) is. So we can instead feed a palette to scale_gradientn(). We feed a vector of colors, picking out our palette using sequential_hcl(palette = "BurgYl",n=10), which gets the colors backwards, hence the %>% rev() statement. Finally, we tell it to pick breaks that play well with log base ten.
MX %>%
ggplot(aes(x = Year + .5,
y = Age + .5,
fill = Male)) +
geom_tile() +
scale_fill_gradientn(colors = sequential_hcl(palette = "BurgYl", n = 10) %>% rev(),
trans = "log10")
And again, making the scientific notation go away. This result is good enough for me. For something more custom you’d need to specify a manual vector of labels. Let’s not waste more time on it now.
library(scales)
MX %>%
ggplot(aes(x = Year + .5,
y = Age + .5,
fill = Male)) +
geom_tile() +
scale_fill_gradientn(colors = sequential_hcl(palette = "BurgYl",n=10) %>% rev(),
trans= "log10",
labels = comma)
Center Lexis cells
Now let’s enforce an equal aspect ration (one year is one age) on the surface, and manage a detail to ensure the squares are centered. To do this, we add \(1/2\) an interval to each year and age. Here it’s an easy half because we’re in single age data, but further thought might be require to center data in other bins.
yrs <- MX$Year %>% range()
MX %>%
ggplot(aes(x = Year + .5,
y = Age + .5,
z = Male,
fill = Male)) +
geom_tile() +
scale_fill_gradientn(colors = sequential_hcl(palette = "BurgYl", n = 10) %>% rev(),
trans= "log10",
labels = comma)+
coord_equal() +
xlim(yrs[1],yrs[2] + 1) +
ylim(0,111) 
Add contours
Adding contours can enhance pattern detection for this sort of gradual smooth surface. We’ll feed a slightly different vector of breaks to get more definition.
my_breaks = 10^seq(0,-7,by = -.5)
MX %>%
ggplot(aes(x = Year + .5,
y = Age + .5,
z = Male,
fill = Male)) +
geom_tile() +
geom_contour(breaks = my_breaks, color = gray(.8), alpha = 50, size = .5) +
scale_fill_gradientn(colors = sequential_hcl(palette = "BurgYl", n = 10) %>% rev(),
trans = "log10",
labels = comma)+
coord_equal() +
xlim(yrs[1],yrs[2] + 1) +
ylim(0,111) 
More details
Finally let’s get rid of the gray background, and clean up axis labels
MX %>%
ggplot(aes(x = Year + .5,
y = Age + .5,
z = Male,
fill = Male)) +
geom_tile() +
geom_contour(breaks = my_breaks, color = gray(.8), alpha = 50, size = .5) +
scale_fill_gradientn(colors = sequential_hcl(palette = "BurgYl", n = 10) %>% rev(),
trans = "log10",
labels = comma)+
coord_equal() +
xlim(yrs[1],yrs[2] + 1) +
ylim(0,111) +
xlab("Year") +
ylab("Age") +
theme_minimal()+
labs(fill="Mortality rate\n(log colors)")
Diverging Lexis plot
Say we want to make a plot of residuals, differences, or ratios (logged, say). The only thing that changes is picking out a divergent palette. Here I’ll calculate year-on-year within-age changes in \(ln(m(x))\).
# shift back before joining
MX2 <-
MX %>%
mutate(Year = Year - 1,
mx_previous = Male) %>%
select(Year, Age, mx_previous)
MX %>%
mutate(mx_now = Male) %>%
select(Year, Age, mx_now) %>%
left_join(MX2, by = c("Year","Age")) %>%
mutate(lrat = log(mx_now / mx_previous),
lrat = case_when(lrat < -.5 ~ -.5,
lrat > .5 ~ .5,
TRUE ~ lrat)) %>%
ggplot(aes(x = Year, y = Age, fill = lrat)) +
geom_tile() +
scale_fill_continuous_diverging("Blue-Red3")+
coord_equal()
Sex ratios are also a good Lexis surface diagnostic sometimes
MX %>%
mutate(lrat = log(Male / Female),
lrat = case_when(lrat < -1 ~ -1,
lrat > 1 ~ 1,
TRUE ~ lrat)) %>%
ggplot(aes(x = Year, y = Age, fill = lrat)) +
geom_tile() +
scale_fill_continuous_diverging("Blue-Red3") +
coord_equal()