docs/notes/duplicate.missing.cases.notes.md

In lindbrook/cholera: Amend, Augment and Aid Analysis of John Snow's Cholera Map

note on duplicate and missing cases

lindbrook 2019-02-07

Duplicate Cases: Cambridge Street

The three pairs of case with duplicate coordinates lie at two different locations on opposite sides of Cambridge Street (91 and 241; 209 and 429; 93 and 214):

Missing Cases: Broad Street

Two of the “missing” cases are located at 40 Broad Street, which lies just southwest (below and to the left) of the Broad Street Pump.

While Snow’s map shows four “bars” or cases,

Dodson and Tobler’s data show only two, cases 32 and 122. Furthermore, the two cases also appear to be farther apart than is seen in nearby stacks:

Missing Case: Noel Street

The third “missing” case is located at 15 Noel Street, which lies north of Broad Street one block south of Oxford Street at the intersection with Berwick Street.

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

But Dodson and Tobler’s data shows only two: 282 and 422. Like the cases on Broad Street, the pair also appears to be farther apart than what is seen in nearby stacks:

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 coordinates 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 cases. They only serve to align bars within a stack. For this reason, there should be no objections to moving this class of bars.

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

Consider 40 Broad Street. On Snow’s map, there are four cases but on Dodson and Tobler’s there are just two. Visual inspection leads me to make 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 could be located just beyond case 122.

The proposed coordinates are illustrated below:

Geometric interpolation: unit distance

To explain how I arrived at these coordinates, I use 40 Broad Street as an example to illustrate the computation of the unit distance in one dimension and then move on to the computation of the two dimensional case. Note that 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:

I then place one “missing” case 1 unit away from 32:

And place the other “missing” case three units away:

Geometric interpolation: in 2 dimensions

Because the 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 in 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).

To compute the coordinates for the “missing” case, I’m literally computing the points of intersection between a circle and line.

The circle is simple: it’s center is the case serving as the reference point (case 32); its radius is the desired multiple of the unit distance, which is 1 in this example.

The line is equally simple, in theory: it’s the line that runs through the stack’s axis.

In practice, computing the line is little complicated because the bars in the stacks are generally not perfectly aligned. To ensure that there is one, unique axis for each stack, I impose two constraints: 1) a stack must lie along the axis that is orthogonal to the road (line segment) where the stack is located[1]; and 2) the axis must pass through 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:

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

Finding the coordinates of the “missing” observations boils down to solving a quadratic equation and picking the appropriate solution.

I do so using with the formulas below:

# 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:

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

Notes

1. For an explanation on how the home road segment is identified, see the “‘Unstacking’ Bars” vignette.

lindbrook/cholera documentation built on Jan. 13, 2025, 3:49 p.m.