# Karhunen–Loève theorem

In the theory of stochastic processes, the **Karhunen–Loève theorem** (named after Kari Karhunen and Michel Loève), also known as the **Kosambi–Karhunen–Loève theorem**^{[1]}^{[2]} is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. Stochastic processes given by infinite series of this form were first^{[3]} considered by Damodar Dharmananda Kosambi.^{[4]} There exist many such expansions of a stochastic process: if the process is indexed over [*a*, *b*], any orthonormal basis of *L*^{2}([*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 sinusoidal functions (that is, sine and cosine functions), the coefficients in the Karhunen–Loève theorem are random variables 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*_{t}}_{t ∈ [a, b]} (where *centered* means that the expectations E(*X*_{t}) are defined and equal to 0 for all values of the parameter *t* in [*a*, *b*]) satisfying a technical continuity condition, *X*_{t} admits a decomposition

where *Z*_{k} are pairwise uncorrelated random variables and the functions *e*_{k} are continuous real-valued functions on [*a*, *b*] that are pairwise orthogonal in *L*^{2}[*a*, *b*]. It is therefore sometimes said that the expansion is *bi-orthogonal* since the random coefficients *Z*_{k} are orthogonal in the probability space while the deterministic functions *e*_{k} are orthogonal in the time domain. The general case of a process *X*_{t} that is not centered can be brought back to the case of a centered process by considering (*X*_{t} − E(*X*_{t})) which is a centered process.

Moreover, if the process is Gaussian, then the random variables *Z*_{k} 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 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 *Hotelling transform*.

## Formulation

- Throughout this article, we will consider a square integrable zero-mean random process
*X*_{t}defined over a probability space (Ω,*F*,**P**) and indexed over a closed interval [*a*,*b*], with covariance function*K*_{X}(*s,t*). We thus have:

- We associate to
*K*_{X}a linear operator*T*_{KX}defined in the following way:

Since*T*_{KX}is a linear operator, it makes sense to talk about its eigenvalues λ_{k}and eigenfunctions*e*_{k}, which are found solving the homogeneous Fredholm integral equation of the second kind

## Statement of the theorem

**Theorem**. Let *X*_{t} 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*_{X}(*s,t*).

Then *K*_{X}(*s,t*) is a Mercer kernel and letting *e*_{k} be an orthonormal basis of *L*^{2}([*a*, *b*]) formed by the eigenfunctions of *T*_{KX} with respective eigenvalues λ_{k}, *X*_{t} admits the following representation

where the convergence is in *L*^{2}, uniform in *t* and

Furthermore, the random variables *Z*_{k} have zero-mean, are uncorrelated and have variance λ_{k}

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*_{X}satisfies the definition of a Mercer kernel. By Mercer's theorem, there consequently exists a set {λ_{k},*e*_{k}(*t*)} of eigenvalues and eigenfunctions of T_{KX}forming an orthonormal basis of*L*^{2}([*a*,*b*]), and*K*_{X}can be expressed as

- The process
*X*_{t}can be expanded in terms of the eigenfunctions*e*_{k}as:

where the coefficients (random variables)*Z*_{k}are given by the projection of*X*_{t}on the respective eigenfunctions

- We may then derive

## 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**_{i} have a joint Gaussian distribution and are stochastically independent if the original process {**X**_{t}}_{t} is Gaussian.

In the Gaussian case, since the variables **Z**_{i} are independent, we can say more:

almost surely.

### The Karhunen–Loève transform decorrelates the process

This is a consequence of the independence of the *Z*_{k}.

### 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*_{k}} of *L*^{2}([*a*, *b*]), we may decompose the process *X*_{t} as:

and we may approximate *X*_{t} by the finite sum 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).

Consider the error resulting from the truncation at the *N*-th term in the following orthonormal expansion:

The mean-square error ε_{N}^{2}(*t*) can be written as:

We then integrate this last equality over [*a*, *b*]. The orthonormality of the *f*_{k} yields:

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*_{k} be normalized. We hence introduce β_{k}, the Lagrangian multipliers associated with these constraints, and aim at minimizing the following function:

Differentiating with respect to *f*_{i}(*t*) and setting the derivative to 0 yields:

which is satisfied in particular when , in other words when the *f*_{k} are chosen to be the eigenfunctions of *T*_{KX}, hence resulting in the KL expansion.

### Explained variance

An important observation is that since the random coefficients *Z*_{k} of the KL expansion are uncorrelated, the Bienaymé formula asserts that the variance of *X*_{t} is simply the sum of the variances of the individual components of the sum:

Integrating over [*a*, *b*] and using the orthonormality of the *e*_{k}, we obtain that the total variance of the process is:

In particular, the total variance of the *N*-truncated approximation is . As a result, the *N*-truncated expansion explains 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 such that .

### The Karhunen–Loève expansion has the minimum representation entropy property

## 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

is a random variable. The approximation from the first vectors of the basis is

The energy conservation in an orthogonal basis implies

This error is related to the covariance of Y defined by

For any vector x[n] we denote by K the covariance operator represented by this matrix,

The error is therefore a sum of the last N-M coefficients of the covariance operator

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 , the approximation error

is minimum if and only if is a Karhunen-Loeve basis ordered by decreasing eigenvalues.

## 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 is approximated with M vectors selected adaptively in an orthonormal basis of . Let be the projection of f over M vectors whose indices are in :

The approximation error is the sum of the remaining coefficients

To minimize this error, the indices in must correspond to the M vectors having the largest inner product amplitude . 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 in decreasing order :. The best non-linear approximation is

It can also be written as inner product thresholding:

The non-linear error is

this error goes quickly to zero as M increases,if the sorted values of have a fast decay as k increases. This decay is quantified by computing the norm of the signal inner products in B:

The following theorem relates the decay of to

**Theorem:(decay of error)** If with then

### 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 :

The random shift P is uniformly distributed on [0,N-1]:

Clearly

and

Since R_{Y} is N periodic, Y is a circular stationary random vector. The covariance operator is a circular convolution with R_{Y} and is therefore diagonalized in the discrete Fourier Karhunen-Loéve basis . The power spectrum is Fourier Transform of R_{Y}:

**Example:** Consider an extreme case where
A theorem stated above guarantees that the Fourier Karhunen-Loéve basis produces a smaller expected approximation error than a canonical basis of Diracs .
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.

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 then the discrete Fourier basis is extremely inefficient 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 gives zero error.
^{[5]}

## Principal component analysis

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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 .

However, when applied to a discrete and finite process , 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 . 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 (where 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:

Rewriting the above integral equation to suit the discrete case, we observe that it turns into:

where 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 , and we write this set of eigenvalues and corresponding eigenvectors, listed in decreasing values of λ_{i}. Let also be the orthonormal matrix consisting of these eigenvectors:

### 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:

In a more compact form, the principal component transform of *X* is defined by:

The *i*-th component of *Y* is , the projection of *X* on and the inverse transform yields the expansion of on the space spanned by the :

As in the continuous case, we may reduce the dimensionality of the problem by truncating the sum at some such that 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).^{[6]}

## 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**_{t} with covariance function

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

and the corresponding eigenvalues are

In order to find the eigenvalues and eigenvectors, we need to solve the integral equation:

differentiating once with respect to *t* yields:

a second differentiation produces the following differential equation:

The general solution of which has the form:

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:

which in turn implies that eigenvalues of *T*_{KX} are:

The corresponding eigenfunctions are thus of the form:

*A* is then chosen so as to normalize *e*_{k}:

This gives the following representation of the Wiener process:

**Theorem**. There is a sequence {*Z*_{i}}_{i} of independent Gaussian random variables with mean zero and variance 1 such that

Note that this representation is only valid for On larger intervals, the increments are not independent. As stated in the theorem, convergence is in the L^{2} norm and uniform in *t*.

### The Brownian bridge

Similarly the Brownian bridge which is a stochastic process with covariance function

can be represented as the series

## Applications

Template:Expand section 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

, |

. |

#### Signal detection in white noise

When the channel noise is white, its correlation function is

and it has constant power spectrum density. In physically practical channel, the noise power is finite, so:

Then the noise correlation function is sinc function with zeros at . Since are uncorrelated and gaussian, they are independent. Thus we can take samples from X(t) with time spacing

Let . We have a total of i.i.d samples to develop the likelihood-ratio test. Define signal , the problem becomes,

The log-likelihood ratio

Then G is the test statistics and the Neyman–Pearson optimum detector is:. As G is gaussian, we can characterize it by finding its mean and variances. Then we get

The false alarm error

And the probability of detection:

#### Signal detection in colored noise

When N(t) is colored (correlated in time) gaussian noise with zero mean and covariance function 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):

where and the orthonormal bases are generated by kernal , i.e., solution to

Do the expansion:

under H and under K. Let , we have

Hence, the log-LR is given by

and the optimum detector is

Define

##### How to find *k*(*t*)

Since

k(t) is the solution to

If N(t)is wide-sense stationary,

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.

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, is a gaussian random variable that can be characterized by its mean and variance.

Hence, we obtain the distributions of *H* and *K*:

The false alarm error is

So the test threshold for the Neyman–Pearson optimum detector is

Its power of detection is

When the noise is white gaussian process, the signal power is

##### Prewhitening

For some type of colored noise, a typical practise is to add a prewhitening 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

The transfer function of prewhitening filter is .

#### 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:

X(t) is a random process with correlation function

The K–L expansion of X(t) is

where

are solutions to

So 's are independent sequence of r.v's with zero mean and variance . Expanding Y(t) and N(t) by , we get

As N(t) is gaussian white noise, 's are i.i.d sequence of r.v with zero mean and variance, then the problem is simplified as follows,

The Neyman–Pearson optimal test:

so the log-likelihood ratio

Since

is just the minimum-mean-square estimate of given 's,

K–L expansion has the following property: If

where

then

So let

Noncausal filter Q(t, s) can be used to get the estimate through

By orthogonality principle, Q(t,s) satisfies

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 . Specifically,

## See also

## Notes

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- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ A wavelet tour of signal processing-Stéphane Mallat
- ↑ X. Tang, “Texture information in run-length matrices,” IEEE Transactions on Image Processing, vol. 7, No. 11, pp. 1602- 1609, Nov. 1998

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## External links

*Mathematica*KarhunenLoeveDecomposition function.*E161: Computer Image Processing and Analysis*notes by Pr. Ruye Wang at Harvey Mudd College [1]