Saturday, May 23, 2015

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Chebyshev Polynomials For Financial Data

Being cognisant of that intellectual giant, Donncha MacNiallis's view of The Pensive Quill, Guest writer Daithi O’Donnabhain enters the cesspit with this viciously anti-Sinn Fein piece.

Daithi O’Donnabhain



Given the recent Bobby Sands anniversary, I was thinking about how many Maths focussed people appreciate poetry, but there is little obvious traffic going the other way. I decided to email Anthony with the idea of testing a Maths focussed article out on audiences not specifically located in this field. The aim of this article is to give an example of analysis with financial data, and ultimately demystify a rather obtuse area, that is usually badly detailed in really expensive books. It wont be to everyone's taste, but then neither is poetry or opinion pieces.



























31 comments :

DaithiD said...

Thanks for including this Anthony. Definitely ( and defiantly!) niche.

AM said...

DaithiD,

Henry Joy or Alfie Gallagher with their range of interests may well understand it but I have not a clue LOL.

I trust your judgement on it and as you know we carry a wide range of material. Thanks for pushing it our way.

larry hughes said...

as soon as I run outa fingers I'm stumped.

DaithiD said...

AM,
Im sure those people (not Alfie or Larry) who have a theory , not just on individual instrument prices, but the whole freaking financial system, will grasp it.Its commendable the platform you provide to people.

With reference to Donncha / SF and the fabled cesspit prescence,I will paraphrase Tom Barry : they had gone in the cesspit to destroy us and our nation and down after them we had to go

Glen Shane-Pass said...

Very interesting article for which many thanks. Back in the stone age of computing Chebyshev polynomials were used to approximate some of the transcendental functions in the ROM of the Sinclair ZX Spectrum.

I suggest the author considers a follow up article on the McLaurin-Taylor series as another method of approximating a function using a finite number of terms.

I would also be interested to hear his take on the use of the Taniyama-Shimura Conjecture by Andrew Wiles in his proof of Fermat's Last Theorem.

More useful to the general readership would be an explanation of the application of mathematics to the theory of politics that allows the Sinn Féin leadership to make 2 + 2 = 5. I fear this might lie beyond the scope of current knowledge.

AM said...

DaithiD,

Donncha might not just be the dunce everybody dismisses him as - he will point to Glen_Shane Pass and claim that both you and he collaborated on this to undermine Sinn Fein LOL.

DaithiD said...

Glen-Shane Pass, thanks for your suggestions, it’s a while since I did MacLaurin-Taylor series stuff but it thought it was approximating an underlying function i.e. you would need to know the polynomial before implementing it? You would still need to indentify an intial model, which is the trickiest part, unless ive misunderstood your point?
I will look into Fermats last theorem stuff.
With regards to Sinn Fein making their members believe 2+2=5, I would imagine it’s a similar model to those that descrive herding behaviour (like in sheep!), probably something like the Ising Lattice , which helpfully was the focus of my third year dissertation.

Robert said...

Anthony,

'Being cognisant of that intellectual giant, Donncha MacNiallis's view of The Pensive Quill..'

Keep an eye on Twitter, having read this he will probaly be calling for that guy 'Chevy Chase's' house to be picketed, his polynomials posing a threat to the peace process!

Alfie Gallagher said...

DaithiD,

This is an interesting article. Indeed, it is particularly relevant to me as I am currently revising polynomial interpolation for my upcoming exam in my OU mathematical modelling course. So thank you for helping my revision!

I probably won't get a chance to give this piece the close reading it deserves until after I sit my exam early next month, but I hope to post a few questions/comments at some stage afterwards.

Anyway, until then, well done. It is to see mathematical ideas discussed on TPQ.

Alfie Gallagher said...

PS. If you are feeling even more adventurous, Daithi, you might consider putting together a layman's guide to some of the mathematical ideas you have used in your article -- namely, the concept of a function and the practical purpose of interpolation. That may help those who have no experience with such concepts understand your article better.

Alfie Gallagher said...

Anthony,

Here is a really good explanation of what a function is. Once you understand functions, you will be able to understand the concept of interpolation -- that is, finding and using a simpler, more manageable function (in Daithi's case, a Chebyshev polynomial) that is a sufficiently accurate approximation of a complicated, awkward function (in Daithi's case, the function concerns how the price of a stock varies with time).

DaithiD said...

Thanks Alfie,
Its hard to know what level to pitch it so it has fluency but is fully detailed. I appreciate your comment helps readers gain a further basis for understanding. Polynomials are not my area of expertise, as far as i can tell they are not suitable for trading system design as they have 'curve fitting' bias, and so lose any predictive quality for unseen data sets. Please get my personal email from Anthony if you have any further questions just incase I miss them on this section at a later date. If you have any ideas you want to monetise : im your guy too!
I have no plans for a follow up article on this theme as of yet, but I agree it good of Anthony to put this out there.

DaithiD said...

PS Alfie, best of wishes for your exam.

Glen Shane-Pass said...

DaithiD: I'm sure you're correct re MacLaurin-Taylor series. It's a long time since I've dome much serious maths.

Simon Singh's book on Fermat is well worth reading.

Herding behaviour (like in sheep!) sounds about right. The dogs circle round and the sheep's response to the possible predators is to bunch together. Once they're in a bunch they can be moved along as a group, and any escapees are immediately chased back in. Sounds about right.

I'm sure there's a witty comment about the Louth TD's knowledge of economics being helped by his expertise in imaginary numbers.

On a more serious note, very sad to hear of the death of John Nash. I'm sure his contributions to game theory were put to good use by SF.

Tain Bo said...

Alfie,

all the very best on your exams and thanks for the input on this one

Tain Bo said...

Daithi,
I laughed thinking Donncha would reply it is as easy as π though reading other articles it appears that cake is more complicated than Pi.

Math language for the better part is foreign a clash between alphanumeric and alphabetical order.

If I say π reads like an odd symbol = C over d the non-math reader would say it does not make any sense and in a sense they would be right as Pi is an irrational number.

3.1415926535897932384626433832795028841971693993751… and on and on makes little sense but is more understandable it just reads as if I randomly hit numbers on the keyboard.

Why Pi might be asked, you mentioned poetry and there are poems in math Pilish writings or piems a combination of poem and Pi..

3 1 4 I 5 9 2 6 5 3
Can. I read a point backwards or forget finer art

There are better examples I just use it as a tool for memory, it is fun, and perhaps people shy away from math, as it appears drab and joyless even though it dominates our lives.

I would have suggested something more fun for the Great mind of Donncha a variation on Buffon’s Needle problem he could amuse the other giant minds with a box of matches and some equally spaced floorboards.
I am prone to the irrational but enjoyed your point though I figure they prefer pie to Pi.

DaithiD said...

Tain, Do you mean literally reading out the Pi as a poem? That sounds like an old Trigger Happy Tv sketch where he destroyed an open mic poetry by just reading out numbers. Truly mind numbing.
Yup its another thing that humbles human minds, we cant relate the circumference of a circle to it diameter, we can only approximate with a symbol. Could get quite spiritual on this topic, about God approximating that which cant be expressed in numbers etc.

Glen Shane, and in an Irish Republican context, the dogs are in the pay of the wolves.

AM said...

DaithiD,

that poem a million was brilliant - I still refer to it when winding people up if they go on too long. Dom Jolly - after 200 they started to realise what was going on!!

DaithiD said...

AM, thats the one!
Its a similar connundrum you are faced with on this site, where the unlimited free expression of one can impinge on all the others. Wonder what number of comments Che will get to Bates/Wilkes before he realises he needs to find another open mic night?

AM said...

DaithiD,

who knows, who cares? He is over with the bonkers crew now. John Gidman's Ghost effectively finished him off. After that we lost interest.

Michael Craig said...

Looks like the design data for a loudspeaker. Why go to all this trouble in economics - just remove the profit motive, it would simplify the equation.

DaithiD said...

Micheal Craig,
What part of a loudspeaker? Sound waves would be just superpositions of Sine/Cosine waves, so quite different to this. Additionally there is no “profit” factored into any of the equations. But even if there was,if you start from a point of individual liberty, having profit motives is good way of allocating resources without the need for central planning, things organise themselves. Like seeds growing toward the light.

Michael Craig said...

'what part of the loudspeaker?', The enclosure.
Ported enclosures use Chebyshev filters to control the pass-band of frequencies emitting from the loudspeaker box. Just an observation that your curves resemble those of the filters, - I'm not a mathematician'

Financial data is used by those who invest in activity. Their only motivation in doing this is to get a reward. I'll have the central planning thanks.

Tain Bo said...

Daithi,
Definitely not for the obvious reason but I did get a laugh thinking about it.
It is simply a word play on Pi (say, the first 100 digits) assigning an equal word value to the corresponding listing of Pi. It depends on memory and that varies mine being not so sharp.

I have a notebook I use occasionally and with a pencil jot down words picking up at the last number. It is just having fun with math and words I apply much the same when playing chess I plot the movement on a graph not only to look at the finished pattern but also to improve my memory when playing against another.

I think I will skip sacred geometry, as cake is sounding as irrational as I am. I did not know some bloke listed a million but got a laugh at Anthony’s use for it.

AM said...

Michael,

central planning might have greater appeal if we could find some way of dispensing with the central planner. He will defeat the purpose of central planning. Socialism has not yet come up with a workable method of having one without the other.

Michael Craig said...

Anthony, I totally agree - one of the reasons why I don't belong to a Marxist party, but given the choice between Laissez-faire capitalism and central planning I'd have to choose the latter.

Daithi said, 'having profit motives is good way of allocating resources'. Well of course it isn't, and this is proven by the vast inequality in incomes today where about 70 of the richest people in the World have more wealth than 3.6 billion of the poorest.

In my book, making money from other peoples' activity is wrong and those who do it are the first to complain about welfare and the health service, claiming that public services 'spend other people's money'.

I'm always open to ideas of how we could have grass roots control of a centrally planned economy.
Of course as a commie my vision is of a moneyless society -each giving according to his/her ability, and taking according to his/her need.

AM said...

Michael,

I prefer neither. Each system creates three types - those at the bottom, the middle and the top. When socialism had its chance to usurp the power grab it replaced it with one of its own. The Market is an ideological fallacy that has never worked for the bulk of people. Yet people might find it difficult to act upon Marx's warning of socialism or barbarism when they have enough experience that socialism as it was practiced was barbarism. The choice of death camp isn't really a good one, whether it be one of Hitler's or one of Stalin's.

Michael Craig said...

There are many reasons why socialism didn't work in the USSR. It never reached what Marx defined as socialism because the ideology got lost in the deformation. Yes there was planning, but as you say it created tiers and everything was from the top down. It suited Stalin to call his system socialism and it also suited the west as it could point out to workers that 'democracy' allowed more personal freedom. In reality socialism hasn't existed. You've read plenty of Marxist stuff so you'll know that 'socialism' is only a transitional system based on paying people according to their activity/work and planning the economy around this.
I think we need a clean sheet especially since there will never be full employment or the need for everyone to work full-time, therefore a starting point might be to find a way of ensuring incomes for all rather than jobs.
Barbarism has arrived. the media can blame religious cults and madmen for the horrific things which are happening to innocent people all around the world, but it can all be laid at the door of capitalist expansion.

Chomsky once asked of the attempts at socialism, ' where is the workers' control?'

AM said...

Which will not cure the healthy scepticism towards socialism when given the opportunities it has had it never managed to exist. I doubt we can blame Stalin alone. Lenin was as much a centralist dictator as Stalin. Once he said that it was dictatorship over the proletariat rather than a dictatorship of the proletariat (and in classic Marxism the dictatorship of the proletariat is indivisible from Socialism and universal suffrage - see Marx on the Paris Commune) he consigned socialism to the dustbin. Socialism it seems is like what a 19th century art critic said of Berlin - always in a state of becoming and never in a state of being. I think the serious problem of barbarism that confronts humanity is the inevitable lust for power of which capitalism is a manifestation rather than a cause.

Michael Craig said...

I agree with your comments about Lenin's part in the failure, and revolutionaries who still hold onto 'democratic centralism' will continue to fail.

DaithiD said...

Micheal Craig, thankyou I never knew that. A very productive comments section here.
With regards to socialism, its like talking religion with AM, my eyes glaze over and I need to check my underwear for accidents.