# 0.16 Nonlinear approximation and wavelet analysis  (Page 2/2)

 Page 2 / 2

Suppose $f$ is piecewise Lipschitz and ${f}_{k}$ ia a piecewise constant.

$|f\left(t\right)-{f}_{k}\left(t\right)|\approx \Delta$

where $\Delta$ is a constant equal to average of $f$ on right and left side of discontinuity in this interval.

$⇒||f-{f}_{k}{||}_{{L}_{2}}^{2}=O\left({k}^{-1}\right)$

where ${k}^{-1}$ is the width of the interval. Notice this rate is quite slow.

This problem naturally suggests the following remedy: use very small intervals near discontinuities and larger intervals insmooth regions. Specifically, suppose we use intervals of width ${k}^{-2\alpha }$ to contain the discontinuities and the intervals ofwidth ${k}^{-1}$ elsewhere. Then accordingly piecewise polynomial approximation ${\stackrel{˜}{f}}_{k}$ satisfies

$||f-{\stackrel{˜}{f}}_{k}{||}_{{L}_{2}}^{2}=O\left({k}^{-2\alpha }\right).$

We can accomplish this need for "adaptive resolution" or "multiresolution" using recursive partitions and trees.

We discussed this idea already in our examination of classification trees. Here is the basic idea again, graphically.

Consider a function $f\in {B}^{\alpha }\left({C}_{\alpha }\right)$ that contains no more than m points of discontinuity, and is ${H}^{\alpha }\left({C}_{\alpha }\right)$ away from these points.

Lemma

Consider a complete RDP with n intervals, then there exists anassociated pruned RDP with $O\left(klogn\right)$ intervals, such that an associated piecewise degree $⌈\alpha ⌉$ polynomial approximation $\stackrel{˜}{\left(}{f\right)}_{k}$ , has a squared approximation error of $O\left(min\left({k}^{-2\alpha },{n}^{-1}\right)\right)$ .

Assume $n>k>m$ . Divide $\left[0,1\right]$ into $k$ intervals. If $f$ is smooth on a particular interval $I$ , then

$|f\left(t\right)-{\stackrel{˜}{f}}_{k}\left(t\right)|=O\left({k}^{-2\alpha }\right)\forall t\in I.$

In intervals that contain a discontinuity, recursively subdivide into two until the discontinuity is contained in an interval ofwidth ${n}^{-1}$ . This process results in at most $lo{g}_{2}n$ addition subintervals per discontinuity, and the squared approximationerror is $O\left(k-2\alpha \right)$ on all of them accept the $m$ intervals of width ${n}^{-1}$ containing the discontinuities where the error is $O\left(1\right)$ at each point.

Thus, the overall squared ${L}_{2}$ norm is

$||f-{\stackrel{˜}{f}}_{k}{||}_{{L}_{2}}^{2}=O\left(min\left({k}^{-2\alpha },{n}^{-1}\right)\right)$

and there are at most $k+lo{g}_{2}n$ intervals in the partition. Since k>m, we can upperbound the number of intervals by $2klo{g}_{2}n$ .

Note that if the initial complete RDP has $n\approx {k}^{2\alpha }$ intervals, then the squared error is $O\left({k}^{-2\alpha }\right)$ .

Thus, we only incur a factor of $2\alpha logk$ additional leafs and achieve the same overall approximation error as in the ${H}^{\alpha }\left({C}_{\alpha }\right)$ case. We will see that this is a small price to pay in order to handle not only smooth functions, but alsopiecewise smooth functions.

## Wavelet approximations

Let $f\in {L}^{2}\left(\left[0,1\right]\right)$ ; $\int {f}^{2}\left(t\right)dt<\infty$ .

A wavelet approximation is a series of the form

$f={c}_{o}+\sum _{j\ge 0}\sum _{k=1}^{{2}^{j}}{\psi }_{j,k}$

where ${c}_{o}$ is a constant $\left({c}_{o}={\int }_{0}^{1}f\left(t\right)dt\right)$ ,

$={\int }_{0}^{1}f\left(t\right){\psi }_{j,k}\left(t\right)dt$

and the basis functions ${\psi }_{j,k}$ are orthonormal, oscillatory signals, each with an associated scale ${2}^{-j}$ and position $k{2}^{-j}$ . ${\psi }_{j,k}$ is called the wavelet at scale ${2}^{-j}$ and position $k{2}^{-j}$ .

## Haar wavelets

${\psi }_{j,k}\left(t\right)={2}^{j/2}\left({\mathbf{1}}_{\left\{t\in \left[{2}^{-j}\left(k-1\right),{2}^{-j}\left(k-1/2\right)\right]\right\}}-{\mathbf{1}}_{\left\{t\in \left[{2}^{-j}\left(k-1/2\right),{2}^{-j}k\right]\right\}}\right)$
${\int }_{0}^{1}{\psi }_{j,k}\left(t\right)dt=0$
${\int }_{0}^{1}{\psi }_{j,k}^{2}\left(t\right)dt={\int }_{\left(k-1\right){2}^{-j}}^{k{2}^{-j}}{2}^{j}dt=1$
${\int }_{0}^{1}{\psi }_{j,k}\left(t\right){\psi }_{l,m}\left(t\right)dt={\delta }_{j,l}.{\delta }_{k,m}$
If $f$ is constant on $\left[{2}^{-j}\left(k-1\right),{2}^{-j}k\right]$ , then
$\int f{\psi }_{j,k}\left(t\right)=0.$

Suppose $f$ is piecewise constant with at most $m$ discontinuities. Let

${f}_{J}={c}_{o}+\sum _{j=0}^{J-1}\sum _{k=1}^{{2}^{j}}{\psi }_{j,k}.$

Then, ${f}_{J}$ has at most $mJ$ non-zero wavelet coefficients; i.e., $=0$ for all but $mJ$ terms, since at most one Haar Wavelet at each scale senses each point of discontinuity. Said another way, allbut at most $m$ of the wavelets at each scale have support over constant regions of $f$ .

${f}_{J}$ itself will be piecewise constant with discontinuities only possible occurring at end points of the intervals $\left[{2}^{-J}\left(k-1\right),{2}^{-J}k\right]$ . Therefore, in this case

$||f-{f}_{J}{||}_{{L}_{2}}^{2}=O\left({2}^{-J}\right).$

Daubechies wavelets are the extension of the Haar wavelet idea. Haar wavelets have one "vanishing moment":

${\int }_{0}^{1}{\psi }_{j,k}=0.$

Daubechies wavelets are "smoother" basis functions with extra vanishing moments. The Daubechies- $N$ wavelet has $N$ vanishing moments.

${\int }_{0}^{1}{t}^{l}{\psi }_{j,k}dt=0forl=0,1,...,N-1.$

The Daubechies-1 wavelet is just the Haar case.

If $f$ is a piecewise degree $\le N$ polynomial with at most m pieces, then using the Daubechies- $N$ wavelet system.

$||f-{f}_{J}{||}_{{L}_{2}}^{2}=O\left({2}^{-J}\right);$

and

${f}_{J}\left(t\right)={c}_{o}+\sum _{j=0}^{J-1}\sum _{k=1}^{{2}^{j}}{\psi }_{j,k}\left(t\right)$

has at most $O\left(mJ\right)$ non-zero wavelet coefficients. ${f}_{J}$ is called the Discrete Wavelet Transform (DWT) approximation of $f$ . The key idea is the same as we saw with trees.

## Sampled data

We can also use DWT's to analyze and represent discrete, sampled functions. Suppose,

$\underline{f}=\left[f\left(1/n\right),f\left(2/n\right),...,f\left(n/n\right)\right]$

then we can write $\underline{f}$ as

$\underline{f}={c}_{o}+\sum _{j=0}^{lo{g}_{2}n-1}\sum _{k=1}^{{2}^{j}}<\underline{f},{\underline{\psi }}_{j,k}>{\underline{\psi }}_{j,k}$

where

${\underline{\psi }}_{j,k}=\left[{\psi }_{j,k}\left(1\right),{\psi }_{j,k}\left(2\right),...,{\psi }_{j,k}\left(n\right)\right]$

is a discrete time analog of the continuous time wavelets we considered before. In particular,

$\sum _{i=1}^{n}{i}^{l}{\psi }_{j,k}\left(i\right)=0,l=0,1,...,N-1$

for the Daubechies- $N$ discrete wavelets.

$<\underline{f},{\underline{\psi }}_{j,k}>={\underline{f}}^{T}{\underline{\psi }}_{j,k}$

Thus, we also have an analogous approximation result: If $\underline{f}$ are samples from a piecewise degree $\le N$ polynomial function with a finite number $m$ of discontinuities, then $\underline{f}$ has $O\left(mJ\right)$ non-zero wavelet coefficients.

## ApproximatingFunctions with wavelets

Suppose $f\in {B}^{\alpha }\left({C}_{\alpha }\right)$ and has a finite number of discontinuities. Let ${f}_{p}$ denote piecewise degree- $N\left(N=⌈\alpha ⌉\right)$ polynomial approximation to $f$ with $O\left(k\right)$ pieces; a uniform partition into $k$ equal length intervals followed by addition splits at the points of discontinuity.

Then

$|f\left(t\right)-{f}_{p}\left(t\right){|}^{2}=O\left({k}^{\left(}-2\alpha \right)\right)\forall t\in \left[0,1\right]$
$⇒|f\left(i/n\right)-{f}_{p}{\left(i/n\right)|}^{2}=O\left({k}^{-2\alpha }\right)i=1,...,n$
$⇒1/n||\underline{f}-{\underline{f}}_{p}{||}_{{L}_{2}}^{2}=O\left({k}^{-2\alpha }\right)\right)$

and ${\underline{f}}_{p}$ has $O\left(klo{g}_{2}n\right)$ non-zero coefficients according to our previous analysis.

## Wavelets in 2-d

Suppose $f$ is a 2-D image that is piecewise polynomial:

A pruned RDP of $k$ squares decorated with polyfits gives

$||f-{f}_{k}{||}_{{L}_{2}}^{2}=O\left({k}^{-1}\right).$

Let $\underline{f}{=\left[f\left(i/k,j/k\right)}_{i,j=1}^{n}$ sample range.

${f}_{n}\left(t\right)=\sum _{i,j=1}^{k}f\left(i/k,j/kk\right){\mathbf{1}}_{\left\{t\in \left[i-1/k,i/k\right)x\left[j-1/k,j/k\right)\right\}}$

then

$||f-{f}_{n}{||}_{{L}_{2}}^{2}=O\left({k}^{-1}\right)$

$O\left(1\right)$ error on $k$ of the ${k}^{2}$ pixels, near zero elsewhere. The DWT of $\underline{f}$ has $O\left(k\right)$ non-zero wavelet coefficients. $O\left({2}^{j}\right)$ at scale ${2}^{-j},j=0,1,...,logn.$

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Raynard
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of psychologys commencement, the traces can be seen in the work of Aristotle, where he talk about soul and body, likewise work in durrant, de anima, all these were somewhere supporting dualism, in which soul could exist separately from body
amaan
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amaan
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edem
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Clara
love messes with the brain, a lot, ergo I believe that Psychology does indeed deal with love
I would like an example as to what and how you think it deals with.
Tyler
how can I discover that this individual has a long-term memory and shot- term memory?
Namuaha
what is synapse
In the central nervous system, a synapse is a small gap at the end of a neuron that allows a signal to pass from one neuron to the next. synapse are found where nerve cells connect with other nerve cells
Najeem
a synapse the connection is where a neuron cell connects to another neuron cell.
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Ibrahim
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Avneet
the only eligibility criteria is that you should have 50% of aggregate in your psychology papers. (bachelors)
syeda
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Avneet
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syeda
To pursue a career as a psychotherapist you'll have to do your bachelors in psychology. (bsc honors is preferable). since there are many fields and you've chosen as a therapist. a masters degree in clinical psychology or therapy and family counseling is preferable.
syeda
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Avneet
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syeda
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syeda
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Vhikkie
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Shilan
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Aakarshan
the only eligibility criteria is that you should have 50% of aggregate in your psychology papers. (bachelors)
syeda
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Vhikkie
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aristatil
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edem
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Aspen
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edem
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sakina
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sakina
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sakina
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Michael
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edem
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Reginald
*believed...sorry for typo
Reginald
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edem
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Reginald
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edem
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ipau
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Roger
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Roger
cognitive development is the growing and development of the brain.
Ecofascism is a theoretical political model in which an authoritarian government would require individuals to sacrifice their own interests to the "organic whole of nature". The term is also used as a rhetorical pejorative to undermine the environmental movement.
ipau
what's the big difference between prejudice and discrimination?
A prejudiced person may not act on their attitude.  Therefore, someone can be prejudiced towards a certain group but not discriminate against them.  Also, prejudice includes all three components of an attitude (affective, behavioral and cognitive), whereas discrimination just involves behavior
Nancy Lee
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Rahul
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Kamohelo
cognitive development is the growing and development of the brain
Jessy
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it dependa on your study. according to what you want to say and explain your result
Pouran
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Jobe
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stages of cognitive development
sensory preoperatinal concrete formal
Rajendra
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Namuaha
Memory is our ability to encode, store, retain and subsequently recall information and past experiences in the human brain. It can be thought of in general terms as the use of past experience to affect or influence current behaviour.
Shilan
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Namuaha
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Jobe
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the study of insecurities and the effect on the host .
Sera
Psychology is the scientific study of behavior & mental processes
Angela
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behaviorosm
Khan
is the study of human behaviour and mental processes
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