Informal Preview of Geo-Spatial Analysis Project

[Nemo]

Where I believe mistakes are being made; is in the determination of the size/dimensions of the 'Standard Deviation Ellipse'. Rather than being constructed to fit pre-determined lengths of major/minor axis; it should be drawn on the basis of two pre-determined foci (points of x-axis deviation), and a pre-determined semi-latus recta (extent of y-axis deviation). Put simply: The lengths of major/minor axis should be 'realized' after the fact; instead of being 'determined' before the fact...

Obvious isn't it

I can definitely say Colin that if I ever see that construct again I will say "Ahhh, a Roberts ellipse!"

I think you have gained the maximum from what little information there is to go by.

I too think it points to the use of the main thoroughfare of the Whitechapel Rd, but I do not think this guarantees the culprit being a local, as somebody travelling into the area, especially via train, would create the same pattern.

I would like to see the probabilities related to which railway station he may have alighted from ie which staion is nearest the epicentre or foci.

The double foci aspect is very interesting. Which lodging houses are nearest these foci? A simple change of lodging would create the foci.

Excellent work and presentation Colin -You should form a lecture on it.

[JRJ]

More Good Stuff

Great Addition Robhouse.

Would be good to see that as an overlay on a street map. Also names, chronology, Kosminski's address.

JRJ

[Sam Flynn]

Before we get dragged into specific hypotheses, shouldn't we take the analysis "raw" first? Much better to take the mathematical approach from "bottom-up", I feel, rather than delve too soon into "top-down" suspect-based thinking. Colin's approach gives us an excellent chance of doing just that - although I did notice him mention Kosminski early on, which was slightly wicked of him

[Chris]

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.

[Sam Flynn]

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.

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?

[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.

[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?

[Mr.Hyde]

Don't think JTR was into maths that much.

[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.

[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.

[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]

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.

[Septic Blue]

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]

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.

[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