Safe Speed Forums

Thanks for that, Steve.

However, I only have a printed graph to go by, and my maths is very rusty - so I can see me burning quite a few candles in the near future.

Cheers
Peter

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler

I said the critical failure. What I wrote was true. Your comments are also valid, but not as a response to my statement.

When we're working on understanding the behaviour of such a huge system, there are bound to be effects and counter effects, not to mention some extraordinarilly rare - even unique - cominations of circumstances. It's sometimes like a fractal thing with infinite complexity at every level.

The trick we're trying to pull off is to examine the data and draw broad and sensible conclusions about how best we should manage the system. Somehow we have to use our judgement to distinguish the data from the noise. We shouldn't be looking for proof - we should be looking for understanding.

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Paul Smith
Our scrap speed cameras petition got over 28,000 sigs
The Safe Speed campaign demands a return to intelligent road safety

I'm still not convinced it logically follows from the fact that speeding events are common but fatal crashes are rare. However, I don't disagree with the conclusion itself, because we also know that speeding is only identified as a causative factor in a minority of serious accidents. The combination of these two facts constitutes a mathematically certain proof that speeding is neither necessary nor sufficient to cause an accident.

Hmmm, rather quiet here lately.

Where have all the JJs gone? (I feel a song coming on)

Cheers
Peter

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler

Just spotted this... I was doing that thing in this thread:

http://www.safespeed.org.uk/forum/viewtopic.php?t=1669

Spreadsheet linked for download. Note Excel's 'PERCENTILE' function which solves most of the problems in one statement.

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Paul Smith
Our scrap speed cameras petition got over 28,000 sigs
The Safe Speed campaign demands a return to intelligent road safety

I have to say, I'm deeply uncomfortable about mixing graphs in the way they are in that thread. Each axis should really have just one definition on any given graph, with the axis definitions clearly stated on the graph. To show probability of death at a given speed, and probability of a particular speed given a death has occurred, on the same graph is just asking for confusion.

I don't think you should be, and not least because there was no conclusion. It's certainly important to explore the relationships looking for good matches and bad matches. We're looking for understanding, remember?

On another level, I share your concern. There's a risk of assuming a relationship where there's none. But we have to guard against unjustified conclusions, not guard against playing with graphs and numbers.

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Paul Smith
Our scrap speed cameras petition got over 28,000 sigs
The Safe Speed campaign demands a return to intelligent road safety

The relationship between conditional probabilities is well understood, and is described in Bayes' Theorem. This is what would be needed to convert between the two distributions in this case.

P (A given B) = P (B given A) * P(A) / P(B)

I think I now better understand Pete's difficulty in using the graphs. It's not a problem of converting between a pdf and cdf, it's one of converting between P(A given B) and P(B given A). To do this requires knowledge of the distributions of A and B, independent of their conditional likelihoods, e.g. you would need to know the distribution of traffic speeds on our roads.

I think that maths is very useful in gaining true understanding. One of the first things they taught us when I did my maths degree is to never state that something is "obvious". If it's "obvious", it should be easy to write a mathematical proof, and if you can't do that, then it can't be "obvious".

I seem to remember that I (we) concluded that the cumulative curve was far more sensitive to the distribution than the underlying function, making it difficult or impossible to work out what the underlying function is without knowledge of the actual distribution.

I suspect that the 4th power curve comes pretty close - and it also matches fairly well with the Minnesota curve, but I'd like to be a bit more confident about it.

Cheers
Peter

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Only when ideology, prejudice and dogma are set aside does the truth emerge - Kepler