Arcsinus law: distinguishing trend from persistency of chance

harrytrader · Dec 7, 2003

Quote from jwlabno:

Distinguishing trend from persistency of chance

Isn't trend something that a statistician would call the persistency of chance? And in any case, why do we need to distinguish between them anyway?

Excellent question and to illustrate it I've even added the regressions lines on each simulation

. I will let you look at them and I will come back tomorrow.

http://www.econometric-wave.com/elitetrader/rw.html
for example (more above as elite forum doesn't accept too many images. I didn't sort them: their order is the original order of simulation)

<img src="http://www.econometric-wave.com/elitetrader/rw_01.gif" >
<img src="http://www.econometric-wave.com/elitetrader/rw_02.gif">
<img src="http://www.econometric-wave.com/elitetrader/rw_03.gif">

franklin · Dec 7, 2003

Quote from harrytrader:

says that even in a VERY LONG GAME of two players which plays a coin toss game with SAME 1/2 PROBABILITY AT EACH BET - that is to say a FAIR game for each player - the periods of gain of each player will tend towards 0 or 1 whereas COMMON SENSE would expect 1/2 that is to say one player will appear as to be practically always the winner and the other always a loser whereas they have exactly the same chance at each bet. Most people will then think that this winner has more ability than the second one whereas in this case it is absolutlety only due to chance !

That's why this law is NOT COMMON SENSE and you can read that it is really not common sense in Hedge Fund Manager Peter Bertein's Book "The Improbable Origins of Modern Wall Street" where he detailed the case (sorry I'm too lazy to translate see p.144 in the french edition or the equivalent in american edition )

If you flip a coin a million times, the chances that it will turn up heads exactly 500,000 times and tails 500,000 times are extremely small. More to the point, the chance that the ratio will be anywhere near 1:1 will be small, since the corresponding piece of the distribution will still be quite small compared to all the places we may end up after a million throws. Seems like common sense to me. Not sure what the big deal is, although I have enjoyed reading the thread. It's just hard to determine who really has skill in a game full of noise.

Mr Subliminal · Dec 7, 2003

Quote from franklin:

More to the point, the chance that the ratio will be anywhere near 1:1 will be small, since the corresponding piece of the distribution will still be quite small compared to all the places we may end up after a million throws.

I don't know what you mean by "anywhere near", but I would expect the ratio to be 1.00 (to 2 decimal places) after a million tosses about 95% of the time, else I'd check the coin. Otherwise, I agree with the gist of your post.

harrytrader · Dec 8, 2003

Perhaps I misunderstood what you mean but it's not about having EXACT number of heads or tails or about counting the number of heads and tails at SNAPSHOT time t= 1000,000, it's about the PERIOD LASTING from time = 0 to time=1000000 where EITHER head will be leading tail OR TAIL will be leading HEAD: you would expect that the most probable case would be that head would be leading tail ABOUT (since EXACT is misplaced in probability context) half of this PERIOD and TAIL would be leading HEAD ABOUT half of this period also by SYMETRY since there is NO SPECIAL CAUSE to favor the one or the other, so that there would be a sort of equilibrium between tail and head. It is in this sense that it would be common sense : but it is misleading in this case, there is a big desequilibrium. So it is about TIME and - not about frequency of head or tails which will tend towards 50% with probability of 1 as n grows per large number law.

Quote from franklin:

If you flip a coin a million times, the chances that it will turn up heads exactly 500,000 times and tails 500,000 times are extremely small. More to the point, the chance that the ratio will be anywhere near 1:1 will be small, since the corresponding piece of the distribution will still be quite small compared to all the places we may end up after a million throws. Seems like common sense to me. Not sure what the big deal is, although I have enjoyed reading the thread. It's just hard to determine who really has skill in a game full of noise.

maxpi · Dec 8, 2003

Quote from harrytrader:

Perhaps I misunderstood what you mean but it's not about having EXACT number of heads or tails or about counting the number of heads and tails at SNAPSHOT time t= 1000,000, it's about the PERIOD LASTING from time = 0 to time=1000000 where EITHER head will be leading tail OR TAIL will be leading HEAD: you would expect that the most probable case would be that head would be leading tail ABOUT (since EXACT is misplaced in probability context) half of this PERIOD and TAIL would be leading HEAD ABOUT half of this period also by SYMETRY since there is NO SPECIAL CAUSE to favor the one or the other, so that there would be a sort of equilibrium between tail and head.

You would expect 50%. The truth is that at times it will get way out of equilibrium. It is surprising. Traders are always surprised when they blow out their accounts too. It's just that the "worst case scenario" for number of consecutive heads or tails will happen eventually and take out the entire account. Thus we have bet sizing based on win/loss ratio and average win to average loss ratio which protects against that. I think perhaps it is better to have enough money in a slower traded account to cover the eventuality of blowing out your heavily margined account and trade like crazy knowing that it will blow out and knowing that you have capital to start again.

Mr Subliminal · Dec 8, 2003

harry, assume that we're about to sit down and play a fair coin toss game for 100 tosses. How can you utilize the arcsinus law to your advantage?

franklin · Dec 9, 2003

Quote from harrytrader:

Perhaps I misunderstood what you mean but it's not about having EXACT number of heads or tails or about counting the number of heads and tails at SNAPSHOT time t= 1000,000, it's about the PERIOD LASTING from time = 0 to time=1000000 where EITHER head will be leading tail OR TAIL will be leading HEAD: you would expect that the most probable case would be that head would be leading tail ABOUT (since EXACT is misplaced in probability context) half of this PERIOD and TAIL would be leading HEAD ABOUT half of this period also by SYMETRY since there is NO SPECIAL CAUSE to favor the one or the other, so that there would be a sort of equilibrium between tail and head. It is in this sense that it would be common sense : but it is misleading in this case, there is a big desequilibrium. So it is about TIME and - not about frequency of head or tails which will tend towards 50% with probability of 1 as n grows per large number law.

If you have someone start walking through a forest in a certain direction (w/o paths, etc.), few people would expect them to come out anywhere near the "projected" point based on the direction in which they were first nudged. Moreover, if we found out that the subject had wandered significantly away from the initial direction by the halfway point, few of us would expect them to magically wander back in the direction of the projected point based on statistical correctness. I think most people would see this as "common sense", and I don't see, from your comments above, that you are saying much more than this.

Mr Subliminal · Dec 9, 2003

Just ran a few sims because I didn't believe you, harry. For a million coin tosses, I would have expected the "time" heads is ahead of tails to be roughly equal to the "time" tails is ahead of heads. Well, the results are as you say, and I agree with you that it is counterintuitive. All I can think of at this late hour is that we are looking at the distribution of

X - (N - X)

where X = no of heads, N = no of tosses

= 2X - N

The mean of 2X - N is 0, and the variance is 4 times the variance of X, which could partially account for the results. Or not. But I see no sine functions anywhere.

Now please explain to me how you and Bernstein progress from this to an equity curve?

OddTrader · Dec 9, 2003

Not sure these links would be relevant:

http://www.chartsmart.com/specify.html

Q
min(#close#,#c1#,#c2#,#c3#) gives the minimum of the last 3 days closes
sqrt(#volume#) square root of the volume
abs() absolute value
asin() arcsine
log10() log 10
log() natural log
mod() remainder from a division operation (e.g. mod(10,3)=1
UQ

http://www.tradingsolutions.com/functions/Arcsine.html

Q
The Arcsine function calculates the arcsine of a value in radians.
UQ

harrytrader · Dec 9, 2003

This just demonstrates that it is not common sense since you don't just understand what it is about: it is not about being far away from the "projected" point, it is about being "nearly always" on the SAME SIDE of the "projected" point "most" of the time whereas you would expect that the OTHER SIDE would also be visited about half of the time which is not true.

Quote from franklin:

If you have someone start walking through a forest in a certain direction (w/o paths, etc.), few people would expect them to come out anywhere near the "projected" point based on the direction in which they were first nudged. Moreover, if we found out that the subject had wandered significantly away from the initial direction by the halfway point, few of us would expect them to magically wander back in the direction of the projected point based on statistical correctness. I think most people would see this as "common sense", and I don't see, from your comments above, that you are saying much more than this.