<< Chapter < Page Chapter >> Page >

Least squares revisited

Armed with the tools of matrix derivatives, let us now proceed to find in closed-form the value of θ that minimizes J ( θ ) . We begin by re-writing J in matrix-vectorial notation.

Given a training set, define the design matrix X to be the m -by- n matrix (actually m -by- n + 1 , if we include the intercept term) that contains the training examples' input values in its rows:

X = x 1 T x 2 T x m T .

Also, let y be the m -dimensional vector containing all the target values from the training set:

y = y 1 y 2 y m .

Now, since h θ ( x ( i ) ) = ( x ( i ) ) T θ , we can easily verify that

X θ - y = x 1 T θ x m T θ - y 1 y m = h θ x 1 - y 1 h θ x m - y m .

Thus, using the fact that for a vector z , we have that z T z = i z i 2 :

1 2 ( X θ - y ) T ( X θ - y ) = 1 2 i = 1 m ( h θ ( x ( i ) ) - y ( i ) ) 2 = J ( θ )

Finally, to minimize J , let's find its derivatives with respect to θ . Combining the second and third equation in  [link] , we find that

A T tr A B A T C = B T A T C T + B A T C


θ J ( θ ) = θ 1 2 ( X θ - y ) T ( X θ - y ) = 1 2 θ θ T X T X θ - θ T X T y - y T X θ + y T y = 1 2 θ tr θ T X T X θ - θ T X T y - y T X θ + y T y = 1 2 θ tr θ T X T X θ - 2 tr y T X θ = 1 2 X T X θ + X T X θ - 2 X T y = X T X θ - X T y

In the third step, we used the fact that the trace of a real number is just the real number; the fourth step used the fact that tr A = tr A T , and the fifth step used Equation  [link] with A T = θ , B = B T = X T X , and C = I , and Equation  [link] . To minimize J , we set its derivatives to zero, and obtain the normal equations :

X T X θ = X T y

Thus, the value of θ that minimizes J ( θ ) is given in closed form by the equation

θ = ( X T X ) - 1 X T y .

Probabilistic interpretation

When faced with a regression problem, why might linear regression, and specifically why might the least-squares cost function J , be a reasonable choice? In this section, we will give a set of probabilistic assumptions, under which least-squares regressionis derived as a very natural algorithm.

Let us assume that the target variables and the inputs are related via the equation

y ( i ) = θ T x ( i ) + ϵ ( i ) ,

where ϵ ( i ) is an error term that captures either unmodeled effects (such as if there are some features very pertinentto predicting housing price, but that we'd left out of the regression), or random noise. Let us further assume that the ϵ ( i ) are distributed IID (independently and identically distributed) accordingto a Gaussian distribution (also called a Normal distribution) with mean zero and some variance σ 2 . We can write this assumption as “ ϵ ( i ) N ( 0 , σ 2 ) .” I.e., the density of ϵ ( i ) is given by

p ( ϵ ( i ) ) = 1 2 π σ exp - ( ϵ ( i ) ) 2 2 σ 2 .

This implies that

p ( y ( i ) | x ( i ) ; θ ) = 1 2 π σ exp - ( y ( i ) - θ T x ( i ) ) 2 2 σ 2 .

The notation “ p ( y ( i ) | x ( i ) ; θ ) ” indicates that this is the distribution of y ( i ) given x ( i ) and parameterized by θ . Note that we should not condition on θ (“ p ( y ( i ) | x ( i ) , θ ) ”), since θ is not a random variable. We can also write the distribution of y ( i ) as as y ( i ) x ( i ) ; θ N ( θ T x ( i ) , σ 2 ) .

Given X (the design matrix, which contains all the x ( i ) 's) and θ , what is the distribution of the y ( i ) 's? The probability of the data is given by p ( y | X ; θ ) . This quantity is typically viewed a function of y (and perhaps X ), for a fixed value of θ . When we wish to explicitly view this as a function of θ , we will instead call it the likelihood function:

L ( θ ) = L ( θ ; X , y ) = p ( y | X ; θ ) .

Questions & Answers

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
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.
anyone know any internet site where one can find nanotechnology papers?
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
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.
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, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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