SERVING THE QUANTITATIVE FINANCE COMMUNITY

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quantyst
Topic Author
Posts: 632
Joined: June 4th, 2008, 5:08 am

### Fish in a Lake

How would you approximate the number of fish in a lake?Give a description of your method.

rralph
Posts: 42
Joined: November 30th, 2005, 1:16 am

### Fish in a Lake

Randomly catch a number of fish, say 100. Tag each one and release them back into the lake.Call the total number of fish in the lake N. Then the proportion of fish in the whole lake that are now tagged is 100/N.Wait a week and then randomly catch 100 fish. Count the number of tagged fish, call this n. If both samples are random, then you would expect the proportion of fish tagged in this sample (n/100) to equal the proportion of tagged fish in the lake as a whole. Thus:100 / N = n / 100N= 100 x 100 / nIf n is too small then the error in your estimate may be too high and you might need to increase the number of fish in each sample to get a better estimate.

quantyst
Topic Author
Posts: 632
Joined: June 4th, 2008, 5:08 am

### Fish in a Lake

QuoteOriginally posted by: rralphRandomly catch a number of fish, say 100. Tag each one and release them back into the lake.Call the total number of fish in the lake N. Then the proportion of fish in the whole lake that are now tagged is 100/N.Wait a week and then randomly catch 100 fish. Count the number of tagged fish, call this n. If both samples are random, then you would expect the proportion of fish tagged in this sample (n/100) to equal the proportion of tagged fish in the lake as a whole. Thus:100 / N = n / 100N= 100 x 100 / nIf n is too small then the error in your estimate may be too high and you might need to increase the number of fish in each sample to get a better estimate.Great description!Any other methods?

rralph
Posts: 42
Joined: November 30th, 2005, 1:16 am

### Fish in a Lake

Couple of ideas:Analyse the entire foodchain (nutrients, available energy, water depth, predator/prey relationship, lifecycle etc) and build a mathematical modelGet a line of boats with fish finder sonar to do a sweep of the whole lake and see what echo returns they getGet the boats to sweep driftnets through the lake and catch all the fish, then count themAsk a local fisherman/fishery management board how many fish are in the lakeDrop sticks of dynamite into the water and see how many fish float to the surfaceAlternatively, you could use poison or electrocution and see what floats up.Drain the entire lake and then count the fish flapping in the mud/sand
Last edited by rralph on December 11th, 2008, 11:00 pm, edited 1 time in total.

MCarreira
Posts: 1724
Joined: July 14th, 2002, 3:00 am

### Fish in a Lake

How do you take into account the (eally small) newborn fish ? They supposedly cannot be caught with methods intended for larger fish.The population should flutuate a lot (Predator-Prey cicles), so your range should be quite wide.

rralph
Posts: 42
Joined: November 30th, 2005, 1:16 am

### Fish in a Lake

QuoteOriginally posted by: MCarreiraHow do you take into account the (eally small) newborn fish ? They supposedly cannot be caught with methods intended for larger fish.The population should flutuate a lot (Predator-Prey cicles), so your range should be quite wide.Small fry - Do you care? I would think that the most interesting question is how many full grown fish there are. You could use some kind of mathematical model to estimate the number of fry. But, is there any reason why you couldn't just use a very fine mesh net? Tagging could be a problem, but should be solvable.Fluctuations - The question asks for an estimate. I think all my solutions would give that. The error bands would depend on the method chosen. The mathematical model might not settle down to a stable population, but I think that this depends on the particular situation. The other methods would give an estimate of the population at a particular point in time.

MCarreira
Posts: 1724
Joined: July 14th, 2002, 3:00 am

### Fish in a Lake

QuoteOriginally posted by: rralphSmall fry - Do you care? I would think that the most interesting question is how many full grown fish there are. You could use some kind of mathematical model to estimate the number of fry. But, is there any reason why you couldn't just use a very fine mesh net? Tagging could be a problem, but should be solvable.Fluctuations - The question asks for an estimate. I think all my solutions would give that. The error bands would depend on the method chosen. The mathematical model might not settle down to a stable population, but I think that this depends on the particular situation. The other methods would give an estimate of the population at a particular point in time.I agree with you, your first answer is a very efficient way of finding the number of fully grown fish ... I just think that the way the question is interpreted should be taken into account.
Last edited by MCarreira on December 11th, 2008, 11:00 pm, edited 1 time in total.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Fish in a Lake

RRalph offers a good portfolio of techniques and the opportunity for a tasty fish dinner in some cases. Yet I also agree with MCarreira that "# of fish" is not a neat integer due to the presence of fry, eggs, and gravid females.One variant of RRalph's solution is to dump poison in the lake and then be confident that the number of fish is exactly zero!

alphaquantum
Posts: 62
Joined: January 9th, 2008, 11:24 pm

### Fish in a Lake

rralph, your methods remind me about similar problem: you have a tiger in the field, and you have to capture it. Describe how would you do that. 1) Shoot it. Any method works fine - gun, rocket, grenade etc. (not an elegant math solution )2) WLOG assume that the field is a square. Divide into 2 equal parts. Locate where the tiger is. Dived that one in two equal parts, and so on. Eventually (read Cantor), you will capture the tiger. 3) Draw a big circle that contains the tiger. Draw another one, smaller, comparable to tiger's size. Make an inverse transformation (symmetric reflection) of big circle into small circle. The tiger will be inside of the small one. you name other methods. Quantyst, are you looking for theoretical methods, or practical ones? While RRalph's method works fine in theory, in practice to sample randomly fishes is almost nonsense. Admit, you have to fish first to understand that
Last edited by alphaquantum on December 12th, 2008, 11:00 pm, edited 1 time in total.

quantyst
Topic Author
Posts: 632
Joined: June 4th, 2008, 5:08 am

### Fish in a Lake

My purpose was to open a discussion of a problem about which I had some ideas of my own, and wondered how the contributors thought about it. Any type of answer, theoretical or practical, would be good. But one criterion that must be maintained is: whatever solution is proposed, it should not change the number of fish in the lake, or at least, it should have the 'least' effect on the number of fish in the lake. For example, killing all fish is not a good idea.rralph's first solution seems the most sensible of all the methods proposed so far, notwithstanding the difficulty of catching the fish.Here's another crazy, but exorbitantly costly (and possibly a scifi), approach: dispatch a squadron (or army) of robotic submarines to photograph all the fish (several times over, if need be), have the photographs analyzed by a computer to ID each unique fish, thereby coming up with a number for the fish present at a particular point in time in the lake.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Fish in a Lake

QuoteOriginally posted by: quantystHere's another crazy, but exorbitantly costly (and possibly a scifi), approach: dispatch a squadron (or army) of robotic submarines to photograph all the fish (several times over, if need be), have the photographs analyzed by a computer to ID each unique fish, thereby coming up with a number for the fish present at a particular point in time in the lake.Photographing all the fish is just the sonar solution in optical frequencies. Neither photography nor sonar will find camouflaged or mud/crevice-dwelling fish (e.g., flounder and eels).Another sci-fi solution is to build a system modeled on various species of electric fish that can generate and detect ultralow frequency electric fields to detect other fish. These fish often dwell in extremely muddy water and use the fields for navigation and locating prey and mates. A robot field-sensor would find mud/crevice-dwelling fish but might suffer from false positives (e.g. turtles, frogs, crayfish) unless the system had sufficient resolution and pattern recognition abilities to discriminate between the different body shapes or field patterns of different species.Although one can use a fleet (not army or squadron ) of submarine robots to exhaustively sample the lake, biologists often use a transect approach in which they sample a slice of the ecosystem and count/survey the organisms within a series of, say, 3 meter x 3 meter squares or a 3 meter x L slice across the space.

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Fish in a Lake

throw a stick of dynamite in the lake and count the number of dead fish that results.Use that to estimate the density

rralph
Posts: 42
Joined: November 30th, 2005, 1:16 am

### Fish in a Lake

QuoteOriginally posted by: ppauperthrow a stick of dynamite in the lake and count the number of dead fish that results.Use that to estimate the densityThink I already had that one.To be honest, only my first solution was intended to be serious. All my later suggestions were attempts to think outside the box. My gut feeling is that no solution which does zero harm can be 100% accurate.

rralph
Posts: 42
Joined: November 30th, 2005, 1:16 am

### Fish in a Lake

QuoteOriginally posted by: alphaquantumrralph, your methods remind me about similar problem: you have a tiger in the field, and you have to capture it. Describe how would you do that. 1) Shoot it. Any method works fine - gun, rocket, grenade etc. (not an elegant math solution )2) WLOG assume that the field is a square. Divide into 2 equal parts. Locate where the tiger is. Dived that one in two equal parts, and so on. Eventually (read Cantor), you will capture the tiger. 3) Draw a big circle that contains the tiger. Draw another one, smaller, comparable to tiger's size. Make an inverse transformation (symmetric reflection) of big circle into small circle. The tiger will be inside of the small one. you name other methods. Quantyst, are you looking for theoretical methods, or practical ones? While RRalph's method works fine in theory, in practice to sample randomly fishes is almost nonsense. Admit, you have to fish first to understand that Tiger in a field:1) Why not use a tranqualiser? I am happy to kill fish in the persuit of science, but killing a Tiger with a rocket launcher seems a little messy (it wouldn't leave you with much of a trophy either)2) If you have a method for finding the tiger, then why not just throw a net over it. I'm sure a sturdy net would be cheaper than all those fences (Tigers are good climbers, so presumably you would have to electrify them too).3) Again, a lot of fences, people or machinary would be needed to form the perimeter and move it. A large hunk of meat suspended over a camoflaged pit would be much cheaper.Fish:I agree that randomly sampling fish is not easy, but certainly not impossible. As with al good statistical problems a good amount of hand waving and talking about "first order approximations" (or is that physics, maybe stats is "correct sampling methodologies") should take care of that. If that doesn't work, then just stick some large error bands around your final estimate and you can't go wrong.
Last edited by rralph on December 14th, 2008, 11:00 pm, edited 1 time in total.

rralph
Posts: 42
Joined: November 30th, 2005, 1:16 am

### Fish in a Lake

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: quantystHere's another crazy, but exorbitantly costly (and possibly a scifi), approach: dispatch a squadron (or army) of robotic submarines to photograph all the fish (several times over, if need be), have the photographs analyzed by a computer to ID each unique fish, thereby coming up with a number for the fish present at a particular point in time in the lake.Photographing all the fish is just the sonar solution in optical frequencies. Neither photography nor sonar will find camouflaged or mud/crevice-dwelling fish (e.g., flounder and eels).Another sci-fi solution...Why not just use your tricorder to measure the life signs within the lake?

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