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The dyadic decision trees studied here are different than classical tree rules, such as CART orC4.5. Those techniques select a tree according to

k ^ = arg min k 1 R ^ n ( f ^ n ( k ) ) + α k ,

for some α > 0 whereas ours was roughly

k ^ = arg min k 1 R ^ n ( f ^ n ( k ) ) + α k ,

for α 3 log 2 2 n . The square root penalty is essential for the risk bound. No such bound exists for CARTor C4.5. Moreover, recent experimental work has shown that the square root penalty often performs better in practice. Finally,recent results show that a slightly tighter bounding procedure for the estimation error can be used to show thatdyadic decision trees (with a slightly different pruning procedure) achieve a rate of

E [ R ( f ^ n T ) ] - R * = O ( n - 1 / 2 ) , as n ,

which turns out to be the minimax optimal rate (i.e., under the boundary assumptions above, no method can achieve a faster rate of convergence tothe Bayes error).

Box counting dimension

The notion of dimension of a sets arises in many aspects of mathematics, and it is particularly relevant to the study offractals (that besides some important applications make really cool t-shirts). The dimension somehow indicates how we shouldmeasure the contents of a set (length, area, volume, etc...). The box-counting dimension is a simple definition of the dimension ofa set. The main idea is to cover the set with boxes with sidelength r . Let N ( r ) denote the smallest number of such boxes, then the box counting dimension is defined as

lim r 0 log N ( r ) - log r .

Although the boxes considered above do not need to be aligned on a rectangulargrid (and can in fact overlap) we can usually consider them over a grid and obtain an upper bound on the box-counting dimension. Toillustrate the main ideas let's consider a simple example, and connect it to the classification scenario considered before.

Let f : [ 0 , 1 ] [ 0 , 1 ] be a Lipschitz function, with Lipschitz constant L ( i.e., | f ( a ) - f ( b ) | L | a - b | , a , b [ 0 , 1 ] ). Define the set

A = { x = ( x 1 , x 2 ) : x 2 = f ( x 1 ) } ,

that is, the set A is the graphic of function f .

Consider a partition with k 2 squared boxes (just like the ones we used in the histograms), the points in set A intersect at most C ' k boxes, with C ' = ( 1 + L ) (and also the number of intersected boxes is greater than k ). The sidelength of the boxes is 1 / k therefore the box-counting dimension of A satisfies

dim B ( A ) lim 1 / k 0 log C ' k - log ( 1 / k ) = lim k log C ' + log ( k ) log ( k ) = 1 .

The result above will hold for any “normal” set A [ 0 , 1 ] 2 that does not occupy any area. For most sets the box-counting dimension is always going to be an integer, butfor some “weird” sets (called fractal sets) it is not an integer. For example, the Koch curvehas box-counting dimension log ( 4 ) / log ( 3 ) = 1 . 26186 ... . This means that it is not quite as small as a 1-dimensional curve, but not as big as a2-dimensional set (hence occupies no area).

To connect these concepts to our classification scenario consider a simple example. Let η ( x ) = P ( Y = 1 | X = x ) and assume η ( x ) has the form

η ( x ) = 1 2 + x 2 - f ( x 1 ) , x ( x 1 , x 2 ) X ,

where f : [ 0 , 1 ] [ 0 , 1 ] is Lipschitz with Lipschitz constant L . The Bayes classifier is then given by

f * ( x ) = 1 { η ( x ) 1 / 2 } 1 { x 2 f ( x 1 ) } .

This is depicted in [link] . Note that this is a special, restricted class of problems. That is, we areconsidering the subset of all classification problems such that the joint distribution P X Y satisfies P ( Y = 1 | X = x ) = 1 / 2 + x 2 - f ( x 1 ) for some function f that is Lipschitz. The Bayes decision boundary is therefore given by

A = { x = ( x 1 , x 2 ) : x 2 = f ( x 1 ) } .

Has we observed before this set has box-counting dimension 1.

Bayes decision boundary for the setup described in Appendix .

Questions & Answers

what is the stm
Brian Reply
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What is STMs full form?
scanning tunneling microscope
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what is differents between GO and RGO?
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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it is a goid question and i want to know the answer as well
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
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