# 10.3 The physics of springs  (Page 7/11)

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## Total least squares

Consider the linear equation $Ax=b$ , where $A\in {\mathbb{R}}^{m×n},b\in {\mathbb{R}}^{m×1},x\in {\mathbb{R}}^{n×1},\phantom{\rule{1.em}{0ex}}m>n$ .

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 ${min}_{x\in {\mathbb{R}}^{n}}{\parallel Ax-b\parallel }^{2}$ . This amounts to saying that the data may be slightly perturbed:

$Ax=b+r,$

where r is some residual noise, and minimizing $\parallel r\parallel$ :

$\underset{r:Ax=b+r}{min}\parallel r\parallel .$

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

Looking more closely, $B={A}^{T}diag\left(A,x\right)$ . We are also not entirely certain of x , which means we are not entirely certain of ${A}^{T}diag\left(A,x\right)$ . 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, ${A}^{T}diag\left(A,x\right)$ in our equation) may be slightly perturbed:

$\left(A,+,E\right)x=b+r,$

where E is some noise in A and r is some noise in b , and minimizing $\parallel \left[E,\phantom{\rule{0.277778em}{0ex}},r\right]\parallel$ :

$\underset{\left[E\phantom{\rule{0.277778em}{0ex}}r\right]:\left(A+E\right)x=b+r}{min}{\parallel \left[E\phantom{\rule{0.277778em}{0ex}}r\right]|}_{F}.$

The last term in the singular value decomposition of $\left[A\phantom{\rule{0.277778em}{0ex}}b\right]$ , $-{s}_{n+1}{u}_{n+1}{v}_{n+1}^{T}$ , is precisely what we want for $\left[E\phantom{\rule{0.277778em}{0ex}}r\right]$ .

At first glance, this exactly what we want. We can find the singular value decomposition of $\left[B\phantom{\rule{0.277778em}{0ex}}f\right]$ , take the last term as $\left[E\phantom{\rule{0.277778em}{0ex}}r\right]$ , 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 $B={A}^{T}diag\left(A,x\right)$ 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.

## 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, $\omega \in \Omega$

Example

Rolling a die is an event.

Flipping a coin is an event.

Definition 2 Random Variable

A mapping from a space of events into the real line, $X:\Omega \to \mathbb{R}$ , or real n -dimensional space, $X:\Omega \to {\mathbb{R}}^{n}$

are nano particles real
yeah
Joseph
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Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
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narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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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
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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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?
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What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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
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