Hawkeye uncertainty and the UDRS
Russell Degnan
Always keen to prove themselves correct, Hawkeye recently released some of the data over the controversial Tendulkar LBW referral. What is missing are the raw numbers, and even the raw images, which would allow us to produce an accurate reconstruction. But we can work with what we do have, with appropriate caveats, to discuss a few pertinent matters in this case.
Reconstructing Hawkeye
I have taken the images from the PDF and imported into a graphics program to make measurements. The pixel at the leftmost point of the ball in each mage has then been taken to get a distance for sideways movement. Although the camera angle foreshortens the distance, the ball seems to cover slightly less ground as it moves closer to the stumps. By taking a linear measurement, therefore, I am making the ball slightly (very slightly, maybe 2 pixels) less likely to hit the stumps.
One thing to note is that the edge defintion is often blurred. What I don't know is whether hakweye has substantially clearer images to take their measurements from. By taking the negative of consecutive images, the ball would be much clearer than in a colour image where it blends with the pitch, but on this evidence it is hard to see how the measurements could be improved by more than half a pixel.
Projecting the prebounce line down, and the postbounce line up registers the bounce at pixel 239. There are then seven points to calculate the projection from at 243,254,264,275,285,295 and 305. In the following frame of the video the ball actually hits the pads and falls to the offside, giving the impression that Tendulkar was hit more centrally than he probably was. Because we are projecting a curve, anything fewer than three points makes the projection impossible  you'd need to assume a linear projection which may or may not be accurate. The data given indicates a framerate of around 150 fps over the 3m between pitching and impact. That however would be problematic for fast bowlers pitching and hitting the batsmen inside 2m. Hawkeye claims to be using technology that can operate at "up to 500 fps" however. Since that would be too much data to analyse quickly, we'll have to assume that they are either a) not showing all the data they analysed in their PDF, or b) only analysing enough data to make a projection (which would be sensible). Giving them the benefit of the doubt that they do the latter, we'll move on.
From the xcoordinates taken above we can see the ball moves 10 or 11 pixels leftwards at each frame, averaging 10.33. From the overhead hawkeye projection and the sideon shots we can calculate the distance the ball travels per frame, the total distance the ball travels from bounce to impact, and impact to the stumps as follows:
Calculated Parameters 

Impact > Stumps  1857mm 

Bounce > Impact  3032mm 

Distance per Frame  424mm 

Bounce > Stumps Frames  11.54 

Sideways movement per frame  14.9mm 

Sideways Impact Point  +0.33mm outside leg stump 

By calculating the number of frames from bounce to stumps (11.83), we can calculate the total sideways movement for a linear projection 14.9mm and add it to the bounce point to find where the lefthand edge passed the stumps pixel 358.23. Because the right hand of the stumps lies at 358 it appears Hawkeye is approximately correct, and the ball does miss the stumps, albeit by (in real terms) 0.33mm. Hawkeye claims more than this, but never actually says how much more. The lesson from that: small changes in the measurement value can have impact the end result by several millimetres.
Error and the UDRS
The difficulty with this projection is that it ignores error. The measuring error at a 1 pixel level (and as noted, that is probably generous given the blur in some shots) is 1.44mm. The standard deviation of the sideways movement measurements is 0.52 pixels or 0.74mm. Across the 5.13 frames of projection, that makes the total error ± 3.82mm just on the projection (there are also measurement errors for the bounce, each frame measurement, and the edge of the stumps).
Taking the projection error alone, and given the projected point where the ball passed the stumps, the probability that the ball actually missed the stumps was a meagre 53%. Still in Tendulkar's favour, but (you'd think) not remotely enough to overturn a decision. Given the closeness of the projection, in order to know with 90% certainty that the ball actually was missing the stumps Hawkeye would need to be some 15 times more accurate than the data they have presented. They probably aren't, given the 2.5m exists as a clumsy attempt to deal with the poor accuracy of projections over that distance. While in pure probability terms the right decision was (probably) arrived at, this almost certainly overstepped the bounds of a remit to correct "obvious" mistakes.
Idle Summers
13th April, 2011 13:08:35
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Comments
Hawkeye uncertainty and the UDRS
I don't think you can add up the errors for the projection like that. You have some discretisation error on every frame, so even at a much higher frame rate you'll have a nonzero standard deviation of the sideways movement measurements. Your method would then multiply that sd by a much larger number than 5.13, which to my intuition would give a projection error larger than what you calculate above.
David Barry 14th April, 2011 21:26:50
Hawkeye uncertainty and the UDRS
David, more frames would also reduce the SD, though by less than the increase in frame rate. Thinking about it further, for a linear projection the frame rate doesn't really matter anyway. From an estimate for the error on each frame (~1mm) the bounds of the projection error is a line between the 1st frame error bounds and the opposite final frame error bounds, projected outwards. That would make it 1.74mm, with the projection missing the stumps by 1.14mm  probability of missing of 74%. That error will blow out substantially on a curve though, where errors will accumulate more akin to what I did above.
Russ 15th April, 2011 09:24:24
Hawkeye uncertainty and the UDRS
Actually, scratch that last point. The error on a curve will be between the minimum curve that fits the inner bounds of the interior points and outer bounds of the end points; and the maximum curve between the outer bounds of the interior points and the inner bounds of the end points. Calculated on a linear projection that will be more than double the error of 1.74mm stated above.
Russ 15th April, 2011 12:36:48
Hawkeye uncertainty and the UDRS
And one final aside, the ball itself has an error margin of about 3mm (the acceptable variation in ball size plus the seam). Stump width is probably roughly the same (maybe more).
Russ 15th April, 2011 12:40:53
Cant they project a shadow of uncertainty?
I mean show something like a football size shadow aligned with the stumps and based on the deviations due to measurement error.
Let the umpires make the decision based on how much of the shadow covers the stumps and how much is outside of it...
marees 20th April, 2011 18:29:59
Hawkeye uncertainty and the UDRS
marees, they could, yes, although I suspect that makes things more confusing for the third umpire  most of the commentators are a lost cause anyway.
Russ 29th April, 2011 19:55:18
