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The scenario: Over a course of 10 minutes 2 security guards patrol a floor and have their movement tracked. They can either be on the left or the right side of the floor, and within each side of the floor are 10 zones. So in total there are 20 zones (left zone 1, right zone 1, left zone 2, right zone 2, left zone 3 right zone 3 etc etc ) As they cross into a new zone the time is recorded.

I am trying to create a chart that has time along the x axis, but I am unsure how to lay out the y axis ??

I had thought about putting the guards on the y axis and having a symbol represent each zone so I could just insert the symbol at the appropriate time as the guard moved into another zone (but that would need 20 different symbols, or just 10 symbols and use different colors for right and left zones)

e.g

 guard 1            


 guard 2   

                0   1  2  3  4  5  6  7  8  9  10
                           time 

I'm not so sure this is the best way to illustrate it. Ultimately this would be done on a large scale to see if there are any trends are evident.

I have searched google images to find a similar structured chart but could find one. I have tried to use Excel to make a similar chart but failed.

any advice as to how people would approach this ? have I got my axis muddled up perhaps ?

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If you only have two-dimensional spatial data (i.e., no vertical movement to track), you might consider visualizing their movement in three dimensions with the third being time, or as an animated 2D diagram. These ideas wouldn't work if you really only have nominal data though, and can't map your zones onto a 2D plane. –  Nick Stauner Jul 11 at 8:16
    
Are zones in some order? If so, which of the left side zones are closest to the right side (and vice-versa? If not, which zones are adjacent to which? –  Glen_b Jul 11 at 8:17
    
Thanks for that suggestion Nick. How would you approach the the axis ? For ease of people reading the chart would you have the security guards on X and the zones on Y ? Perosnally i see it as begin either /or, but I always welcome an outside opinion on charts. –  B.Miller Jul 11 at 8:20
    
Hi Glen_b, the zones are in a mirror image . so I could split the floor down the middle (length ways) so left zone 1 is adjacent to right zone 1. The floor (in a rough way ) is best thought of as a rectangle. Lowest zone numbers at the south end and they get higher as you approach the north end of the floor –  B.Miller Jul 11 at 8:24
    
If you want me or Nick to see your comments, you need to put an at symbol in front of our usernames, like so: @NickStauner (though @Nick would work). On the other hand, when you're responding to an answer rather then a comment, the @ is unnecessary. –  Glen_b Jul 11 at 11:49
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4 Answers 4

Three variables need to be visualized: time, zone, and side. We should capitalize on the two Cartesian coordinates of the plot to map two of these. Then some graphical quality--symbol, color, lightness, or orientation--will be needed to symbolize the third.

To help the eye follow the temporal sequence it can help to connect the symbols with faint line segments. We can obtain a little more information by erasing any segments that seem to correspond to breaks in the series.

The first solution uses symbol type and color to distinguish the sides, vertical position to identify zones, and horizontal position for time. It is designed to display the progress of zone and side over time to help visualize the transitions. To clarify overlaps, the symbol for side 2 is positioned above its nominal location and the symbol for side 1 is positioned below its nominal location.

Figure 1

This figure makes it immediately apparent that Guard 2 prefers the blue side (side 1) over the red (side 2) and that she moves around the zones more and in a more regular fashion.

The use of identically scaled and oriented time axes in these parallel plots enables visual comparison of the guard's patrol patterns during any time interval. With many guards, this construction lends itself well to a "small multiple" display showing all data simultaneously.

The second solution uses symbol color to distinguish times, vertical position to identify sides, and horizontal position for zone. It is a map-like display (which would generalize to a more complex spatial layout). This could be used to study the frequencies with which each space are entered by each guard and, in a more limited way, to visualize the movements among the spaces.

Figure 2

The different frequencies with which the zones and sides were visited are clearly displayed in this figure. The failure of Guard 2 to visit side 2 of zone 10 is immediately apparent, whereas it was not evident in the first figure.


These figures were produced in R. The input data structure is a list of parallel vectors for the times, zones, and sides: one per guard. The following code begins by generating some sample data randomly. Two functions to make the plot for a given guard are provided, corresponding to the figures.

#
# Side transition matrices
#
transition.side <- function(left=1/2, right=1/2) {
  rbind(c(left, 1-left), c(1-right, right))
}
#
# Zone transition matrices
#
transition.zone <- function(n, up=1/2, stay=0, down=(1-up-stay)) {
  x <- rep(c(down, stay, up, rep(0, n-2)), n)
  q <- matrix(x[-c(1, 2:n+n^2)], n)
  q <- q / apply(q, 1, sum)
  return (q)
}
n.zones <- 10
guards <- list(list(side=transition.side(1/2,1/2),
                    zone=transition.zone(n.zones,1/2,0)),
               list(side=transition.side(3/4,1/4),
                    zone=transition.zone(n.zones,1/8,3/4)))
#
# Create Markov chain walks for all guards.
#
n.steps <- 500
walks <- list()
for (g in guards) {
  zone <- integer(n.steps)
  side <- integer(n.steps)

  # Random starting location
  zone[1] <- sample.int(n.zones, 1)
  side[1] <- sample.int(2, 1)

  for (i in 2:n.steps) {
    zone[i] <- sample.int(n.zones, 1, prob=g$zone[zone[i-1],])
        side[i] <- sample.int(2, 1, prob=g$side[side[i-1],])
  }
  s <- cumsum(sample(c(rexp(n.steps-3), rexp(3, 10/n.steps))))
  walks <- c(walks, list(list(zone=zone, side=side, time=s/max(s))))
}
#
# Display a walk.
#
plot.walk <- function(walk, ...) {
  n <- length(walk$zone)
      #
      # Find outlying time differences.
      #
      d <- diff(walk$time)
  q <- quantile(d, c(1/4, 1/2, 3/4))
  threshold <- q[2] + 5 * (q[3]-q[1])
  breaks <- unique(c(which(d > threshold), n))
  #
  # Plot the data.
  #
  sym <- c(0, 19)
  col <- c("#2020d080", "#d0202080")
  plot(walk$time, walk$zone, type="n", xlab="Time", ylab="Zone", ...)
  j <- 1
  for (i in breaks) {
    lines(walk$time[j:(i-1)], walk$zone[j:(i-1)], col="#00000040")
    j <- i+1
  }
  points(walk$time, walk$zone+0.2*(walk$side-3/2), pch=sym[walk$side], col=col[walk$side],
             cex=min(1,sqrt(200/n)))
    }
    plot.walk2 <- function(walk, n.zones=10, n.sides=2, ...) {
      n <- length(walk$zone)
      #
      # Find outlying time differences.
      #
      d <- diff(walk$time)
      q <- quantile(d, c(1/4, 1/2, 3/4))
      threshold <- q[2] + 5 * (q[3]-q[1])
      breaks <- unique(c(which(d > threshold), n))
      #
      # Plot the reference map
      #
      col <- "#3050b0"
      plot(c(1/2, n.zones+1/2), c(1/2, n.sides+1/2), type="n", bty="n", tck=0, 
           fg="White",
           xaxp=c(1,n.zones,n.zones-1), yaxp=c(1,n.sides,n.sides-1), 
           xlab="Zone", ylab="Side", ...)
      polygon(c(1/2,n.zones+1/2,n.zones+1/2,1/2,1/2), c(1/2,1/2,n.sides+1/2,n.sides+1/2,1/2),
              border=col, col="#fafafa", lwd=2)
      for (i in 2:n.zones) lines(rep(i-1/2,2), c(1/2, n.sides+1/2), col=col)
      for (i in 2:n.sides) lines(c(1/2, n.zones+1/2), rep(i-1/2,2), col=col)
      #
      # Plot the data.
      #         
      col <- terrain.colors(n, alpha=1/2)
      x <- walk$zone + runif(n, -1/3, 1/3)
      y <- walk$side + runif(n, -1/3, 1/3)
  j <- 1
  for (i in breaks) {
    lines(x[j:(i-1)], y[j:(i-1)], col="#00000020")
    j <- i+1
  }
  points(x, y, pch=19, cex=min(1,sqrt(200/n)), col=col)
}
par(mfcol=c(length(guards), 1))
i <- 1
for (g in walks) {
  plot.walk(g, main=paste("Guard", i))
  i <- i+1
}
i <- 1
for (g in walks) {
  plot.walk2(g, main=paste("Guard", i))
  i <- i+1
}
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As a way of displaying the information, perhaps I'd try to do something like this, with zone on the y-axis and time on the x-axis:

enter image description here

... but what kind of display is best depends on what you're trying to get out of it.

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As always, the best visual(s) depends on what question(s) you want to answer. Possible questions that would lead to different visuals:

  • Is a guard's pattern is predictable?
  • Is a guard stationary too often?
  • Are both guards are in the same zone too often?
  • Are there zones that are never or rarely visited?
  • How are the guards different?
  • What's the average time spent in each zone?
  • What's the average/maximum time between visits for each zone?

Since you did ask for a visual over time, here's a heat map time view showing the time between visits to each zone over time. That is, for each time/zone combination the cell color indicates how long it's been since the zone was visited. If there is a special cut-off for acceptable time between visits, you can adjust the color scale to indicate it.

enter image description here

I'm using whuber's simulated random walk data which has time as a continuous measure, which causes some artifacts as a heat map (some cells have no data (white stripes) and some have multiple data). The original question suggests discrete time data which will work better in a heat map.

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I don't think I understand how to read these plots. According to the legend, it seems that white spaces correspond to times when a zone is occupied by a guard. How, then, is it possible for one guard to occupy all 20 rooms at once, such as is shown between times 0.2 and 0.26? (Incidentally, in the question times are explicitly continuous, rather than discrete, as indicated in the first paragraph: they record events controlled by the guards themselves.) –  whuber Jul 17 at 22:18
    
@whuber The pure white bands are periods of no-data and only show because of my crudely coercing the data into a regular heat map. Each zone should be getting darker during those times (and they are darker on the other side of the bands). –  xan Jul 18 at 0:35
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Depends on the goals of the trends, are you trying to ensure they are covering the area? However I would abstract time and do something like this-

enter image description here

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