Informal Preview of Geo-Spatial Analysis Project

Rossmo's 'CGT' designates a rectangular 'search area' that is divided into 40,000 'cells'; to each of which a proprietary 'distance-decay' function (including empirically derived constants and exponents) is applied.

My impression is that these functions containing empirically determined parameters will be appropriate (if at all) only to modern cases in which the killer travels over a large area by car, and won't tell us anything at all about a 19th-century case in which he most likely travelled over a small area on foot.

I agree wholeheartedly, Chris! When one considers the possibility that thousands of 'local' residents, like Charles Lechmere (a.k.a. "Charles Cross"), traversed the entire 'killing field' as a matter of daily routine; the application of an extremely complex 'distance-decay' function - utilizing empirically derived constants and exponents - to each of some 40,000 rectangular 'cells', on the basis of just five-or-six data points, smacks of 'milking' what little information we have for infinitely more than it is worth.

And; as I have already stated:

While I truly believe that the Probability Distribution, which I have depicted, and the 'Geo-Profile' Probability Distribution that I have actually created, both afford invaluable perspectives; I must acknowledge that they may simply be 'profiling' the residences and 'activity spaces' of the victims, while indicating very little about the killer(s).

[John Bennett]

I have just been catching up on this thread and from looking at the analysis of x and y axis data points, empirically determined parameters, standard deviation circles, mean centres, least aggregate dispersions, distance decay functions, cumulative distribution functions and foci eccentricity, I have been able to draw a singular conclusion:

And that is that my brain hurts.

[Chris]

Rossmo's 'CGT' designates a rectangular 'search area' that is divided into 40,000 'cells'; to each of which a proprietary 'distance-decay' function (including empirically derived constants and exponents) is applied.

My impression is that these functions containing empirically determined parameters will be appropriate (if at all) only to modern cases in which the killer travels over a large area by car, and won't tell us anything at all about a 19th-century case in which he most likely travelled over a small area on foot.

It seems to me that the Goulston Street graffito location should not just be taken into consideration as a data point. From the layman's perspective, and according to common sense, the location of the GSG indicates more than just a data point, since it indicates the direction walked by the perpetrator after the Eddowes murder, and likely suggests the direction of the killer's residence. It seems to me that any mathematical model would take this into consideration... admittedly I have no idea HOW this would be incorporated into a geo profile model such as this...

Hi Rob,

I think the spirit of objectivity would invariably dictate that the apron 'deposit' be perceived as a component of the Eddowes 'observation', i.e. the Eddowes 'data point'; such that any 'weight' of factorization given to the Goulston Street doorway, would have to be taken away from the Mitre Square murder-site.

On the other hand; these distributions can be assembled with as much subjectivity as one deems applicable. The doorway in Goulston Street could be added to the distribution as a complete 'observation' / 'data point' in itself; and the axes of ellipse-orientation could be rotated to accommodate the direction of 'egress' that the apron 'deposit' seems to suggest. But again; this would be very subjective in its application, and would therefore, not be suitable for my purposes.

… I am unclear why an ellipse is being used at all, as opposed to a circle or any other shape. It seems rather arbitrary. Is it because the spatial distribution of sites is longer than it is high (so to speak)?

Figure 1: Standard Deviation Circle / Ellipse (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

The 'Mean-Center' (i.e. the murder-site epicenter) and 'Standard Deviation Circle' provide the most efficient description of the murder-site distribution, attainable by way of just two 'statistics'. I will be explaining the concept (and methods of calculation) of 'Standard Deviation' in my forthcoming formal presentation; using of course, layman-terminology. Hopefully then; this will make sense.

A shortcoming however, of the 'Standard Deviation Circle', is its inability to account for any directional bias or 'skew' that might be evident in the distribution of 'observations'. Specifically, in this case (as you rightfully noted); the "distribution of sites is longer than it is high (so to speak)". It is also 'tilted'; but the 'Circle' accounts for neither of these two attributes.

The use of the 'Standard Deviation Ellipse' is conceptually akin to stretching the corresponding 'Circle' in its two directions of 'greatest aggregate dispersion' (in this case: 54.47°; 234.47°), while watching it shrink accordingly from its two directions of 'least aggregate dispersion' (in this case: 144.47°; 324.47°); in order to make it describe the distribution more accurately. As such; the 'Ellipse' (i.e. the actual curve) depicts a constant degree or 'magnitude' of deviation from the murder-site epicenter, regardless of applicable distance. In other words: Any point on the green ellipse can be seen conceptually as being of 'equal' deviation from the murder-site epicenter.

It seems the actual spatial distribution of the sites is somewhat lopsided (weighted to the west)... I am picturing a more amorphous blob.

If we were to analyze the distribution of murder sites as a set of two-to-three clusters or 'kernels', such that a similar number of ellipses of varying size and orientation were 'merged', in order to describe the entire distribution; then we would in fact, be seeing an "amorphous blob".

The complex models used by geographic profilers such as Kim Rossmo, David Canter, Ned Levine (CrimeStat), et al …, would also each generate an "amorphous blob".


'Screen-Capture' (photograph of television screen) from "Revealed" Jack the Ripper: The First Serial Killer (2006)

A Portion of Kim Rossmo's 'Criminal Geographic Target' (i.e. his proprietary geographic profile), for the 'MacNaghten-Five' Victims of 'Jack the Ripper'

Rossmo's 'CGT' designates a rectangular 'search area' that is divided into 40,000 'cells'; to each of which a proprietary 'distance-decay' function (including empirically derived constants and exponents) is applied. Each 'cell' is thereby assessed as to its likelihood of having played host to the residence of 'Jack the Ripper'. The 'probability distribution' in this case, is indeed an "amorphous blob".

Note the three 'peaks' of relatively high probability density (red):

- The vicinity of the southeast corner of New Goulston Street / Middlesex Street, Parish of St. Mary Whitechapel

- The vicinity of the southwest corner (Lolesworth Buildings) and northeast corner of George Street / Thrawl Street, Parish of Christ Church Spitalfields ***

- The vicinity of the northeast corner of Osborn Place / Brick Lane, (Boundary) Parishes of St. Mary Whitechapel & Christ Church Spitalfields

*** Rossmo described this, the 'highest' of the three 'peaks', in the following manner:

"… the peak area, where the profile is falling on, covers Flower & Dean Street, Fashion Street, Thrawl Street, …"

We have somehow interpreted this to mean that, according to Rossmo: "'Jack the Ripper' probably lived on Flower & Dean Street".

This is a significant misrepresentation of Rossmo's 'conclusions'; especially in light of the fact that the red portion of this 'peak' barely comes into contact with Flower & Dean Street – doing so, only in the immediate vicinity of #5, a large doss house on the south side of the thoroughfare, opposite #56 (the 'White House').

Rossmo's model would actually suggest that the 'highest' point (i.e. the immediate vicinity of Lolesworth Buildings), within this, the 'highest peak'; would be the single most likely residence of 'Jack the Ripper'. That is a 'far cry' from suggesting that he probably lived there.

Note also, the two 'valleys' of relatively low probability density (blue):

- The vicinity of the intersection of Dorset Street / Crispin Street, Parish of Christ Church Spitalfields

- The vicinity of Whitechapel Road; from Osborn Street / Church Lane –to- Black Lion Yard / Fieldgate Street, Parish of St. Mary Whitechapel

[String]

Just a couple of questions.
Are these distances as the crow flies?
I thought that the distances would be skewed a bit if we took into account actual walking distances on the ground. This would include alley ways that may have been used as short cuts.
Could you also not then use walking times rather than distances as the murderer might be more mindful of the amount of time that he was abroad rather than distance he has travelled? I know if I'm out walking I judge it by time rather than by distance as I can measure time more easily.
Interesting stuff BTW.

[robhouse]

It seems to me that the Goulston Street graffito location should not just be taken into consideration as a data point. From the layman's perspective, and according to common sense, the location of the GSG indicates more than just a data point, since it indicates the direction walked by the perpetrator after the Eddowes murder, and likely suggests the direction of the killer's residence. It seems to me that any mathematical model would take this into consideration... admittedly I have no idea HOW this would be incorporated into a geo profile model such as this... I am not a mathematician, and I am not really following the whole discussion here.

Also I am unclear why an ellipse is being used at all, as opposed to a circle or any other shape. It seems rather arbitrary. Is it because the spatial distribution of sites is longer than it is high (so to speak)? It seems the actual spatial distribution of the sites is somewhat lopsided (weighted to the west)... I am picturing a more amorphous blob. But then again, I dont do math.

Rob H

[Sam Flynn]

While the 'Elliptical Perspective' could certainly suggest the possibility that the murderer operated from two different bases; I believe that I would follow that 'avenue' by establishing two distinct murder-site epicenters (using two distinct sub-distributions), rather than establishing a 'best-fit' ellipse for the entire distribution.

I'd agree with that, Colin - hence my earlier comment about "overlapping catchment areas". And thanks for another detailed and interesting post.

To me the major axis of the last and thus far culmative "Deviations from Murder Site Epicenter (Elliptical)" suggests the killer was very comfortable in using the Whitechapel road (or any parallel throughfare closely northward i.e., Wentworth/Montague St.) at least to reach his crime scenes if not to egress from them; ...

The elliptical perspective implies a certain degree of 'mobility' provided by Aldgate High Street / Whitechapel High Street / Whitechapel Road (or as you suggest: Wentworth Street / Old Montague Street); as well as any role that the thoroughfare(s) might have played as 'barriers' in the minds of the victims or killer (or both). This "degree of 'mobility'" may have been specifically 'utilized' by the killer(s); or it may simply have affected the daily routines and activities of the victims, such that each one of them died in an area, in which she was likely to be found.

While I truly believe that the Probability Distribution, which I have depicted, and the 'Geo-Profile' Probability Distribution that I have actually created, both afford invaluable perspectives; I must acknowledge that they may simply be 'profiling' the residences and 'activity spaces' of the victims, while indicating very little about the killer(s).

... further suggesting in my mind that the man was local or at least extremely conversant with the ins and outs of the "Whitechapel Murder District."

Conversely; the directional bias or 'skew' of the murder-site distribution could be indicative of a killer who felt uncomfortable veering too far off of the main 'arteries', for lack of an acute familiarity with the area.

I would suggest the following:

1) That you include the location of Eddowes Apron on Goulston Street as a data point, ...

I first considered doing so when I began work on this endeavor (over a year ago). I have pondered the idea again, since reading your suggestion, but I must stand by my initial conclusion: That the apron 'deposit' in Goulston Street was not an observed 'event' in itself; rather it was a component of the Eddowes 'event'. Therefore, any factorization of this 'sub-event' into the overall 'equation' would have to be 'weighted' (e.g. 25% of a data point); thereby reducing the weight of the Eddowes murder-site accordingly (i.e. to 75% of a data point).

3) There seems to be a growing though not universal concensus that Liz Stride was not a Ripper victim (i.e., Ripper Podcast: Anything But Your Prayers: The Murder of Elizabeth Stride). As she seems to be the least likely Ripper Victim of the 6, it might be useful to create a parallel study without her. (or if not too difficult parallel studies for all possible permutations beyond the ‘3 certain’ victims …).

"… it might be useful to create a parallel study without her. (or if not too difficult parallel studies for all possible permutations beyond the ‘3 certain’ victims …)."

Certain 'permutations' (some including 'weighted' observations/murder-sites) are on the 'back-burner'.

I agree, JRJ - if so, and bearing in mind that we may have an elliptical "orbit" to contend with, it's just possible that Jack had two bases in the region of both foci; or, if you like, he had two circular "catchment areas" that overlapped. If so, he may have moved lodgings between murders, settling more-or-less in the western part of the district after the Chapman murder until the end of the series, prior to which he lived more towards Mile End.

It's worth bearing in mind that an elliptical probability distribution like this doesn't have well-defined foci - only a centre and axes.

Colin has illustrated the shape of the distribution by picking a particular elliptical contour to show graphically, but he might just as well have picked an ellipse half as big or one twice as big, and of course those would have quite different foci.

It's not an ellipse of best fit, but a probability density distribution whose peak is at the centre, and whose contours are ellipses.

Here's a graphic look at the Distribution Density Function (i.e. the 'Probability Density Function') ('one-tailed') for six data points, i.e. five 'degrees of freedom'. As with any distribution density function; one standard deviation marks a point of inflection in the density's 'curvature'. As Chris has rightly observed; the density's peak is at 0.00 standard deviations, i.e. the murder-site epicenter.



This is a graphic depiction of the Cumulative Distribution Function (i.e. the 'Accumulated Probability Function') for six data points, i.e. five 'degrees of freedom' (blue), which suggests accumulated 'probabilities' at incremental multiples of 'standard deviation' from the murder-site epicenter, that the impending subsequent murder would occur within; and the corresponding 'delayed' version of the same function (yellow), which suggests accumulated 'probabilities' at incremental multiples of 'standard deviation' from the murder-site epicenter, that the killer has operated from within. Remember; one standard deviation is the '50%-threshold', i.e. the point, at which there is a perceived 'probability' of 63.68% that the murderer would continue to kill within (assuming he were to continue); and therefore a 31.84% perceived 'probability' that the he would be found to be living within (assuming he were to be found).

The following are graphic depictions of the actual distributions (0.00% - 99.50%), as seen from the 'Circular Perspective':


Figure 1: Incremental Probability Distribution (0.00% - 99.50%) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009


Figure 2: Incremental 'Geo-Profile' Probability Distribution (0.00% - 99.50%) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

I have not progressed in my work to the point of being able to exhibit the corresponding 'Elliptical Perspective' of each distribution. Those graphics are being prepared; but will require several weeks to complete.

I can, however, 'reiterate' this graphic, which should help to clarify Chris's observation.

Again:

It's worth bearing in mind that an elliptical probability distribution like this doesn't have well-defined foci - only a centre and axes.

Colin has illustrated the shape of the distribution by picking a particular elliptical contour to show graphically, but he might just as well have picked an ellipse half as big or one twice as big, and of course those would have quite different foci.

Figure 3: Deviations from Murder Site Epicenter (Elliptical) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Each ellipse should be seen as 'containing' a certain accumulated portion of each distribution; i.e. [the perceived 'probability' that the impending subsequent murder would occur within] / [the perceived 'probability' that the killer would be found to be living within].

e.g.:

Tabram: 0.15 Standard Deviations; 11.34% / 4.96%

One Standard Deviation (Red): 1.00 Standard Deviations; 63.68% / 31.84%

Nichols: 1.22 Standard Deviations; 72.32% / 38.18%

Again; each ellipse should be seen as 'containing' a certain accumulated portion of each distribution. And as Chris has observed: Each ellipse, while sharing a common center, a common axis-orientation, and common major/minor-axis proportions; does in fact, have a unique set of foci.

While the 'Elliptical Perspective' could certainly suggest the possibility that the murderer operated from two different bases; I believe that I would follow that 'avenue' by establishing two distinct murder-site epicenters (using two distinct sub-distributions), rather than establishing a 'best-fit' ellipse for the entire distribution.

One thing I wondered was whether the calculated eccentricity is statistically significant, given the small sample size. Even if there were an underlying centrally symmetrical distribution, sampling would produce an asymmetrical distribution, and therefore an eccentric ellipse. The smaller the sample, the larger this artefactual eccentricity would be.

I believe that 'sampling error' could affect both foci-eccentricity and axis-orientation.

I am also skeptical of the practicality in using the 'Elliptical Perspective' to depict a distribution beyond ~two standard deviations. As such; I am contemplating a model, in which foci-eccentricity actually increases (to a specified limit) below the 'threshold' of one standard deviation, and decreases (to '0') beyond the same 'threshold'.

Thanks to everyone who has contributed to this discussion. Your feedback will invariably benefit my preparations of a formal presentation, which I hope to begin on the message boards (albeit in a 'dedicated' thread), in four-to-five weeks. The process of making the formal presentation could require several months to complete; so, please bear with me. Also; I promise to provide a wealth of applicable data, as well as explanations of my analysis in layman-terms.

In the meantime; please keep the discussion underway. I probably won't be able to participate very often; but I will eagerly anticipate additional contributions to this thread.

[Sam Flynn]

But I am still a bit of a sceptic about these geographical analyses anyway.

So am I, Chris - inasmuch as one might hope they'd be able to pinpoint the killer's residence. However, I'm not so pessimistic about narrowing things down to a "fuzzy" zone, within reasonable tolerance. I would suggest, however, that such tolerances would be rather broad with a sample size this small. In the context of the area we're talking about, that might mean several thousand residents.

Not that I don't find the whole concept intriguing, and potentially useful in a general sense - on the contrary, I do. I certainly look forward to seeing more of Colin's typically thorough and interesting work.

[Chris]

Interesting, Chris - I was going to suggest a parallel with electron clouds, but thought I'd better stick at Bode's planetary "Law". That said, doesn't the "central focus" idea assume that he had a single base throughout the series of murders? If he had two, then wouldn't a pair of overlapping probability densities produce much the same effect?

It probably would, to the extent that we can tell from five or six sample points.

One thing I wondered was whether the calculated eccentricity is statistically significant, given the small sample size. Even if there were an underlying centrally symmetrical distribution, sampling would produce an asymmetrical distribution, and therefore an eccentric ellipse. The smaller the sample, the larger this artefactual eccentricity would be. I suppose a statistician would be able to tell us whether the observed eccentricity is likely to be real rather than a sampling artefact.

But I am still a bit of a sceptic about these geographical analyses anyway.

[Sam Flynn]

Don't think JTR was into maths that much.

He was into maths as much as most other humans are - he operated in a physical world, and mathematics is pretty fundamental to that.

[Mr.Hyde]

Mr.Hyde

Interesting, Chris - I was going to suggest a parallel with electron clouds, but thought I'd better stick at Bode's planetary "Law". That said, doesn't the "central focus" idea assume that he had a single base throughout the series of murders? If he had two, then wouldn't a pair of overlapping probability densities produce much the same effect?

Yep!Two.
CV1 & CV4.

[Mr.Hyde]

Don't think JTR was into maths that much.

[Sam Flynn]

It's not an ellipse of best fit, but a probability density distribution whose peak is at the centre, and whose contours are ellipses.

Interesting, Chris - I was going to suggest a parallel with electron clouds, but thought I'd better stick at Bode's planetary "Law". That said, doesn't the "central focus" idea assume that he had a single base throughout the series of murders? If he had two, then wouldn't a pair of overlapping probability densities produce much the same effect?

[Chris]

I misunderstood. I thought the point was to calculate where the ellipse of best fit should be - an equivalent exercise to predicting the orbit of a planet based on Bode's Law. I'm not sure whether that would be possible, but if it can be done for circles, why not ellipses?

It's not an ellipse of best fit, but a probability density distribution whose peak is at the centre, and whose contours are ellipses.