Задача оптимизации
формируется следующим образом.

Имеется система
автоматического управления заданной
структуры. Необходимо так выбрать
параметры этой системы, чтобы получить
минимум среднеквадратичной ошибки при
заданных статистических характеристиках
полезного сигнала и помехи.

Обозначим

 –
выходной сигнал системы, оптимальной
по критерию минимума среднеквадратичной
ошибки.

Если

 –
выходной сигнал любой другой системы
данного класса (неоптимальной), то


, (4.5)

где

 –
требуемый выходной сигнал (аргумент

для простоты записи опущен).

Добавляя

в
скобки правой части неравенства (4.5),
получим


. (4.6)

Следовательно, из
(4.5) и (4.6):


, (4.7)

где


. (4.7)

Из равенства (4.7)
следует, что для выполнения неравенства
(4.5) необходимо, чтобы


(4.8)


. (4.9)

Так как первое
слагаемое в правой части равенства
(4.7) всегда положительно, то для выполнения
требования (4.8) достаточно приравнять
нулю второе слагаемое равенства (4.7):


. (4.10)

Требование (4.9) по
физическому смыслу говорит о том, что
величина среднеквадратичной ошибки
оптимальной системы всегда конечна.
Формально это значит, что в состав
сигнала

не
должен входить белый шум.

Равенство (4.10)
вместе с ограничением по виду выходного
сигнала оптимальной системы является
общим
условием минимума

среднеквадратичной ошибки.

26. Уравнение оптимальной линейной системы.

Получим уравнение,
определяющее оптимальную линейную
систему.

Примем за динамическую
характеристику линейной оптимальной
системы весовую функцию

.

Выходной сигнал
оптимальной системы определяется
формулой


(4.11)

где

 –
входной сигнал системы;

 –
время наблюдения входного сигнала
(здесь и далее полагается

).

Весовую функцию
произвольной линейной системы представим
в виде


. (4.12)

Тогда сигнал на
выходе произвольной системы будет равен


. (4.13)

Критерием
оптимальности данной системы будем
считать минимум среднеквадратичной
ошибки.

В соответствии с
полученным условием (4.10) с учётом (4.13)
запишем


. (4.14)

После умножения
выражения в круглых скобках на

и выполнения операции математического
ожидания получаем

,

(4.15)

где обозначено


. (4.16)

В общем случае,
когда

и функция

,
равенство (4.15) будет удовлетворяться
при условии, что равно нулю выражение
в скобках, т.е.


. (4.17)

Выражение (4.17) и
есть уравнение, определяющее оптимальную
линейную систему. Оно называется
интегральным уравнением Винера-Хонфа
и получено из условия (4.10).

  1. Определение весовой функции оптимальной линейной системы

Предполагается,
что мы умеем найти

– весовую функцию системы, преобразующую
данную случайную функцию X(t)
в белый шум V(t).

При этом известна
весовая функция обратной системы

,
формирующей X(t)
из белого шума V(t),
т.е.


(4.28)

Обозначим весовую
функцию оптимальной линейной системы
для белого шума V(t)
на входе
через

.

Искомая оптимальная
система представляет собой последовательное
соединение линейных систем с весовыми
функциями

и

(рис 4.2).

Рис. 4.2. Оптимальная
система

Ранее была получена
формула для

.


, (4.29)

где

 –
интенсивность белого шума V(t).

По формуле для
последовательного соединения систем,
описываемых весовыми функциями, имеем

В данном случае


 –
переменная интегрирования

,


 –
момент действия на систему функции
(входного сигнала).

Существуют
аналитические методы решения задачи,
когда функция X(t)
стационарна и имеет дробно-рациональную
спектральную плотность.

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In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In the Bayesian setting, the term MMSE more specifically refers to estimation with quadratic loss function. In such case, the MMSE estimator is given by the posterior mean of the parameter to be estimated. Since the posterior mean is cumbersome to calculate, the form of the MMSE estimator is usually constrained to be within a certain class of functions. Linear MMSE estimators are a popular choice since they are easy to use, easy to calculate, and very versatile. It has given rise to many popular estimators such as the Wiener–Kolmogorov filter and Kalman filter.

Motivation[edit]

The term MMSE more specifically refers to estimation in a Bayesian setting with quadratic cost function. The basic idea behind the Bayesian approach to estimation stems from practical situations where we often have some prior information about the parameter to be estimated. For instance, we may have prior information about the range that the parameter can assume; or we may have an old estimate of the parameter that we want to modify when a new observation is made available; or the statistics of an actual random signal such as speech. This is in contrast to the non-Bayesian approach like minimum-variance unbiased estimator (MVUE) where absolutely nothing is assumed to be known about the parameter in advance and which does not account for such situations. In the Bayesian approach, such prior information is captured by the prior probability density function of the parameters; and based directly on Bayes theorem, it allows us to make better posterior estimates as more observations become available. Thus unlike non-Bayesian approach where parameters of interest are assumed to be deterministic, but unknown constants, the Bayesian estimator seeks to estimate a parameter that is itself a random variable. Furthermore, Bayesian estimation can also deal with situations where the sequence of observations are not necessarily independent. Thus Bayesian estimation provides yet another alternative to the MVUE. This is useful when the MVUE does not exist or cannot be found.

Definition[edit]

Let x be a n\times 1 hidden random vector variable, and let y be a m\times 1 known random vector variable (the measurement or observation), both of them not necessarily of the same dimension. An estimator {\hat {x}}(y) of x is any function of the measurement y. The estimation error vector is given by e={\hat {x}}-x and its mean squared error (MSE) is given by the trace of error covariance matrix

{\displaystyle \operatorname {MSE} =\operatorname {tr} \left\{\operatorname {E} \{({\hat {x}}-x)({\hat {x}}-x)^{T}\}\right\}=\operatorname {E} \{({\hat {x}}-x)^{T}({\hat {x}}-x)\},}

where the expectation \operatorname {E} is taken over x conditioned on y. When x is a scalar variable, the MSE expression simplifies to {\displaystyle \operatorname {E} \left\{({\hat {x}}-x)^{2}\right\}}. Note that MSE can equivalently be defined in other ways, since

{\displaystyle \operatorname {tr} \left\{\operatorname {E} \{ee^{T}\}\right\}=\operatorname {E} \left\{\operatorname {tr} \{ee^{T}\}\right\}=\operatorname {E} \{e^{T}e\}=\sum _{i=1}^{n}\operatorname {E} \{e_{i}^{2}\}.}

The MMSE estimator is then defined as the estimator achieving minimal MSE:

{\displaystyle {\hat {x}}_{\operatorname {MMSE} }(y)=\operatorname {argmin} _{\hat {x}}\operatorname {MSE} .}

Properties[edit]

  • When the means and variances are finite, the MMSE estimator is uniquely defined[1] and is given by:
{\displaystyle {\hat {x}}_{\operatorname {MMSE} }(y)=\operatorname {E} \{x\mid y\}.}
In other words, the MMSE estimator is the conditional expectation of x given the known observed value of the measurements. Also, since {\displaystyle {\hat {x}}_{\mathrm {MMSE} }} is the posterior mean, the error covariance matrix C_{e}is equal to the posterior covariance {\displaystyle C_{X|Y}} matrix,

{\displaystyle C_{e}=C_{X|Y}}.
  • The MMSE estimator is unbiased (under the regularity assumptions mentioned above):
{\displaystyle \operatorname {E} \{{\hat {x}}_{\operatorname {MMSE} }(y)\}=\operatorname {E} \{\operatorname {E} \{x\mid y\}\}=\operatorname {E} \{x\}.}
  • The MMSE estimator is asymptotically unbiased and it converges in distribution to the normal distribution:
{\displaystyle {\sqrt {n}}({\hat {x}}_{\operatorname {MMSE} }-x)\xrightarrow {d} {\mathcal {N}}\left(0,I^{-1}(x)\right),}
where I(x) is the Fisher information of x. Thus, the MMSE estimator is asymptotically efficient.
{\displaystyle \operatorname {E} \{({\hat {x}}_{\operatorname {MMSE} }-x)g(y)\}=0}
for all g(y) in closed, linear subspace {\displaystyle {\mathcal {V}}=\{g(y)\mid g:\mathbb {R} ^{m}\rightarrow \mathbb {R} ,\operatorname {E} \{g(y)^{2}\}<+\infty \}} of the measurements. For random vectors, since the MSE for estimation of a random vector is the sum of the MSEs of the coordinates, finding the MMSE estimator of a random vector decomposes into finding the MMSE estimators of the coordinates of X separately:

{\displaystyle \operatorname {E} \{(g_{i}^{*}(y)-x_{i})g_{j}(y)\}=0,}
for all i and j. More succinctly put, the cross-correlation between the minimum estimation error {\displaystyle {\hat {x}}_{\operatorname {MMSE} }-x} and the estimator {\hat {x}} should be zero,

{\displaystyle \operatorname {E} \{({\hat {x}}_{\operatorname {MMSE} }-x){\hat {x}}^{T}\}=0.}

Linear MMSE estimator[edit]

In many cases, it is not possible to determine the analytical expression of the MMSE estimator. Two basic numerical approaches to obtain the MMSE estimate depends on either finding the conditional expectation {\displaystyle \operatorname {E} \{x\mid y\}} or finding the minima of MSE. Direct numerical evaluation of the conditional expectation is computationally expensive since it often requires multidimensional integration usually done via Monte Carlo methods. Another computational approach is to directly seek the minima of the MSE using techniques such as the stochastic gradient descent methods; but this method still requires the evaluation of expectation. While these numerical methods have been fruitful, a closed form expression for the MMSE estimator is nevertheless possible if we are willing to make some compromises.

One possibility is to abandon the full optimality requirements and seek a technique minimizing the MSE within a particular class of estimators, such as the class of linear estimators. Thus, we postulate that the conditional expectation of x given y is a simple linear function of y, {\displaystyle \operatorname {E} \{x\mid y\}=Wy+b}, where the measurement y is a random vector, W is a matrix and b is a vector. This can be seen as the first order Taylor approximation of {\displaystyle \operatorname {E} \{x\mid y\}}. The linear MMSE estimator is the estimator achieving minimum MSE among all estimators of such form. That is, it solves the following optimization problem:

{\displaystyle \min _{W,b}\operatorname {MSE} \qquad {\text{s.t.}}\qquad {\hat {x}}=Wy+b.}

One advantage of such linear MMSE estimator is that it is not necessary to explicitly calculate the posterior probability density function of x. Such linear estimator only depends on the first two moments of x and y. So although it may be convenient to assume that x and y are jointly Gaussian, it is not necessary to make this assumption, so long as the assumed distribution has well defined first and second moments. The form of the linear estimator does not depend on the type of the assumed underlying distribution.

The expression for optimal b and W is given by:

b={\bar {x}}-W{\bar {y}},
W=C_{XY}C_{Y}^{-1}.

where {\displaystyle {\bar {x}}=\operatorname {E} \{x\}}, {\displaystyle {\bar {y}}=\operatorname {E} \{y\},} the C_{{XY}} is cross-covariance matrix between x and y, the C_{{Y}} is auto-covariance matrix of y.

Thus, the expression for linear MMSE estimator, its mean, and its auto-covariance is given by

{\displaystyle {\hat {x}}=C_{XY}C_{Y}^{-1}(y-{\bar {y}})+{\bar {x}},}
{\displaystyle \operatorname {E} \{{\hat {x}}\}={\bar {x}},}
C_{\hat {X}}=C_{XY}C_{Y}^{-1}C_{YX},

where the C_{{YX}} is cross-covariance matrix between y and x.

Lastly, the error covariance and minimum mean square error achievable by such estimator is

C_{e}=C_{X}-C_{\hat {X}}=C_{X}-C_{XY}C_{Y}^{-1}C_{YX},
{\displaystyle \operatorname {LMMSE} =\operatorname {tr} \{C_{e}\}.}

Univariate case[edit]

For the special case when both x and y are scalars, the above relations simplify to

{\displaystyle {\hat {x}}={\frac {\sigma _{XY}}{\sigma _{Y}^{2}}}(y-{\bar {y}})+{\bar {x}}=\rho {\frac {\sigma _{X}}{\sigma _{Y}}}(y-{\bar {y}})+{\bar {x}},}
{\displaystyle \sigma _{e}^{2}=\sigma _{X}^{2}-{\frac {\sigma _{XY}^{2}}{\sigma _{Y}^{2}}}=(1-\rho ^{2})\sigma _{X}^{2},}

where {\displaystyle \rho ={\frac {\sigma _{XY}}{\sigma _{X}\sigma _{Y}}}} is the Pearson’s correlation coefficient between x and y.

The above two equations allows us to interpret the correlation coefficient either as normalized slope of linear regression

{\displaystyle \left({\frac {{\hat {x}}-{\bar {x}}}{\sigma _{X}}}\right)=\rho \left({\frac {y-{\bar {y}}}{\sigma _{Y}}}\right)}

or as square root of the ratio of two variances

{\displaystyle \rho ^{2}={\frac {\sigma _{X}^{2}-\sigma _{e}^{2}}{\sigma _{X}^{2}}}={\frac {\sigma _{\hat {X}}^{2}}{\sigma _{X}^{2}}}}.

When \rho =0, we have {\displaystyle {\hat {x}}={\bar {x}}} and {\displaystyle \sigma _{e}^{2}=\sigma _{X}^{2}}. In this case, no new information is gleaned from the measurement which can decrease the uncertainty in x. On the other hand, when {\displaystyle \rho =\pm 1}, we have {\displaystyle {\hat {x}}={\frac {\sigma _{XY}}{\sigma _{Y}}}(y-{\bar {y}})+{\bar {x}}} and {\displaystyle \sigma _{e}^{2}=0}. Here x is completely determined by y, as given by the equation of straight line.

Computation[edit]

Standard method like Gauss elimination can be used to solve the matrix equation for W. A more numerically stable method is provided by QR decomposition method. Since the matrix C_{Y} is a symmetric positive definite matrix, W can be solved twice as fast with the Cholesky decomposition, while for large sparse systems conjugate gradient method is more effective. Levinson recursion is a fast method when C_{Y} is also a Toeplitz matrix. This can happen when y is a wide sense stationary process. In such stationary cases, these estimators are also referred to as Wiener–Kolmogorov filters.

Linear MMSE estimator for linear observation process[edit]

Let us further model the underlying process of observation as a linear process: y=Ax+z, where A is a known matrix and z is random noise vector with the mean {\displaystyle \operatorname {E} \{z\}=0} and cross-covariance C_{XZ}=0. Here the required mean and the covariance matrices will be

{\displaystyle \operatorname {E} \{y\}=A{\bar {x}},}
C_{Y}=AC_{X}A^{T}+C_{Z},
C_{XY}=C_{X}A^{T}.

Thus the expression for the linear MMSE estimator matrix W further modifies to

W=C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}.

Putting everything into the expression for {\hat {x}}, we get

{\hat {x}}=C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}(y-A{\bar {x}})+{\bar {x}}.

Lastly, the error covariance is

C_{e}=C_{X}-C_{\hat {X}}=C_{X}-C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}AC_{X}.

The significant difference between the estimation problem treated above and those of least squares and Gauss–Markov estimate is that the number of observations m, (i.e. the dimension of y) need not be at least as large as the number of unknowns, n, (i.e. the dimension of x). The estimate for the linear observation process exists so long as the m-by-m matrix (AC_{X}A^{T}+C_{Z})^{-1} exists; this is the case for any m if, for instance, C_{Z} is positive definite. Physically the reason for this property is that since x is now a random variable, it is possible to form a meaningful estimate (namely its mean) even with no measurements. Every new measurement simply provides additional information which may modify our original estimate. Another feature of this estimate is that for m < n, there need be no measurement error. Thus, we may have C_{Z}=0, because as long as AC_{X}A^{T} is positive definite, the estimate still exists. Lastly, this technique can handle cases where the noise is correlated.

Alternative form[edit]

An alternative form of expression can be obtained by using the matrix identity

C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}A^{T}C_{Z}^{-1},

which can be established by post-multiplying by (AC_{X}A^{T}+C_{Z}) and pre-multiplying by (A^{T}C_{Z}^{-1}A+C_{X}^{-1}), to obtain

W=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}A^{T}C_{Z}^{-1},

and

C_{e}=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}.

Since W can now be written in terms of C_{e} as W=C_{e}A^{T}C_{Z}^{-1}, we get a simplified expression for {\hat {x}} as

{\hat {x}}=C_{e}A^{T}C_{Z}^{-1}(y-A{\bar {x}})+{\bar {x}}.

In this form the above expression can be easily compared with weighed least square and Gauss–Markov estimate. In particular, when C_{X}^{-1}=0, corresponding to infinite variance of the apriori information concerning x, the result W=(A^{T}C_{Z}^{-1}A)^{-1}A^{T}C_{Z}^{-1} is identical to the weighed linear least square estimate with C_{Z}^{-1} as the weight matrix. Moreover, if the components of z are uncorrelated and have equal variance such that C_{Z}=\sigma ^{2}I, where I is an identity matrix, then W=(A^{T}A)^{-1}A^{T} is identical to the ordinary least square estimate.

Sequential linear MMSE estimation[edit]

In many real-time applications, observational data is not available in a single batch. Instead the observations are made in a sequence. One possible approach is to use the sequential observations to update an old estimate as additional data becomes available, leading to finer estimates. One crucial difference between batch estimation and sequential estimation is that sequential estimation requires an additional Markov assumption.

In the Bayesian framework, such recursive estimation is easily facilitated using Bayes’ rule. Given k observations, y_{1},\ldots ,y_{k}, Bayes’ rule gives us the posterior density of x_{k} as

{\displaystyle {\begin{aligned}p(x_{k}|y_{1},\ldots ,y_{k})&\propto p(y_{k}|x,y_{1},\ldots ,y_{k-1})p(x_{k}|y_{1},\ldots ,y_{k-1})\\&=p(y_{k}|x_{k})p(x_{k}|y_{1},\ldots ,y_{k-1}).\end{aligned}}}

The {\displaystyle p(x_{k}|y_{1},\ldots ,y_{k})} is called the posterior density, {\displaystyle p(y_{k}|x_{k})} is called the likelihood function, and {\displaystyle p(x_{k}|y_{1},\ldots ,y_{k-1})} is the prior density of k-th time step. Here we have assumed the conditional independence of y_{k} from previous observations {\displaystyle y_{1},\ldots ,y_{k-1}} given x as

{\displaystyle p(y_{k}|x_{k},y_{1},\ldots ,y_{k-1})=p(y_{k}|x_{k}).}

This is the Markov assumption.

The MMSE estimate \hat{x}_k given the k-th observation is then the mean of the posterior density {\displaystyle p(x_{k}|y_{1},\ldots ,y_{k})}. With the lack of dynamical information on how the state x changes with time, we will make a further stationarity assumption about the prior:

{\displaystyle p(x_{k}|y_{1},\ldots ,y_{k-1})=p(x_{k-1}|y_{1},\ldots ,y_{k-1}).}

Thus, the prior density for k-th time step is the posterior density of (k-1)-th time step. This structure allows us to formulate a recursive approach to estimation.

In the context of linear MMSE estimator, the formula for the estimate will have the same form as before: {\hat {x}}=C_{XY}C_{Y}^{-1}(y-{\bar {y}})+{\bar {x}}. However, the mean and covariance matrices of X and Y will need to be replaced by those of the prior density {\displaystyle p(x_{k}|y_{1},\ldots ,y_{k-1})} and likelihood {\displaystyle p(y_{k}|x_{k})}, respectively.

For the prior density {\displaystyle p(x_{k}|y_{1},\ldots ,y_{k-1})}, its mean is given by the previous MMSE estimate,

{\displaystyle {\bar {x}}_{k}=\mathrm {E} [x_{k}|y_{1},\ldots ,y_{k-1}]=\mathrm {E} [x_{k-1}|y_{1},\ldots ,y_{k-1}]={\hat {x}}_{k-1}},

and its covariance matrix is given by the previous error covariance matrix,

{\displaystyle C_{X_{k}|Y_{1},\ldots ,Y_{k-1}}=C_{X_{k-1}|Y_{1},\ldots ,Y_{k-1}}=C_{e_{k-1}},}

as per by the properties of MMSE estimators and the stationarity assumption.

Similarly, for the linear observation process, the mean of the likelihood {\displaystyle p(y_{k}|x_{k})} is given by {\displaystyle {\bar {y}}_{k}=A{\bar {x}}_{k}=A{\hat {x}}_{k-1}} and the covariance matrix is as before

{\displaystyle {\begin{aligned}C_{Y_{k}|X_{k}}&=AC_{X_{k}|Y_{1},\ldots ,Y_{k-1}}A^{T}+C_{Z}=AC_{e_{k-1}}A^{T}+C_{Z}.\end{aligned}}}.

The difference between the predicted value of Y_{k}, as given by {\displaystyle {\bar {y}}_{k}=A{\hat {x}}_{k-1}}, and its observed value y_{k} gives the prediction error {\displaystyle {\tilde {y}}_{k}=y_{k}-{\bar {y}}_{k}}, which is also referred to as innovation or residual. It is more convenient to represent the linear MMSE in terms of the prediction error, whose mean and covariance are {\displaystyle \mathrm {E} [{\tilde {y}}_{k}]=0} and {\displaystyle C_{{\tilde {Y}}_{k}}=C_{Y_{k}|X_{k}}}.

Hence, in the estimate update formula, we should replace {\bar {x}} and C_{X} by {\displaystyle {\hat {x}}_{k-1}} and {\displaystyle C_{e_{k-1}}}, respectively. Also, we should replace {\bar {y}} and C_{Y} by {\displaystyle {\bar {y}}_{k-1}} and {\displaystyle C_{{\tilde {Y}}_{k}}}. Lastly, we replace C_{{XY}} by

{\displaystyle {\begin{aligned}C_{X_{k}Y_{k}|Y_{1},\ldots ,Y_{k-1}}&=C_{e_{k-1}{\tilde {Y}}_{k}}=C_{e_{k-1}}A^{T}.\end{aligned}}}

Thus, we have the new estimate as new observation y_{k} arrives as

{\displaystyle {\begin{aligned}{\hat {x}}_{k}&={\hat {x}}_{k-1}+C_{e_{k-1}{\tilde {Y}}_{k}}C_{{\tilde {Y}}_{k}}^{-1}(y_{k}-{\bar {y}}_{k})\\&={\hat {x}}_{k-1}+C_{e_{k-1}}A^{T}(AC_{e_{k-1}}A^{T}+C_{Z})^{-1}(y_{k}-A{\hat {x}}_{k-1})\end{aligned}}}

and the new error covariance as

{\displaystyle C_{e_{k}}=C_{e_{k-1}}-C_{e_{k-1}}A^{T}(AC_{e_{k-1}}A^{T}+C_{Z})^{-1}AC_{e_{k-1}}.}

From the point of view of linear algebra, for sequential estimation, if we have an estimate {\hat {x}}_{1} based on measurements generating space Y_{1}, then after receiving another set of measurements, we should subtract out from these measurements that part that could be anticipated from the result of the first measurements. In other words, the updating must be based on that part of the new data which is orthogonal to the old data.

The repeated use of the above two equations as more observations become available lead to recursive estimation techniques. The expressions can be more compactly written as

{\displaystyle W_{k}=C_{e_{k-1}}A^{T}(AC_{e_{k-1}}A^{T}+C_{Z})^{-1},}
{\displaystyle {\hat {x}}_{k}={\hat {x}}_{k-1}+W_{k}(y_{k}-A{\hat {x}}_{k-1}),}
{\displaystyle C_{e_{k}}=(I-W_{k}A)C_{e_{k-1}}.}

The matrix W_{k} is often referred to as the Kalman gain factor. The alternative formulation of the above algorithm will give

{\displaystyle C_{e_{k}}^{-1}=C_{e_{k-1}}^{-1}+A^{T}C_{Z}^{-1}A,}
{\displaystyle W_{k}=C_{e_{k}}A^{T}C_{Z}^{-1},}
{\displaystyle {\hat {x}}_{k}={\hat {x}}_{k-1}+W_{k}(y_{k}-A{\hat {x}}_{k-1}),}

The repetition of these three steps as more data becomes available leads to an iterative estimation algorithm. The generalization of this idea to non-stationary cases gives rise to the Kalman filter. The three update steps outlined above indeed form the update step of the Kalman filter.

Special case: scalar observations[edit]

As an important special case, an easy to use recursive expression can be derived when at each k-th time instant the underlying linear observation process yields a scalar such that {\displaystyle y_{k}=a_{k}^{T}x_{k}+z_{k}}, where a_{k} is n-by-1 known column vector whose values can change with time, x_{k} is n-by-1 random column vector to be estimated, and z_{k} is scalar noise term with variance \sigma_k^2. After (k+1)-th observation, the direct use of above recursive equations give the expression for the estimate \hat{x}_{k+1} as:

{\displaystyle {\hat {x}}_{k+1}={\hat {x}}_{k}+w_{k+1}(y_{k+1}-a_{k+1}^{T}{\hat {x}}_{k})}

where y_{{k+1}} is the new scalar observation and the gain factor w_{k+1} is n-by-1 column vector given by

{\displaystyle w_{k+1}={\frac {C_{e_{k}}a_{k+1}}{\sigma _{k+1}^{2}+a_{k+1}^{T}C_{e_{k}}a_{k+1}}}.}

The {\displaystyle C_{e_{k+1}}} is n-by-n error covariance matrix given by

{\displaystyle C_{e_{k+1}}=(I-w_{k+1}a_{k+1}^{T})C_{e_{k}}.}

Here, no matrix inversion is required. Also, the gain factor, w_{k+1}, depends on our confidence in the new data sample, as measured by the noise variance, versus that in the previous data. The initial values of {\hat {x}} and C_{e} are taken to be the mean and covariance of the aprior probability density function of x.

Alternative approaches: This important special case has also given rise to many other iterative methods (or adaptive filters), such as the least mean squares filter and recursive least squares filter, that directly solves the original MSE optimization problem using stochastic gradient descents. However, since the estimation error e cannot be directly observed, these methods try to minimize the mean squared prediction error {\displaystyle \mathrm {E} \{{\tilde {y}}^{T}{\tilde {y}}\}}. For instance, in the case of scalar observations, we have the gradient {\displaystyle \nabla _{\hat {x}}\mathrm {E} \{{\tilde {y}}^{2}\}=-2\mathrm {E} \{{\tilde {y}}a\}.} Thus, the update equation for the least mean square filter is given by

{\displaystyle {\hat {x}}_{k+1}={\hat {x}}_{k}+\eta _{k}\mathrm {E} \{{\tilde {y}}_{k}a_{k}\},}

where \eta _{k} is the scalar step size and the expectation is approximated by the instantaneous value {\displaystyle \mathrm {E} \{a_{k}{\tilde {y}}_{k}\}\approx a_{k}{\tilde {y}}_{k}}. As we can see, these methods bypass the need for covariance matrices.

[edit]

In many practical applications, the observation noise is uncorrelated. That is, C_{Z} is a diagonal matrix. In such cases, it is advantageous to consider the components of y as independent scalar measurements, rather than vector measurement. This allows us to reduce computation time by processing the m\times 1 measurement vector as m scalar measurements. The use of scalar update formula avoids matrix inversion in the implementation of the covariance update equations, thus improving the numerical robustness against roundoff errors. The update can be implemented iteratively as:

{\displaystyle w_{k+1}^{(\ell )}={\frac {C_{e_{k}}^{(\ell )}A_{k+1}^{(\ell )T}}{C_{Z_{k+1}}^{(\ell )}+A_{k+1}^{(\ell )}C_{e_{k}}^{(\ell )}(A_{k+1}^{(\ell )T})}}}
{\displaystyle C_{e_{k+1}}^{(\ell )}=(I-w_{k+1}^{(\ell )}A_{k+1}^{(\ell )})C_{e_{k}}^{(\ell )}}
{\displaystyle {\hat {x}}_{k+1}^{(\ell )}={\hat {x}}_{k}^{(\ell -1)}+w_{k+1}^{(\ell )}(y_{k+1}^{(\ell )}-A_{k+1}^{(\ell )}{\hat {x}}_{k}^{(\ell -1)})}

where {\displaystyle \ell =1,2,\ldots ,m}, using the initial values {\displaystyle C_{e_{k+1}}^{(0)}=C_{e_{k}}} and {\displaystyle {\hat {x}}_{k+1}^{(0)}={\hat {x}}_{k}}. The intermediate variables {\displaystyle C_{Z_{k+1}}^{(\ell )}} is the \ell -th diagonal element of the m\times m diagonal matrix {\displaystyle C_{Z_{k+1}}}; while {\displaystyle A_{k+1}^{(\ell )}} is the \ell -th row of m\times n matrix A_{{k+1}}. The final values are {\displaystyle C_{e_{k+1}}^{(m)}=C_{e_{k+1}}} and {\displaystyle {\hat {x}}_{k+1}^{(m)}={\hat {x}}_{k+1}}.

Examples[edit]

Example 1[edit]

We shall take a linear prediction problem as an example. Let a linear combination of observed scalar random variables {\displaystyle z_{1},z_{2}} and {\displaystyle z_{3}} be used to estimate another future scalar random variable {\displaystyle z_{4}} such that {\displaystyle {\hat {z}}_{4}=\sum _{i=1}^{3}w_{i}z_{i}}. If the random variables {\displaystyle z=[z_{1},z_{2},z_{3},z_{4}]^{T}} are real Gaussian random variables with zero mean and its covariance matrix given by

{\displaystyle \operatorname {cov} (Z)=\operatorname {E} [zz^{T}]=\left[{\begin{array}{cccc}1&2&3&4\\2&5&8&9\\3&8&6&10\\4&9&10&15\end{array}}\right],}

then our task is to find the coefficients w_{i} such that it will yield an optimal linear estimate {\displaystyle {\hat {z}}_{4}}.

In terms of the terminology developed in the previous sections, for this problem we have the observation vector {\displaystyle y=[z_{1},z_{2},z_{3}]^{T}}, the estimator matrix W=[w_{1},w_{2},w_{3}] as a row vector, and the estimated variable {\displaystyle x=z_{4}} as a scalar quantity. The autocorrelation matrix C_{Y} is defined as

{\displaystyle C_{Y}=\left[{\begin{array}{ccc}E[z_{1},z_{1}]&E[z_{2},z_{1}]&E[z_{3},z_{1}]\\E[z_{1},z_{2}]&E[z_{2},z_{2}]&E[z_{3},z_{2}]\\E[z_{1},z_{3}]&E[z_{2},z_{3}]&E[z_{3},z_{3}]\end{array}}\right]=\left[{\begin{array}{ccc}1&2&3\\2&5&8\\3&8&6\end{array}}\right].}

The cross correlation matrix C_{{YX}} is defined as

{\displaystyle C_{YX}=\left[{\begin{array}{c}E[z_{4},z_{1}]\\E[z_{4},z_{2}]\\E[z_{4},z_{3}]\end{array}}\right]=\left[{\begin{array}{c}4\\9\\10\end{array}}\right].}

We now solve the equation C_{Y}W^{T}=C_{YX} by inverting C_{Y} and pre-multiplying to get

{\displaystyle C_{Y}^{-1}C_{YX}=\left[{\begin{array}{ccc}4.85&-1.71&-0.142\\-1.71&0.428&0.2857\\-0.142&0.2857&-0.1429\end{array}}\right]\left[{\begin{array}{c}4\\9\\10\end{array}}\right]=\left[{\begin{array}{c}2.57\\-0.142\\0.5714\end{array}}\right]=W^{T}.}

So we have {\displaystyle w_{1}=2.57,} {\displaystyle w_{2}=-0.142,} and w_{{3}}=.5714
as the optimal coefficients for {\displaystyle {\hat {z}}_{4}}. Computing the minimum
mean square error then gives {\displaystyle \left\Vert e\right\Vert _{\min }^{2}=\operatorname {E} [z_{4}z_{4}]-WC_{YX}=15-WC_{YX}=.2857}.[2] Note that it is not necessary to obtain an explicit matrix inverse of C_{Y} to compute the value of W. The matrix equation can be solved by well known methods such as Gauss elimination method. A shorter, non-numerical example can be found in orthogonality principle.

Example 2[edit]

Consider a vector y formed by taking N observations of a fixed but unknown scalar parameter x disturbed by white Gaussian noise. We can describe the process by a linear equation y=1x+z, where 1=[1,1,\ldots ,1]^{T}. Depending on context it will be clear if 1 represents a scalar or a vector. Suppose that we know [-x_{0},x_{0}] to be the range within which the value of x is going to fall in. We can model our uncertainty of x by an aprior uniform distribution over an interval [-x_{0},x_{0}], and thus x will have variance of \sigma _{X}^{2}=x_{0}^{2}/3.. Let the noise vector z be normally distributed as N(0,\sigma _{Z}^{2}I) where I is an identity matrix. Also x and z are independent and C_{XZ}=0. It is easy to see that

{\displaystyle {\begin{aligned}&\operatorname {E} \{y\}=0,\\&C_{Y}=\operatorname {E} \{yy^{T}\}=\sigma _{X}^{2}11^{T}+\sigma _{Z}^{2}I,\\&C_{XY}=\operatorname {E} \{xy^{T}\}=\sigma _{X}^{2}1^{T}.\end{aligned}}}

Thus, the linear MMSE estimator is given by

{\begin{aligned}{\hat {x}}&=C_{XY}C_{Y}^{-1}y\\&=\sigma _{X}^{2}1^{T}(\sigma _{X}^{2}11^{T}+\sigma _{Z}^{2}I)^{-1}y.\end{aligned}}

We can simplify the expression by using the alternative form for W as

{\displaystyle {\begin{aligned}{\hat {x}}&=\left(1^{T}{\frac {1}{\sigma _{Z}^{2}}}I1+{\frac {1}{\sigma _{X}^{2}}}\right)^{-1}1^{T}{\frac {1}{\sigma _{Z}^{2}}}Iy\\&={\frac {1}{\sigma _{Z}^{2}}}\left({\frac {N}{\sigma _{Z}^{2}}}+{\frac {1}{\sigma _{X}^{2}}}\right)^{-1}1^{T}y\\&={\frac {\sigma _{X}^{2}}{\sigma _{X}^{2}+\sigma _{Z}^{2}/N}}{\bar {y}},\end{aligned}}}

where for y=[y_{1},y_{2},\ldots ,y_{N}]^{T} we have {\bar {y}}={\frac {1^{T}y}{N}}={\frac {\sum _{i=1}^{N}y_{i}}{N}}.

Similarly, the variance of the estimator is

\sigma _{\hat {X}}^{2}=C_{XY}C_{Y}^{-1}C_{YX}={\Big (}{\frac {\sigma _{X}^{2}}{\sigma _{X}^{2}+\sigma _{Z}^{2}/N}}{\Big )}\sigma _{X}^{2}.

Thus the MMSE of this linear estimator is

{\displaystyle \operatorname {LMMSE} =\sigma _{X}^{2}-\sigma _{\hat {X}}^{2}={\Big (}{\frac {\sigma _{Z}^{2}}{\sigma _{X}^{2}+\sigma _{Z}^{2}/N}}{\Big )}{\frac {\sigma _{X}^{2}}{N}}.}

For very large N, we see that the MMSE estimator of a scalar with uniform aprior distribution can be approximated by the arithmetic average of all the observed data

{\displaystyle {\hat {x}}={\frac {1}{N}}\sum _{i=1}^{N}y_{i},}

while the variance will be unaffected by data \sigma _{\hat {X}}^{2}=\sigma _{X}^{2}, and the LMMSE of the estimate will tend to zero.

However, the estimator is suboptimal since it is constrained to be linear. Had the random variable x also been Gaussian, then the estimator would have been optimal. Notice, that the form of the estimator will remain unchanged, regardless of the apriori distribution of x, so long as the mean and variance of these distributions are the same.

Example 3[edit]

Consider a variation of the above example: Two candidates are standing for an election. Let the fraction of votes that a candidate will receive on an election day be x\in [0,1]. Thus the fraction of votes the other candidate will receive will be 1-x. We shall take x as a random variable with a uniform prior distribution over [0,1] so that its mean is {\bar {x}}=1/2 and variance is \sigma _{X}^{2}=1/12. A few weeks before the election, two independent public opinion polls were conducted by two different pollsters. The first poll revealed that the candidate is likely to get y_{1} fraction of votes. Since some error is always present due to finite sampling and the particular polling methodology adopted, the first pollster declares their estimate to have an error z_{1} with zero mean and variance \sigma _{Z_{1}}^{2}. Similarly, the second pollster declares their estimate to be y_{2} with an error z_{2} with zero mean and variance \sigma _{Z_{2}}^{2}. Note that except for the mean and variance of the error, the error distribution is unspecified. How should the two polls be combined to obtain the voting prediction for the given candidate?

As with previous example, we have

{\begin{aligned}y_{1}&=x+z_{1}\\y_{2}&=x+z_{2}.\end{aligned}}

Here, both the {\displaystyle \operatorname {E} \{y_{1}\}=\operatorname {E} \{y_{2}\}={\bar {x}}=1/2}. Thus, we can obtain the LMMSE estimate as the linear combination of y_{1} and y_{2} as

{\hat {x}}=w_{1}(y_{1}-{\bar {x}})+w_{2}(y_{2}-{\bar {x}})+{\bar {x}},

where the weights are given by

{\begin{aligned}w_{1}&={\frac {1/\sigma _{Z_{1}}^{2}}{1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}+1/\sigma _{X}^{2}}},\\w_{2}&={\frac {1/\sigma _{Z_{2}}^{2}}{1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}+1/\sigma _{X}^{2}}}.\end{aligned}}

Here, since the denominator term is constant, the poll with lower error is given higher weight in order to predict the election outcome. Lastly, the variance of {\hat {x}} is given by

\sigma _{\hat {X}}^{2}={\frac {1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}}{1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}+1/\sigma _{X}^{2}}}\sigma _{X}^{2},

which makes \sigma _{\hat {X}}^{2} smaller than \sigma _{X}^{2}. Thus, the LMMSE is given by

{\displaystyle \mathrm {LMMSE} =\sigma _{X}^{2}-\sigma _{\hat {X}}^{2}={\frac {1}{1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}+1/\sigma _{X}^{2}}}.}

In general, if we have N pollsters, then {\displaystyle {\hat {x}}=\sum _{i=1}^{N}w_{i}(y_{i}-{\bar {x}})+{\bar {x}},} where the weight for i-th pollster is given by {\displaystyle w_{i}={\frac {1/\sigma _{Z_{i}}^{2}}{\sum _{j=1}^{N}1/\sigma _{Z_{j}}^{2}+1/\sigma _{X}^{2}}}} and the LMMSE is given by {\displaystyle \mathrm {LMMSE} ={\frac {1}{\sum _{j=1}^{N}1/\sigma _{Z_{j}}^{2}+1/\sigma _{X}^{2}}}.}

Example 4[edit]

Suppose that a musician is playing an instrument and that the sound is received by two microphones, each of them located at two different places. Let the attenuation of sound due to distance at each microphone be a_{1} and a_{2}, which are assumed to be known constants. Similarly, let the noise at each microphone be z_{1} and z_{2}, each with zero mean and variances \sigma _{Z_{1}}^{2} and \sigma _{Z_{2}}^{2} respectively. Let x denote the sound produced by the musician, which is a random variable with zero mean and variance \sigma _{X}^{2}. How should the recorded music from these two microphones be combined, after being synced with each other?

We can model the sound received by each microphone as

{\begin{aligned}y_{1}&=a_{1}x+z_{1}\\y_{2}&=a_{2}x+z_{2}.\end{aligned}}

Here both the {\displaystyle \operatorname {E} \{y_{1}\}=\operatorname {E} \{y_{2}\}=0}. Thus, we can combine the two sounds as

y=w_{1}y_{1}+w_{2}y_{2}

where the i-th weight is given as

{\displaystyle w_{i}={\frac {a_{i}/\sigma _{Z_{i}}^{2}}{\sum _{j}a_{j}^{2}/\sigma _{Z_{j}}^{2}+1/\sigma _{X}^{2}}}.}

See also[edit]

  • Bayesian estimator
  • Mean squared error
  • Least squares
  • Minimum-variance unbiased estimator (MVUE)
  • Orthogonality principle
  • Wiener filter
  • Kalman filter
  • Linear prediction
  • Zero-forcing equalizer

Notes[edit]

  1. ^ “Mean Squared Error (MSE)”. www.probabilitycourse.com. Retrieved 9 May 2017.
  2. ^ Moon and Stirling.

Further reading[edit]

  • Johnson, D. “Minimum Mean Squared Error Estimators”. Connexions. Archived from Minimum Mean Squared Error Estimators the original on 25 July 2008. Retrieved 8 January 2013.
  • Jaynes, E.T. (2003). Probability Theory: The Logic of Science. Cambridge University Press. ISBN 978-0521592710.
  • Bibby, J.; Toutenburg, H. (1977). Prediction and Improved Estimation in Linear Models. Wiley. ISBN 9780471016564.
  • Lehmann, E. L.; Casella, G. (1998). “Chapter 4”. Theory of Point Estimation (2nd ed.). Springer. ISBN 0-387-98502-6.
  • Kay, S. M. (1993). Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall. pp. 344–350. ISBN 0-13-042268-1.
  • Luenberger, D.G. (1969). “Chapter 4, Least-squares estimation”. Optimization by Vector Space Methods (1st ed.). Wiley. ISBN 978-0471181170.
  • Moon, T.K.; Stirling, W.C. (2000). Mathematical Methods and Algorithms for Signal Processing (1st ed.). Prentice Hall. ISBN 978-0201361865.
  • Van Trees, H. L. (1968). Detection, Estimation, and Modulation Theory, Part I. New York: Wiley. ISBN 0-471-09517-6.
  • Haykin, S.O. (2013). Adaptive Filter Theory (5th ed.). Prentice Hall. ISBN 978-0132671453.

    Анализ систем на основе критерия минимума среднеквадратичной ошибки [c.189]

    Если ковариационные функции процессов Х( ) и У(/) известны точно, то можно воспользоваться винеровским критерием минимума среднеквадратичной ошибки Этот критерий утверждает, что функция /г(и) должна быть выбрана так, чтобы минимизировать среднеквадратичную ошибку шумовой компоненты, т е [c.190]

    Синтез следящих систем. Одно из первых инженерных применений анализа на основе критерия минимума среднеквадратичной ошибки было сделано при синтезировании следящих систем для зенитных орудий и в радиолокационных следящих системах [5] Например, от радиолокационной следящей системы требуется, чтобы она следила за самолетом несмотря на возмущения отра- [c.189]

    Другое применение критерий минимума среднеквадратичной ошибки находит в задаче об идентификации системы В этом случае в распоряжении имеются входной сигнал и соответствующий ему выходной сигнал от некоторой системы, требуется вывести линейное приближение к этой системе для дальнейшего его использования при управлении или моделировании Предположим, например, что система представляет собой черный ящик (рис 5 7). Если вход является реализацией случайного процесса Х 1), то выход можно рассматривать как реализацию случайного процесса У(0< где [c.190]

    Оптимальное в любом смысле корреляционное окно, например (7 2 3), будет зависеть от неизвестного спектра Гхх(/). Этот недостаток свойствен не только спектральному анализу Вообще говоря, справедливо правило, согласно которому наилучший план действий должен опираться на некоторые представления об истинном положении вещей Следовательно, очень валено проводить четкое различие между планированием спектрального анализа до сбора данных и самим анализом данных, после того как они собраны Мы хотели бы использовать критерии минимума среднеквадратичной ошибки или какой-нибудь аналогичный критерий до проведения спектрального анализа, чтобы решить, например, какой длины нужно взять запись Но после того как данные собраны, могло бы оказаться, что наши представления относительно Гхх (/) были абсолютно неправильны [c.26]

    Если плотность вероятности ге ([х у) симметрична относительно среднего значения гпг [х у и унимодальна (т. е. монотонно невозрастающая функция [х — /П] [х у ), то байесовская оценка (5.18) совпадает с оценкой по критерию максимума апостериорной плотности вероятности. В этом случае функция С (х) не должна быть обязательно выпуклой, а лишь монотонно неубывающей функцией х (см. приложение С). Так как нормальная плотность вероятности унимодальна, то всегда, когда плотность вероятности ге ([х I у) нормальна, оценка по максимуму апостериорной плотности вероятности совпадает с широким классом байесовских оценок, который включает оценки по минимуму среднеквадратичной ошибки (или минимуму дисперсии). [c.158]


Оптимальная фильтрация по критерию минимума среднеквадратичной ошибки.



При решении задач измерения параметров сигнала, необходимо получать минимально искажённую информацию. Для этого применяют фильтры. Сглаживающий фильтр позволяет выделить сигнал на фоне шумов с минимальными искажениями. Чтобы предсказать поведение сигала во времени, применяют прогнозирующие фильтры. Для количественной оценки работы фильтра используют критерий минимума среднеквадратической ошибки:
clip_image215, где clip_image217– оценка сигнала в момент времени t, Δ – интервал прогнозирования сигнала.
Очевидно, при Δ > 0, оценка сигала clip_image219 даёт возможность предсказать значение сигнала S(t) на временной интервал Δ вперёд. При этом ошибку предсказания можно определить если известны корреляционные функции сигнала и шума.
Предположим на вход линейного фильтра действует смесь сигналов S(t) и аддитивного шума n(t). Необходимо определить характеристики фильтра, выходной сигнал которого минимально отличался бы от истинного значения сигнала в момент времени (t+Δ). Получим, что при Δ=0 имеет место сглаживающий фильтр; при Δ≠0 и n(t)=0 — прогнозирующий фильтр, при Δ≠0 и n(t) ≠0 – сглаживающе-прогнозирующий фильтр.
Частотный коэффициент передачи такого фильтра:
clip_image221,

где Sвх(ω) – спектральная плотность сигнала S(t), Wвх(ω) – спектр мощности помехи, Δ – интервал прогнозирования.
Выражение справедливо если сигнал и помеха независимы.
Т.о. можно создать фильтр, позволяющий спрогнозировать сигнал на определённый интервал времени Δ.

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