<< Chapter < Page Chapter >> Page >

Three elements of statistical data analysis

  • of learning from data and prediction problems.
    • concentration inequalities
    • uniform deviation bounds
    • approximation theory
    • rates of convergence
  • that run in polynomial time (e.g., decision trees, wavelet methods, support vector machines).

Learning from data

To formulate the basic learning from data problem, we must specify several basic elements: data spaces, probability measures, loss functions, andstatistical risk.

Data spaces

Learning from data begins with a specification of two spaces:

X Input Space
Y Output Space .

The input space is also sometimes called the “feature space” or “signal domain.” The output space is also called the “class label space,”“outcome space,” “response space,” or “signal range.”

X = R d d -dimensional Euclidean space of ``feature vectors''
Y = { 0 , 1 } two classes or ``class labels''
X = R one-dimensional signal domain (e.g., time-domain)
Y = R real-valued signal

A classic example is estimating a signal f in noise:

Y = f ( X ) + W

where X is a random sample point on the real line and W is a noise independent of X .

Probability measure and expectation

Define a joint probability distribution on X × Y denoted P X , Y . Let ( X , Y ) denote a pair of random variables distributed according to P X , Y . We will also have use for marginal and conditional distributions. Let P X denote the marginal distribution on X , and let P Y | X denote the conditional distribution of Y given X . For any distribution P , let p denote its density function with respect to the corresponding dominating measure; e.g., Lebesgue measure for continuous random variables or counting measure for discrete random variables.

Define the expectation operator:

E X , Y [ f ( X , Y ) ] f ( x , y ) d P X , Y ( x , y ) = f ( x , y ) p X , Y ( x , y ) d x d y .

We will also make use of corresponding marginal and conditional expectations such as E X and E Y | X .

Wherever convenient and obvious based on context, we may drop the subscripts (e.g., E instead of E X , Y ) for notational ease.

Loss functions

A loss function is a mapping

: Y × Y R .

In binary classification problems, Y = { 0 , 1 } . The 0 / 1 loss function is usually used: ( y 1 , y 2 ) = 1 y 1 y 2 , where 1 A is the indicator function which takes a value of 1 if condition A is true and zero otherwise. We typically will compare a true label y with a prediction y ^ , in which case the 0 / 1 loss simply counts misclassifications.

In regression or estimation problems, Y = R . The squared error loss function is often employed: ( y 1 , y 2 ) = ( y 1 - y 2 ) 2 , the square of the difference between y 1 and y 2 . In application, we are interested in a true value y in comparison to an estimate y ^ .

Statistical risk

The basic problem in learning is to determine a mapping f : X Y that takes an input x X and predicts the corresponding output y Y . The performance of a given map f is measured by its expected loss or risk :

R ( f ) E X , Y ( f ( X ) , Y ) .

The risk tells us how well, on average, the predictor f performs with respect to the chosen loss function. A key quantity of interestis the mininum risk value, defined as

R * = inf f R ( f )

where the infinum is taking over all measurable functions.

The learning problem

Suppose that ( X , Y ) are distributed according to P X , Y ( ( X , Y ) P X , Y for short). Our goal is to find a map so that f ( X ) Y with high probability. Ideally, we would chose f to minimize the risk R ( f ) = E [ ( f ( X ) , Y ) ] . However, in order to compute the risk (and hence optimize it) we need to know the jointdistribution P X , Y . In many problems of practical interest, the joint distribution is unknown, and minimizing the risk is notpossible.

Suppose that we have some exemplary samples from the distribution. Specifically, consider n samples X i , Y i i = 1 n distributed independently and identically (iid) according to the otherwise unknown P X , Y . Let us call these samples training data , and denote the collection by D n X i , Y i i = 1 n . Let's also define a collection of candidate mappings F . We will use the training data D n to pick a mapping f n F that we hope will be a good predictor. This is sometimes called the Model Selection problem. Note that the selected model f n is a function of the training data:

f n ( X ) = f ( X ; D n ) ,

which is what the subscript n in f n refers to. The risk of f n is given by

R ( f n ) = E X , Y [ ( f n ( X ) , Y ) ] .

Note that since f n depends on D n in addition to a new random pair ( X , Y ) , the risk is a random variable (i.e., a function of the training data D n ). Therefore, we are interested in the expected risk , computed over random realizations of the training data:

E D n [ R ( f n ) ] .

We hope that f n produces a small expected risk.

The notion of expected risk can be interpreted as follows. We would like to define an algorithm (a model selection process) that performswell on average, over any random sample of n training data. The expected risk is a measure of the expected performance of thealgorithm with respect to the chosen loss function. That is, we are not gauging the risk of a particular map f F , but rather we are measuring the performance of the algorithm that takes any realizationof training data and selects an appropriate model in F .

This course is concerned with determining “good” model spaces F and useful and effective model selection algorithms.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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 .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistical learning theory' conversation and receive update notifications?