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Learning theory

Bias/variance tradeoff

When talking about linear regression, we discussed the problem of whether to fit a “simple” model such as the linear“ y = θ 0 + θ 1 x ,” or a more “complex” model such as the polynomial “ y = θ 0 + θ 1 x + θ 5 x 5 .” We saw the following example:

Data set with a best fit line going through it
Data set with a better fit curve going through it
Data set with a best fit curve going through it touching each point along the way

Fitting a 5th order polynomial to the data (rightmost figure) did not result in a good model. Specifically, even though the 5th order polynomial did a very good jobpredicting y (say, prices of houses) from x (say, living area) for the examples in the training set, we do not expect the model shown to be a good one for predictingthe prices of houses not in the training set. In other words, what's has been learned from the training set does not generalize well to other houses. The generalization error (which will be made formal shortly) of a hypothesis is its expected error on examples not necessarily in thetraining set.

Both the models in the leftmost and the rightmost figures above have large generalization error. However, the problems that the two models suffer fromare very different. If the relationship between y and x is not linear, then even if we were fitting a linear model to a very large amount of trainingdata, the linear model would still fail to accurately capture the structure in the data. Informally, we define the bias of a model to be the expected generalization error even if we were to fit it to a very (say,infinitely) large training set. Thus, for the problem above, the linear model suffers from large bias, and may underfit (i.e., fail to capturestructure exhibited by) the data.

Apart from bias, there's a second component to the generalization error, consisting of the variance of a model fitting procedure. Specifically, when fitting a 5th order polynomial as in the rightmost figure, there is alarge risk that we're fitting patterns in the data that happened to be present in our small, finite training set, but that do not reflect thewider pattern of the relationship between x and y . This could be, say, because in thetraining set we just happened by chance to get a slightly more-expensive-than-average house here, and a slightly less-expensive-than-average house there, and so on.By fitting these “spurious” patterns in the training set, we might again obtain a model with large generalization error. In this case, we say themodel has large variance. In these notes, we will not try to formalize the definitions of bias and variance beyond this discussion.While bias and variance are straightforward to define formally for, e.g., linear regression,there have been several proposals for the definitions of bias and variance for classification, and there is as yet no agreement on what isthe “right” and/or the most useful formalism.

Often, there is a tradeoff between bias and variance. If our model is too “simple” and has very few parameters, then it may have largebias (but small variance); if it is too “complex” and has very many parameters, then it may suffer from large variance (but have smaller bias). In the example above, fitting a quadratic function does better than either of the extremes of a firstor a fifth order polynomial.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
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