---
title: "Duplicate and Missing Cases"
author: "lindbrook"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Duplicate and Missing Cases}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r, echo = FALSE, message = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
library(cholera)
library(HistData)
```
## Overview

John Snow's map of the 1854 cholera outbreak in London is a canonical example of data visualization:^[The map was originally published in Snow's 1855 book, "On The Mode Of Communication Of Cholera", and was reprinted as John Snow et. al., 1936. _Snow on Cholera: Being a Reprint of Two Papers_. New York: The Common Wealth Fund. You can also find the map online (a high resolution version is available at http://www.ph.ucla.edu/epi/snow/highressnowmap.html) and in many books, including Edward Tufte's 1997 "Visual Explanations: Images and Quantities, Evidence and Narrative".]

![](msu-snows-mapB.jpg)

In 1992, Rusty Dodson and Waldo Tobler digitized the map. While the original data are no longer available,^[http://www.ncgia.ucsb.edu/pubs/snow/snow.html] they are preserved in Michael Friendly's 'HistData' package. Those data are plotted below:

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}
street.list <- split(Snow.streets[, c("x", "y")], Snow.streets$street)
plot(Snow.deaths$x, Snow.deaths$y, pch = 20, cex = 0.5,
  xlim = range(Snow.streets$x), ylim = range(Snow.streets$y),
  xlab = "x", ylab = "y", asp = 1)
invisible(lapply(street.list, lines, lwd = 0.75))
points(Snow.pumps$x, Snow.pumps$y, pch = 17, col = "blue")
```

However, there are two apparent coding errors, which stem from three misplaced cases.

While the data record 578 bars, only 575 of them have a unique x-y coordinate.^[There is a lack of consensus about the actual number of cases represented in Snow's map. For what it's worth, I manually recounted the data on Snow's map and the result I got matches Dodson and Tobler's count.] Three pairs have identical coordinates: 1) 93 and 214; 2) 91 and 241; and 3) 209 and 429. Within the scheme of stacking bars to represent the fatality count, this cannot occur. All bars should have a unique x-y coordinate. For this reason, duplicates are likely to be coding errors.

```{r}
duplicates <- HistData::Snow.deaths[(duplicated(HistData::Snow.deaths[,
  c("x", "y")])), ]

duplicates.id <- lapply(duplicates$x, function(i) {
  HistData::Snow.deaths[HistData::Snow.deaths$x == i, "case"]
})

HistData::Snow.deaths[unlist(duplicates.id), ]
```
Fortunately, a careful comparison of Snow's map and the map generated by Dodson and Tobler's data reveals that there are three "missing" bars in the latter. An expedient "fix" would be to simply use the duplicate observations to fill in for the "missing" bars:

```{r}
fatalities <- HistData::Snow.deaths

fix <- data.frame(x = c(12.56974, 12.53617, 12.33145), y = c(11.51226, 11.58107, 14.80316))

fatalities[c(91, 93, 209), c("x", "y")] <- fix
```

This fixed data set is available as _fatalities_ in this package and as _Snow.deaths2_ in \code{HistData} (>= ver. 0.7-8). For those interested, an explanation of how I arrived at those values follows.

## Duplicate Cases: Cambridge Street

The three pairs of duplicates lie at two different locations on opposite sides of Cambridge Street:

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}
street.list <- split(Snow.streets[, c("x", "y")], Snow.streets$street)

# Cambridge Street #
street.name <- "Cambridge Street"
cam.data <- roads[roads$name == street.name, ]
cam.data <- cam.data[order(cam.data$x), ]
d1 <- cam.data[-nrow(cam.data), c("x", "y")]
d2 <- cam.data[-1, c("x", "y")]

intercept.slope <- lapply(seq_len(nrow(cam.data) - 1), function(i) {
  coef(lm(y ~ x, data = rbind(d1[i, ], d2[i, ])))
})

sel <- 2
cam.angle <- atan(intercept.slope[[sel]][2]) * 180L / pi
cam.x <- mean(cam.data[2:3, "x"])
cam.y <- intercept.slope[[sel]][1] + intercept.slope[[sel]][2] * cam.x

# Broad Street #
street.name <- "Broad Street"
broad.data <- roads[roads$name == street.name, ]
broad.list <- street.list[paste(unique(broad.data$street))]
broad.list <- lapply(broad.list, function(df) {
  df[order(df$x, decreasing = TRUE), ]
})

broad.pts.data <- do.call(rbind, broad.list)
broad.pts.data <- broad.pts.data[seq_len(nrow(broad.pts.data)) %% 2 != 0, ]

segment.ols <- lapply(broad.list, function(x) {
  coef(lm(y ~ x, data = x))
})

sel <- "216"
broad.angle <- atan(segment.ols$`216`["x"]) * 180 / pi
broad.x <- median(broad.pts.data[5:6, "x"])
broad.y <- segment.ols[[sel]][1] + segment.ols[[sel]][2] * broad.x

# Plot #
plot(Snow.deaths[, c("x", "y")], xlim = c(12, 13), ylim = c(10.9, 11.9),
     pch = NA, cex = 0.5, col = "blue", asp = 1)
invisible(lapply(street.list, lines, col = "lightgray"))
points(Snow.pumps[, c("x", "y")], col = "blue", pch = 2, cex = 1)
text(Snow.pumps[, c("x", "y")], col = "blue", labels = "Broad Street Well",
     pos = 4, cex = 0.85)
points(Snow.deaths[-unlist(duplicates.id), c("x", "y")])
title(main = "The Three Pairs of Duplicate Cases")

invisible(lapply(duplicates.id, function(x) {
  points(Snow.deaths[c(x[1], x[2]), c("x", "y")], pch = 1:0, cex = c(1, 1.75),
    col = c("black", "red"))
}))

id.data <- do.call(rbind, duplicates.id)
top.data <- Snow.deaths[id.data[, 1], c("x", "y")]
bottom.data <- Snow.deaths[id.data[, 2], c("x", "y")]
text(top.data, labels = rownames(top.data), pos = c(3, 2), cex = 0.9,
     col = "red")
text(bottom.data[1:2, ], labels = rownames(bottom.data)[1:2], pos = 1,
     cex = 0.9)
text(bottom.data[3, c("x", "y")] + c(0.03, -0.03),
     labels = rownames(bottom.data)[3], cex = 0.9)
legend(x = "bottomleft",
       legend = c("Well", "Duplicate", "Case"),
       col = c("blue", "red", "black"),
       pch = c(2, 0, 1),
       bg = "white",
       cex = 0.8,
       title = "Key")

text(cam.x, cam.y, "Cambridge St", srt = cam.angle)
text(broad.x, broad.y, "Broad St", srt = broad.angle)
```

## Missing Cases: Broad Street

40 Broad Street, which lies just southwest (below and left) of the Broad Street pump and which is the accepted home of the index case, is the location of two of the "missing" cases.

Snow's map shows four "bars" or cases:

![](broad.street.A.png)

But Dodson and Tobler's data show only two, cases 32 and 122. Furthermore, the two cases also appear to be farther apart than what we see in neighboring stacks:

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}

broad.40 <- c(32, 122)

plot(Snow.deaths[, c("x", "y")], xlim = c(12, 13), ylim = c(10.9, 11.9),
     pch = NA, cex = 0.5, col = "blue", asp = 1)
invisible(lapply(street.list, lines, col = "lightgray"))

points(Snow.deaths[-c(unlist(duplicates.id), broad.40), c("x", "y")])

points(Snow.pumps[, c("x", "y")], col = "blue", pch = 2, cex = 1)
text(Snow.pumps[, c("x", "y")], col = "blue", labels = "Broad Street Well",
     pos = 4, cex = 0.85)

points(Snow.deaths[broad.40, c("x", "y")], col = "#009E73")
text(Snow.deaths[broad.40, c("x", "y")], labels = Snow.deaths$case[broad.40],
     cex = 0.9, pos = 4, col = "#009E73")

title(main = "40 Broad Street:\nTwo (32 & 122) Rather than Four Cases")

invisible(lapply(duplicates.id, function(x) {
  points(Snow.deaths[x[1], c("x", "y")])
}))

legend(x = "bottomleft",
       legend = c("Well", "Case", "40 Broad St"),
       col = c("blue", "black", "#009E73"),
       pch = c(2, 1, 1),
       bg = "white",
       cex = 0.8,
       title = NULL)

text(cam.x, cam.y, "Cambridge St", srt = cam.angle)
text(broad.x, broad.y, "Broad St", srt = broad.angle)
```

## Missing Cases: Noel Street

15 Noel Street, which lies north of Broad Street one block south of Oxford Street at the intersection with Berwick Street, is the location of the third "missing" case.

Snow's map shows three cases at the end of Noel Street:

![](noel.street.png)

But Dodson and Tobler's data shows only two, cases 282 and 422. They also appear to be farther apart than what we see in neighboring stacks:

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}

# Noel Street #
street.name <- "Noel Street"
noel.data <- roads[roads$name == street.name, ]
noel.list <- street.list[paste(unique(noel.data$street))]
noel.list <- lapply(noel.list, function(df) {
  df[order(df$x, decreasing = TRUE), ]
})

segment.ols <- lapply(noel.list, function(x) {
  coef(lm(y ~ x, data = x))
})

sel <- "118"

noel.angle <- atan(segment.ols[[sel]]["x"]) * 180 / pi
noel.x <- abs(noel.list[[sel]][1, "x"] - noel.list[[sel]][2, "x"]) / 3 +
  noel.list[[sel]][2, "x"]
noel.y <- segment.ols[[sel]][1] + segment.ols[[sel]][2] * noel.x

# Poland Street #
street.name <- "Poland Street"
poland.data <- roads[roads$name == street.name, ]
poland.list <- street.list[paste(unique(poland.data$street))]
poland.list <- lapply(poland.list, function(df) {
  df[order(df$x, decreasing = TRUE), ]
})

segment.ols <- lapply(poland.list, function(x) {
  coef(lm(y ~ x, data = x))
})

sel <- "140"

poland.angle <- atan(segment.ols[[sel]]["x"]) * 180 / pi
poland.x <- abs(poland.list[[sel]][1, "x"] - poland.list[[sel]][2, "x"]) /
            4 + poland.list[[sel]][2, "x"]
poland.y <- segment.ols[[sel]][1] + segment.ols[[sel]][2] * poland.x

# Plot #
noel.street <- c(282, 422)

plot(Snow.deaths[, c("x", "y")], xlim = c(11.75, 12.75), ylim = c(14.25, 15.25),
     asp = 1, pch = NA)
invisible(lapply(street.list, lines, col = "lightgray"))

points(Snow.deaths[-noel.street, c("x", "y")])

points(Snow.deaths[noel.street, c("x", "y")], col = "#009E73")
text(Snow.deaths[noel.street, c("x", "y")], cex = 0.9, pos = 4,
     labels = Snow.deaths$case[noel.street], col = "#009E73")

title(main = paste0("15 Noel Street:", "\n",
                    " Two (282 & 422) Rather than Three Cases"))

legend(x = "bottomleft",
       legend = c("Case", "15 Noel St"),
       col = c("black", "#009E73"),
       pch = 1,
       bg = "white",
       cex = 0.8,
       title = NULL)

text(noel.x, noel.y, "Noel St", srt = noel.angle)
text(poland.x, poland.y, "Poland St", srt = poland.angle)
```

## A Solution

One solution to these problems is to use the three duplicate observations from Cambridge Street to fill in for the three "missing" observations on Broad and Noel Streets. This is both plausible and reasonable.

What makes it plausible is that with the exception of the x-y coordinate of the case at the base of a stack, the remaining coordinates, which make up 44% (257/578) of all bars and which include the proposed locations for the three "missing" observations, do not represent the location of a case. They only serve to align bars within a stack. For this reason, there should be no objections to moving this category of bars.

What makes it reasonable is that I do not relocate bars by hand. Instead, I use simple geometric interpolation. I use existing observations in a stack to determine the "standard" _unit_ distance between bars and then use multiples of that _unit_ distance to position "missing" observations.

Consider 40 Broad Street. On Snow's map, there are four cases but in Dodson and Tobler there are just two. Visual inspection leads me to three claims: 1) the distance between the two observed cases, 32 and 122, is greater than that at neighboring stacks; 2) there appears to be room for one "missing" case between 32 and 122; and 3) the other "missing" case should be located just beyond case 122.

The proposed coordinates are illustrated below:

```{r, echo = FALSE}

# Equation of line for the "home" road segment
segmentOLS <- function(segment) {
  lm(y ~ x, data = street.list[[segment]])
}

# Slope for orthogonal stack axis
orthogonalSlope <- function(segment) {
  ols <- segmentOLS(segment)
  -1 / coef(ols)[2]
}

# Intercept for orthogonal stack axis
orthogonalIntercept <- function(case, segment = sel) {
  Snow.deaths[case, "y"] - orthogonalSlope(segment) * Snow.deaths[case, "x"]
}

# Euclidean distance between observed points
unitDistance <- function(case1, case2) {
  dist(Snow.deaths[c(case1, case2), c("x", "y")])
}

# Quadratic equation
quadratic <- function(a, b, c) {
  root1 <- (-b + sqrt(b^2 - 4 * a * c)) / (2 * a)
  root2 <- (-b - sqrt(b^2 - 4 * a * c)) / (2 * a)
  c(root1, root2)
}

# Geometric interpolation
interpolatedPoints <- function(case1, case2, segment = sel, multiplier = 0.5) {
  p <- Snow.deaths[case1, "x"]
  q <- Snow.deaths[case1, "y"]
  radius <- multiplier * unitDistance(case1, case2)
  m <- orthogonalSlope(segment)
  b <- orthogonalIntercept(case1)
  A <- (m^2 + 1)
  B <- 2 * (m * b - m * q - p)
  C <- (q^2 - radius^2 + p^2 - 2 * b * q + b^2)
  quadratic(A, B, C)
}

```

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}

sel <- "216"  # street list element, one with 2 endpoints

plot(Snow.deaths[, c("x", "y")], xlim = c(12, 13), ylim = c(10.9, 11.9),
     pch = NA, cex = 0.5, col = "blue", asp = 1)

invisible(lapply(street.list[names(street.list) != sel], lines,
                 col = "lightgray"))

points(Snow.pumps[, c("x", "y")], col = "blue", pch = 2, cex = 1)

text(Snow.pumps[, c("x", "y")], col = "blue", labels = "Broad Street Well",
     pos = 4, cex = 0.85)

points(Snow.deaths[-c(broad.40, duplicates$case), c("x", "y")], col = "gray")
points(Snow.deaths[broad.40, c("x", "y")], col = "#009E73")

text(Snow.deaths[broad.40, c("x", "y")], labels = Snow.deaths$case[broad.40],
     cex = 0.9, pos = 4, col = "#009E73")

seg.data <- unlist(street.list[[sel]])

segments(seg.data["x1"], seg.data["y1"], seg.data["x2"], seg.data["y2"],
         col = "#009E73")

title(main = paste0("Geometric Interpolation: Candidate Locations for",
                    "\n", "the Two Missing Cases at 40 Broad Street"))

ols <- segmentOLS(sel)
segment.slope <- coef(ols)[2]
segment.intercept <- coef(ols)[1]
orthogonal.slope <- orthogonalSlope(sel)

orthogonal.intercept.32 <- orthogonalIntercept(32)
orthogonal.intercept.122 <- orthogonalIntercept(122)

move.candidates <- c(91, 93, 209)
stay.in.place <- c(214, 241, 429)
candidates <- c(move.candidates, stay.in.place)

x.out1 <- max(interpolatedPoints(32, 122))
x.out2 <- max(interpolatedPoints(32, 122, multiplier = 1.5))
y.out1 <- orthogonal.slope * x.out1 + orthogonal.intercept.32
y.out2 <- orthogonal.slope * x.out2 + orthogonal.intercept.32

points(x.out1, y.out1, col = "red", pch = 4)
points(x.out2, y.out2, col = "red", pch = 4)
text(x.out1, y.out1, col = "red", labels = "one missing case", pos = 4,
     cex = 0.8)
text(x.out2, y.out2, col = "red", labels = "other missing case", pos = 4,
     cex = 0.8)

text(cam.x, cam.y, "Cambridge St", srt = cam.angle)
text(broad.x, broad.y, "Broad St", srt = broad.angle)

legend(x = "bottomleft",
       legend = c("Well", "Broad St", "Candidate", "Others"),
       col = c("blue", "#009E73", "red", "gray"),
       pch = c(2, 1, 4, 1),
       bg = "white",
       cex = 0.8,
       title = "Key")

```

## Geometric interpolation: _unit_ distance

To explain how I got these coordinates, I illustrate the computation of the _unit_ distance in one dimension and then move to the two dimensional case.

I use 40 Broad Street as the example. I rotate the data so that the stack lies along a single (horizontal) dimension. To fit one "missing" case between cases 32 and 122, I set the _unit_ distance between the two observed cases to 2:

```{r, fig.width = 5, fig.height = 3.5, echo = FALSE}

plot(0:3, rep(0, 4), xlim = c(0, 3.6), ylim = c(0, 1), pch = NA, xaxt = "n",
     yaxt = "n", bty = "n", ann = FALSE)
axis(1, seq(0.3, 3.3, 1), labels = 0:3)
points(c(0.3, 2.3), rep(0.5, 2), col = "#009E73")
text(c(0.3, 2.3), rep(0.5, 2), labels = c(32, 122), pos = 1, cex = 0.9,
     col = "#009E73")
arrows(0.3, 0.6, 2.3, 0.6, code = 3, length = 0.067, angle = 90, col = "blue")
arrows(0.3, 0.6, 2.3, 0.6, code = 3, length = 0.1, col = "blue")
text(1.3, 0.7, "2 units", col = "blue")
title(main = "Broad Street Unit Distance")

```

I then place one "missing" case 1 _unit_ away from 32:

```{r, fig.width = 5, fig.height = 3.5, echo = FALSE}
plot(0:3, rep(0, 4), xlim = c(0, 3.6), ylim = c(0, 1), pch = NA, xaxt = "n",
     yaxt = "n", bty = "n", ann = FALSE)
axis(1, seq(0.3, 3.3, 1), labels = 0:3)
points(c(0.3, 2.3), rep(0.5, 2), col = "#009E73")
text(c(0.3, 2.3), rep(0.5, 2), labels = c(32, 122), pos = 1, cex = 0.9,
     col = "#009E73")
points(1.3, 0.5, col = "red", pch = 4)
text(1.3, 0.5, labels = "missing", pos = 1, col = "red")
arrows(0.3, 0.6, 1.3, 0.6, code = 3, length = 0.067, angle = 90, col = "blue")
arrows(0.3, 0.6, 1.3, 0.6, length = 0.1, col = "blue")
text(0.8, 0.7, "1 unit", col = "blue")
title(main = "Location for One Missing Case")
```

And place the other "missing" case three _units_ away:

```{r, fig.width = 5, fig.height = 3.5, echo = FALSE}
plot(0:3, rep(0, 4), xlim = c(0, 3.6), ylim = c(0, 1), pch = NA, xaxt = "n",
     yaxt = "n", bty = "n", ann = FALSE)
axis(1, seq(0.3, 3.3, 1), labels = 0:3)
points(c(0.3, 2.3), rep(0.5, 2), col = "#009E73")
text(c(0.3, 2.3), rep(0.5, 2), labels = c(32, 122), pos = 1, cex = 0.9,
     col = "#009E73")
points(1.3, 0.5, pch = 4)
points(3.3, 0.5, col = "red", pch = 4)
text(3.3, 0.5, labels = "missing", pos = 1, col = "red")
arrows(0.3, 0.6, 3.3, 0.6, code = 3, length = 0.067, angle = 90, col = "blue")
arrows(0.3, 0.6, 3.3, 0.6, length = 0.1, col = "blue")
text(1.8, 0.7, "3 units", col = "blue")
title(main = "Location for Other Missing Case")

```

## Geometric interpolation: in 2 dimensions

Because stacks lie in two dimensional space and because we need to respect geographical features like roads, the actual interpolation is a bit more complicated.

Consider the example of the "missing" observation that lies between cases 32 and 122. Just as with the one dimensional example, the "missing" observation should be located one _unit_ away from case 32. The difference in two dimensions is that it needs to be located one _unit_ away from case 32 (at the red "x") as measured along the line that runs through the stack's axis (black dotted line).

```{r, echo = FALSE}

sel <- "216"  # street list element, one with 2 endpoints

ols <- segmentOLS(sel)
segment.slope <- coef(ols)[2]
segment.intercept <- coef(ols)[1]
orthogonal.slope <- orthogonalSlope(sel)
orthogonal.intercept.32 <- orthogonalIntercept(32)

angle <- atan(orthogonal.slope) * 180 / pi

x.pt <- (orthogonal.intercept.32 - segment.intercept) /
  (segment.slope - orthogonal.slope)

y.pt <- segment.slope * x.pt + segment.intercept

x.out2 <- max(interpolatedPoints(32, 122, multiplier = 2))
y.out2 <- orthogonal.slope * x.out2 + orthogonal.intercept.32

meter0 <- function(x, y, radius = 0.025) {
  p <- x
  q <- y
  m <- segment.slope
  b <- q - p * segment.slope
  A <- (m^2 + 1)
  B <- 2 * (m * b - m * q - p)
  C <- (q^2 - radius^2 + p^2 - 2 * b * q + b^2)
  c(max(quadratic(A, B, C)), b)
}

meter <- function(case, radius = 0.025) {
  p <- Snow.deaths[case, "x"]
  q <- Snow.deaths[case, "y"]
  m <- segment.slope
  b <- q - p * segment.slope
  A <- (m^2 + 1)
  B <- 2 * (m * b - m * q - p)
  C <- (q^2 - radius^2 + p^2 - 2 * b * q + b^2)
  c(max(quadratic(A, B, C)), b)
}

meterB <- function(x, y, radius = 0.025) {
  p <- x
  q <- y
  m <- segment.slope
  b <- q - p * segment.slope
  A <- (m^2 + 1)
  B <- 2 * (m * b - m * q - p)
  C <- (q^2 - radius^2 + p^2 - 2 * b * q + b^2)
  c(quadratic(A, B, C), b)
}

```

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}

center <- Snow.deaths[32, c("x", "y")]
delta <- 0.1

plot(Snow.deaths[, c("x", "y")], xlim = c(center$x - delta, center$x + delta),
     ylim = c(center$y - delta, center$y + delta),
     pch = NA, col = "blue", asp = 1)

abline(a = orthogonal.intercept.32, b = orthogonal.slope, lty = "dotted",
       lwd = 1.5)
points(Snow.deaths[broad.40, c("x", "y")], col = "#009E73")
text(Snow.deaths[broad.40, c("x", "y")], labels = Snow.deaths$case[broad.40],
     pos = 2, col = "#009E73")

x.out1 <- max(interpolatedPoints(32, 122))
y.out1 <- orthogonal.slope * x.out1 + orthogonal.intercept.32
points(x.out1, y.out1, pch = 4, col = "red", lwd = 2)
text(x.out1, y.out1, labels = "missing observation", pos = 2, col = "red")

x1 <- meter("32", 0.0125)[1]
y1 <- segment.slope * x1 + meter("32")[2]
x2 <- meter0(x.out1, y.out1, 0.0125)[1]
y2 <- segment.slope * x2 + meter0(x.out1, y.out1, 0.0125)[2]
arrows(x1, y1, x2, y2, code = 3, length = 0.05, angle = 90, col = "blue")
arrows(x1, y1, x2, y2, length = 0.1, col = "blue")

x1.label <- meter0(x1, y1, 0.0125)[1]
y1.label <- segment.slope * x1.label + meter0(x1, y1, 0.0125)[2]
x2.label <- meter0(x2, y2, 0.0125)[1]
y2.label <- segment.slope * x2.label + meter0(x2, y2, 0.0125)[2]
text(mean(c(x1.label, x2.label)), mean(c(y1.label, y2.label)), labels = "1 unit",
     srt = angle, col = "blue")

abline(ols)

```

To compute the coordinates for the "missing" case, I find the points of intersection between a circle and line. The circle is simple: its center is the case or reference point (case 32), and its radius is the desired multiple of the _unit_ distance, which is 1 in this example. The line is more complicated: it's the line that runs through the stack's axis.

What makes the line complicated is that the bars in the stacks in Dodson and Tobler are generally _not_ perfectly aligned. To ensure that there is only one unique axis for any stack, I impose two constraints: 1) a stack must lie along an axis that is orthogonal to the road (line segment) where the stack is located^[For an explanation on how the home road segment is identified, see the "'Unstacking' Bars" vignette.]; and 2) the axis must pass through the reference point, the bar at the base of the stack (i.e., the bar closest to the stack's home road segment).

As a consequence, the line drawn through the stack's axis will pass through the circle's center:

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}
plot(Snow.deaths[, c("x", "y")], xlim = c(center$x - delta, center$x + delta),
     ylim = c(center$y - delta, center$y + delta),
     pch = NA, col = "blue", asp = 1)

abline(a = orthogonal.intercept.32, b = orthogonal.slope, lty = "dotted",
       lwd = 1.5)
points(Snow.deaths[broad.40, c("x", "y")], col = "#009E73")
text(Snow.deaths[broad.40, c("x", "y")], labels = Snow.deaths$case[broad.40],
     pos = 2, col = "#009E73")

broad.df <- Snow.deaths[broad.40, c("x", "y")]

r <- dist(broad.df) / 2  # radius
unit.base <- 100
unit.radians <- 2 * pi / unit.base
circumference.x <- Snow.deaths[32, "x"] + r * cos(0:unit.base * unit.radians)
circumference.y <- Snow.deaths[32, "y"] + r * sin(0:unit.base * unit.radians)
lines(circumference.x, circumference.y, col = "#009E73")

arrows(broad.df["32", "x"],
       broad.df["32", "y"],
       broad.df["32", "x"] + r,
       broad.df["32", "y"],
       col = "blue", length = 0.1)

text(broad.df["32", "x"] + 0.5 * r, broad.df["32", "y"], pos = 3,
     labels = "1 unit", col = "blue", cex = 0.9)

abline(ols)
```

There will always be two points of intersection (red squares):

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}
plot(Snow.deaths[, c("x", "y")], xlim = c(center$x - delta, center$x + delta),
     ylim = c(center$y - delta, center$y + delta),
     pch = NA, col = "blue", asp = 1)

abline(a = orthogonal.intercept.32, b = orthogonal.slope, lty = "dotted", lwd = 1.5)
points(Snow.deaths[broad.40, c("x", "y")], col = "#009E73")
text(Snow.deaths[broad.40, c("x", "y")], labels = Snow.deaths$case[broad.40],
     pos = 2, col = "#009E73")

x.out1 <- max(interpolatedPoints(32, 122))
y.out1 <- orthogonal.slope * x.out1 + orthogonal.intercept.32
points(x.out1, y.out1, pch = 15, col = "red")
text(x.out1, y.out1, labels = "point 2", pos = 4, col = "red")

x.out1 <- min(interpolatedPoints(32, 122))
y.out1 <- orthogonal.slope * x.out1 + orthogonal.intercept.32
points(x.out1, y.out1, pch = 15, col = "red")
text(x.out1, y.out1, labels = "point 1", pos = 2, col = "red")

broad.df <- Snow.deaths[broad.40, c("x", "y")]

r <- dist(broad.df) / 2  # radius
unit.base <- 100
unit.radians <- 2 * pi / unit.base
circumference.x <- Snow.deaths[32, "x"] + r * cos(0:unit.base * unit.radians)
circumference.y <- Snow.deaths[32, "y"] + r * sin(0:unit.base * unit.radians)
lines(circumference.x, circumference.y, col = "#009E73")

arrows(broad.df["32", "x"],
       broad.df["32", "y"],
       broad.df["32", "x"] + r,
       broad.df["32", "y"],
       col = "blue", length = 0.125)

text(broad.df["32", "x"] + 0.5 * r, broad.df["32", "y"], pos = 3,
     labels = "1 unit", col = "blue")

abline(ols)
```

So, finding the coordinates of the "missing" observations boils down to solving a quadratic equation and picking the appropriate solution.

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}

plot(Snow.deaths[, c("x", "y")], xlim = c(center$x - delta, center$x + delta),
     ylim = c(center$y - delta, center$y + delta),
     pch = NA, col = "blue", asp = 1)

abline(a = orthogonal.intercept.32, b = orthogonal.slope, lty = "dotted",
       lwd = 1.5)
points(Snow.deaths[broad.40, c("x", "y")], col = "#009E73")
text(Snow.deaths[broad.40, c("x", "y")], labels = Snow.deaths$case[broad.40],
     pos = 2, col = "#009E73")

x.out1 <- max(interpolatedPoints(32, 122))
y.out1 <- orthogonal.slope * x.out1 + orthogonal.intercept.32
points(x.out1, y.out1, pch = 4, col = "red", lwd = 2)
text(x.out1, y.out1, labels = "missing observation", pos = 4, col = "red",
     cex = 0.95)

broad.df <- Snow.deaths[broad.40, c("x", "y")]

r <- dist(broad.df) / 2  # radius
unit.base <- 100
unit.radians <- 2 * pi / unit.base
circumference.x <- Snow.deaths[32, "x"] + r * cos(0:unit.base * unit.radians)
circumference.y <- Snow.deaths[32, "y"] + r * sin(0:unit.base * unit.radians)
lines(circumference.x, circumference.y, col = "#009E73")

arrows(broad.df["32", "x"],
       broad.df["32", "y"],
       broad.df["32", "x"] + r,
       broad.df["32", "y"],
       col = "blue", length = 0.125)

text(broad.df["32", "x"] + 0.5 * r, broad.df["32", "y"], pos = 3,
     labels = "1 unit", col = "blue", cex = 0.9)

abline(ols)

```

I do so using with the formulas below:
```{r, eval = FALSE}
# Equation of line for the "home" road segment
# "segment" is the numeric ID of one of the 528 sets of line segments in Dodson and Tobler
segmentOLS <- function(segment) {
  lm(y ~ x, data = street.list[[segment]])
}

# Slope for orthogonal stack axis
orthogonalSlope <- function(segment) {
  ols <- segmentOLS(segment)
  -1 / coef(ols)[2]
}

# Intercept for orthogonal stack axis
# "case" if the reference point of a stack
orthogonalIntercept <- function(case, segment) {
  Snow.deaths[case, "y"] - orthogonalSlope(segment) * Snow.deaths[case, "x"]
}

# unit distance is a function of the Euclidean distance between "case1" and "case2"
unitDistance <- function(case1, case2) {
  dist(Snow.deaths[c(case1, case2), c("x", "y")])
}

# Quadratic equation
quadratic <- function(a, b, c) {
  root1 <- (-b + sqrt(b^2 - 4 * a * c)) / (2 * a)
  root2 <- (-b - sqrt(b^2 - 4 * a * c)) / (2 * a)
  c(root1, root2)
}

# Geometric interpolation
# "multiplier" scales and defines the unit distance
interpolatedPoints <- function(case1, case2, segment, multiplier = 0.5) {
  p <- Snow.deaths[case1, "x"]
  q <- Snow.deaths[case1, "y"]
  radius <- multiplier * unitDistance(case1, case2)
  m <- orthogonalSlope(segment)
  b <- orthogonalIntercept(case1)
  A <- (m^2 + 1)
  B <- 2 * (m * b - m * q - p)
  C <- (q^2 - radius^2 + p^2 - 2 * b * q + b^2)
  quadratic(A, B, C)
}

```

## Proposed coordinates

Using this approach, I get the following results. For the two "missing" cases at 40 Broad Street, I move two duplicates, cases 91 and 93, from Cambridge Street:

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}

sel <- "216"  # street list element, one with 2 endpoints

plot(Snow.deaths[, c("x", "y")], xlim = c(12, 13), ylim = c(10.9, 11.9),
     pch = NA, cex = 0.5, col = "blue", asp = 1)

invisible(lapply(street.list[names(street.list) != sel], lines,
                 col = "lightgray"))

points(Snow.pumps[, c("x", "y")], col = "blue", pch = 2, cex = 1)

text(Snow.pumps[, c("x", "y")], col = "blue", labels = "Broad Street Well",
     pos = 4, cex = 0.85)

points(Snow.deaths[-c(broad.40, unlist(duplicates.id)), c("x", "y")])
points(Snow.deaths[broad.40, c("x", "y")], col = "#009E73")

points(bottom.data)
points(top.data[3, ], col = "red", pch = 0, cex = 1.75)

text(Snow.deaths[broad.40, c("x", "y")], labels = Snow.deaths$case[broad.40],
     cex = 0.9, pos = 4)

text(bottom.data[1:2, ], labels = rownames(bottom.data)[1:2], pos = 1,
     cex = 0.9)
text(bottom.data[3, c("x", "y")] + c(0.03, -0.03), cex = 0.9,
     labels = rownames(bottom.data)[3])
text(top.data[3, ], labels = rownames(top.data)[3], pos = 3, cex = 0.9,
     col = "red")

seg.data <- unlist(street.list[[sel]])

segments(seg.data["x1"], seg.data["y1"], seg.data["x2"], seg.data["y2"],
         col = "#009E73")

title(main = paste0("Two Duplicate Cases (91 & 93) as Substitutes for ", "\n",
                    "the Two Missing Cases at 40 Broad Street"))

ols <- segmentOLS(sel)
segment.slope <- coef(ols)[2]
segment.intercept <- coef(ols)[1]
orthogonal.slope <- orthogonalSlope(sel)

orthogonal.intercept.32 <- orthogonalIntercept(32)
orthogonal.intercept.122 <- orthogonalIntercept(122)

move.candidates <- c(91, 93, 209)
stay.in.place <- c(214, 241, 429)
candidates <- c(move.candidates, stay.in.place)

x.out1 <- max(interpolatedPoints(32, 122))
x.out2 <- max(interpolatedPoints(32, 122, multiplier = 1.5))

y.out1 <- orthogonal.slope * x.out1 + orthogonal.intercept.32
y.out2 <- orthogonal.slope * x.out2 + orthogonal.intercept.32

broad40.32 <- Snow.deaths[32, c("x", "y")]
broad40.122 <- Snow.deaths[122, c("x", "y")]

broad.df <- data.frame(x = c(broad40.32$x, broad40.122$x),
                       y = c(broad40.32$y, broad40.122$y))

r <- dist(broad.df) / 2  # radius
unit.base <- 100
unit.radians <- 2 * pi / unit.base
circumference.x <- Snow.deaths[32, "x"] + r * cos(0:unit.base * unit.radians)
circumference.y <- Snow.deaths[32, "y"] + r * sin(0:unit.base * unit.radians)
lines(circumference.x, circumference.y, col = "#009E73")

r <- dist(broad.df) * 1.5  # radius
unit.base <- 100
unit.radians <- 2 * pi / unit.base
circumference.x <- Snow.deaths[32, "x"] + r * cos(0:unit.base * unit.radians)
circumference.y <- Snow.deaths[32, "y"] + r * sin(0:unit.base * unit.radians)
lines(circumference.x, circumference.y, col = "#009E73")

x.pt <- (orthogonal.intercept.32 - segment.intercept) /
        (segment.slope - orthogonal.slope)

y.pt <- segment.slope * x.pt + segment.intercept

segments(x.pt, y.pt, x.out2, y.out2, col = "#009E73")

points(x.out1, y.out1, pch = 0, col = "red")
points(x.out2, y.out2, pch = 0, col = "red")

ordered.pair1 <- paste0("(", round(x.out1[1], 3), ", ", round(y.out1[1], 3),
                        ")")

ordered.pair2 <- paste0("(", round(x.out2[1], 3), ", ", round(y.out2[1], 3),
                        ")")

arrows(Snow.deaths[93, "x"], Snow.deaths[93, "y"], x.out1, y.out1,
       col = "black", length = 1/8)

arrows(Snow.deaths[91, "x"], Snow.deaths[91, "y"], x.out2, y.out2,
       col = "black", length = 1/8)

text(x.out1, y.out1, pos = 2, col = "red", cex = 0.9,
     labels = paste(move.candidates[2], ordered.pair1))
text(x.out2, y.out2, pos = 1, col = "red", cex = 0.9,
     labels = paste(move.candidates[1], ordered.pair2))

legend(x = "bottomleft",
       legend = c("Well", "Duplicate", "Case", "40 Broad St"),
       col = c("blue", "red", "black", "#009E73"),
       pch = c(2, 0, 1, 1),
       bg = "white",
       cex = 0.8,
       title = "Key")

text(cam.x, cam.y, "Cambridge St", srt = cam.angle)
text(broad.x, broad.y, "Broad St", srt = broad.angle)
```

For the missing case at 15 Noel Street, I move the remaining duplicate, case 209, from Cambridge Street:

```{r, fig.width = 5.25, fig.height = 5.25, fig.align = "center", echo = FALSE}

noel.street <- c(282, 422)

sel <- "118" # street list element, one with 2 endpoints

plot(Snow.deaths[, c("x", "y")], xlim = c(11.75, 12.75), ylim = c(14.25, 15.25),
     asp = 1, pch = NA)

invisible(lapply(street.list[names(street.list) != sel], lines,
                 col = "lightgray"))

seg.data <- unlist(street.list[[sel]])
segments(seg.data["x1"], seg.data["y1"], seg.data["x2"], seg.data["y2"],
         col = "#009E73")

points(Snow.deaths[-noel.street, c("x", "y")])

points(Snow.deaths[noel.street, c("x", "y")], col = "#009E73")
text(Snow.deaths[noel.street, c("x", "y")], cex = 0.9, pos = 4,
     labels = Snow.deaths$case[noel.street], col = "#009E73")

title(main = paste0("Duplicate Observation (209) as Substitute", "\n",
                    "for Missing Case at 15 Noel Street"))

ols <- segmentOLS(sel)
segment.slope <- coef(ols)[2]
segment.intercept <- coef(ols)[1]
orthogonal.slope <- orthogonalSlope(sel)

orthogonal.intercept.282 <- orthogonalIntercept(282)
orthogonal.intercept.422 <- orthogonalIntercept(422)

x.out <- max(interpolatedPoints(282, 422))
y.out <- orthogonal.slope * x.out + orthogonal.intercept.282

noel.282 <- Snow.deaths[282, c("x", "y")]
noel.422 <- Snow.deaths[422, c("x", "y")]
noel.df <- data.frame(x = c(noel.282$x, noel.422$x),
                      y = c(noel.282$y, noel.422$y))

r <- dist(noel.df) * 0.5  # radius
unit.base <- 100
unit.radians <- 2 * pi / unit.base
circumference.x <- Snow.deaths[282, "x"] + r * cos(0:unit.base * unit.radians)
circumference.y <- Snow.deaths[282, "y"] + r * sin(0:unit.base * unit.radians)
lines(circumference.x, circumference.y, col = "#009E73")

x.pt <- (orthogonal.intercept.282 - segment.intercept) /
  (segment.slope - orthogonal.slope)
y.pt <- segment.slope * x.pt + segment.intercept
segments(x.pt, y.pt, x.out, y.out, col = "#009E73")

ordered.pair <- paste0("(", round(x.out[1], 3), ", ", round(y.out[1], 3), ")")

points(x.out[1], y.out[1], pch = 0, col = "red")
text(x.out[1], y.out[1], labels = paste("209", ordered.pair), col = "red",
     cex = 0.9, pos = 2)

arrows(Snow.deaths[209, "x"], Snow.deaths[209, "y"], x.out[1], y.out[1],
       col = "black", length = 1/8)

legend(x = "bottomleft",
       legend = c("Duplicate", "Case", "15 Noel St"),
       col = c("red", "black", "#009E73"),
       pch = c(0, 1, 1),
       bg = "white",
       cex = 0.8,
       title = "Key")

text(noel.x, noel.y, "Noel St", srt = noel.angle)
text(poland.x, poland.y, "Poland St", srt = poland.angle)
```

## Notes
