|
|
| Line 1: |
Line 1: |
| {{technical|date=June 2012}}
| | Roberto is what's written in his birth certificate rather he never really adored that name. Managing people is literally where his primary profit coming in comes from. [http://Www.adobe.com/cfusion/search/index.cfm?term=&Base+jumping&loc=en_us&siteSection=home Base jumping] is something that he been doing for a number of. Massachusetts has always been his lifestyle place and his people loves it. Go with regard to his website to hit upon out more: http://circuspartypanama.com<br><br>Feel free to surf to my homepage :: clash of clans hack cydia ([http://circuspartypanama.com http://circuspartypanama.com/]) |
| In the theory of [[stochastic process]]es, the '''Karhunen–Loève theorem''' (named after [[Kari Karhunen]] and [[Michel Loève]]), also known as the '''Kosambi–Karhunen–Loève theorem'''<ref name="sapatnekar">{{Citation |last=Sapatnekar |first=Sachin |title= Overcoming variations in nanometer-scale technologies|journal= IEEE Journal on Emerging and Selected Topics in Circuits and Systems|volume= 1|year= 2011 |issue= 1|pages= 5–18}}</ref><ref name="ghoman">{{Citation |last=Ghoman |first=Satyajit |last2= Wang|first2= Zhicun|last3=Chen |first3=PC |last4=Kapania|first4=Rakesh|title= A POD-based Reduced Order Design Scheme for Shape Optimization of Air Vehicles|booktitle=Proc of 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA-2012-1808, Honolulu, Hawaii |year=2012 }}</ref> is a representation of a stochastic process as an infinite linear combination of [[orthogonal function]]s, analogous to a [[Fourier series]] representation of a function on a bounded interval. Stochastic processes given by infinite series of this form were first<ref name="Raju">{{Citation |first=C.K. |last=Raju |title=Kosambi the Mathematician |journal=Economic and Political Weekly |volume=44 |year=2009 |issue=20 |pages=33–45 }}</ref> considered by [[Damodar Dharmananda Kosambi]].<ref name="Kosambi">{{Citation |first=D. D. |last=Kosambi |title=Statistics in Function Space |journal=Journal of the Indian Mathematical Society |volume=7 |year=1943 |issue= |pages=76–88 |id={{MathSciNet|9816}} }}.</ref> There exist many such expansions of a stochastic process: if the process is indexed over [''a'', ''b''], any [[orthonormal basis]] of ''L''<sup>2</sup>([''a'', ''b'']) yields an expansion thereof in that form. The importance of the Karhunen–Loève theorem is that it yields the best such basis in the sense that it minimizes the total [[mean squared error]].
| |
|
| |
| In contrast to a Fourier series where the coefficients are real numbers and the expansion basis consists of [[trigonometric function|sinusoidal functions]] (that is, [[sine]] and [[cosine]] functions), the coefficients in the Karhunen–Loève theorem are [[random variable]]s and the expansion basis depends on the process. In fact, the orthogonal basis functions used in this representation are determined by the [[covariance function]] of the process. One can think that the Karhunen–Loève transform adapts to the process in order to produce the best possible basis for its expansion.
| |
| | |
| In the case of a ''centered'' stochastic process {''X''<sub>''t''</sub>}<sub>''t'' ∈ [''a'', ''b'']</sub> (where ''centered'' means that the expectations E(''X''<sub>''t''</sub>) are defined and equal to 0 for all values of the parameter ''t'' in [''a'', ''b'']) satisfying a technical continuity condition, ''X''<sub>''t''</sub> admits a decomposition
| |
| :<math> X_t = \sum_{k=1}^\infty Z_k e_k(t)</math>
| |
| where ''Z''<sub>''k''</sup> are pairwise [[uncorrelated]] random variables and the functions ''e''<sub>''k''</sub> are continuous real-valued functions on [''a'', ''b''] that are pairwise [[orthogonal]] in ''L''<sup>2</sup>[''a'', ''b'']. It is therefore sometimes said that the expansion is ''bi-orthogonal'' since the random coefficients ''Z''<sub>''k''</sup> are orthogonal in the probability space while the deterministic functions ''e''<sub>''k''</sub> are orthogonal in the time domain. The general case of a process ''X''<sub>''t''</sub> that is not centered can be brought back to the case of a centered process by considering (''X''<sub>''t''</sub> − E(''X''<sub>''t''</sub>)) which is a centered process.
| |
| | |
| Moreover, if the process is [[Gaussian process|Gaussian]], then the random variables ''Z''<sub>''k''</sub> are Gaussian and [[stochastically independent]]. This result generalizes the ''Karhunen–Loève transform''. An important example of a centered real stochastic process on [0,1] is the [[Wiener process]]; the Karhunen–Loève theorem can be used to provide a canonical orthogonal representation for it. In this case the expansion consists of sinusoidal functions.
| |
| | |
| The above expansion into uncorrelated random variables is also known as the ''Karhunen–Loève expansion'' or ''Karhunen–Loève decomposition''. The [[statistic|empirical]] version (i.e., with the coefficients computed from a sample) is known as the ''Karhunen–Loève transform'' (KLT), ''[[principal component analysis]]'', ''proper orthogonal decomposition (POD)'', ''[[Empirical orthogonal functions]]'' (a term used in [[meteorology]] and [[geophysics]]), or the ''[[Harold Hotelling|Hotelling]] transform''.
| |
| | |
| == Formulation ==
| |
| *Throughout this article, we will consider a square integrable zero-mean random process ''X''<sub>''t''</sub> defined over a probability space (Ω,''F'','''P''') and indexed over a closed interval [''a'', ''b''], with covariance function ''K''<sub>''X''</sub>(''s,t''). We thus have:
| |
| ::<math>\forall t\in [a,b], X_t\in L^2(\Omega,\mathcal{F},\mathrm{P}),</math>
| |
| ::<math>\forall t\in [a,b], \mathrm{E}[X_t]=0,</math>
| |
| ::<math>\forall t,s \in [a,b], K_X(s,t)=\mathrm{E}[X_s X_t].</math>
| |
| | |
| *We associate to ''K''<sub>''X''</sub> a [[linear operator]] ''T''<sub>''K''<sub>''X''</sub></sub> defined in the following way:
| |
| :<math>
| |
| \begin{array}{rrl}
| |
| T_{K_X}: L^2([a,b]) &\rightarrow & L^2([a,b])\\
| |
| f(t) & \mapsto & \int_{[a,b]} K_X(s,t) f(s) ds
| |
| \end{array}
| |
| </math><br>Since ''T''<sub>''K''<sub>''X''</sub></sub> is a linear operator, it makes sense to talk about its eigenvalues λ<sub>''k''</sub> and eigenfunctions ''e''<sub>''k''</sub>, which are found solving the homogeneous Fredholm [[integral equation]] of the second kind
| |
| :<math>\int_{[a,b]} K_X(s,t) e_k(s)\,ds=\lambda_k e_k(t)</math>
| |
| | |
| == Statement of the theorem ==
| |
| '''Theorem'''. Let ''X''<sub>''t''</sub> be a zero-mean square integrable stochastic process defined over a probability space (Ω,''F'','''P''') and indexed over a closed and bounded interval [''a'', ''b''], with continuous covariance function ''K''<sub>''X''</sub>(''s,t'').
| |
| | |
| Then ''K''<sub>''X''</sub>(''s,t'') is a [[Mercer's theorem|Mercer kernel]] and letting ''e''<sub>''k''</sub> be an orthonormal basis of ''L''<sup>2</sup>([''a'', ''b'']) formed by the eigenfunctions of ''T''<sub>''K''<sub>''X''</sub></sub> with respective eigenvalues λ<sub>''k''</sub>, ''X''<sub>''t''</sub> admits the following representation
| |
| | |
| :<math>
| |
| X_t=\sum_{k=1}^\infty Z_k e_k(t)
| |
| </math>
| |
| | |
| where the convergence is in [[Convergence of random variables#Convergence in mean|''L''<sup>2</sup>]], uniform in ''t'' and
| |
| | |
| :<math>
| |
| Z_k=\int_{[a,b]} X_t e_k(t)\, dt
| |
| </math>
| |
| | |
| Furthermore, the random variables ''Z''<sub>''k''</sub> have zero-mean, are uncorrelated and have variance λ<sub>''k''</sub>
| |
| | |
| :<math>
| |
| \mathrm{E}[Z_k]=0,~\forall k\in\mathbb{N} \quad\quad\mbox{and}\quad\quad \mathrm{E}[Z_i Z_j]=\delta_{ij} \lambda_j,~\forall i,j\in \mathbb{N}
| |
| </math>
| |
| | |
| Note that by generalizations of Mercer's theorem we can replace the interval [''a'', ''b''] with other compact spaces ''C'' and the Lebesgue measure on [''a'', ''b''] with a Borel measure whose support is ''C''.
| |
| | |
| ==Proof==
| |
| | |
| *The covariance function ''K''<sub>''X''</sub> satisfies the definition of a Mercer kernel. By [[Mercer's theorem]], there consequently exists a set {λ<sub>''k''</sub>,''e''<sub>''k''</sub>(''t'')} of eigenvalues and eigenfunctions of T<sub>''K''<sub>''X''</sub></sub> forming an orthonormal basis of ''L''<sup>2</sup>([''a'',''b'']), and ''K''<sub>''X''</sub> can be expressed as
| |
| :<math>K_X(s,t)=\sum_{k=1}^\infty \lambda_k e_k(s) e_k(t) </math>
| |
| | |
| *The process ''X''<sub>''t''</sub> can be expanded in terms of the eigenfunctions ''e''<sub>''k''</sub> as:
| |
| :<math>X_t=\sum_{k=1}^\infty Z_k e_k(t)</math><br> where the coefficients (random variables) ''Z''<sub>''k''</sub> are given by the projection of ''X''<sub>''t''</sub> on the respective eigenfunctions
| |
| :<math>Z_k=\int_{[a,b]} X_t e_k(t) \,dt</math>
| |
| | |
| *We may then derive
| |
| :<math>\mathrm{E}[Z_k]=\mathrm{E}\left[\int_{[a,b]} X_t e_k(t) \,dt\right]=\int_{[a,b]} \mathrm{E}[X_t] e_k(t) dt=0</math><br>and:
| |
| :<math>
| |
| \begin{array}[t]{rl}
| |
| \mathrm{E}[Z_i Z_j]&=\mathrm{E}\left[ \int_{[a,b]}\int_{[a,b]} X_t X_s e_j(t)e_i(s) dt\, ds\right]\\
| |
| &=\int_{[a,b]}\int_{[a,b]} \mathrm{E}\left[X_t X_s\right] e_j(t)e_i(s) dt\, ds\\
| |
| &=\int_{[a,b]}\int_{[a,b]} K_X(s,t) e_j(t)e_i(s) dt \, ds\\
| |
| &=\int_{[a,b]} e_i(s)\left(\int_{[a,b]} K_X(s,t) e_j(t) dt\right) ds\\
| |
| &=\lambda_j \int_{[a,b]} e_i(s) e_j(s) ds\\
| |
| &=\delta_{ij}\lambda_j
| |
| \end{array}
| |
| </math><br>where we have used the fact that the ''e''<sub>''k''</sub> are eigenfunctions of ''T''<sub>''K''<sub>''X''</sub></sub> and are orthonormal.
| |
| | |
| *Let us now show that the convergence is in ''L''<sup>2</sup>:<br>let <math>S_N=\sum_{k=1}^N Z_k e_k(t)</math>.
| |
| :<math>
| |
| \begin{align}
| |
| \mathrm{E}[|X_t-S_N|^2]&=\mathrm{E}[X_t^2]+\mathrm{E}[S_N^2]-2\mathrm{E}[X_t S_N]\\
| |
| &=K_X(t,t)+\mathrm{E}\left[\sum_{k=1}^N \sum_{l=1}^N Z_k Z_l e_k(t)e_l(t) \right] -2\mathrm{E}\left[X_t\sum_{k=1}^N Z_k e_k(t)\right]\\
| |
| &=K_X(t,t)+\sum_{k=1}^N \lambda_k e_k(t)^2 -2\mathrm{E}\left[\sum_{k=1}^N \int_a^b X_t X_s e_k(s) e_k(t) ds\right]\\
| |
| &=K_X(t,t)-\sum_{k=1}^N \lambda_k e_k(t)^2
| |
| \end{align}
| |
| </math><br>which goes to 0 by Mercer's theorem.
| |
| | |
| == Properties of the Karhunen–Loève transform ==
| |
| | |
| === Special case: Gaussian distribution ===
| |
| Since the limit in the mean of jointly Gaussian random variables is jointly Gaussian, and jointly Gaussian random (centered) variables are independent if and only if they are orthogonal, we can also conclude:
| |
| | |
| '''Theorem'''. The variables '''Z'''<sub>''i''</sub> have a joint Gaussian distribution and are stochastically independent if the original process {'''X'''<sub>''t''</sub>}<sub>''t''</sub> is Gaussian.
| |
| | |
| In the gaussian case, since the variables '''Z'''<sub>''i''</sub> are independent, we can say more:
| |
| | |
| :<math> \lim_{N \rightarrow \infty} \sum_{i=1}^N e_i(t) Z_i(\omega) = X_t(\omega) </math>
| |
| almost surely.
| |
| | |
| === The Karhunen–Loève transform decorrelates the process ===
| |
| This is a consequence of the independence of the ''Z''<sub>''k''</sub>.
| |
| | |
| === The Karhunen–Loève expansion minimizes the total mean square error ===
| |
| | |
| In the introduction, we mentioned that the truncated Karhunen–Loeve expansion was the best approximation of the original process in the sense that it reduces the total mean-square error resulting of its truncation. Because of this property, it is often said that the KL transform optimally compacts the energy.
| |
| | |
| More specifically, given any orthonormal basis {''f''<sub>''k''</sub>} of ''L''<sup>2</sup>([''a'', ''b'']), we may decompose the process ''X''<sub>''t''</sub> as:
| |
| | |
| :<math>X_t(\omega)=\sum_{k=1}^\infty A_k(\omega) f_k(t)</math>
| |
| | |
| where <math>A_k(\omega)=\int_{[a,b]} X_t(\omega) f_k(t)\,dt</math>
| |
| | |
| and we may approximate ''X''<sub>''t''</sub> by the finite sum <math>\hat{X}_t(\omega)=\sum_{k=1}^N A_k(\omega) f_k(t)</math> for some integer ''N''.
| |
| | |
| '''Claim'''.
| |
| Of all such approximations, the KL approximation is the one that minimizes the total mean square error (provided we have arranged the eigenvalues in decreasing order).
| |
| | |
| <div class="NavFrame collapsed">
| |
| <div class="NavHead">[Proof]</div>
| |
| <div class="NavContent" style="text-align:left">
| |
| Consider the error resulting from the truncation at the ''N''-th term in the following orthonormal expansion:
| |
| :<math>\epsilon_N(t)=\sum_{k=N+1}^\infty A_k(\omega) f_k(t)</math>
| |
| The mean-square error ε<sub>''N''</sub><sup>2</sup>(''t'') can be written as:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \varepsilon_N^2(t)&=\mathrm{E}\left[\sum_{i=N+1}^\infty \sum_{j=N+1}^\infty A_i(\omega) A_j(\omega) f_i(t) f_j(t)\right]\\
| |
| &=\sum_{i=N+1}^\infty \sum_{j=N+1}^\infty \mathrm{E}\left[\int_{[a, b]}\int_{[a, b]} X_t X_s f_i(t)f_j(s) ds\, dt\right] f_i(t) f_j(t)\\
| |
| &=\sum_{i=N+1}^\infty \sum_{j=N+1}^\infty f_i(t) f_j(t) \int_{[a, b]}\int_{[a, b]}K_X(s,t) f_i(t)f_j(s) ds\, dt | |
| \end{align}
| |
| </math>
| |
| | |
| We then integrate this last equality over [''a'', ''b'']. The orthonormality of the ''f''<sub>k</sub> yields:
| |
| | |
| :<math>
| |
| \int_{[a, b]} \varepsilon_N^2(t) dt=\sum_{k=N+1}^\infty \int_{[a, b]}\int_{[a, b]} K_X(s,t) f_k(t)f_k(s) ds\, dt
| |
| </math>
| |
| | |
| The problem of minimizing the total mean-square error thus comes down to minimizing the right hand side of this equality subject to the constraint that the ''f''<sub>''k''</sub> be normalized. We hence introduce β<sub>k</sub>, the Lagrangian multipliers associated with these constraints, and aim at minimizing the following function:
| |
| | |
| :<math>
| |
| Er[f_k(t),k\in\{N+1,\ldots\}]=\sum_{k=N+1}^\infty \int_{[a, b]}\int_{[a, b]} K_X(s,t) f_k(t)f_k(s) ds dt-\beta_k \left(\int_{[a, b]} f_k(t) f_k(t) dt -1\right)
| |
| </math>
| |
| | |
| Differentiating with respect to ''f''<sub>''i''</sub>(''t'') and setting the derivative to 0 yields:
| |
| | |
| :<math>
| |
| \frac{\partial Er}{\partial f_i(t)}=\int_{[a, b]} \left(\int_{[a, b]} K_X(s,t) f_i(s) ds -\beta_i f_i(t)\right)dt=0
| |
| </math>
| |
| | |
| which is satisfied in particular when <math>\int_{[a, b]} K_X(s,t) f_i(s) \,ds =\beta_i f_i(t)</math>, in other words when the ''f''<sub>''k''</sub> are chosen to be the eigenfunctions of ''T''<sub>''K''<sub>''X''</sub></sub>, hence resulting in the KL expansion.
| |
| </div>
| |
| </div>
| |
| | |
| === Explained variance ===
| |
| An important observation is that since the random coefficients ''Z''<sub>''k''</sub> of the KL expansion are uncorrelated, the [[Variance#Sum of uncorrelated variables .28Bienaym.C3.A9 formula.29|Bienaymé formula]] asserts that the variance of ''X''<sub>''t''</sub> is simply the sum of the variances of the individual components of the sum:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \mbox{Var}[X_t]&=\sum_{k=0}^\infty e_k(t)^2 \mbox{Var}[Z_k]=\sum_{k=1}^\infty \lambda_k e_k(t)^2
| |
| \end{align}
| |
| </math>
| |
| | |
| Integrating over [''a'', ''b''] and using the orthonormality of the ''e''<sub>''k''</sub>, we obtain that the total variance of the process is:
| |
| | |
| :<math>
| |
| \int_{[a,b]} \mbox{Var}[X_t] dt=\sum_{k=1}^\infty \lambda_k
| |
| </math>
| |
| | |
| In particular, the total variance of the ''N''-truncated approximation is <math>\sum_{k=1}^N \lambda_k</math>. As a result, the ''N''-truncated expansion explains <math>\sum_{k=1}^N \lambda_k/\sum_{k=1}^\infty \lambda_k</math> of the variance; and if we are content with an approximation that explains, say, 95% of the variance, then we just have to determine an <math>N\in\mathbb{N}</math> such that <math>\sum_{k=1}^N \lambda_k/\sum_{k=1}^\infty \lambda_k \geq 0.95</math>.
| |
| | |
| === The Karhunen–Loève expansion has the minimum representation entropy property ===
| |
| {{Expand section|date=May 2011}}
| |
| | |
| == Linear Karhunen-Loeve Approximations ==
| |
| | |
| Let us consider a whole class of signals we want to approximate over the first M vectors of a basis. These signals are modeled as realizations of a random vector Y[n] of size N. To optimize the approximation we design a basis that minimizes the average approximation error. This section proves that optimal bases are karhunen-loeve bases that diagonalize the covariance matrix of Y. The random vector Y can be decomposed in an orthogonal basis <math>{{\left\{ {{g}_{m}} \right\}}_{0\le m\le N}}:</math>
| |
| | |
| <math>Y=\sum\limits_{m=0}^{N-1}{\left\langle Y,{{g}_{m}} \right\rangle {{g}_{m}},}</math>
| |
| | |
| where each
| |
| <math>\left\langle Y,{{g}_{m}} \right\rangle =\sum\limits_{n=0}^{N-1}{Y\left[ n \right]}.g_{m}^{*}\left[ n \right]</math>
| |
| | |
| is a random variable. The approximation from the first <math>M\le N</math> vectors of the basis is
| |
|
| |
| <math>{{Y}_{M}}=\sum\limits_{m=0}^{M-1}{\left\langle Y,{{g}_{m}} \right\rangle {{g}_{m}}}</math>
| |
| | |
| The energy conservation in an orthogonal basis implies
| |
| | |
| <math>\varepsilon \left[ M \right]=E\left\{ {{\left\| Y-{{Y}_{M}} \right\|}^{2}} \right\}=\sum\limits_{m=M}^{N-1}{E\left\{ {{\left| \left\langle Y,{{g}_{m}} \right\rangle \right|}^{2}} \right\}}</math>
| |
| | |
| This error is related to the covariance of Y defined by
| |
|
| |
| <math>R\left[ n,m \right]=E\left\{ Y\left[ n \right]{{Y}^{*}}\left[ m \right] \right\}</math>
| |
| | |
| For any vector x[n] we denote by K the covariance operator represented by this matrix,
| |
| | |
| <math>E\left\{ {{\left| \left\langle Y,x \right\rangle \right|}^{2}} \right\}=\left\langle Kx,x \right\rangle =\sum\limits_{n=0}^{N-1}{\sum\limits_{m=0}^{N-1}{R\left[ n,m \right]x\left[ n \right]{{x}^{*}}\left[ m \right]}}</math>
| |
| | |
| The error <math>\varepsilon \left[ M \right]</math> is therefore a sum of the last N-M coefficients of the covariance operator
| |
| | |
| <math>\varepsilon \left[ M \right]=\sum\limits_{m=M}^{N-1}{\left\langle K{{g}_{m}},{{g}_{m}} \right\rangle }</math>
| |
| | |
| The covariance operator K is Hermitian and Positive and is thus diagonalized in an orthogonal basis called a Karhunen-Loeve basis. The following theorem states that a Karhunen-Loeve basis is optimal for linear approximations.
| |
| | |
| '''Theorem:(Optimality of Karhunen-Loeve basis)''' Let K be acovariance operator. For all <math>M\ge 1</math>, the approximation error
| |
| | |
| <math>\varepsilon \left[ M \right]=\sum\limits_{m=M}^{N-1}{\left\langle K{{g}_{m}},{{g}_{m}} \right\rangle }</math>
| |
| | |
| is minimum if and only if <math>{{\left\{ {{g}_{m}} \right\}}_{0\le m<N}}</math> is a Karhunen-Loeve basis ordered by decreasing eigenvalues.
| |
| | |
| <math>\left\langle K{{g}_{m}},{{g}_{m}} \right\rangle \ge \left\langle K{{g}_{m+1}},{{g}_{m+1}} \right\rangle for,0\le m<N-1</math>
| |
| | |
| == Non-Linear Approximation in Bases ==
| |
| | |
| Linear approximations project the signal on M vectors a priori. The approximation can be made more precise by choosing the M orthogonal vectors depending on the signal properties. This section analyzes the general performance of these non-linear approximations.
| |
| A signal <math>f\in \Eta </math> is approximated with M vectors selected adaptively in an orthonormal basis <math>\Beta ={{\left\{ {{g}_{m}} \right\}}_{m\in \mathbb{N}}}</math> of <math>\Eta </math>. Let <math>{{f}_{M}}</math> be the projection of f over M vectors whose indices are in <math>{{I}_{M}}</math>:
| |
| | |
| <math>{{f}_{M}}=\sum\limits_{m\in {{I}_{M}}}^{{}}{\left\langle f,{{g}_{m}} \right\rangle {{g}_{m}}}</math>
| |
| | |
| The approximation error is the sum of the remaining coefficients
| |
| | |
| <math>\varepsilon \left[ M \right]=\left\{ {{\left\| f-{{f}_{M}} \right\|}^{2}} \right\}=\sum\limits_{m\notin {{I}_{M}}}^{N-1}{\left\{ {{\left| \left\langle f,{{g}_{m}} \right\rangle \right|}^{2}} \right\}}</math>
| |
| | |
| To minimize this error, the indices in <math>{{I}_{M}}</math> must correspond to the M vectors having the largest inner product amplitude
| |
| <math>\left| \left\langle f,{{g}_{m}} \right\rangle \right|</math>. These are the vectors that best correlate f. They can thus be interpreted as the main features of f. The resulting error is necessarily smaller than the error of a linear approximation which selects the M approximation vectors independently of f.
| |
| Let us sort <math>{{\left\{ \left| \left\langle f,{{g}_{m}} \right\rangle \right| \right\}}_{m\in \mathbb{N}}}</math> in decreasing order :<math>\left| \left\langle f,{{g}_{{{m}_{k}}}} \right\rangle \right|\ge \left| \left\langle f,{{g}_{{{m}_{k+1}}}} \right\rangle \right|</math>. The best non-linear approximation is
| |
| | |
| <math>{{f}_{M}}=\sum\limits_{k=1}^{M}{\left\langle f,{{g}_{{{m}_{k}}}} \right\rangle {{g}_{{{m}_{k}}}}}</math>
| |
| | |
| It can also be written as inner product thresholding:
| |
| | |
| <math>{{f}_{M}}=\sum\limits_{m=0}^{\infty }{{{\theta }_{T}}\left( \left\langle f,{{g}_{m}} \right\rangle \right){{g}_{m}}}</math>
| |
| | |
| with <math>T=\left| \left\langle f,{{g}_{{{m}_{M}}}} \right\rangle \right|</math> and
| |
| | |
| <math>{{\theta }_{T}}\left( x \right)=\left\{ \begin{align}
| |
| & x,if\left| x \right|\ge T \\
| |
| & 0,if\left| x \right|<T \\
| |
| \end{align} \right.</math>
| |
| | |
| The non-linear error is
| |
| | |
| <math>\varepsilon \left[ M \right]=\left\{ {{\left\| f-{{f}_{M}} \right\|}^{2}} \right\}=\sum\limits_{k=M+1}^{\infty }{\left\{ {{\left| \left\langle f,{{g}_{{{m}_{k}}}} \right\rangle \right|}^{2}} \right\}}</math>
| |
| | |
| this error goes quickly to zero as M increases,if the sorted values of <math>\left| \left\langle f,{{g}_{{{m}_{k}}}} \right\rangle \right|</math> have a fast decay as k increases. This decay is quantified by computing the <math>{{\Iota }^{\Rho }}</math> norm of the signal inner products in B:
| |
| | |
| <math>{{\left\| f \right\|}_{\Beta ,p}}={{\left( \sum\limits_{m=0}^{\infty }{{{\left| \left\langle f,{{g}_{m}} \right\rangle \right|}^{p}}} \right)}^{1/p}}</math>
| |
| | |
| The following theorem relates the decay of <math>\varepsilon \left[ M \right]</math> to <math>{{\left\| f \right\|}_{\Beta ,p}}</math>
| |
| | |
| '''Theorem:(decay of error)''' If <math>{{\left\| f \right\|}_{\Beta ,p}}<+\infty </math> with <math>p<2</math> then
| |
| | |
| <math>\varepsilon \left[ M \right]\le \frac{\left\| f \right\|_{\Beta ,p}^{2}}{\frac{2}{p}-1}{{M}^{1-\frac{2}{p}}}</math>
| |
| | |
| and <math>\varepsilon \left[ M \right]=o\left( {{M}^{1-\frac{2}{p}}} \right)</math>. Conversely, if <math>\varepsilon \left[ M \right]=o\left( {{M}^{1-\frac{2}{p}}} \right)</math> then
| |
| | |
| <math>{{\left\| f \right\|}_{\Beta ,q}}<+\infty </math> for any <math>q>p</math>.
| |
| | |
| === Non-optimality of Karhunen-Loéve Bases ===
| |
| | |
| To further illustrate the differences between linear and non -linear approximations, we study the decomposition of a simple non-Gaussian random vector in a Karhunen-Loéve basis. Processes whose realizations have a random translation are stationary. The Karhunen-Loéve basis is then a Fourier basis and we study its performance.
| |
| To simplify the analysis, consider a random vector Y[n] of size N that is random shift modulo N of a deterministic signal f[n] of zero mean <math>\sum\nolimits_{n=0}^{N-1}{f\left[ n \right]=0}</math>:
| |
| | |
| <math>Y\left[ n \right]=f\left[ \left( n-p \right)\bmod N \right]</math>
| |
| | |
| The random shift P is uniformly distributed on [0,N-1]:
| |
| | |
| <math>\Pr \left( P=p \right)=\frac{1}{N}for,0\le p<N</math>
| |
| | |
| Clearly
| |
| | |
| <math>E\left\{ Y\left[ n \right] \right\}=\frac{1}{N}\sum\limits_{p=0}^{N-1}{f\left[ \left( n-p \right)\bmod N \right]}=0</math>
| |
| | |
| and
| |
| | |
| <math>R\left[ n,k \right]=E\left\{ Y\left[ n \right]Y\left[ k \right] \right\}=\frac{1}{N}\sum\limits_{p=0}^{N-1}{f\left[ \left( n-p \right)\bmod N \right]}f\left[ \left( k-p \right)\bmod N \right]</math>
| |
| | |
| <math>=\frac{1}{N}f\Theta \bar{f}\left[ n-k \right]with,\bar{f}\left[ n \right]=f\left[ -n \right]</math>
| |
| | |
| Hence <math>R\left[ n,k \right]={{R}_{Y}}\left[ n-k \right]with,</math>
| |
| | |
| <math>{{R}_{Y}}\left[ k \right]=\frac{1}{N}f\Theta \bar{f}\left[ k \right]</math>
| |
| | |
| Since R<sub>Y</sub> is N periodic, Y is a circular stationary random vector. The covariance operator is a circular convolution with R<sub>Y</sub> and is therefore diagonalized in the discrete Fourier Karhunen-Loéve basis <math>{{\left\{ \frac{1}{\sqrt{N}}{{e}^{\frac{i2\pi mn}{N}}} \right\}}_{0\le m<N}}</math>. The power spectrum is Fourier Transform of R<sub>Y</sub>:
| |
| | |
| <math>{{P}_{Y}}\left[ m \right]={{{\hat{R}}}_{Y}}\left[ m \right]=\frac{1}{N}{{\left| \hat{f}\left[ m \right] \right|}^{2}}</math>
| |
| | |
| '''Example:''' Consider an extreme case where <math>f\left[ n \right]=\delta \left[ n \right]-\delta \left[ n-1 \right]</math>
| |
| A theorem stated above guarantees that the Fourier Karhunen-Loéve basis produces a smaller expected approximation error than a canonical basis of Diracs <math>{{\left\{ {{g}_{m}}\left[ n \right]=\delta \left[ n-m \right] \right\}}_{0\le m<N}}</math>.
| |
| Indeed we do not know a priori the abscissa of the non-zero coefficients of Y, so there is no particular Dirac that is better adapted to perform the approximation . But the Fourier vectors cover the whole support of Y and thus absorb a part of the signal energy.
| |
| | |
| <math>E\left\{ {{\left| \left\langle Y\left[ n \right],\frac{1}{\sqrt{N}}{{e}^{\frac{i2\pi mn}{N}}} \right\rangle \right|}^{2}} \right\}={{P}_{Y}}\left[ m \right]=\frac{4}{N}{{\sin }^{2}}\left( \frac{\pi k}{N} \right)</math>
| |
|
| |
| Selecting higher frequency Fourier coefficients yields a better mean-square approximation than choosing a priori a few Dirac vectors to perform the approximation.
| |
| The situation is totally different for non-linear approximations. If <math>f\left[ n \right]=\delta \left[ n \right]-\delta \left[ n-1 \right]</math> then the discrete Fourier basis is extremely inefficient because because f and hence Y have an energy that is almost uniformly spread among all Fourier vectors. In contrast, since f has only two non-zero coefficients in the Dirac basis, a non-linear approximation of Y with <math>M\ge 2</math> gives zero error.
| |
| <ref>A wavelet tour of signal processing-Stéphane Mallat</ref>
| |
| | |
| == Principal component analysis ==
| |
| {{Main|Principal component analysis}}
| |
| | |
| We have established the Karhunen–Loève theorem and derived a few properties thereof. We also noted that one hurdle in its application was the numerical cost of determining the eigenvalues and eigenfunctions of its covariance operator through the Fredholm integral equation of the second kind <math>\int_{[a,b]} K_X(s,t) e_k(s)\,ds=\lambda_k e_k(t)</math> .
| |
| | |
| However, when applied to a discrete and finite process <math>\left(X_n\right)_{n\in\{1,\ldots,N\}}</math>, the problem takes a much simpler form and standard algebra can be used to carry out the calculations.
| |
| | |
| Note that a continuous process can also be sampled at ''N'' points in time in order to reduce the problem to a finite version.
| |
| | |
| We henceforth consider a random ''N''-dimensional vector <math>X=\left(X_1~X_2~\ldots~X_N\right)^T</math>. As mentioned above, ''X'' could contain ''N'' samples of a signal but it can hold many more representations depending on the field of application. For instance it could be the answers to a survey or economic data in an econometrics analysis.
| |
| | |
| As in the continuous version, we assume that ''X'' is centered, otherwise we can let <math>X:=X-\mu_X</math> (where <math>\mu_X</math> is the [[mean vector]] of ''X'') which is centered.
| |
| | |
| Let us adapt the procedure to the discrete case.
| |
| | |
| === Covariance matrix ===
| |
| Recall that the main implication and difficulty of the KL transformation is computing the eigenvectors of the linear operator associated to the covariance function, which are given by the solutions to the integral equation written above.
| |
| | |
| Define Σ, the covariance matrix of ''X''. Σ is an ''N'' by ''N'' matrix whose elements are given by:
| |
| :<math>\Sigma_{ij}=E[X_i X_j],\qquad \forall i,j \in \{1,\ldots,N\}</math>
| |
| | |
| Rewriting the above integral equation to suit the discrete case, we observe that it turns into:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| &\sum_{i=1}^N \Sigma_{ij} e_j=\lambda e_i\\
| |
| \Leftrightarrow \quad& \Sigma e=\lambda e
| |
| \end{align}
| |
| </math>
| |
| | |
| where <math>e=(e_1~e_2~\ldots~e_N)^T</math> is an ''N''-dimensional vector.
| |
| | |
| The integral equation thus reduces to a simple matrix eigenvalue problem, which explains why the PCA has such a broad domain of applications.
| |
| | |
| Since Σ is a positive definite symmetric matrix, it possesses a set of orthonormal eigenvectors forming a basis of <math>\R^N</math>, and we write <math>\{\lambda_i,\phi_i\}_{i\in\{1,\ldots,N\}}</math> this set of eigenvalues and corresponding eigenvectors, listed in decreasing values of λ<sub>''i''</sub>. Let also <math>\Phi</math> be the orthonormal matrix consisting of these eigenvectors:
| |
| :<math>
| |
| \begin{align}
| |
| \Phi &:=\left(\phi_1~\phi_2~\ldots~\phi_N\right)^T\\
| |
| \Phi^T \Phi &=I
| |
| \end{align}
| |
| </math>
| |
| | |
| === Principal component transform ===
| |
| It remains to perform the actual KL transformation, called the ''principal component transform'' in this case. Recall that the transform was found by expanding the process with respect to the basis spanned by the eigenvectors of the covariance function. In this case, we hence have:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| X &=\sum_{i=1}^N \langle \phi_i,X\rangle \phi_i\\
| |
| &=\sum_{i=1}^N \phi_i^T X \phi_i
| |
| \end{align}
| |
| </math>
| |
| | |
| In a more compact form, the principal component transform of ''X'' is defined by:
| |
| :<math>
| |
| \left\{
| |
| \begin{array}{rl}
| |
| Y&=\Phi^T X\\
| |
| X&=\Phi Y
| |
| \end{array}
| |
| \right.
| |
| </math>
| |
| | |
| The ''i''-th component of ''Y'' is <math>Y_i=\phi_i^T X</math>, the projection of ''X'' on <math>\phi_i</math> and the inverse transform <math>X=\Phi Y</math> yields the expansion of <math>X</math> on the space spanned by the <math>\phi_i</math>:
| |
| | |
| :<math>
| |
| X=\sum_{i=1}^N Y_i \phi_i=\sum_{i=1}^N \langle \phi_i,X\rangle \phi_i
| |
| </math>
| |
| | |
| As in the continuous case, we may reduce the dimensionality of the problem by truncating the sum at some <math>K\in\{1,\ldots,N\}</math> such that <math>\frac{\sum_{i=1}^K \lambda_i}{\sum_{i=1}^N \lambda_i}\geq \alpha</math> where α is the explained variance threshold we wish to set.
| |
| | |
| We can also reduce the dimensionality through the use of multilevel dominant eigenvector estimation (MDEE).<ref>X. Tang, “Texture information in run-length matrices,” IEEE Transactions on Image Processing, vol. 7, No. 11, pp. 1602- 1609, Nov. 1998</ref>
| |
| | |
| == Examples ==
| |
| | |
| === The Wiener process ===
| |
| There are numerous equivalent characterizations of the [[Wiener process]] which is a mathematical formalization of [[Brownian motion]]. Here we regard it as the centered standard Gaussian process '''W'''<sub>''t''</sub> with covariance function
| |
| :<math> K_{W}(t,s) = \operatorname{Cov}(W_t,W_s) = \min (s,t). </math>
| |
| | |
| We restrict the time domain to [''a'',''b'']=[0,1] without loss of generality.
| |
| | |
| The eigenvectors of the covariance kernel are easily determined. These are
| |
| :<math> e_k(t) = \sqrt{2} \sin \left( \left(k - \textstyle\frac{1}{2}\right) \pi t \right)</math>
| |
| and the corresponding eigenvalues are
| |
| :<math> \lambda_k = \frac{1}{(k -\frac{1}{2})^2 \pi^2}. </math>
| |
| | |
| <div class="NavFrame collapsed">
| |
| <div class="NavHead">[Proof]</div>
| |
| <div class="NavContent" style="text-align:left">
| |
| In order to find the eigenvalues and eigenvectors, we need to solve the integral equation:
| |
| | |
| :<math> | |
| \begin{align}
| |
| \int_{[a,b]} K_W(s,t) e(s)ds&=\lambda e(t)\qquad \forall t, 0\leq t\leq 1\\
| |
| \int_0^1\min(s,t) e(s)ds&=\lambda e(t)\qquad \forall t, 0\leq t\leq 1 \\
| |
| \int_0^t s e(s) ds + t \int_t^1 e(s) ds &= \lambda e(t) \qquad \forall t, 0\leq t\leq 1
| |
| \end{align}
| |
| </math>
| |
| | |
| differentiating once with respect to ''t'' yields:
| |
| | |
| :<math>
| |
| \int_{t}^1 e(s) ds=\lambda e'(t)
| |
| </math>
| |
| | |
| a second differentiation produces the following differential equation:
| |
| :<math>
| |
| -e(t)=\lambda e''(t)
| |
| </math>
| |
| | |
| The general solution of which has the form:
| |
| :<math>
| |
| e(t)=A\sin\left(\frac{t}{\sqrt{\lambda}}\right)+B\cos\left(\frac{t}{\sqrt{\lambda}}\right)
| |
| </math>
| |
| | |
| where ''A'' and ''B'' are two constants to be determined with the boundary conditions. Setting ''t''=0 in the initial integral equation gives ''e''(0)=0 which implies that ''B''=0 and similarly, setting ''t''=1 in the first differentiation yields ''e' ''(1)=0, whence:
| |
| | |
| :<math>\cos\left(\frac{1}{\sqrt{\lambda}}\right)=0</math>
| |
| | |
| which in turn implies that eigenvalues of ''T''<sub>''K''<sub>''X''</sub></sub> are:
| |
| | |
| :<math>\lambda_k=\left(\frac{1}{(k-\frac{1}{2})\pi}\right)^2,\qquad k\geq 1</math>
| |
| | |
| The corresponding eigenfunctions are thus of the form:
| |
| | |
| :<math>e_k(t)=A \sin\left((k-\frac{1}{2})\pi t\right),\qquad k\geq 1</math>
| |
| | |
| ''A'' is then chosen so as to normalize ''e''<sub>''k''</sub>:
| |
| | |
| :<math>\int_0^1 e_k^2(t) dt=1\quad \implies\quad A=\sqrt{2}</math>
| |
| </div>
| |
| </div>
| |
| | |
| This gives the following representation of the Wiener process:
| |
| | |
| '''Theorem'''. There is a sequence {''Z''<sub>''i''</sub>}<sub>''i''</sub> of independent Gaussian random variables with mean zero and variance 1 such that
| |
| :<math> W_t = \sqrt{2} \sum_{k=1}^\infty Z_k \frac{\sin \left(\left(k - \frac{1}{2}\right) \pi t\right)}{ \left(k - \frac{1}{2}\right) \pi}. </math>
| |
| Note that this representation is only valid for <math> t\in[0,1]. </math> On larger intervals, the increments are not independent. As stated in the theorem, convergence is in the L<sup>2</sup> norm and uniform in ''t''.
| |
| | |
| === The Brownian bridge ===
| |
| Similarly the [[Brownian bridge]] <math>B_t=W_t-tW_1</math> which is a [[stochastic process]] with covariance function
| |
| :<math>K_B(t,s)=\min(t,s)-ts</math>
| |
| can be represented as the series
| |
| :<math>B_t = \sum_{k=1}^\infty Z_k \frac{\sqrt{2} \sin(k \pi t)}{k \pi}</math>
| |
| | |
| == Applications ==
| |
| {{Expand section|date=July 2010}}
| |
| [[Adaptive optics]] systems sometimes use K–L functions to reconstruct wave-front phase information (Dai 1996, JOSA A).
| |
| Karhunen–Loève expansion is closely related to the [[Singular Value Decomposition]]. The latter has myriad applications in image processing, radar, seismology, and the like. If one has independent vector observations from a vector valued stochastic process then the left singular vectors are [[maximum likelihood]] estimates of the ensemble KL expansion.
| |
| | |
| === Applications in signal estimation and detection===
| |
| | |
| ====Detection of a known continuous signal S(t)====
| |
| In communication, we usually have to decide whether a signal from a noisy channel contains valuable information. The following hypothesis testing is used for detecting continuous signal s(t) from channel output X(t), N(t) is the channel noise, which is usually assumed zero mean gaussian process with correlation function <math>R_{N} (t, s) = E[N(t)N(s)]</math>
| |
| {|
| |
| |-
| |
| |<math>H: X(t) = N(t)</math>,
| |
| |-
| |
| |<math>K: X(t) = N(t)+s(t), t\in(0,T)</math>.
| |
| |}
| |
| | |
| ====Signal detection in white noise====
| |
| When the channel noise is white, its correlation function is
| |
| | |
| :<math>R_{N}(t) = \frac{N_0}{2} \delta (t)</math>,
| |
| | |
| and it has constant power spectrum density. In physically practical channel, the noise power is finite, so:
| |
| | |
| :<math>S_{N}(f) = \frac{N_{0}}{2} \text{ for } |f|<w, 0 \text{ for } |f|>w</math>.
| |
| | |
| Then the noise correlation function is sinc function with zeros at <math>\frac{n}{2\omega}, n = ...-1,0,1,...</math> .
| |
| Since are uncorrelated and gaussian, they are independent. Thus we can take samples from X(t) with time spacing
| |
| | |
| :<math> \Delta t = \frac{n}{2\omega}</math> within (0,T).
| |
| | |
| Let <math>X_i = X(i\Delta t)</math>. We have a total of <math>n = \frac{T}{\Delta t} = T(2\omega) = 2\omega T</math> i.i.d samples <math>\{X_1, X_2,...,X_n\}</math> to develop the likelihood-ratio test. Define signal <math>S_i = S(i\Delta t)</math>, the problem becomes,
| |
| | |
| :<math>H: X_i = N_i</math>,
| |
| | |
| :<math>K: X_i = N_i + S_i, i = 1,2...n.</math>
| |
| | |
| The log-likelihood ratio
| |
| | |
| :<math>\mathcal{L}(\underline{x}) = \log\frac{\sum^n_{i=1} (2S_i x_i - S_i^2)}{2\sigma^2} \Leftrightarrow \Delta t \Sigma ^n_{i = 1} S_i x_i = \sum^n_{i=1} S(i\Delta t)x(i\Delta t)\Delta t \gtrless \lambda_2</math>.
| |
| | |
| As <math> t \rightarrow 0, \text{ let } G = \int^T_0 S(t)x(t)dt</math>.
| |
| | |
| Then G is the test statistics and the [[Neyman–Pearson lemma|Neyman–Pearson optimum detector]] is:<math>G(\underline{x}) > G_0 \Rightarrow K, < G_0 \Rightarrow H</math>. As G is gaussian, we can characterize it by finding its mean and variances. Then we get
| |
| | |
| :<math>H: G \sim N(0,\frac{N_{0}E}{2})</math>
| |
| | |
| :<math>K: G \sim N(E,\frac{N_{0}E}{2})</math>, | |
| | |
| where <math>E = \int^T_{0} S^2(t)dt</math> is the signal energy.
| |
| | |
| The false alarm error
| |
| | |
| :<math>\alpha = \int^{\infty}_{G_{0}} N(0,\frac{N_{0}E}{2})dG \Rightarrow G_0 = \sqrt{\frac{N_0 E}{2}} \Phi^{-1}(1-\alpha)</math>
| |
| | |
| And the probability of detection:
| |
| | |
| :<math>\beta = \int^{\infty}_{G_0} N(E, \frac{N_0 E}{2})dG = 1-\Phi(\frac{G_0 - E}{\sqrt{\frac{N_0 E}{2}}}) = \Phi [\sqrt{\frac{2E}{N_0}} - \Phi^{-1}(1-\alpha)] , \Phi(\cdot)</math> is the cdf of standard normal gaussian variable.
| |
| | |
| ====Signal detection in colored noise ====
| |
| When N(t) is colored (correlated in time) gaussian noise with zero mean and covariance function <math>R_N(t,s) = E[X(t)X(s)],</math> we cannot sample independent discrete observations by evenly spacing the time. Instead, we can use K–L expansion to uncorrelate the noise process and get independent gaussian observation 'samples'.
| |
| The K–L expansion of N(t):
| |
| | |
| :<math>N(t) = \sum^{\infty}_{i=1} N_i \Phi_i(t), 0<t<T</math>,
| |
| | |
| where <math>N_i =\int N(t)\Phi_i(t)dt</math> and the orthonormal bases <math>\{\Phi_i{t}\} </math>are generated by kernal <math>R_N(t,s)</math>, i.e., solution to
| |
| | |
| :<math> \int ^T _0 R_N(t,s)\Phi_i(s)ds = \lambda_i \Phi_i(t), var[N_i] = \lambda_i</math>.
| |
| | |
| Do the expansion:
| |
| | |
| :<math>S(t) = \sum^{\infty}_{i = 1}S_i\Phi_i(t)</math>,
| |
| | |
| where <math>S_i = \int^T _0 S(t)\Phi_i(t)dt, 0<t<T.</math>, then
| |
| | |
| :<math>X_i = \int^T _0 X(t)\Phi_i(t) dt = N_i</math>
| |
| | |
| under H and <math>N_i + S_i</math> under K. Let <math>\overline{X} = \{X_1,X_2,\dots\}</math>, we have
| |
| | |
| :<math>{N_i}</math> are independent gaussian r.v's with variance <math>\lambda_i</math>
| |
| | |
| :under H: <math>\{X_i\}</math> are independent gaussian r.v's. <math>f_H[x(t)|0<t<T] = f_H(\underline{x}) = \prod^{\infty} _{i=1} \frac{1}{\sqrt{2\pi \lambda_i}}exp[-\frac{x_i^2}{2 \lambda_i}]</math>
| |
| | |
| :under K: <math>\{X_i - S_i\}</math> are independent gaussian r.v's. <math>f_K[x(t)|0<t<T] = f_K(\underline{x}) = \prod^{\infty} _{i=1} \frac{1}{\sqrt{2\pi \lambda_i}}exp[-\frac{(x_i - S_i)^2}{2 \lambda_i}]</math>
| |
| | |
| Hence, the log-LR is given by
| |
| | |
| :<math>\mathcal{L}(\underline{x}) = \sum^{\infty}_{i=1} \frac{2S_i x_i - S_i^2}{2\lambda_i}</math>
| |
| | |
| and the optimum detector is
| |
| | |
| :<math>G = \sum^{\infty}_{i=1} S_i x_i \lambda_i > G_0 \Rightarrow K, < G_0 \Rightarrow H.</math>
| |
| | |
| Define
| |
| | |
| :<math>k(t) = \sum^{\infty}_{i=1} \lambda_i S_i \Phi_i(t), 0<t<T,</math>
| |
| | |
| then <math>G = \int^T _0 k(t)x(t)dt</math>.
| |
| | |
| =====How to find ''k''(''t'')=====
| |
| Since
| |
| | |
| :<math>\int^T_0 R_N(t,s)k(s)ds = \sum^{\infty}_{i=1} \lambda_i S_i \int^T _0 R_N(t,s)\Phi_i (s) ds = \sum^{\infty}_{i=1} S_i \Phi_i(t) = S(t)</math>,
| |
| | |
| k(t) is the solution to
| |
| | |
| :<math>\int^T_0 R_N(t,s)k(s)ds = S(t)</math>.
| |
| | |
| If N(t)is wide-sense stationary,
| |
| | |
| :<math>\int^T_0 R_N(t-s)k(s)ds = S(t) </math>,
| |
| | |
| which is known as the [[Wiener–Hopf equation]]. The equation can be solved by taking fourier transform, but not practically realizable since infinite spectrum needs spatial factorization.
| |
| A special case which is easy to calculate k(t) is white gaussian noise.
| |
| | |
| :<math>\int^T_0 \frac{N_0}{2}\delta(t-s)k(s)ds = S(t) \Rightarrow k(t) = C S(t), 0<t<T</math>.
| |
| | |
| The corresponding impulse response is h(t) = k(T-t) = C S(T-t). Let C = 1, this is just the result we arrived at in previous section for detecting of signal in white noise.
| |
| | |
| =====Test threshold for Neyman–Pearson detector=====
| |
| Since X(t)is gaussian process, <math>G = \int^T_0 k(t)x(t)dt</math> is a gaussian random variable that can be characterized by its mean and variance.
| |
| | |
| :<math>E[G|H] = \int^T_0 k(t)E[x(t)|H]dt = 0</math>
| |
| | |
| :<math>E[G|K] = \int^T_0 k(t)E[x(t)|K]dt = \int^T_0 k(t)S(t)dt \equiv \rho</math>
| |
| | |
| :<math>E[G^2|H] = \int^T_0\int^T_0 k(t)k(s) R_N(t,s)dtds = \int^T_0 k(t)(\int^T_0 k(s)R_N(t,s)ds)=\int^T_0 k(t)S(t)dt = \rho</math>
| |
| | |
| :<math>var[G|H] = E[G^2|H] - (E[G|H])^2 = \rho</math>
| |
| | |
| :<math>E[G^2|K]=\int^T_0\int^T_0k(t)k(s)E[x(t)x(s)]dtds = \int^T_0\int^T_0k(t)k(s)(R_N(t,s) +S(t)S(s))dtds = \rho + \rho^2</math>
| |
| | |
| :<math>var[G|K] = E[G^2|K] - (E[G|K])^2 = \rho + \rho^2 -\rho^2 = \rho. </math>
| |
| | |
| Hence, we obtain the distributions of ''H'' and ''K'':
| |
| | |
| :<math>H: G \sim N(0,\rho)</math>
| |
| | |
| :<math>K: G \sim N(\rho, \rho)</math>
| |
| | |
| The false alarm error is
| |
| | |
| :<math>\alpha = \int^{\infty}_{G_0} N(0,\rho)dG = 1 - \Phi(\frac{G_0}{\sqrt{\rho}}).</math>
| |
| | |
| So the test threshold for the Neyman–Pearson optimum detector is
| |
| | |
| :<math>G_0 = \sqrt{\rho} \Phi^{-1} (1-\alpha)</math>.
| |
| | |
| Its power of detection is
| |
| | |
| :<math>\beta = \int^{\infty}_{G_0} N(\rho, \rho)dG = \Phi [\sqrt{\rho} - \Phi^{-1}(1 - \alpha)]</math>.
| |
| | |
| When the noise is white gaussian process, the signal power is
| |
| | |
| :<math>\rho = \int^T_0 k(t)S(t)dt = \int^T_0 S(t)^2 dt = E</math>.
| |
| | |
| =====Prewhitening=====
| |
| For some type of colored noise, a typical practise is to add a prewhiterning filter before the matched filter to transform the colored noise into white noise. For example, N(t) is a wide-sense stationary colored noise with correlation function
| |
| | |
| :<math>\R_N(\tau) = \frac{B N_0}{4} e^{-B|\tau|}</math>
| |
| | |
| :and <math>S_N(f) = \frac{N_0}{2(1+(\frac{w}{B})^2)}</math>.
| |
| | |
| The transfer function of prewhitening filter is <math>H(f) = 1 + j \frac{w}{B}</math>.
| |
| | |
| ====Detection of a gaussian random signal in AWGN====
| |
| When the signal we want to detect from the noisy channel is also random, for example, a white gaussian process X(t), we can still implement K–L expansion to get independent sequence of observation. In this case, the detection problem is described as follows:
| |
| | |
| :<math>H_0 : Y(t) = N(t)</math>
| |
| | |
| :<math>H_1 : Y(t) = N(t) + X(t), 0<t<T. </math>
| |
| | |
| X(t) is a random process with correlation function <math>R_X(t,s) = E\{X[t]X[s]\}</math>
| |
| | |
| The K–L expansion of X(t) is
| |
| | |
| :<math>X(t) = \sum^{\infty}_{i=1} X_i \Phi_i(t)</math>,
| |
| | |
| where
| |
| | |
| :<math>X_i =\int^T_0 X(t) \Phi_i(t). \Phi(t)</math>
| |
| | |
| are solutions to
| |
| | |
| :<math> \int^T_0 R_X(t,s)\Phi_i(s)ds= \lambda_i \Phi_i(t)</math>.
| |
| | |
| So <math>X_i</math>'s are independent sequence of r.v's with zero mean and variance <math>\lambda_i</math>. Expanding Y(t) and N(t) by <math>\Phi_i(t)</math>, we get
| |
| | |
| :<math>Y_i = \int^T_0 Y(t)\Phi_i(t)dt = \int^T_0 [N(t) + X(t)]\Phi_i(t) = N_i + X_i</math>,
| |
| | |
| where <math>N_i = \int^T_0 N(t)\Phi_i(t)dt.</math>
| |
| | |
| As N(t) is gaussian white noise, <math>N_i</math>'s are i.i.d sequence of r.v with zero mean and variance<math>\frac{N_0}{2}</math>, then the problem is simplified as follows,
| |
| | |
| :<math>H_0: Y_i = N_i</math>
| |
| | |
| :<math>H_1: Y_i = N_i + X_i</math>
| |
| | |
| The Neyman–Pearson optimal test:
| |
| | |
| :<math>\Lambda = \frac{f_Y|H_1}{f_Y|H_0} = Ce^{-\sum^{\infty}_{i=1}\frac{y_i^2}{2} \frac{\lambda_i}{\frac{N_0}{2}(\frac{N_0}{2} + \lambda_i)}}</math>,
| |
| | |
| so the log-likelihood ratio
| |
| | |
| :<math>\mathcal{L} = ln(\Lambda) = K -\sum^{\infty}_{i=1}\frac{y_i^2}{2} \frac{\lambda_i}{\frac{N_0}{2}(\frac{N_0}{2} + \lambda_i)}</math>.
| |
| | |
| Since
| |
| | |
| :<math>\hat{X_i} = \frac{\lambda_i}{\frac{N_0}{2}(\frac{N_0}{2} + \lambda_i)}</math>
| |
| | |
| is just the minimum-mean-square estimate of <math>X_i</math> given <math>Y_i</math>'s,
| |
| | |
| :<math>\mathcal{L} = K + \frac{1}{N_0} \sum^{\infty}_{i=1} Y_i \hat{X_i}</math>.
| |
| | |
| K–L expansion has the following property: If
| |
| | |
| :<math>f(t) = \sum f_i \Phi_i(t), g(t) = \sum g_i \Phi_i(t)</math>,
| |
| | |
| where
| |
| | |
| :<math>f_i = \int_0^T f(t) \Phi_i(t), g_i = \int_0^T g(t)\Phi_i(t).</math>,
| |
| | |
| then
| |
| | |
| :<math>\sum^{\infty}_{i=1} f_i g_i = \int^T_0 g(t)f(t)dt</math>.
| |
| | |
| So let
| |
| | |
| :<math>\hat{X(t|T)} = \sum^{\infty}_{i=1} \hat{X_i}\Phi_i(t)</math>, <math>\mathcal{L} = K + \frac{1}{N_0} \int^T_0 Y(t) \hat{X(t|T)}dt</math>.
| |
| | |
| Noncausal filter Q(t, s) can be used to get the estimate through
| |
| | |
| :<math>\hat{X(t|T)} = \int^T_0 Q(t,s)Y(s)ds</math>.
| |
| | |
| By [[orthogonality principle]], Q(t,s) satisfies
| |
| | |
| :<math>\int^T_0 Q(t,s)R_X(s,t)ds + \frac{N_0}{2} Q(t, \lambda) = R_X(t, \lambda), 0 < \lambda < T, 0<t<T. </math>.
| |
| | |
| However for practical reason, it's necessary to further derive the causal filter h(t, s), where h(t, s) = 0 for s > t, to get estimate <math>\hat{X(t|t)}</math>.
| |
| Specifically,
| |
| | |
| :<math>Q(t,s) = h(t,s) + h(s, t) - \int^T_0 h(\lambda, t)h(s, \lambda)d\lambda</math>.
| |
| | |
| ==See also==
| |
| *[[Principal component analysis]]
| |
| *[[Proper orthogonal decomposition]]
| |
| *[[Polynomial chaos]]
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| * {{cite book
| |
| |first1=Henry
| |
| |last1=Stark
| |
| |first2=John W.
| |
| |last2=Woods
| |
| |title=Probability, Random Processes, and Estimation Theory for Engineers
| |
| |publisher=Prentice-Hall, Inc
| |
| |year=1986
| |
| |isbn=0-13-711706-X
| |
| |url = http://openlibrary.org/books/OL21138080M/Probability_random_processes_and_estimation_theory_for_engineers
| |
| }}
| |
| *{{cite book
| |
| |first1=Roger
| |
| |last1=Ghanem
| |
| |first2=Pol
| |
| |last2=Spanos
| |
| |publisher = Springer-Verlag
| |
| |isbn = 0-387-97456-3
| |
| |title = Stochastic finite elements: a spectral approach
| |
| |url = http://openlibrary.org/books/OL1865197M/Stochastic_finite_elements
| |
| |year = 1991
| |
| }}
| |
| * {{cite book
| |
| |first1=I.
| |
| |last1=Guikhman
| |
| |first2=A.
| |
| |last2=Skorokhod
| |
| |title=Introduction a la Théorie des Processus Aléatoires
| |
| |publisher=Éditions MIR
| |
| |year=1977
| |
| }}
| |
| * {{cite book
| |
| |first1=B.
| |
| |last1=Simon
| |
| |title=Functional Integration and Quantum Physics
| |
| |publisher=Academic Press
| |
| |year=1979
| |
| }}
| |
| * {{cite journal
| |
| |last1=Karhunen
| |
| |first1=Kari
| |
| |title=Über lineare Methoden in der Wahrscheinlichkeitsrechnung
| |
| |journal=Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys.
| |
| |year=1947
| |
| |volume=37
| |
| |pages=1–79
| |
| }}
| |
| * {{cite book
| |
| |first1=M.
| |
| |last1=Loève
| |
| |title=Probability theory.'' Vol. II, 4th ed.
| |
| |series=Graduate Texts in Mathematics
| |
| |volume=46
| |
| |publisher=Springer-Verlag
| |
| |year=1978
| |
| |isbn=0-387-90262-7
| |
| }}
| |
| * {{cite journal
| |
| |first1=G.
| |
| |last1=Dai
| |
| |title=Modal wave-front reconstruction with Zernike polynomials and Karhunen–Loeve functions
| |
| |journal=JOSA A
| |
| |volume=13
| |
| |issue=6
| |
| |page=1218
| |
| |year=1996
| |
| |doi=10.1364/JOSAA.13.001218
| |
| |bibcode=1996JOSAA..13.1218D
| |
| }}
| |
| *Wu B., Zhu J., Najm F.(2005) "A Non-parametric Approach for Dynamic Range Estimation of Nonlinear Systems". In Proceedings of Design Automation Conference(841-844) 2005
| |
| *Wu B., Zhu J., Najm F.(2006) "Dynamic Range Estimation". IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 25 Issue:9 (1618-1636) 2006
| |
| * {{cite arXiv
| |
| |title=Entropy Encoding, Hilbert Space and Karhunen–Loeve Transforms
| |
| |first1=Palle E. T.
| |
| |last1=Jorgensen
| |
| |first2=Myung-Sin
| |
| |last2=Song
| |
| |eprint=math-ph/0701056
| |
| |year=2007
| |
| |bibcode=2007JMP....48j3503J
| |
| }}
| |
| <!--
| |
| * {{cite arXiv
| |
| |title=Synthesis of Taylor Phase Screens with Karhunen–Loeve Basis Functions
| |
| |first1=Richard J.
| |
| |last1=Mathar
| |
| |eprint=0705.1700
| |
| |class=astro-ph
| |
| |year=2007
| |
| |bibcode=2007arXiv0705.1700M
| |
| }}
| |
| -->
| |
| * {{cite journal
| |
| |first1=Richard J.
| |
| |last1=Mathar
| |
| |journal=Baltic Astronomy
| |
| |title=Karhunen–Loeve basis functions of Kolmogorov turbulence in the sphere
| |
| |year=2008
| |
| |volume=17
| |
| |issue=3/4
| |
| |pages=383–398
| |
| |bibcode=2008BaltA..17..383M
| |
| |arxiv = 0805.3979 }}
| |
| * {{cite arXiv
| |
| |first1=Richard J.
| |
| |last1=Mathar
| |
| |title=Modal decomposition of the von-Karman covariance of atmospheric turbulence in the circular entrance pupil
| |
| |eprint=0911.4710
| |
| |class=astro-ph.IM
| |
| |year=2009
| |
| |bibcode=2009arXiv0911.4710M
| |
| }}
| |
| * {{cite journal
| |
| |first1=Richard J.
| |
| |last1=Mathar
| |
| |journal=Waves in Random and Complex Media
| |
| |title=Karhunen–Loeve basis of Kolmogorov phase screens covering a rectangular stripe
| |
| |volume=20
| |
| |issue=1
| |
| |doi=10.1080/17455030903369677
| |
| |year=2010
| |
| |pages=23–35
| |
| |bibcode=2020WRCM...20...23M
| |
| }}
| |
| | |
| ==External links==
| |
| * ''Mathematica'' [http://reference.wolfram.com/mathematica/ref/KarhunenLoeveDecomposition.html KarhunenLoeveDecomposition] function.
| |
| * ''E161: Computer Image Processing and Analysis'' notes by Pr. Ruye Wang at [[Harvey Mudd College]] [http://fourier.eng.hmc.edu/e161/lectures/klt/klt.html]
| |
| | |
| {{DEFAULTSORT:Karhunen-Loeve theorem}}
| |
| [[Category:Estimation theory]]
| |
| [[Category:Probability theorems]]
| |
| [[Category:Signal processing]]
| |
| [[Category:Stochastic processes]]
| |
| [[Category:Statistical theorems]]
| |
| | |
| [[fr:Transformée de Karhunen-Loève]]
| |
| [[ru:Теорема Кархунена-Лоэва]]
| |
| [[sv:Karhunen-Loeve-transformen]]
| |