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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

Example

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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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Abhi
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ninjadapaul
20/(×-6^2)
Salomon
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ninjadapaul
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ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
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I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
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ninjadapaul
so you not have an equal sign anywhere in the original equation?
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Commplementary angles
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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what is system testing?
AMJAD
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Azam
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Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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