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Posts tagged with central limit theorem

In smile modelling or pictures of the term structure of options or bonds, one speaks of a “volatility landscape” or “risk landscape”.

http://www.science.uva.nl/research/scs/edu/imcs/research/cf-03.png
http://m.rgbimg.com/cache1roJ8k/users/m/mi/micromoth/600/nfUPWLi.jpg
http://www.timera-energy.com/wp-content/uploads/2012/04/Vol-Concepts.png

http://www.math.uni-trier.de/~sachs/scientific_interests/projects/volsurface.png

That is assigning numbers to price-points and time-points; contingencies form a “surface”.

http://upload.wikimedia.org/wikipedia/commons/b/b1/Ivsrf.gif
http://4.bp.blogspot.com/-gwQjSSHR5RY/TxsIHBjijgI/AAAAAAAAAh0/rR0nzyNYT_Y/s1600/farm-flooded%2Brice%2Bfield.jpg
http://www.timera-energy.com/wp-content/uploads/2012/04/Volatility-Surface.jpg

I tend to forget that for farmers, the actual landscape—the actual (sur)face of the Earth is itself the risk landscape.

http://static.auctionservices.com/images/1325775/DCP_9266.JPG
http://www.lavocedelnordest.eu/wp-content/uploads/2014/01/storm-and-rain-over-green-field-jennifer-brindley.jpg
http://images.secondspace.com/p/SUP/5F/C6/93/0F/31/7CXL_14.jpg
http://images.china.cn/attachement/jpg/site1007/20120613/001422373e19114278d84a.jpg

  • Hillocks get more sun (could be good or bad depending on the cooling-degree days
    New Hampshirel CDD 1895-2009
    , the chance of frost, and the abundance of rain)
  • Dells and ravines get more water—which could be good if it’s dry,
    http://www.livemint.com/rf/Image-621x414/LiveMint/Period1/2013/05/23/Photos/drought1--621x414.JPG
    http://news.bbc.co.uk/media/images/38159000/jpg/_38159395_farmer_afp_300.jpg
    http://www.oxfam.org/sites/www.oxfam.org/files/imagecache/space_fullwidth/media/reactions/48865scr-climate-rice-paddy-620.jpg
    http://newsimg.bbc.co.uk/media/images/45437000/jpg/_45437836_farmerindiaafpgetty466.jpg
    or catastrophic in case of flood.
    http://cache.boston.com/universal/site_graphics/blogs/bigpicture/iowa_06_17/iowa9.jpg
    http://assets.inhabitat.com/wp-content/blogs.dir/1/files/2013/01/hay-in-a-flooded-field-537x356.jpg
    http://media.al.com/wire/photo/9657830-large.jpg
    http://4.bp.blogspot.com/_H91cnPsZsBw/TJmZCpXaUiI/AAAAAAAACWk/mOciwQsWgRs/s320/floodedfield-Apaj.JPG
    http://www.johnharveyphoto.com/Life/10_2009/FloodedFieldOfPumpkins_300W.jpg
  • Of course that depends on the crop type. Rice wants to be flooded. Even I know that.
    http://www.vosizneias.com/assets/uploads/news_photos/thumbnails/800_omagpvuyjmnichh6ug5qf8lqjg8vhwkf.jpg
    http://s1.reutersmedia.net/resources/r/?m=02&d=20111107&t=2&i=528335754&w=580&fh=&fw=&ll=&pl=&r=img-2011-11-07T103647Z_01_NOOTR_RTRMDNC_0_India-603600-1
    http://www.vosizneias.com/assets/uploads/news_photos/thumbnails/800_p0gqgzk8813x741owemvapg4pdleoa4g.jpg

  • And just like derivatives, agriculture has its term contingencies. Water in autumn is too late to grow the baby saplings but, too, a flood might not be as bad for the granary as it was for spring's seedlings.
    http://wuppenif.files.wordpress.com/2012/03/eastern-ontario-farmland-soaked-in-spring-runoff.jpg
    http://bloximages.chicago2.vip.townnews.com/santamariatimes.com/content/tncms/assets/v3/editorial/4/8a/48aa336a-551a-11e0-91ef-001cc4c03286/4d8998f8c2a5c.preview-620.jpg
  • Symbiosis between “funded” (planted) neighbours could result in a “value-added merger” if, for example, the bugs which are attracted to one plant fend off another plant’s predators.
  • A monogenetic crop could all be wiped out by the same disease.
  • Diversification, then, would seem to mirror finance as one wants to invest fully in the “cash crop” (let’s say a junk bond), but risks increase as eggs are concentrated in one basket.
  • Or say you wish that lucrative bridge loan’s IRR were applied to your entire portfolio—perhaps this is like a plant with rare seeds, or a plant that only takes in exactly perfect parts of your land.
  • If a farmer could get “negative correlated assets” (half the plants do better in dry; half do better in wet), that would reduce the “portfolio variation”.
  • Is there anything in finance that, like alfalfa, regenerates the “soil” for the next year’s crop?
  • We speak of “exposure” in finance—well, furrows in la terre literally change the exposure to the sun over the course of its chariot ride across the sky!
    http://timjacob.com/wp-content/uploads/2013/01/FEBRUARY-FIELD-36X60.jpg

So you convolve the crop type with the weather it receives localised to its exact spot in the ground. (its place in the "field" — oh! I mean its place in the field!)

Is it possible, then, to apply the lessons of modern portfolio theory to crop selection?

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i.e. think about the number of rearrangements of AAAAABBBBBBB and then revalue A as + and B as − — or whatever.
Requires knowing that combinatorics can be thought of in terms of counting injections, bijections, etc. rather than real-life examples like cards or coin flips.

i.e. think about the number of rearrangements of AAAAABBBBBBB and then revalue A as + and B as − — or whatever.

Requires knowing that combinatorics can be thought of in terms of counting injections, bijections, etc. rather than real-life examples like cards or coin flips.


hi-res




Central Limit Theorem
A nice illustration of the Central Limit Theorem by convolution.in R:
Heaviside <- function(x) {      ifelse(x>0,1,0) }HH <- convolve( Heaviside(x), rev(Heaviside(x)),        type = "open"   )HHHH <- convolve(HH, rev(HH),   type = "open"   )HHHHHHHH <- convolve(HHHH, rev(HHHH),   type = "open"   )etc.
What I really like about this dimostrazione is that it’s not a proof, rather an experiment carried out on a computer.
This empiricism is especially cool since the Bell Curve, 80/20 Rule, etc, have become such a religion.NERD NOTE:  Which weapon is better, a 1d10 longsword, or a 2d4 oaken staff? Sometimes the damage is written as 1-10 longsword and 2-8 quarterstaff. However, these ranges disregard the greater likelihood of the quarterstaff scoring 4,5,6 damage than 1,2,7,8. The longsword’s distribution 1d10 ~Uniform[1,10], while 2d4 looks like a Λ.
(To see this another way, think of the combinatorics.)

Central Limit Theorem

A nice illustration of the Central Limit Theorem by convolution.

in R:

Heaviside <- function(x) {      ifelse(x>0,1,0) }
HH <- convolve( Heaviside(x), rev(Heaviside(x)),        type = "open"   )
HHHH <- convolve(HH, rev(HH),   type = "open"   )
HHHHHHHH <- convolve(HHHH, rev(HHHH),   type = "open"   )
etc.


What I really like about this dimostrazione is that it’s not a proof, rather an experiment carried out on a computer.

This empiricism is especially cool since the Bell Curve, 80/20 Rule, etc, have become such a religion.




NERD NOTE:  Which weapon is better, a 1d10 longsword, or a 2d4 oaken staff? Sometimes the damage is written as 1-10 longsword and 2-8 quarterstaff. However, these ranges disregard the greater likelihood of the quarterstaff scoring 4,5,6 damage than 1,2,7,8. The longsword’s distribution 1d10 ~Uniform[1,10], while 2d4 looks like a Λ.

(To see this another way, think of the combinatorics.)


hi-res