# 0.11 Decision trees  (Page 5/5)

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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

$\stackrel{^}{k}=arg\underset{k\ge 1}{min}\left\{{\stackrel{^}{R}}_{n},\left({\stackrel{^}{f}}_{n}^{\left(k\right)}\right),+,\alpha ,k\right\},$

for some $\alpha >0$ whereas ours was roughly

$\stackrel{^}{k}=arg\underset{k\ge 1}{min}\left\{{\stackrel{^}{R}}_{n},\left({\stackrel{^}{f}}_{n}^{\left(k\right)}\right),+,\alpha ,\sqrt{k}\right\},$

for $\alpha \approx \sqrt{\frac{3log2}{2n}}$ . 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

$E\left[R\left({\stackrel{^}{f}}_{n}^{T}\right)\right]-{R}^{*}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}O\left({n}^{-1/2}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to \infty ,$

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 $N\left(r\right)$ denote the smallest number of such boxes, then the box counting dimension is defined as

$\underset{r\to 0}{lim}\frac{logN\left(r\right)}{-logr}.$

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 $f:\left[0,1\right]\to \left[0,1\right]$ be a Lipschitz function, with Lipschitz constant $L$ ( i.e., $|f\left(a\right)-f\left(b\right)|\le L|a-b|,\phantom{\rule{4pt}{0ex}}\forall a,b\in \left[0,1\right]$ ). Define the set

$A=\left\{x=\left({x}_{1},{x}_{2}\right):{x}_{2}=f\left({x}_{1}\right)\right\},$

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 ${C}^{\text{'}}k$ boxes, with ${C}^{\text{'}}=\left(1+⌈L⌉\right)$ (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

$\begin{array}{ccc}\hfill {dim}_{B}\left(A\right)& \le & \underset{1/k\to 0}{lim}\frac{log{C}^{\text{'}}k}{-log\left(1/k\right)}\hfill \\ & =& \underset{k\to \infty }{lim}\frac{log{C}^{\text{'}}+log\left(k\right)}{log\left(k\right)}\hfill \\ & =& 1.\hfill \end{array}$

The result above will hold for any “normal” set $A\subseteq {\left[0,1\right]}^{2}$ 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 $log\left(4\right)/log\left(3\right)=1.26186...$ . 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 $\eta \left(x\right)=P\left(Y=1|X=x\right)$ and assume $\eta \left(x\right)$ has the form

$\eta \left(x\right)=\frac{1}{2}+{x}_{2}-f\left({x}_{1}\right),\phantom{\rule{1.em}{0ex}}\forall x\equiv \left({x}_{1},{x}_{2}\right)\in \mathcal{X},$

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

${f}^{*}\left(x\right)={\mathbf{1}}_{\left\{\eta \left(x\right)\ge 1/2\right\}}\equiv {\mathbf{1}}_{\left\{{x}_{2}\ge f\left({x}_{1}\right)\right\}}.$

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 ${P}_{XY}$ satisfies $P\left(Y=1|X=x\right)=1/2+{x}_{2}-f\left({x}_{1}\right)$ for some function $f$ that is Lipschitz. The Bayes decision boundary is therefore given by

$A=\left\{x=\left({x}_{1},{x}_{2}\right):{x}_{2}=f\left({x}_{1}\right)\right\}.$

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

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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