Informal Preview of Geo-Spatial Analysis Project

Do I understand correctly that the geographical profilers weren't using a rotated ellipse, but one aligned with the x and y axes?

The rotation of the 'Standard Deviation Ellipse', to accommodate the directional bias or 'skew' of a distribution, would appear to be standard procedure in all practical applications of geo-spatial analysis: Geographic profiling being no exception.

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.


Figure 4: Standard Deviation Ellipse !!! (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

[robhouse]

Would Geo Spatial Analyisis applied to Ted Bundy Anywhere in his sequence result in a positive identification .. ok .. obtuse but my point is clear

Hard to say, since Bundy moved a lot. In my opinion, geographic profiling as a concept is valid... whether or not the actual methodology used is sound, I dont really know. Geographic profiling, from what I understand, was rather accurate in assessing the residence of the Yorkshire Ripper... ie. in narrowing the field. This despite the fact that Sutcliffe was essentially caught by pure luck.

And I dont think geographic profiling would always be used to determine a killer's residence per se. In the beltway sniper case in the US for example, I believe most of the people were shot near onramps for the interstate highway. This would suggest something about how the killers got around (in a car, via the highway) and their methodology... but nothing about their residence... etc.

Rob H

[Philbee]

Would Geo Spatial Analyisis applied to Ted Bundy Anywhere in his sequence result in a positive identification .. ok .. obtuse but my point is clear

[The Grave Maurice]

I wish I understood a fraction of what is going on here. It makes my head hurt. Although, fortunately, I love maps, so thanks very much, Colin. I shall await further developments.

[Chris]

What bollocks.

Thank you for your courteous comment.

But seriously, as I have said, I'm not a fan of geographical profiling. But by training I am a mathematician, so at least I know how to diagonalise a matrix. That's the kind of thing that used to be taught in school. It probably isn't now, but I hope it's still taught fairly early in a degree course.

Whatever anyone thinks about the pros and cons of geographical profiling, if it's the case that its leading proponents don't understand this kind of thing, that's quite revealing.

[Simon Wood]

Hi All,

"The angle of clockwise rotation is given as . . ."

"Would someone please verify their accessibility? I am able to see them whether they are 'locked' or 'unlocked'; and therefore cannot be certain that they are accessible to others."

"Wouldn't the correct procedure be to form a matrix of the sums of x squared, x times y and y squared, and then find its principal axes?"

"The murder-site distribution is the product of a simultaneous x-axis/y-axis standard deviation from its epicenter."

What bollocks.

Regards,

Simon

[Chris]

The angle of clockwise rotation is given as ...

Yes - that's one of the principal axes of the matrix I mentioned.

Do I understand correctly that the geographical profilers weren't using a rotated ellipse, but one aligned with the x and y axes?

OOOOOPS !!!

I forgot to 'unlock' the seven flickr images in my most recent post.

Would someone please verify their accessibility? I am able to see them whether they are 'locked' or 'unlocked'; and therefore cannot be certain that they are accessible to others.

Wouldn't the correct procedure be to form a matrix of the sums of x squared, x times y and y squared, and then find its principal axes?

The angle of clockwise rotation is given as ...



... where each (x - x) represents x-axis deviation, and each (y - y) represents y-axis deviation.

[Mr.Hyde]

Mr.Hyde

Wouldn't the correct procedure be to form a matrix of the sums of x squared, x times y and y squared, and then find its principal axes?

Anyone making "book" on this?
What odds-Flower and Dean?
Chris,think he primarily used knifes,not axes.Only kidding-keep up the great work!
Hint-do not look within.Look at the boundaries.

[Chris]

Wouldn't the correct procedure be to form a matrix of the sums of x squared, x times y and y squared, and then find its principal axes?

The interesting thing about ellipses, of course, is that they have two foci, ... Would it be possible to factor this into your work, … ?

Funny you should ask, Gareth! I actually believe that I may have stumbled onto a new approach to the 'construction' of the 'Standard Deviation Ellipse'; one in which the foci are just that: The foci.


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

In order to determine the orientation of the major axis of the 'Standard Deviation Ellipse', we must utilize a system of Cartesian coordinates. The longitudinal axis is dubbed x'; while the latitudinal axis is labeled y'.

We must then determine the degree of clockwise rotation that minimizes the sum of the squared distances between each of the six murder-sites and the two axes. This requires a sort of 'trigonometric least-squares regression', which I will describe at some later juncture.

As the determined angle of clockwise rotation is greater than 45.00°, i.e. 54.47°, it is more practical to think in terms of a counter-clockwise rotation of 35.53°.


Figure 2: Landscape Rotation / Standard Deviation (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red Color-Shaded Area: 'Standard Deviation Circle'

In order to 'counter' the visual effect of the 35.53° counter-clockwise axis-rotation, the landscape is rotated 35.53° clockwise.

The four corners of the 'Standard Deviation Rectangle' are the points of simultaneous x-axis/y-axis standard deviation; and each coincides with a point on the 'Standard Deviation Circle'.


Figure 3: Standard Deviation Ellipse ??? (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Until fairly recently (i.e. sometime within the last twenty years), it would seem that the prevailing 'conventional wisdom' dictated that the 'Standard Deviation Ellipse' fit inside the 'Standard Deviation Rectangle'; such that the length of its major axis would be equal to two x-axis standard deviations, while the length of its minor axis would be equal to two y-axis standard deviations.

WRONG! WRONG! WRONG!

I was able to see that immediately!

The murder-site distribution is the product of a simultaneous x-axis/y-axis standard deviation from its epicenter. Yet this ellipse only coincides with each point of x-axis standard deviation in isolation, i.e. in the absence of any degree of y-axis deviation (and vice versa). In other words; the yellow ellipse does not coincide with the four corners of the 'Standard Deviation Rectangle', so it is not the 'Standard Deviation Ellipse'.

It would seem that some statisticians have tried in recent years to 'right this wrong' by adding certain 'inflation' factors to their calculations of 'Standard Deviation'. The only example, which I have been able to decipher, is that of Ned Levine (CrimeStat), who multiplies a distribution's 'Variance' by a very simple factor of '2', before calculating its square root, in order to determine its 'Standard Deviation'.

And, guess what? The CrimeStat 'Standard Deviation Ellipse', in this case, coincides with the four corners of the 'Standard Deviation Rectangle'. But, it's not quite right! It's ever-so slightly too circular! In other words; its major axis is too short, while its minor axis is too long: Albeit, discrepancies of less than 0.5%, in each case.

But, these disparities are more than mere 'rounding error'. And the factor of '2' seems too convenient, and all too random. I tend to believe that millions of iterations went into the determination of that factor.

And what about the foci? Shouldn’t an ellipse be 'constructed' around its foci? …


Figure 4: Standard Deviation 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 two points of x-axis standard deviation are the foci. The four 'extents' of y-axis standard deviation - from the two foci - are the semi-latus recta. It's that simple!

My ellipse (someday - the 'Roberts Ellipse' – I hope!) coincides with the four corners of the 'Standard Deviation Rectangle' by definition. Every point on the ellipse is the same summed-distance from the two points of x-axis standard deviation, i.e. the two foci; and that "summed-distance" (depicted by the two green lines) is a function of the fact that the ellipse coincides with the four points of simultaneous x-axis/y-axis standard deviation.

The 'Roberts Ellipse' (I have seen nary a reference to this particular construct) is built around its foci. CrimeStat's ellipse is not!


Figure 5: Comparison (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009


Figure 6: Landscape Re-Orientation (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009


Figure 7: 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

Red: 'Standard Deviation Ellipse'

Each ellipse is a 'multiple' of the 'Standard Deviation Ellipse'.

Forgive me if you've thought of doing so already!

No apology needed, Gareth.

[Sam Flynn]

Elliptical reasoning on my part

Hi Colin

The interesting thing about ellipses, of course, is that they have two foci, one towards either "pole" (if you like) of the major axis of the ellipse. Would it be possible to factor this into your work, as this may have an interesting bearing on epicentres and the like? Forgive me if you've thought of doing so already!

Hi All,

Many Thanks! Your feedback and compliments are truly appreciated.

I can't reiterate enough, that at this point I am merely providing an informal preview of the project, and that detailed explanations (in 'layman terms') will be forthcoming in my formal presentations.

***Also; please note that the flickr beneath each 'frame' is a direct link, specifically to that image.***

… one thing that struck me is that the site of Tabram's murder is so close to the "epicentre" that the predicted distribution would be reasonably similar if she were excluded. Unfortunately the same won't be true of Stride.

Quite right, Chris! The removal of Tabram would shift the epicenter a grand total of 20.53 yards east/northeast to the western 'mouth' of Old Montague Street; whereas the removal of Stride would shift the epicenter north/northwest to the vicinity of the 'White House', on Flower & Dean Street.

The real 'eye-opener' regarding Stride, can be seen in the progression from the circular perspective …


Figure 2: Deviations from Murder Site Epicenter (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

… to the elliptical perspective.


Figure 9: 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

Notice that the relative positions of Stride and Eddowes are switched, in going from 'circular' to 'elliptical'.

The elliptical perspective, which accounts for the 'mobility' provided by Aldgate High Street / Whitechapel High Street / Whitechapel Road, as well as role that the thoroughfare might have played as a 'barrier' in the minds of the victims or killer (or both); suggests that even though Dutfield's Yard was in closer proximity to the murder-site epicenter than was Mitre Square, it should have been viewed as a less likely venue for the impending subsequent murder, following the death of Mary Jane Kelly.

My observation on Yalford Street, by the way, was intended only as a comment on Macnaghten's statement - not as any indication that Aaron Kozminski is likely to have been the murderer.

I didn't mean to suggest otherwise.

[smezenen]

Wonderfull work. Being new to the serious research of this subject this is the first time i have seen the murder sites layed out in this way (all on one map). It was very surprising for me to see how small the area of operation actually is. I had always just assumed it was much larger (Miles between vitims).

[Stewart P Evans]

Fan

I'm not particularly a fan of geographical profiling, but one thing that struck me is that the site of Tabram's murder is so close to the "epicentre" that the predicted distribution would be reasonably similar if she were excluded. Unfortunately the same won't be true of Stride.

I don't think you will find that Colin is a particular fan of geographical profiling either.