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Total least squares

Consider the linear equation A x = b , where A R m × n , b R m × 1 , x R n × 1 , m > n .

This equation is overdetermined and has no precise answer. The simplest approach to finding x is a least-squares fitting model, which finds the curve with the least difference between the value of the curve at a point and the value of the data at that point; i.e., it solves min x R n A x - b 2 . This amounts to saying that the data may be slightly perturbed:

A x = b + r ,

where r is some residual noise, and minimizing r :

min r : A x = b + r r .

When we compare this to our equation B k = f , we see that this is an appropriate method: we are not entirely confident of f , and can perturb it slightly.

Looking more closely, B = A T diag A x . We are also not entirely certain of x , which means we are not entirely certain of A T diag A x . This is best reflected in the total least squares approach, in which both the data ( b in the simple equation, f in our equation) and the matrix ( A in the simple equation, A T diag A x in our equation) may be slightly perturbed:

A + E x = b + r ,

where E is some noise in A and r is some noise in b , and minimizing E r :

min [ E r ] : ( A + E ) x = b + r [ E r ] | F .

The last term in the singular value decomposition of [ A b ] , - s n + 1 u n + 1 v n + 1 T , is precisely what we want for [ E r ] .

At first glance, this exactly what we want. We can find the singular value decomposition of [ B f ] , take the last term as [ E r ] , and solve for k . When we implement this method, however, we get worse results compared to the measured data. Standard least squares returns a k with only 182.04% percent error (See [link] ); total least squares returns a k with 269.17% percent error (See [link] ). Looking at the structure of B = A T diag A x and E gives a hint as to why. The adjacency matrix, A , encodes information about the structure of the network, so it has a very specific pattern of zeros, which is reflected in B . There are no similar restrictions on E , allowing zeros in inappropriate places. This is physically equivalent to sprouting a new spring between two nodes, an absurdity. [link] below compares the structure of B ( [link] ) and E ( [link] ). Light green entires correspond to a zero; everything else corresponds to a nonzero entry. E has many non-zero entries where there should not be any. Note the scale for the colorbar on the right: the entries of E are two orders of magnitude smaller than the entries in B . Though they are small, they represent connections between nodes and springs that do not exist, throwing off the entire result. Requiring that particular entries equal zero makes the problem combinatorally harder.

Results from Least Squares and Total Least Squares
Total Least Squares: Structure of B and E

Statistical approaches

Statistical background

Because we would like to use statistical inference, it is important to have a basic understanding of several statistical concepts.

Definition 1 Probability Space

A space, Ω , of all possible events, ω Ω


Rolling a die is an event.

Flipping a coin is an event.

Loading forces onto the spring network is an event.

Definition 2 Random Variable

A mapping from a space of events into the real line, X : Ω R , or real n -dimensional space, X : Ω R n

Questions & Answers

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
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
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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