thus spake a.k.tag:www.thusspakeak.com,2013-04-24://1Movable Type 5.2.13Further Still On A Calculus Of Differences - studenttag:www.thusspakeak.com,2017:/student//12.2392017-09-15T19:00:00Z2017-09-15T19:01:15Zstudent
For some time now my fellow students and I have been whiling away our spare time considering the similarities of the relationships between sequences and series and those between the derivatives and integrals of functions. Having defined differential and integral operators for a sequence s_{n} with
Δ s_{n} = s_{n} - s_{n-1}
and
n
Δ^{-1}s_{n} =
Σ
s_{i}
i = 1
where Σ is the summation sign, we found analogues for the product rule, the quotient rule and the rule of integration by parts, as well as formulae for the derivatives and integrals of monomial sequences, being those whose terms are non-negative integer powers of their indices, and higher order, or repeated, derivatives and integrals in general.
We have since spent some time considering how we might solve equations relating sequences to their derivatives, known as differential equations when involving functions, and it is upon our findings that I shall now report.
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The Best Laid Schemata - a.k.tag:www.thusspakeak.com,2017:/ak//9.2382017-09-01T19:00:00Z2017-09-01T19:18:22Z
We have seen how we can exploit a simple model of biological evolution, known as a genetic algorithm, to search for global maxima of functions, being those points at which they return their greatest values.
This model treated the function being optimised as a non-negative measure of the fitness of individuals to survive and reproduce, replacing negative results with zero, and represented their chromosomes with arrays of bits which were mapped onto its arguments by treating subsets of them as integers that were linearly mapped to floating point numbers with given lower and upper bounds. It simulated sexual reproduction by splitting pairs of the chromosomes of randomly chosen individuals at a randomly chosen position and swapping their bits from it to their ends, and mutations by flipping randomly chosen bits from the chromosomes of randomly chosen individuals. Finally, and most crucially, it set the probability that an individual would be copied into the next generation to its fitness as a proportion of the total fitness of the population, ensuring that that total fitness would tend to increase from generation to generation.
I concluded by noting that, whilst the resulting algorithm was reasonably effective, it had some problems that a theoretical analysis would reveal and that is what we shall look into in this post.
a.k.
We have seen how we can exploit a simple model of biological evolution, known as a genetic algorithm, to search for global maxima of functions, being those points at which they return their greatest values.
This model treated the function being optimised as a non-negative measure of the fitness of individuals to survive and reproduce, replacing negative results with zero, and represented their chromosomes with arrays of bits which were mapped onto its arguments by treating subsets of them as integers that were linearly mapped to floating point numbers with given lower and upper bounds. It simulated sexual reproduction by splitting pairs of the chromosomes of randomly chosen individuals at a randomly chosen position and swapping their bits from it to their ends, and mutations by flipping randomly chosen bits from the chromosomes of randomly chosen individuals. Finally, and most crucially, it set the probability that an individual would be copied into the next generation to its fitness as a proportion of the total fitness of the population, ensuring that that total fitness would tend to increase from generation to generation.
I concluded by noting that, whilst the resulting algorithm was reasonably effective, it had some problems that a theoretical analysis would reveal and that is what we shall look into in this post.
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Share And Share Alike - baron m.tag:www.thusspakeak.com,2017:/baron_m//11.2372017-08-18T19:00:00Z2017-08-18T18:59:30Z
Sir R----- my fine fellow! Come join me in quenching this summer eve's thirst with a tankard of cold ale! Might I presume that your thirst for wager is as pressing as that for refreshment?
I am gladdened to hear it Sir! Gladdened to hear it indeed!
This day's sweltering heat has put me in mind of the time that I found myself temporarily misplaced in the great Caloris rainforest on Mercury. I had been escorting the Velikovsky expedition, which had secured the patronage of the Russian Imperial court for its mission to locate the source of the Amazon, and on one particularly close evening our encampment was attacked by a band of Salamanders which, unlike their diminutive Earthly cousins, stood some eight feet tall and wielded vicious looking barbed spears.
baron m.
Sir R----- my fine fellow! Come join me in quenching this summer eve's thirst with a tankard of cold ale! Might I presume that your thirst for wager is as pressing as that for refreshment?
I am gladdened to hear it Sir! Gladdened to hear it indeed!
This day's sweltering heat has put me in mind of the time that I found myself temporarily misplaced in the great Caloris rainforest on Mercury. I had been escorting the Velikovsky expedition, which had secured the patronage of the Russian Imperial court for its mission to locate the source of the Amazon, and on one particularly close evening our encampment was attacked by a band of Salamanders which, unlike their diminutive Earthly cousins, stood some eight feet tall and wielded vicious looking barbed spears.
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It's All In The Genes - a.k.tag:www.thusspakeak.com,2017:/ak//9.2362017-08-04T19:00:00Z2017-08-04T19:05:22Z
Last time we took a look at the simulated annealing global minimisation algorithm which searches for points at which functions return their least possible values and which drew its inspiration from the metallurgical process of annealing which minimises the energy state of the crystalline structure of metals by first heating and then slowly cooling them.
Now as it happens, physics isn't the only branch of science from which we can draw inspiration for global optimisation algorithms. For example, in biology we have the process of evolution through which the myriad species of life on Earth have become extraordinarily well adapted to their environments. Put very simply this happens because offspring differ slightly from their parents and differences that reduce the chances that they will survive to have offspring of their own are less likely to be passed down through the generations than those that increase those chances.
Noting that extraordinarily well adapted is more or less synonymous with near maximally adapted, it's not unreasonable to suppose that we might exploit a mathematical model of evolution to search for global maxima of functions, being those points at which they return their greatest possible values.
a.k.
Last time we took a look at the simulated annealing global minimisation algorithm which searches for points at which functions return their least possible values and which drew its inspiration from the metallurgical process of annealing which minimises the energy state of the crystalline structure of metals by first heating and then slowly cooling them.
Now as it happens, physics isn't the only branch of science from which we can draw inspiration for global optimisation algorithms. For example, in biology we have the process of evolution through which the myriad species of life on Earth have become extraordinarily well adapted to their environments. Put very simply this happens because offspring differ slightly from their parents and differences that reduce the chances that they will survive to have offspring of their own are less likely to be passed down through the generations than those that increase those chances.
Noting that extraordinarily well adapted is more or less synonymous with near maximally adapted, it's not unreasonable to suppose that we might exploit a mathematical model of evolution to search for global maxima of functions, being those points at which they return their greatest possible values.
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On Divisions - studenttag:www.thusspakeak.com,2017:/student//12.2352017-07-21T19:00:00Z2017-07-21T20:19:50Z
The Baron's game most recent game consisted of a series of some six wagers upon the toss of an unfair coin that turned up one side nine times out of twenty and the other eleven times out of twenty at a cost of one fifth part of a coin. Sir R----- was to wager three coins from his purse upon the outcome of each toss, freely divided between heads and tails, and was to return to it twice the value he wagered correctly.
Clearly, our first task in reckoning the fairness of this game is to figure Sir R-----'s optimal strategy for placing his coins. To do this we shall need to know his expected winnings in any given round for any given placement of his coins.
student
The Baron's game most recent game consisted of a series of some six wagers upon the toss of an unfair coin that turned up one side nine times out of twenty and the other eleven times out of twenty at a cost of one fifth part of a coin. Sir R----- was to wager three coins from his purse upon the outcome of each toss, freely divided between heads and tails, and was to return to it twice the value he wagered correctly.
Clearly, our first task in reckoning the fairness of this game is to figure Sir R-----'s optimal strategy for placing his coins. To do this we shall need to know his expected winnings in any given round for any given placement of his coins.
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Annealing Down - a.k.tag:www.thusspakeak.com,2017:/ak//9.2342017-07-07T19:00:00Z2017-07-07T19:11:16Z
A few years ago we saw how we could search for a local minimum of a function, being a point for which it returns a lesser value than any in its immediate vicinity, by taking random steps and rejecting those that lead uphill; an algorithm that we dubbed the blindfolded hill climber. Whilst we have since seen that we could progress towards a minimum much more rapidly by choosing the directions in which to step deterministically, there is a way that we can use random steps to yield better results.
a.k.
A few years ago we saw how we could search for a local minimum of a function, being a point for which it returns a lesser value than any in its immediate vicinity, by taking random steps and rejecting those that lead uphill; an algorithm that we dubbed the blindfolded hill climber. Whilst we have since seen that we could progress towards a minimum much more rapidly by choosing the directions in which to step deterministically, there is a way that we can use random steps to yield better results.
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Further On A Calculus Of Differences - studenttag:www.thusspakeak.com,2017:/student//12.2332017-06-16T19:00:00Z2017-06-16T19:02:42Zstudent
As I have previously reported, my fellow students and I have found our curiosity drawn to the calculus of sequences, in which we define analogues of the derivatives and integrals of functions for a sequence s_{n} with the operators
Δ s_{n} = s_{n} - s_{n-1}
and
n
Δ^{-1}s_{n} =
Σ
s_{i}
i = 1
respectively, where Σ is the summation sign, for which we interpret all non-positively indexed elements as zero.
I have already spoken of the many and several fascinating similarities that we have found between the derivatives of sequences and those of functions and shall now describe those of their integrals, upon which we have spent quite some mental effort these last few months.
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Copulating Normally - a.k.tag:www.thusspakeak.com,2017:/ak//9.2322017-06-02T19:00:00Z2017-06-10T03:44:14Z
Last year we took a look at multivariate uniformly distributed random variables, which generalise uniform random variables to multiple dimensions with random vectors whose elements are independently uniformly distributed. We have now seen how we can similarly generalise normally distributed random variables with the added property that the normally distributed elements of their vectors may be dependent upon each other; specifically that they may be correlated.
As it turns out, we can generalise this dependence to arbitrary sets of random variables with a fairly simple observation.
a.k.
Last year we took a look at multivariate uniformly distributed random variables, which generalise uniform random variables to multiple dimensions with random vectors whose elements are independently uniformly distributed. We have now seen how we can similarly generalise normally distributed random variables with the added property that the normally distributed elements of their vectors may be dependent upon each other; specifically that they may be correlated.
As it turns out, we can generalise this dependence to arbitrary sets of random variables with a fairly simple observation.
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Divisions - baron m.tag:www.thusspakeak.com,2017:/baron_m//11.2302017-05-19T19:00:00Z2017-05-19T19:00:27Zbaron m.
Greetings Sir R-----! I trust that I find you in good spirit? Will you join me in a draught of this rather fine Cognac and perchance some sporting diversion?
Good man!
I propose a game that ever puts me in mind of an adventure of mine in the town of Bağçasaray, where I was posted after General Lacy had driven Khan Fetih Giray out from therein. I had received word that the Khan was anxious to retake the town and been given orders to hold it at all costs.
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The Cumulative Distribution Unction - a.k.tag:www.thusspakeak.com,2017:/ak//9.2292017-05-05T19:00:00Z2017-05-26T23:56:29Za.k.
We have previously seen how we can generalise normally distributed random variables to multiple dimensions by defining vectors with elements that are linear functions of independent standard normally distributed random variables, having means of zero and standard deviations of one, with
Z' = L × Z + μ
where L is a constant matrix, Z is a vector whose elements are the independent standard normally distributed random variables and μ is a constant vector.
So far we have derived and implemented the probability density function and the characteristic function of the multivariate normal distribution that governs such random vectors but have yet to do the same for its cumulative distribution function since it's a rather more difficult task and thus requires a dedicated treatment, which we shall have in this post.
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On Turnabout Is Fair Play - studenttag:www.thusspakeak.com,2017:/student//12.2272017-04-21T19:00:00Z2017-04-21T19:04:40Z
Last time they met, the Baron challenged Sir R----- to turn a square of twenty five coins, all but one of which the Baron had placed heads up, to tails by flipping vertically or horizontally adjacent pairs of heads.
As I explained to the Baron, although I'm not at all sure that he was following me, this is essentially the mutilated chess board puzzle and can be solved by exactly the same argument. Specifically, we need simply imagine that the game were played upon a five by five checker board...
student
Last time they met, the Baron challenged Sir R----- to turn a square of twenty five coins, all but one of which the Baron had placed heads up, to tails by flipping vertically or horizontally adjacent pairs of heads.
As I explained to the Baron, although I'm not at all sure that he was following me, this is essentially the mutilated chess board puzzle and can be solved by exactly the same argument. Specifically, we need simply imagine that the game were played upon a five by five checker board...
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Multiple Multiply Normal Functions - a.k.tag:www.thusspakeak.com,2017:/ak//9.2262017-04-07T19:00:00Z2017-05-26T23:53:27Za.k.
Last time we took a look at how we could define multivariate normally distributed random variables with linear functions of multiple independent standard univariate normal random variables.
Specifically, given a Z whose elements are independent standard univariate normal random variables, a constant vector μ and a constant matrix L
Z' = L × Z + μ
has linearly dependent normally distributed elements, a mean vector of μ and a covariance matrix of
Σ' = L × L^{T}
where L^{T} is the transpose of L in which the rows and columns are switched.
We got as far as deducing the characteristic function and the probability density function of the multivariate normal distribution, leaving its cumulative distribution function and its complement aside until we'd implemented both them and the random variable itself, which we shall do in this post.
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On A Calculus Of Differences - studenttag:www.thusspakeak.com,2017:/student//12.2252017-03-17T19:00:00Z2017-03-17T20:06:04Zstudent
The interest of my fellow students and I has been somewhat piqued of late by a curious similarity of the relationship between sequences and series to that between the derivatives and integrals of functions. Specifically, for a function f taking a non-negative argument x, we have
x
F(x) =
∫
f(x) dx
0
f(x) =
d
F(x)
dx
and for a sequence s having terms
s_{1}, s_{2}, s_{3}, ...
we can define a series S with terms
n
S_{n} = s_{1} + s_{2} + s_{3} + ... + s_{n} =
Σ
s_{i}
i = 1
where Σ is the summation sign, from which we can recover the terms of the sequence with
s_{n} = S_{n} - S_{n-1}
using the convention that S_{0} equals zero.
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.
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Every Which Way Is Normal - a.k.tag:www.thusspakeak.com,2017:/ak//9.2242017-03-03T20:00:00Z2017-03-03T20:02:01Z
A few months ago we saw how we could generalise the concept of a random variable to multiple dimensions by generating random vectors rather than numbers. Specifically we took a look at the multivariate uniform distribution which governs random vectors whose elements are independently uniformly distributed.
Whilst it demonstrated that we can find multivariate versions of distribution functions such as the probability density function, the cumulative distribution function and the characteristic function, the uniform distribution is fairly trivial and so, for a more interesting example, this time we shall look at generalising the normal distribution to multiple dimensions.
a.k.
A few months ago we saw how we could generalise the concept of a random variable to multiple dimensions by generating random vectors rather than numbers. Specifically we took a look at the multivariate uniform distribution which governs random vectors whose elements are independently uniformly distributed.
Whilst it demonstrated that we can find multivariate versions of distribution functions such as the probability density function, the cumulative distribution function and the characteristic function, the uniform distribution is fairly trivial and so, for a more interesting example, this time we shall look at generalising the normal distribution to multiple dimensions.
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Turnabout Is Fair Play - baron m.tag:www.thusspakeak.com,2017:/baron_m//11.2232017-02-17T20:00:00Z2017-02-17T20:04:31Z
Why, you look chilled to the bone Sir R-----! Come sit by the hearth and warm yourself whilst I fetch you a medicinal glass of brandy.
To your very good health sir! Will you join me in a wager whilst you recover?
Good show!
I propose a game that I learned upon the banks of the river Styx whilst my fellow travellers and I were waiting for the ferry. This being the third time that I had died, I was quite accustomed to the appalling service quality of the Hadean public transport system and so was most appreciative of a little sport to pass the time.
baron m.
Why, you look chilled to the bone Sir R-----! Come sit by the hearth and warm yourself whilst I fetch you a medicinal glass of brandy.
To your very good health sir! Will you join me in a wager whilst you recover?
Good show!
I propose a game that I learned upon the banks of the river Styx whilst my fellow travellers and I were waiting for the ferry. This being the third time that I had died, I was quite accustomed to the appalling service quality of the Hadean public transport system and so was most appreciative of a little sport to pass the time.
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