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Minimum mean square error

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
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
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

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

where the expectation $$\operatorname{E}$$ is taken over $$x$$ conditioned on $$y$$. When $$x$$ is a scalar variable, the MSE expression simplifies to $$\operatorname{E} \left\{ (\hat{x} - x)^2 \right\}$$. Note that MSE can equivalently be defined in other ways, since
 * $$\operatorname{tr} \left\{ \operatorname{E}\{ee^T \} \right\} = \operatorname{E} \left\{ \operatorname{tr}\{ee^T \} \right\} = \operatorname{E}\{e^T e \} = \sum_{i=1}^n \operatorname{E}\{e_i^2\}.$$

The MMSE estimator is then defined as the estimator achieving minimal MSE:
 * $$\hat{x}_{\operatorname{MMSE}}(y) = \operatorname{argmin}_{\hat{x}} \operatorname{MSE}.$$

Properties

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

 * The MMSE estimator is unbiased (under the regularity assumptions mentioned above):
 * $$\operatorname{E}\{\hat{x}_{\operatorname{MMSE}}(y)\} = \operatorname{E}\{\operatorname{E}\{x\mid y\}\} = \operatorname{E}\{x\}.$$

 * The MMSE estimator is asymptotically unbiased and it converges in distribution to the normal distribution:
 * $$ \sqrt{n}(\hat{x}_{\operatorname{MMSE}} - x) \xrightarrow{d} \mathcal{N}\left(0, I^{-1}(x)\right),$$
 * where $$I(x)$$ is the Fisher information of $$x$$. Thus, the MMSE estimator is asymptotically efficient.

 * The orthogonality principle: When $$x$$ is a scalar, an estimator constrained to be of certain form $$\hat{x}=g(y)$$ is an optimal estimator, i.e. $$\hat{x}_{\operatorname{MMSE}}=g^*(y),$$ if and only if
 * $$\operatorname{E} \{ (\hat{x}_{\operatorname{MMSE}}-x) g(y) \} = 0$$
 * for all $$g(y)$$ in closed, linear subspace $$\mathcal{V} = \{g(y)\mid g:\mathbb{R}^m \rightarrow \mathbb{R}, \operatorname{E}\{g(y)^2\} < + \infty \}$$ 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:
 * $$\operatorname{E} \{ (g_i^*(y)-x_i) g_j(y) \} = 0,$$
 * for all i and j. More succinctly put, the cross-correlation between the minimum estimation error $$\hat{x}_{\operatorname{MMSE}}-x$$ and the estimator $$\hat{x}$$ should be zero,
 * $$\operatorname{E} \{ (\hat{x}_{\operatorname{MMSE}}-x)\hat{x}^T \} = 0.$$

 * If $$x$$ and $$y$$ are jointly Gaussian, then the MMSE estimator is linear, i.e., it has the form $$Wy+b$$ for matrix $$W$$ and constant $$b$$. This can be directly shown using the Bayes theorem. As a consequence, to find the MMSE estimator, it is sufficient to find the linear MMSE estimator.

Linear MMSE estimator
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 $$\operatorname{E}\{x\mid y\}$$ 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 $$y$$, $$\operatorname{E}\{x\mid y\} = W y + b$$, 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 $$\operatorname{E}\{x\mid y\}$$. The linear MMSE estimator is the estimator achieving minimum MSE among all estimators of such form. That is, it solves the following optimization problem:
 * $$\min_{W,b} \operatorname{MSE} \qquad \text{s.t.} \qquad \hat{x} = W y + b.$$

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:
 * $$b = \bar{x} - W \bar{y},$$
 * $$ W = C_{XY}C^{-1}_{Y}.$$

where $$\bar{x} = \operatorname{E}\{x\}$$, $$\bar{y} = \operatorname{E}\{y\},$$ 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
 * $$\hat{x} = C_{XY}C^{-1}_{Y}(y-\bar{y}) + \bar{x},$$
 * $$\operatorname{E}\{\hat{x}\} = \bar{x},$$
 * $$C_{\hat{X}} = C_{XY} C^{-1}_Y C_{YX},$$

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

 * $$C_e = C_X - C_{\hat{X}} = C_X - C_{XY} C^{-1}_Y C_{YX},$$
 * $$\operatorname{LMMSE} = \operatorname{tr} \{C_e\}.$$

Let us have the optimal linear MMSE estimator given as $$\hat{x} = Wy+b$$, where we are required to find the expression for $$W$$ and $$b$$. It is required that the MMSE estimator be unbiased. This means,

 * $$\operatorname{E}\{\hat{x}\} = \operatorname{E}\{x\}.$$

Plugging the expression for $$\hat{x}$$ in above, we get

 * $$b = \bar{x} - W \bar{y},$$

where $$\bar{x} = \operatorname{E}\{x\}$$ and $$\bar{y} = \operatorname{E}\{y\}$$. Thus we can re-write the estimator as

 * $$\hat{x} = W(y-\bar{y}) + \bar{x}$$

and the expression for estimation error becomes

 * $$\hat{x} - x = W(y-\bar{y}) - (x-\bar{x}).$$

From the orthogonality principle, we can have $$\operatorname{E} \{ (\hat{x}-x) (y-\bar{y})^T\} = 0$$, where we take $$g(y) = y - \bar{y}$$. Here the left-hand-side term is

\begin{align} \operatorname{E} \{ (\hat{x}-x)(y - \bar{y})^T\} &= \operatorname{E} \{ (W(y-\bar{y}) - (x-\bar{x})) (y - \bar{y})^T \} \\ &= W \operatorname{E} \{(y-\bar{y})(y-\bar{y})^T \} - \operatorname{E} \{ (x-\bar{x})(y-\bar{y})^T \} \\ &= WC_Y - C_{XY}. \end{align} $$

When equated to zero, we obtain the desired expression for $$W$$ as

 * $$ W = C_{XY}C^{-1}_Y .$$

The $$C_{XY}$$ is cross-covariance matrix between X and Y, and $$C_{Y}$$ is auto-covariance matrix of Y. Since $$C_{XY}=C^T_{YX}$$, the expression can also be re-written in terms of $$C_{YX}$$ as

 * $$W^T = C^{-1}_Y C_{YX} .$$

Thus the full expression for the linear MMSE estimator is

 * $$\hat{x} = C_{XY} C^{-1}_Y (y-\bar{y}) + \bar{x}.$$

Since the estimate $$\hat{x}$$ is itself a random variable with $$\operatorname{E}\{\hat{x}\} = \bar{x}$$, we can also obtain its auto-covariance as

\begin{align} C_{\hat{X}} &= \operatorname{E}\{(\hat x - \bar x)(\hat x - \bar x)^T\} \\ &= W \operatorname{E}\{(y-\bar{y})(y-\bar{y})^T\} W^T \\ &= W C_Y W^T .\\ \end{align} $$

Putting the expression for $$W$$ and $$W^T$$, we get

 * $$C_{\hat{X}} = C_{XY} C^{-1}_Y C_{YX}.$$

Lastly, the covariance of linear MMSE estimation error will then be given by

\begin{align} C_e &= \operatorname{E}\{(\hat x - x)(\hat x - x)^T\} \\ &= \operatorname{E}\{(\hat x - x)(W(y-\bar{y}) - (x-\bar{x}))^T\} \\ &= \underbrace{\operatorname{E}\{(\hat x - x)(y-\bar{y})^T \}}_0 W^T - \operatorname{E}\{(\hat x - x)(x-\bar{x})^T\} \\ &= - \operatorname{E}\{(W(y-\bar{y}) - (x-\bar{x}))(x-\bar{x})^T\} \\ &= \operatorname{E}\{(x-\bar{x})(x-\bar{x})^T\} - W \operatorname{E}\{(y-\bar{y})(x-\bar{x})^T\} \\ &= C_X - WC_{YX}. \end{align} $$

The first term in the third line is zero due to the orthogonality principle. Since $$W = C_{XY}C^{-1}_Y$$, we can re-write $$C_e$$ in terms of covariance matrices as

 * $$C_e = C_X - C_{XY} C^{-1}_Y C_{YX} .$$

This we can recognize to be the same as $$C_e = C_X - C_{\hat{X}}.$$ Thus the minimum mean square error achievable by such a linear estimator is

 * $$\operatorname{LMMSE} = \operatorname{tr}\{C_e\} $$.

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

 * $$ \hat{x} = \frac{\sigma_{XY}}{\sigma_Y^2}(y-\bar{y}) + \bar{x} = \rho \frac{\sigma_{X}}{\sigma_Y}(y-\bar{y}) + \bar{x},$$
 * $$\sigma^2_e = \sigma_X^2 - \frac{\sigma_{XY}^2}{\sigma_Y^2} = (1 - \rho^2)\sigma_X^2,$$

where $$\rho = \frac{\sigma_{XY}}{\sigma_X \sigma_Y}$$ 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

 * $$ \left(\frac{\hat{x} - \bar{x}}{\sigma_{X}}\right) = \rho \left(\frac{y-\bar{y}}{\sigma_Y}\right)$$

or as square root of the ratio of two variances

 * $$\rho^2 = \frac{\sigma_X^2 - \sigma_e^2}{\sigma_X^2} = \frac{\sigma^2_{\hat{X}}}{\sigma^2_X}$$.

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

Computation
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
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 $$\operatorname{E}\{z\}=0$$ and cross-covariance $$C_{XZ} = 0$$. Here the required mean and the covariance matrices will be

 * $$\operatorname{E}\{y\} = A\bar{x},$$
 * $$C_Y = AC_XA^T + C_Z,$$
 * $$C_{XY} = C_X A^T .$$

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

 * $$W = C_X A^T(AC_XA^T + C_Z)^{-1} .$$

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

 * $$\hat{x} = C_X A^T(AC_XA^T + C_Z)^{-1}(y-A\bar{x}) + \bar{x}.$$

Lastly, the error covariance is

 * $$C_e = C_X - C_{\hat{X}} = C_X - C_X A^T(AC_XA^T + C_Z)^{-1}AC_X .$$

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 $$(AC_XA^T + C_Z)^{-1}$$ 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 $$AC_XA^T$$ is positive definite, the estimate still exists. Lastly, this technique can handle cases where the noise is correlated.

Alternative form
An alternative form of expression can be obtained by using the matrix identity
 * $$C_X A^T(AC_XA^T + C_Z)^{-1} = (A^TC_Z^{-1}A + C_X^{-1})^{-1} A^T C_Z^{-1},$$

which can be established by post-multiplying by $$(AC_XA^T + C_Z)$$ and pre-multiplying by $$(A^TC_Z^{-1}A + C_X^{-1}),$$ to obtain

 * $$W = (A^TC_Z^{-1}A + C_X^{-1})^{-1} A^TC_Z^{-1},$$

and
 * $$C_e = (A^TC_Z^{-1}A + C_X^{-1})^{-1}.$$

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 ridge regression, 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 $$W = (A^TC_Z^{-1}A)^{-1} A^TC_Z^{-1}$$ 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 $$W = (A^TA)^{-1}A^T$$ is identical to the ordinary least square estimate. When apriori information is available as $$C_X^{-1}=\lambda I$$ and the $$z$$ are uncorrelated and have equal variance, we have $$W = (A^TA + \lambda I)^{-1}A^T$$, which is identical to ridge regression solution.

Sequential linear MMSE estimation
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

\begin{align} 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{align} $$

The $$p(x_k|y_1, \ldots, y_k)$$ is called the posterior density, $$p(y_k|x_k)$$ is called the likelihood function, and $$p(x_k|y_1, \ldots, y_{k-1})$$ is the prior density of k-th time step. Here we have assumed the conditional independence of $$y_k$$ from previous observations $$y_1, \ldots, y_{k-1}$$ given $$x$$ as

 * $$p(y_k|x_k,y_1,\ldots,y_{k-1}) = p(y_k|x_k).$$

This is the Markov assumption.

The MMSE estimate $$\hat{x}_k$$ given the k-th observation is then the mean of the posterior density $$p(x_k|y_1,\ldots, y_k)$$. With the lack of dynamical information on how the state $$x$$ changes with time, we will make a further stationarity assumption about the prior:

 * $$p(x_k|y_1, \ldots, y_{k-1}) = p(x_{k-1}|y_1, \ldots, y_{k-1}).$$

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: $$\hat{x} = C_{XY}C^{-1}_{Y}(y-\bar{y}) + \bar{x}.$$ However, the mean and covariance matrices of $$X$$ and $$Y$$ will need to be replaced by those of the prior density $$p(x_k|y_1,\ldots, y_{k-1})$$ and likelihood $$p(y_k|x_k)$$, respectively.

For the prior density $$p(x_k|y_1, \ldots, y_{k-1})$$, its mean is given by the previous MMSE estimate,

 * $$\bar{x}_{k}=\mathrm{E}[x_k|y_1,\ldots,y_{k-1}]=\mathrm{E}[x_{k-1}|y_1,\ldots,y_{k-1}]=\hat{x}_{k-1}$$,

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

 * $$C_{X_k|Y_1,\ldots,Y_{k-1}} = C_{X_{k-1}|Y_1,\ldots,Y_{k-1}} = C_{e_{k-1}},$$

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

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

\begin{align} C_{Y_k|X_k} &= A C_{X_k|Y_1,\ldots,Y_{k-1}} A^T + C_Z = A C_{e_{k-1}} A^T + C_Z. \end{align} $$.

The difference between the predicted value of $$Y_k$$, as given by $$\bar{y}_k = A\hat{x}_{k-1}$$, and its observed value $$y_k$$ gives the prediction error $$\tilde{y}_k = y_k - \bar{y}_k$$, 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 $$\mathrm{E}[\tilde{y}_k] = 0$$ and $$C_{\tilde{Y}_k} = C_{Y_k|X_k}$$.

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

\begin{align} 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{align} $$

Thus, we have the new estimate as new observation $$y_k$$ arrives as

\begin{align} \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{align} $$ and the new error covariance as
 * $$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

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

The matrix $$W_k$$ is often referred to as the Kalman gain factor. The alternative formulation of the above algorithm will give

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

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
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 $$y_k = a_k^T x_k + z_k$$, 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:
 * $$\hat{x}_{k+1} = \hat{x}_k + w_{k+1}(y_{k+1} - a^T_{k+1} \hat{x}_k)$$

where $$y_{k+1}$$ is the new scalar observation and the gain factor $$w_{k+1}$$ is n-by-1 column vector given by
 * $$w_{k+1} = \frac{C_{e_k} a_{k+1}}{\sigma^2_{k+1} + a^T_{k+1}C_{e_k} a_{k+1}}.$$

The $$C_{e_{k+1}}$$ is n-by-n error covariance matrix given by
 * $$C_{e_{k+1}} = (I - w_{k+1}a^T_{k+1})C_{e_k} .$$

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 $$\mathrm{E}\{\tilde{y}^T \tilde{y}\}$$. For instance, in the case of scalar observations, we have the gradient $$\nabla_{\hat{x}} \mathrm{E}\{\tilde{y}^2\} = -2 \mathrm{E}\{\tilde{y} a\}.$$ Thus, the update equation for the least mean square filter is given by
 * $$\hat{x}_{k+1} = \hat{x}_k + \eta_k \mathrm{E}\{\tilde{y}_k a_k\},$$

where $$\eta_k$$ is the scalar step size and the expectation is approximated by the instantaneous value $$\mathrm{E}\{a_k \tilde{y}_k\} \approx a_k \tilde{y}_k$$. As we can see, these methods bypass the need for covariance matrices.

Special Case: vector observation with uncorrelated noise
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:

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

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

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

\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}$$ such that it will yield an optimal linear estimate $$\hat z_{4}$$.

In terms of the terminology developed in the previous sections, for this problem we have the observation vector $$y = [z_1, z_2, z_3]^T$$, the estimator matrix $$W = [w_1, w_2, w_3]$$ as a row vector, and the estimated variable $$x = z_4$$ as a scalar quantity. The autocorrelation matrix $$C_Y$$ is defined as
 * $$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}$$ is defined as
 * $$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
 * $$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 $$w_1=2.57,$$ $$w_2=-0.142,$$ and $$w_{3}=.5714$$ as the optimal coefficients for $$\hat z_4$$. Computing the minimum mean square error then gives $$\left\Vert e\right\Vert _{\min}^2=\operatorname{E}[z_4 z_4]-WC_{YX}=15-WC_{YX}=.2857$$. 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
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 $$N(0,\sigma_Z^2I)$$ where $$I$$ is an identity matrix. Also $$x$$ and $$z$$ are independent and $$C_{XZ} = 0$$. It is easy to see that

\begin{align} & \operatorname{E}\{y\} = 0, \\ & C_Y = \operatorname{E}\{yy^T\} = \sigma_X^2 11^T + \sigma_Z^2I, \\ & C_{XY} = \operatorname{E}\{xy^T\} = \sigma_X^2 1^T. \end{align} $$

Thus, the linear MMSE estimator is given by

\begin{align} \hat{x} &= C_{XY}C_Y^{-1} y \\ &= \sigma_X^2 1^T(\sigma_X^2 11^T + \sigma_Z^2I)^{-1} y. \end{align} $$ We can simplify the expression by using the alternative form for $$W$$ as

\begin{align} \hat{x} &= \left(1^T \frac{1}{\sigma_Z^2}I 1 + \frac{1}{\sigma_X^2}\right)^{-1} 1^T \frac{1}{\sigma_Z^2} I y \\ &= \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{align} $$

where for $$y = [y_1,y_2,\ldots,y_N]^T $$ we have $$\bar{y} = \frac{1^Ty}{N} = \frac{\sum_{i=1}^N y_i}{N}.$$

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
 * $$\operatorname{LMMSE} = \sigma_X^2 - \sigma_{\hat{X}}^2 = \Big(\frac{\sigma_Z^2}{\sigma_X^2 + \sigma_Z^2/N}\Big) \frac{\sigma_X^2} N.$$

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
 * $$\hat{x} = \frac 1 N \sum_{i=1}^N y_i,$$

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
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{align} y_1 &= x + z_1 \\ y_2 &= x + z_2. \end{align} $$

Here, both the $$\operatorname{E}\{y_1\} = \operatorname{E}\{y_2\} = \bar{x} = 1/2$$. 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{align} 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{align} $$ Here, since the denominator term is constant, the poll with lower error is given higher weight in order to predict the election outcome. Lastly, the variance of $$\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.$$ Thus, the LMMSE is given by
 * $$\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 $$\hat{x} = \sum_{i=1}^N w_i (y_i - \bar{x}) + \bar{x},$$ where the weight for i-th pollster is given by $$w_i = \frac{1/\sigma_{Z_i}^2}{\sum_{j=1}^N 1/\sigma_{Z_j}^2 + 1/\sigma_X^2}$$ and the LMMSE is given by $$\mathrm{LMMSE} = \frac{1}{\sum_{j=1}^N 1/\sigma_{Z_j}^2 + 1/\sigma_X^2}.$$

Example 4
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{align} y_1 &= a_1 x + z_1 \\ y_2 &= a_2 x + z_2. \end{align} $$

Here both the $$\operatorname{E}\{y_1\} = \operatorname{E}\{y_2\} = 0$$. 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
 * $$w_i = \frac{a_i/\sigma_{Z_i}^2}{\sum_j a_j^2/\sigma_{Z_j}^2 + 1/\sigma_{X}^2}.$$

title: Minimum mean square error


text:
IRCA

IRCA may refer to:

 * First Nations Media Australia, formerly Indigenous Remote Communications Association
 * Immigration Reform and Control Act of 1986
 * Institut royal de la culture amazighe
 * International Ragdoll Cat Association
 * International Railways of Central America, a Guatemalan railway
 * International Register of Certificated Auditors
 * IRCA (football club), an association football club in Guatemala belonging to International Railways of Central America

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: IRCA


text:
Talk:Aspiette

Made the link Autistic Spectrum Disorder point to the Autistic Spectrum page which exists MttJocy 00:20, 19 Apr 2005 (UTC) Thinking again here, as this is more of a simple definition would it not be better placed on wikidictionary? MttJocy 00:29, 19 Apr 2005 (UTC)
 * This is a rare word, I have never heard of it outside Wikipedia. See also discussion in Talk:Asperger's syndrome. (Actually you discussed there...)

 * Words that tell the gender of an Asperger have been opposed in some circles because there are the same criteria for both male and female Aspergers. It would make an artificial distinction. There is also possibility for feminist critique (Critique on rare words? Yup, you can have such.) because both aspies (male) and aspiettes are together called aspies, which makes a male aspie a standard. The standard is even emphasized, since aspiette is derived from aspie. And it is gender binary etc. But this is predictable critique, and the word is so rare that you can't have an article on it. Move it to Wiktionary and mark it there as a rare word, too. -Hapsiainen 11:29, Jun 1, 2005 (UTC)
 * To a certain degree I would have to disaggree with "because there are the same criteria for both male and female Aspergers. It would make an artificial distinction." Although this is of course true the conditions tend to be exhibited in a different way, but of course this is simply an affectionate term not of an official nature MttJocy 11:19, 12 July 2005 (UTC)

Hi, I am the inventor of the word Aspiette. Just because my word isn't popular yet, it doesn't mean you can change the original definition that I made of it! Geez - since when you do have the control over the word that I invented? It was, after all, a natural progression from the nickname Aspie to adding the 'ette' onto the end of it. If that doesn't sound like a logical step to invent a new word, then I really really don't know what is! From Kitkatsavvy - the inventor of Aspiette (as seen in #chatautism at irc.chatautism.com)... ALSO, OY, stop changing MY word definition! *http://en.wikipedia.org/wiki/Talk:Asperger's_syndrome Geez, in wikipedia MY word definition STILL exists just the same as I originally made it to be. I have changed THIS definition to the same one as I have made above - now leave it alone! This word was invented by me, and now just because it wasn't in google, SOMEONE decides to change the definition! If it changes one more time, I will complain to the wiki people! Kitkatsavvy 10:27, 12 Jul 2005 (UTC)
 * http://en.wikipedia.org/wiki/Aspiette
 * I would still stick by my origional statement on 19 Apr 2005 that as you have stated this is a definition and I see little means by which this can be expanded, and has not been in the 2 months in which it has existed so therefore cannot be defined as a stub article. This therefore is not actually content which should be in wikipedia as stated 1.2 Wikipedia is not a dictionary MttJocy 11:19, 12 July 2005 (UTC)

Seeing as the word Aspie (Gender non-specfic) as any given person with Aspergere's Syndrome or related HFA diagnosis or symptoms (male or female) was first used by Damon Matthew Wise, any misconstrued female only version thereof derived must refer to the original creation of the word and its context, and not solely to a derivation, without reference to the original: http://en.wikipedia.org/wiki/Talk:Aspies

title: Talk:Aspiette


text:
Mobile Homer

"Mobile Homer" is the thirteenth episode of the sixteenth season of the American animated television series The Simpsons. It was first broadcast on the Fox network in the United States on March 20, 2005. In the episode, Marge saves money for life insurance, worried about Homer after a near-fatal incident and his bad medical history. Angered by his wife's new measures to cut back financially, Homer spends the savings on a motor home, in which he spends most of his time, causing a rift between them.

The episode was written by Tim Long, and was the first to be directed by Raymond S. Persi.

Plot
Marge takes the children on a leisurely Sunday afternoon drive while Homer stays home to clean the garage. A series of incidents results in Homer being suffocated by the garage door, and his life is saved when the family returns and Lisa and Bart apply first aid. That night, Marge insists that Homer apply for life insurance, but he is deemed uninsurable due to his poor medical history. After watching a melodramatic "inspired by real-life" made-for-TV film, Marge fears for the family's financial well-being and begins purchasing cheap, low-quality products, keeping the savings in a large jar. Homer argues that as the family's breadwinner, he has the right to spend the money. Angered by Marge's excessive measures, he secretly and spitefully purchases a motorhome, which he parks in the backyard and begins living in. Marge withdraws her word from him, and the parents begin competing for Bart and Lisa's loyalty.

While refueling the motorhome, Homer meets a group of fellow motorhome owners and invites them to settle in his backyard. Their late-night merriment further aggravates Marge, who cuts off their electricity and causes Homer's newfound friends to ditch him. Bart and Lisa, bothered by their parents' escalating argument, decide to return the motorhome to the dealership themselves while their parents are distracted. Homer agrees to return the motorhome, but he and Marge discover the note their children have left, and they give chase in their car. Although they demonstrate to Bart and Lisa that they have made up, the children are unable to reach the motorhome's brakes, and it plunges off an incomplete runaway truck ramp onto a Turkish freighter leaving the port. Marge persuades the captain to turn around by offering him several cans of mushroom soup. Homer's attempt to return the motorhome to land sends it sinking into the harbor. The family stays for dinner on the freighter, and Marge expresses a lack of tension over their financial loss due to the small amount of hashish laced in her food.

Song
During the scene with the other RV owners in his backyard, a member of the RV convoy sings a version of Geoff Mack's "I've Been Everywhere".

"Istanbul (Not Constantinople)" is featured at the end of the episode, referring to the Turks that picked them up at the end on their cargo ship.

Reception
In its original broadcast, "Mobile Homer" acquired a Nielsen rating of 8.6, and was viewed in 8.6 million households.

Walter J. Keegan Jr. of TV Squad thought that the episode did not have enough laughs, but did have enough subtle Simpsons humor about SUVs, Turkish sailors, and evil religious icons. He also thought that the idea of Marge filling the viewers in on what Homer does at work (since he is not seen there a lot anymore) was good, while his most puzzling moment was Bart's drawing of Homer.

title: Mobile Homer


text:
Talk:Spencer Fox

Disliked?
Should it also be mentioned that he is very hated by his peers? I mean if it has his hobbies, why not include that? Why is he even listed here? He was born and raised in New York. Would the person that keeps adding him, please remove him or give us a good reason why he should be recognized as a prominent person who contributed to Kirkland Lake, Ontario, Canada? Kirklandlake (talk) 05:30, 2 October 2008 (UTC)

His Aunt trained him into voice acting
His aunt was the one that actually trained him into voice acting in the beginning. He also stated in an interview he is also a solo musician as well in his career, a few month back, another IP user actually reverted and removed it, vandalizing it, that's why. It's information was here when the article was first created, It shouldn't have been removed by that IP user, I will soon be posting a link here and show the original person that removed it in this article's edit history. It's best we should discuss this here instead of reverting edits back and forth over and over, because reverting edits will cause problems and won't solve anything. 50.252.251.149 (talk) 19:20, 3 June 2016 (UTC)

Here is the link of the article's edit history and of the original IP user that removed it, https://en.wikipedia.org/w/index.php?title=Spencer_Fox&diff=710108823&oldid=704509948 50.252.251.149 (talk) 19:24, 3 June 2016 (UTC)

title: Talk:Spencer Fox


text:
Retraining

Retraining or refresher training is the process of learning a new or the same old skill or trade for the same group of personnel. Retraining is required to be provided on a regular basis to avoid personnel obsolescence due to technological changes and the individuals' memory capacity. This short-term instruction course shall serve to re-acquaint personnel with skills previously learnt (recall to retain the potentials) or to bring their knowledge or skills up-to-date (latest) so that skills stay sharp. This kind of training could be provided annually or more frequently as maybe required, based on the importance of consistency of the task of which the skill is involved. Examples of refreshers are cGMP, GDP, HSE trainings. Retraining (repetition of a training conducted earlier) shall also be conducted for an employee, when the employee is rated as ‘not qualified’ for a skill or knowledge, as determined based on the assessment of answers in the training questionnaire of the employee.

Affected people
The need to retrain workers is often thought to apply to older members of the workforce, many of whom saw their occupations disappear and their skills lose value as technology, outsourcing and a weak economy combined to erode their ability to make a living. While older Americans do not face as high a rate of unemployment as the country’s teenagers and young adults, when they do find themselves unemployed, they remain unemployed for more than twice as long as teenagers.

While the stereotype for retraining needs is the older worker, youth in the United States and across the European Community (OECD) and Africa suffer from the same problem. The gap between the skills they possess and those that employers are actively seeking is significant and stagnating to their employment prospects. Currently in the United States, psychology, history and the performing arts make up 22% of college degrees earned. Demand for skilled employees, however, is in the areas of technology and engineering, currently at 5% of conferred degrees. “In both Britain and the United States, many people with expensive liberal arts degrees are finding it impossible to get decent jobs,” reports the Economist in its April 27, 2013 issue, adding that in northern Africa, job applicants with degrees face an unemployment level twice that of non-degreed candidates.

While technology anxiety and a nervousness about learning new processes and acquiring new skill sets has impacted older workers, younger job seekers are also facing a deficit of “applied soft skills” such as work ethic, social skills, communication and leadership.

Policy issues
The need for greater partnership and transfer of information between institutions of higher education is essential in reducing the skills gap for old and young people alike. Expanded internships, returnships, and post-hiring training can help from the employers’ perspective and upgraded and more authentic technical training will help close the gap on the side of educators. The World Bank's 2019 World Development Report on the future of work explains that flexible learning opportunities at universities and adult learning programs that allow workers to retrain and retool are vital in order for labor markets to adjust to the future of work.

There is some controversy surrounding the use of retraining to offset economic changes caused by free trade and automation. For example, most studies show that displaced factory workers in the United States on the average have lower wages after retraining to other positions when a factory is closed due to offshoring. A similar issue surrounds movement from technical jobs to liaison jobs due to offshore outsourcing. Such changes may also favor certain personality types over others, due to the changing tasks and skills required. Other research estimates that one academic year of such retraining at a community college increases the long-term earnings by about 8 percent for older males and by about 10 percent for older females.

Government policy may make a difference in employability and motivation for retraining and re-entry into the workforce for older workers. In economies with greater regulations surrounding the hiring, termination and wages, reductions in unemployment were difficult to achieve. The very groups harmed by continued higher unemployment were those that the regulations sought to protect.

Retraining is sometimes offered as part of workfare programs, which may include support for transportation, childcare, or an internship.

Health effects of unemployment
As difficult and controversial as it may seem, retraining older and younger workers alike to prepare them to be part of a changing workforce will have a lasting impact on workers across the globe. Unemployed workers are at significantly greater risk for poor physical health, greater stress, alcoholism, marital problems and even suicide. Among young workers, beginning their careers with extended bouts of joblessness results in lower overall earnings and more unemployment throughout their careers.

title: Retraining


text: User talk:67.8.223.42

title: User talk:67.8.223.42


text:
Wikipedia:Articles for deletion/Faf

 This page is an archive of the discussion about the proposed deletion of the article below. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was delete. – ABCD 19:17, 25 Apr 2005 (UTC)

Faf
Page was vanity article. By the way, I am the game's primary coder, although I did not create the Wiki article.
 * Delete. Seems to have been blanked, original page in history did not show notability. Andrew Lenahan - St ar bli nd  00:54, Apr 19, 2005 (UTC)
 * Delete, not an encyclopedia article. Author uses the word "we" at least four times within the first paragraph. Mgm|(talk) 08:21, Apr 19, 2005 (UTC)
 * Del per Starblind. (But not per Mgm: "we" is grounds for cleanup-tone not vfd.)  —msh210 16:39, 19 Apr 2005 (UTC)
 * Delete as per Starblind. Master Thief Garrett 04:45, 23 Apr 2005 (UTC)

 * This page is now preserved as an archive of the debate and, like some other VfD subpages, is no longer 'live'. Subsequent comments on the issue, the deletion, or the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.

title: Wikipedia:Articles for deletion/Faf


text:
Norwich railway station

Norwich railway station (formerly Norwich Thorpe) is the northern terminus of the Great Eastern Main Line in the East of England, serving the cathedral city of Norwich, Norfolk. It is 114 mi down the main line (measured via Ipswich) from London Liverpool Street, the western terminus.

It is also the terminus of numerous secondary lines: the Breckland Line to Cambridge; the Bittern Line to Sheringham; and the Wherry Lines to Great Yarmouth and Lowestoft.

The station is currently managed by Greater Anglia, which also operates the majority of the trains that serve the station. East Midlands Railway operates the services to Liverpool Lime Street via Nottingham, Sheffield and Manchester Piccadilly.

History
At one time, there were three railway stations in Norwich. Norwich Thorpe is the current and only remaining station and still known locally as "Thorpe station". Norwich Victoria was the terminus for some passenger services from London until 1916, as well as being a goods station until its demolition in the 1970s. Norwich City was the terminus of the Midland and Great Northern Joint Railway line from Melton Constable until it was closed to passengers in 1959.

Early history
The original station was opened by the Yarmouth & Norwich Railway (Y&NR), which was the earliest railway in Norfolk. The YNR's Act of Parliament of 18 June 1842 authorised the issue of £200,000 worth of shares to build a line between the two towns, via Reedham and the Yare valley. The chairman was George Stephenson and the chief engineer was his son, Robert Stephenson. Construction started in April 1843 and the 20+1/2 mi line was completed within a year. There was an inspection and inaugural run on 12 April 1844 and a ceremonial opening on 30 April 1844, followed the next day by the beginning of regular passenger services.

On 18 May 1844, 17 days after the Y&NR started running train services, Parliament gave the Royal Assent to the Norwich & Brandon Railway (N&BR). This was part of a plan to link the Y&NR with London, by linking up with the Eastern Counties Railway (ECR) being built from Newport, Essex, to Brandon, Suffolk. Work started quickly during 1844 and went on into 1845. On 30 June 1845, a Bill authorising the amalgamation of the Y&NR with the N&BR came into effect and Norwich station became a Norfolk Railway asset.

The N&BR line arrived at the station on 15 December 1845, which offered a route to Shoreditch in London via Cambridge and Bishop's Stortford. The Eastern Union Railway (EUR) was building a line towards Norwich and that led to great rivalry between the EUR and the ECR. The ECR trumped the EUR by taking over the Norfolk Railway, including Norwich Station, on 8 May 1848. The following year, the EUR started services to Norwich Victoria. The opening of Norwich Victoria on 12 December 1849 led to the ECR naming its station Norwich Thorpe. On 27 August 1851, EUR services from Ipswich started serving the better-placed Thorpe station.

By the 1860s, the railways in East Anglia were in financial trouble and most were leased to the Eastern Counties Railway. They wanted to amalgamate them formally, but government agreement could not be obtained until an Act of Parliament on 7 August 1862, when the Great Eastern Railway (GER) was formed by the amalgamation. Actually, Norwich Thorpe and Norwich Victoria became GER stations on 1 July 1862, when the GER took over the ECR and the EUR before the Bill had received the Royal Assent.

Great Eastern Railway (1862-1922)
A decade after the GER was formed, the latter promoted a new line from Norwich to Cromer. That line was opened on 20 October 1874 and a new station was constructed at the junction of the Cromer line and Yarmouth & Norwich line. The new station, Whitlingham, stood between Norwich Thorpe and Brundall on the Yarmouth line.

With traffic growing, it was apparent a new station was required in Norwich. It was built to the north of the original station, opening on 3 May 1886 and is the structure surviving today. The old terminus became part of the expanded goods facilities.

The new station was built, at the cost of £60,000, by Messrs Youngs and Son, of Norwich, from designs by Messrs J Wilson and W. N. Ashbee, the company's engineer and architect respectively. The attractive station building was constructed around a central clock tower (the clock was supplied by Dixons and Co of London Street, Norwich) with two-storey matching wings either side. A portico was built onto the clock-tower section. There was a circulating area with a high ceiling and the roof was supported by ironwork supplied by contractor Barnard Bishop and Barnard. The roof extended partly down the platforms, which were then covered by canopies for part of their length. There were initially five platforms, with engine-release roads between platforms 2 and 3, and 4 and 5, which allowed locomotives to be detached from trains without the need to shunt the carriages out of the station.

The GER and Norwich Thorpe changed little for the next 30 years. On 22 May 1916, the GER closed Trowse station as a wartime economy measure. That meant the first station south of Thorpe on the Ipswich line was Swainsthorpe and the next station west of Thorpe on the Ely line was Hethersett. On 1 April 1919, five months after the end of the war, the GER reopened Trowse station. The GER went out of existence following the creation of the "Big Four" railway companies in 1923.

London and North Eastern Railway (1923-1947)
On 1 January 1923, the GER amalgamated with several other railways to form the London and North Eastern Railway (LNER). This was as a result of the Railways Act 1921, which saw most of the 120 railway companies grouped into four main companies, in an effort to stem their losses. Norwich Thorpe became an LNER asset.

During World War II the station was bombed in June 1940 and April 1942.

Following the Transport Act 1947 the Big Four railway companies, including the LNER, were amalgamated into the nationalised British Railways (BR).

British Railways (1948-1994)
On 1 January 1948, the nationalisation of Britain's railways saw the operation of Norwich Thorpe station pass to British Railways (Eastern Region).

Platform 6 was added in 1954 and a modern booking hall was built in 1955.

During the late 1950s, steam locomotives were phased out across the East Anglian network and replaced by diesel-powered trains.

After Norwich City station was closed as part of the Beeching cuts, British Rail decided to revert the name of the station to Norwich, which took effect on 5 May 1969.

When the station closed briefly for electrification works in 1986, Trowse, a disused suburban station, was put back into service as the temporary terminus of the line. It closed again when Norwich re-opened. The signalling was also modernised at that time and the track layout simplified. On completion of the electrification project, Norwich-London InterCity trains switched from being hauled by Class 47 diesel locomotives to Class 86 electric locomotives.

The privatisation era (1994-present)
On 1 April 1994, under the Railways Act 1993, ownership of the station passed to a new private company, Railtrack, which was transferred to the state-controlled non-profit Network Rail in October 2002 after experiencing financial difficulties.

Train services to Norwich were later privatised, with most services passing to Anglia Railways in January 1997. Services towards the West Midlands were taken over by Central Trains in March 1997. Anglia trains handed over their franchise to National Express East Anglia in 2004. Three years later, on 11 November 2007, the Central Trains franchise was broken up and West Midlands services to Norwich were taken over by East Midlands Trains. The National Express East Anglia franchise passed to Abellio Greater Anglia on 5 February 2012. All services operated by East Midlands Trains were transferred to East Midlands Railway in August 2019, after EMT's franchise expired.

Layout

 * Platform 1: Greater Anglia inter-city services to London Liverpool Street and used occasionally for Breckland Line services to Cambridge and Stansted Airport; East Midlands Railway services to Liverpool Lime Street via Nottingham.
 * Platform 2: Greater Anglia inter-city services to London Liverpool Street.
 * Platform 3: Greater Anglia inter-city services to London Liverpool Street and services to Cambridge and Stansted Airport; East Midlands Railway services to Liverpool Lime Street via Nottingham.
 * Platforms 4-6: Greater Anglia rural services to Great Yarmouth, Lowestoft and Sheringham. Platform 4 is also used occasionally at peak times for inter-city services to London Liverpool Street. Platform 6 is also infrequently used for services to Cambridge and Stansted Airport.

Services
The station is served by Greater Anglia and East Midlands Railway.

The typical off-peak service frequency (Monday to Saturday) is:
 * 1 tph to Liverpool Lime Street
 * 2tph to London Liverpool Street, of which 1tph makes additional calls at Stowmarket, Chelmsford and Stratford
 * 1tph to Stansted Airport
 * 1tph to Sheringham
 * 1tph to Lowestoft, of which 1tp2h (1 train every 2 hours) calls at Oulton Broad North only and 1tpd serves Buckenham
 * 1tph to Great Yarmouth, of which:
 * 2tpd (trains per day) run via Cantley, Reedham and Berney Arms
 * The remaining trains run via Lingwood and Acle

Accidents and incidents

 * On 10 September 1874, the Thorpe rail accident, East Anglia's worst train crash, occurred at Thorpe St Andrew between Norwich Thorpe and Brundall, killing 25 people and injuring 75.
 * On 21 January 1881, two passenger trains collided at Norwich Thorpe junction just beyond Carrow Road Bridge. Both trains had their locomotive and leading carriage derailed. There were, however, no casualties.
 * On 21 July 2013, in the early hours, a passenger train ran into another which was stabled in Norwich's platform 6, injuring eight people. An investigation blamed driver fatigue.

Engine sheds
Norwich engine shed was located to the south west of the station. This depot closed in 1982 and was replaced by a new facility at Crown Point which, in 2015, is responsible for the maintenance of the main line electric fleet and local diesel multiple units.

Miscellaneous
Before carriages were lit by electric lighting they were lit by gas. Norwich had an oil gas works and carriages north of a line from Harwich to Cambridge were supplied with oil gas. The gas was distributed to other stations in a dedicated fleet of ten tank wagons. Use of the facility declined in the 1930s although up until the 1950s catering vehicles were still supplied.

Children's author Arthur Ransome set the opening paragraph of Coot Club (1934) at Norwich Thorpe station. It also appears in the 1971 film The Go-Between.

title: Norwich railway station


text: User talk:68.85.200.254

title: User talk:68.85.200.254


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Picnic of Love

Picnic of Love is the fifth album by Anal Cunt. It was released on July 21, 1998. The album was released as a joke and a parody of love songs, as well as of the band itself. The album is characterized by the opposite of everything that the band represents. While the normal Anal Cunt album consists of 40 to 50 tracks, mainly under a minute in length, Picnic of Love contains only eleven comparatively long tracks. The downtuned distorted guitars, rapid drumming and screeched vocals of other Anal Cunt albums is replaced by soft acoustic guitar and vocalist Seth Putnam singing in gentle falsetto. The most apparent change is the lyrical content: all of the songs are about inoffensive, sweet subjects as opposed to the band's more typical deliberately offensive or provocative lyrics.

Content observation
Allmusic's review of the album erroneously claims that Picnic of Love "is a collection of ballads that never ceases to have a tragic and violent ending by the end of each song", but none of the songs actually do feature a climax in this manner. The entire album is almost completely devoid of screamed vocals, with the latter half of the final track, "In My Heart There's a Star Named After You" being the only exception.

Personnel

 * "Sensitive" Seth Putnam – vocals, mixing
 * Allison Dunn – vocals (1, 5)
 * "Gentle" Josh Martin – acoustic guitar
 * Mike Livingston – recording, engineering, mixing

title: Picnic of Love


text:
F. B. Meyer

Frederick Brotherton Meyer (8 April 1847 – 28 March 1929), a contemporary and friend of D. L. Moody and A. C. Dixon, was a Baptist pastor and evangelist in England involved in ministry and inner city mission work on both sides of the Atlantic. Author of numerous religious books and articles, many of which remain in print today, he was described in an obituary as The Archbishop of the Free Churches.

Introduction
Frederick Meyer was born in London. He attended Brighton College and graduated from the University of London in 1869. He studied theology at Regent's Park College.

Meyer was part of the Higher Life movement and preached often at the Keswick Convention. He was known as a crusader against immorality. He preached against drunkenness and prostitution. He is said to have brought about the closing of hundreds of saloons and brothels.

While in York in the early 1870s F. B. Meyer met the American evangelist Dwight L. Moody, whom he introduced to other chapels, churches, and ministers in England, and by exchange was invited to make several trips to minister in America. The two preachers became lifelong friends.

Meyer played an important part in the beginnings of the Welsh revival, holding Keswick meetings in Wales in 1903 at which Evan Roberts, among others, pledged to spend at least one day a month praying for revival.

In 1908, Meyer and his wife made a tour of South Africa, where they met Mohandas Gandhi spending several days with him in May. Meyer expressed a cautious sympathy with Gandhi's movement of passive resistance.

In June 1916, together with Hubert Peet, a Quaker, he visited British conscientious objectors in France, to report upon their position in the light of news that 42 resisting men had been forcibly transported there. The visit took place shortly before 35 of the men were court-martialled and formally sentenced to death, but immediately reprieved.

F. B. Meyer wrote over 75 books, including Christian biographies and devotional commentaries on the Bible. In 1918, Meyer, along with seven other clergymen, was also a signatory to the London Manifesto asserting that the Second Coming was imminent.

An illustrated biography of his life was published in 1929, with a new edition a few years later. 2007 saw the release of a new biography of Meyer, F.B. Meyer: If I had a hundred lives, written by Professor Bob Holman and published by Christian Focus Publications.

Associated chapels
Meyer began pastoring churches in 1870; his first pastorate was at Pembroke Baptist Chapel in Liverpool, his second at Priory Street Baptist Church in York, 1872. Other chapels and churches he pastored were:
 * Victoria Road Church in Leicester (1874–1878)
 * Melbourne Hall in Leicester (1878/80-1888) - founded by Meyer
 * Regent's Park Chapel in London (1888–1892 and 1909–1915)
 * Christ Church in London (1892–1909 and 1915–1921)

Of these, Melbourne Hall and Christ Church are perhaps most closely associated with his independence of approach.

Melbourne Hall, Leicester : Melbourne Hall has been described as F. B. Meyer's abiding monument; it was initiated in 1878 as a 'Church of Christ' with a small band of believers who fund-raised for, built, and opened the premises in 1880. As an entirely new, independent venture, designed to evangelise the people lying outside ordinary Christian agencies as a local mission, with every member a 'worker' active in the local community, it was decided not to name it a 'chapel' or a 'church', nor a 'tabernacle', and not the old nonconformist term of 'meeting house'; but simply a 'hall'.

Meyer's mission centre attracted great interest - visitors included national figures such as Hudson Taylor as well as local people; ''Melbourne Hall became a centre as well as a sphere. All sorts of people visited... it became, in fact, the Church of the Cordial Welcome, and as a consequence a place of pilgrimage and a centre of evangelical and missionary influence in Leicester and far beyond.'' His 'Farewell Meeting' in 1888 was presided over by the Mayor of Leicester.

Meyer nevertheless decided to move on to other pastorates in London - Regent's Park Chapel and Christ Church.

Christ Church, London : In 1892, Christopher Newman Hall was due to retire from the Christ Church complex in Lambeth, and invited Meyer to leave the Baptist's Regent's Park Chapel and its wealthy church-going district, to become his successor at the non-denominational institution, the successor to Rowland Hill and James Sherman's Surrey Chapel from where many welfare societies and services operated for the largely working class and slum district. Meyer wrote to his people at Regent's Park Chapel ''Shall I devote the remaining years of my manhood to the service of a section of the Church of Christ, or accept a position that is equally in touch with all sections of Evangelical Christians ? '' and after careful consideration, and successfully negotiating that a Baptistery would be provided, he decided to take on the role. Meyer left Regent's Park Chapel and entered upon his new charge in September 1892.

This being the year that Charles Spurgeon died, leading to unrest at the nearby baptist Metropolitan Tabernacle, Meyer was able to attract a considerable number of its former members to migrate to Christ Church. Frederick Meyer stayed there until 1902, when Dr. A. T. Pierson was asked to undertake his duties during two prolonged periods of travel abroad. Returning from his sabbaticals to Christ Church, Meyer continued as its pastor until 1909. In September of that year he returned to Regent's Park Chapel for nearly 6 years, coming back to Christ Church as sole minister from May 1915 until 1921.

Final days
Frederick Meyer spent the last few years of his life working as a pastor in England's churches, but still made trips to North America, including one he made at age 80 (his earlier evangelistic tours had included South Africa and Asia, as well as the United States and Canada). A few days before his death, Meyer wrote the following words to a friend: "I have just heard, to my great surprise, that I have but a few days to live. It may be that before this reaches you, I shall have entered the palace. Don't trouble to write. We shall meet in the morning." Following F. B. Meyer's death in 1929, an English newspaper, The Daily Telegraph, described him as The Archbishop of the Free Churches. Across the Atlantic, he had earlier been described in The New York Observer as a man of international fame whose services are constantly sought by churches over the wide and increasing empire of Christendom. In 2007 Stephen Timms wrote of him as a man with enduring popularity, dubbed virtually a Christian socialist.

Works

 * Israel: a Prince with God - The Story of Jacob Re-Told (1887)
 * Christian Living (1888)
 * Elijah and the Secret of His Power (1888)
 * The Shepherd Psalm (1889)
 * Abraham: The Obedience of Faith (1890)
 * John the Baptist (1890)
 * The Prophet of Hope: Studies in Zechariah (1890)
 * Joshua and the Land of Promise (1893)
 * Moses: The Servant of God (1893)
 * The Way Into the Holiest: Expositions on the Epistle to the Hebrews (1893)
 * Christ in Isaiah (1895)
 * David: Shepherd, Psalmist, King (1895)
 * The Secret of Guidance (1896)
 * A Castaway, and other addresses (1897)
 * A Good Start (1897)
 * Love to the Uttermost (1897)
 * Paul: A Servant of Jesus Christ (1897)
 * Saved and Kept: Counsels to Young Believers and Christian Endeavourers (1897)
 * Our Daily Homily (1898)
 * Tried by Fire (1900)
 * Back to Bethel: Separation from Sin, and Fellowship with God (1901)
 * Jottings and Hints for Lay Preachers (1903)
 * In Defence of the Faith (1907)
 * Peter: Fisherman, Disciple, Apostle (1919)
 * Samuel: The Prophet (prior to 1924)
 * The Gospel of John (1950?)
 * Expository Preaching; Plans and Methods
 * Joseph: Beloved-Hated-Exalted

title: F. B. Meyer


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Wikipedia:Articles for deletion/The Allied Republic

 This page is an archive of the discussion about the proposed deletion of the article below. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was speedy deleted by Postdlf (nonsense) --cesarb 01:21, 24 Apr 2005 (UTC)

The Allied Republic
This article is just nonsense. It's obviously made up by some kid. Is this allowed to be speedily deleted or do I need to put it here? I'm putting it here just to be safe, since I'm unfamiliar with speedy deletion policies. It appears to be made in conjunction with The Great Hill, so I'm putting that here too. LDan 00:29, 19 Apr 2005 (UTC)
 * Delete another typically goofy micronation article. Andrew Lenahan - St ar bli nd 00:52, Apr 19, 2005 (UTC)
 * Delete as nonsense. I love the fact that the word "piedmont" got in there though. Rje 00:59, Apr 19, 2005 (UTC)
 * Ctrl-Atl and Delete Total nonsense. Delete this article and all articles linked from this. If the user recreates the page ban the user / IP. --DuKot 01:02, 19 Apr 2005 (UTC)
 * So you think we should delete nation, president, monarchy, etc.? Be careful what you say. Eric119 02:16, 19 Apr 2005 (UTC)
 * Delete the hoax article, but not nation, president, or monarchy. Well, maybe president. I'll think about it. -- 8^D gab 02:36, 2005 Apr 19 (UTC)
 * I already speedy deleted this once. I shall do so again.  Postdlf 03:20, 19 Apr 2005 (UTC)

 * This page is now preserved as an archive of the debate and, like some other VfD subpages, is no longer 'live'. Subsequent comments on the issue, the deletion, or the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.


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The New Odd Couple

The New Odd Couple is an American sitcom television series that aired on ABC from 1982 to 1983, and was an updated version of the 1970s television series The Odd Couple. The New Odd Couple was the second attempt to remake a series of one of Neil Simon's plays with a primarily African-American and European-American cast. The first was Barefoot in the Park (which also premiered on September 24, 1970, the same day as the original Odd Couple series).

Synopsis
In this series, Felix and Oscar were both African-American college buddies who met in the 1950s. Felix was portrayed by Ron Glass and Oscar was portrayed by Demond Wilson. The characterizations were still the same, as Felix was a prissy neatfreak and Oscar was a fun-loving and sloppy character. John Schuck also appeared as Murray the Cop, who was kept Caucasian, as was the character of Roy, who was played by Bart Braverman.

The show ran for 18 episodes. When production on the series began, a Hollywood writers strike was underway; as a result, eight of the early episodes recycled scripts from the original series. By the time the writers began producing new scripts, it was too late, as the show never found an audience. The series was cancelled in 1983.

Cast

 * Ron Glass as Felix Unger
 * Demond Wilson as Oscar Madison
 * Sheila Anderson as Cecily Pigeon
 * Bart Braverman as Roy
 * Ronalda Douglas as Gwendolyn Pigeon
 * Jo Marie Payton as Mona
 * John Schuck as Murray
 * Liz Torres as Maria
 * Telma Hopkins as Frances

Production
The New Odd Couple was executive produced by Garry and Anthony W. Marshall, and produced by Paramount Television. The theme music was still the same as the original series (composed by Neal Hefti), but it was played with a more updated urban arrangement.

This marked the second time Ron Glass and Demond Wilson collaborated; nearly a decade prior, they appeared together in two episodes of Sanford and Son: "Card Sharps" (S2, E6) and "Once a Thief" (S4, E15).

Syndication
Reruns of the series aired on BET in the early 1990s. TV Land occasionally aired reruns when they were airing reruns of the original 1970s series.

title: The New Odd Couple


text:
Wikipedia:Articles for deletion/The Great Hill

 This page is an archive of the discussion about the proposed deletion of the article below. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was REDIRECTED. Postdlf 23:24, 30 Apr 2005 (UTC)

The Great Hill
Part of The Allied Republic nonsense above. Rje 01:02, Apr 19, 2005 (UTC)
 * Redirect to Central Park, where "The Great Hill" is a popular gathering place. -- 8^D gab 02:41, 2005 Apr 19 (UTC)
 * I speedy deleted the hoax article and recreated it as a redirect, per BD2412. Postdlf 03:22, 19 Apr 2005 (UTC)

 * This page is now preserved as an archive of the debate and, like some other VfD subpages, is no longer 'live'. Subsequent comments on the issue, the deletion, or the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.


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Albert Vielfaure

Albert Vielfaure (born April 6, 1923 – December 12, 2007) was a politician in Manitoba, Canada. He was a Liberal member of the Legislative Assembly of Manitoba from 1962 to 1969.

Vielfaure was born and educated at La Broquerie (completing Grade 11), and worked as a farmer and bulk oil salesman. In 1951, he married Solange Desrosiers. Vielfaure served as a member of the Chamber of Commerce. He also was active in the Knights of Columbus.

He was first elected to the Manitoba legislature in the 1962 provincial election, defeating Progressive Conservative Rene Préfontaine by 308 votes in the rural francophone riding of La Vérendrye. He was re-elected by a much greater margin in the 1966 election. The Liberals were in opposition during this period, and Vielfaure spent his entire legislative career on the opposition benches. He did not run for re-election in 1969, after his riding was significantly changed by redistribution.

Vielfaure returned to his farm after leaving politics, focusing on hog and corn production. He served on the Manitoba Hog Marketing Commission from 1970 to 1975, and was an advisor to the federal Minister of Agriculture from 1970 to 1979. From 1986 to 1990, he was an adviser for farm programming on the Canadian Broadcasting Corporation's French network. He served with the Fédération des Caisses Populaires du Manitoba from 1971 to 1982, and received the "Prix Riel" in 2000 for his lifetime achievements in the francophone community. He retired from active farming in 1999, and has been recognized as a member of the Manitoba Agricultural Hall of Fame.

On the night of December 11, 2007, after a deteriorating condition, Vielfaure suffered a serious heart attack which slowed his breathing down and caused him to die in the early morning.

title: Albert Vielfaure


text: User talk:68.227.254.224

title: User talk:68.227.254.224


text: User talk:65.201.172.123

title: User talk:65.201.172.123


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III Army Corps (Wehrmacht)

III Army Corps was a corps level formation of the German Army during World War II.

III Army Corps
The III Corps was formed in October 1934 as '''III. Armeekorps'''. The corps took part in Fall Weiss, the 1939 invasion of Poland as a part of Army Group North. It then took part in Fall Gelb as a part of Army Group A, participating in the assault through the Ardennes. In March 1941, the corps was upgraded to a motorised corps status and redesignated III Armeekorps (mot). The Corps was attached to Army Group South for Operation Barbarossa, the invasion of the Soviet Union. The corps advanced through Ukraine and took part in the Battle of Brody, Battle of Kiev, Battle of Rostov, Battle of Kharkov and Battle of Uman.

III Panzer Corps
III Panzer Corps was formed in June 1942 from III Army Corps and attached to Army Group A, the formation tasked with capturing the Caucasus as a part of Fall Blau.

In mid-1943, following the loss of the 6th Army at the Battle of Stalingrad, III Panzer Corps took part in the battles around Kharkov as part of Army Group Don. During Operation Citadel, the Corps was the striking force of Army Detachment Kempf as they attempted to protect the right flank of the 4th Panzer Army. It also took part in Operation Roland, and was involved in the retreat from Belgorod to the Dniepr.

At the beginning of 1944, the Corps participated in the relief of the forces trapped in the Korsun-Cherkassy Pocket. In March the Corps was encircled in the Kamenets-Podolsky pocket, along with the rest of the 1st Panzer Army. III Corps drove the breakout and escape. Due to heavy losses, from November 1944 to January 1945, the corps was redesignated as Gruppe Breith, after its commander Hermann Breith.

In late 1944, III Panzerkorps participated in Operation Konrad, the failed attempts to relieve the German and Hungarian garrison at Budapest. The corps then took part in Operation Spring Awakening in Hungary. After its failure, the corps retreated through Austria, surrendering to the U.S. Army on 8 May 1945.

Commanders

 * Curt Haase (1 Sep 1939 - 13 Nov 1940)
 * Kurt von Greiff (13 Nov 1940 - 15 Jan 1941)
 * Eberhard von Mackensen (15 Jan 1941 - 31 Mar 1942)
 * Leo Freiherr Geyr von Schweppenburg (31 Mar 1942 - 20 July 1942)
 * Eberhard von Mackensen (20 July 1942 - 2 Jan 1943)
 * Hermann Breith (2 Jan 1943 - 20 Oct 1943)
 * Heinz Ziegler (20 Oct 1943 - 25 Nov 1943)
 * Friedrich Schulz (25 Nov 1943 - 9 Jan 1944)
 * Hermann Breith (9 Jan 1944 - 31 May 1944)
 * Dietrich von Saucken (31 May 1944 - 29 June 1944)
 * Hermann Breith (29 June 1944 - 8 May 1945)

Others

 * Friedrich Foertsch, member of the general staff of III Army Corps 1938–39; later Inspector General of the Bundeswehr 1961–63.

Orders Of Battle
III.Armeekorps, May 1940 - Fall Gelb

 * Stab der Korps
 * Arko 3
 * Korps-Nachrichten-Abteilung 43
 * Korps-Nachschubtruppen 403
 * Feldgendarmerie-Trupp 403
 * 3. Infanterie Division
 * 23. Infanterie Division
 * 52. Infanterie Division

III.Armeekorps (mot), September 1941 - Operation Barbarossa
 * Stab der Korps
 * Arko 3
 * Korps-Nachrichten-Abteilung 43
 * Korps-Nachschubtruppen 403
 * Feldgendarmerie-Trupp 403
 * 14.Panzer Division
 * 60.Infanterie Division (mot)
 * 13.Panzer Division
 * SS-Division Wiking
 * 198.Infanterie Division

III.Panzerkorps, July 1943 - Operation Citadel
 * Corps Troops (Stab der Korps)
 * Artillery Commander 3 (Arko 3)
 * 43rd Panzer Signal Battalion
 * Korps-Nachrichten-Abteilung 43
 * Korps-Nachschubtruppen 403
 * Feldgendarmerie-Trupp 403
 * Corps attached units
 * Sturmgeschütz-Abteilung 228
 * Sturmgeschütz-Abteilung 905
 * Sturmgeschütz-Batterie 393
 * schwere-Panzer-Abteilung 503
 * 6.Panzer-Division under Major General Walther von Hünersdorff
 * 11.Panzer-Regiment (with 86 tanks)
 * 4.Panzergrenadier-Regiment
 * 114.Panzergrenadier-Regiment
 * 76.Panzerartillerie-Regiment
 * 7.Panzer-Division under Lieutenant General Hans Freiherr von Funck
 * 25.Panzer-Regiment (with 87 tanks)
 * 6.Panzergrenadier-Regiment
 * 7.Panzergrenadier-Regiment
 * 78.Panzerartillerie-Regiment
 * 19.Panzer-Division under Lieutenant General Gustav Schmidt
 * 27.Panzer-Regiment (with 70 tanks)
 * 73.Panzergrenadier-Regiment
 * 74.Panzergrenadier-Regiment
 * 19.Panzerartillerie-Regiment
 * 168.Infanterie-Division under Major General Walter Chales de Beaulieu
 * 417.Infanterie-Regiment
 * 429.Infanterie-Regiment
 * 442.Infanterie-Regiment
 * 248.Artillerie-Regiment

III Panzerkorps, March 1944 - Korsun-Cherkassy Pocket
 * Stab der Korps
 * Arko 3
 * Korps-Nachrichten-Abteilung 43
 * Korps-Nachschubtruppen 403
 * Feldgendarmerie-Trupp 403
 * 1.Panzer Division
 * 16.Panzer Division
 * 17.Panzer Division
 * 1.SS Panzer Division Leibstandate SS Adolf Hitler
 * schwere Panzer Abteilung 503

III Panzerkorps, March 1945 - Operation Spring Awakening
 * Stab der Korps
 * Arko 3
 * Korps-Nachrichten-Abteilung 43
 * Korps-Nachschubtruppen 403
 * Feldgendarmerie-Trupp 403
 * schwere-Panzer-Abteilung 509
 * 3.Panzer Division
 * 1.Panzer Division
 * 23.Panzer Division

title: III Army Corps (Wehrmacht)


text:
Talk:HMS Tireless (P327)

USN subs
Why is this in the USN sub category? 132.205.15.43 00:35, 19 Apr 2005 (UTC)

title: Talk:HMS Tireless (P327)


text:
Shakib Arslan

Shakib Arslan (شَكيب أَرْسَلان; 25 December 1869 – 9 December 1946) was an Arab writer, poet, historian, politician, and Emir in Lebanon. A prolific writer, he produced some 20 books and 2,000 articles, as well as two collections of poetry and a "prodigious correspondence". He was known as Amir al-Bayān (أَميرُ البَيان) due to his influential writings.

Biography
Influenced by the ideas of Jamal al-Din al-Afghani and Muhammad Abduh, Arslan became a strong supporter of the pan-Islamic policies of Abdul Hamid II. As an Arab nationalist, Arslan was an advocate of pan-Maghrebism (the unification of Algeria, Tunisia and Morocco). He also argued that the survival of the Ottoman Empire was the only guarantee against the division of the ummah and its occupation by the European imperial powers. For Arslan, Ottomanism and Islam were inseparable, and reform of Islam would naturally lead to the revival of the Ottoman Empire.

Exiled from his homeland by the French Mandate authorities, Arslan spent most of the inter-war years in Geneva, where he served as the unofficial representative of Syria and Palestine to the League of Nations and wrote a steady stream of articles for the periodical press of Arab countries. There he cofounded and edited a newspaper entitled La Nation Arabe. His partner in this activity was Ihsan Al Jabri, a Syrian exile. Arslan was also a contributor to Barid Al Sharq, a propaganda newspaper published in Berlin, Nazi Germany. However, Arslan did not personally agree with Nazism, instead viewing them as a tool to break the other colonial powers. In 1939, he wrote to Daniel Guérin that if the Germans proved to be no better to the Arabs, "they would have only changed masters." In his diary, he remarked that the Italians would simply turn Palestine into an Italian colony.

Advocacy
Amir Shakib proposed an interpretation of Islam imbued with a sense of political power and moral courage. He sought to rebuild the bonds of Islamic unity, urging Muslims from Morocco to Iraq to remember their common commitment to Islam despite their individual differences. Shakib believed that recognising and acting upon this common bond could lead to liberation from their existing subjugation. He also saw this unity as a way of reviving what he saw as their illustrious history. Arslan's work inspired anti-imperialist propaganda campaigns, much to the irritation of the British and French authorities in the Arab world.

He defended Islam as an essential component of social morality. His message, with its call to action and defence of traditional values in a time of great uncertainty, was well received and attracted widespread attention in the 1920s and 1930s. It was during this period that he wrote his most famous work, Our Decline: Its Causes and Remedies, which described what Arslan believed to be the reasons for the weakness of existing Muslim governments.

He contributed to Muhib Al Din Al Khatib's Cairo-based magazine Al Fath, a modernist Salafi publication.

Personal life
Born into a Druze family, he always tried to combine his faith with mainstream Islam, but later converted to Sunni Islam, "establishing himself as an orthodox Muslim serving the interests of Sunni Islam".

He married Suleima Alkhas Hatog, a Jordanian of Circassian descent. They had a son, Ghalib (born 1917 in Lebanon) and two daughters, May (1928–2013) and Nazima (born 1930 in Switzerland). His daughter May married the Lebanese Druze politician Kamal Jumblatt, making the Lebanese politician Walid Jumblatt a grandson of Arslan.

Arslan died on 9 December 1946, three months after returning to Lebanon.

title: Shakib Arslan


text:
Oerlikon FF

The FF were a series of 20mm autocannon introduced by Oerlikon in the late 1920s. The name comes from the German term Flügel Fest, meaning wing mounted, fixed, being one of the first 20mm guns to be small and light enough to fit into a fighter aircraft's wing. The FF series inspired many 20mm cannon used in World War II, including the Hispano-Suiza HS.404 (adopted by the French, British and U.S. forces), the German MG FF, and the Japanese IJNAS's Type 99 cannon.

Design
The basic design was based on the original Oerlikon 20 mm cannon, in turn based on the German Empire's Becker 20 mm cannon of World War I. In the post-war period, Oerlikon had developed the original design to produce three autocannons with increasingly powerful rounds; the original model, later known as the "F", used the Becker 20x70mm round, the newer "L" model used a 20x100 round, and finally the "S" model used a 20x110. All of these weapons used the API blowback operational principle.

In 1935 Oerlikon started development of versions suitable for use as wing-mounted guns. In addition to various changes needed for remote firing and cartidge ejection, the new series focused primarily on rate-of-fire. The original "FF" used a slightly upgraded 128 gram 20x72mm round with a muzzle velocity of 600 m/s at a cyclic rate of 520 rounds per minute, almost double that of the original F and AF models. The gun weighed only 24 kg. Similar improvements led to the 30 kg FF L using a longer 20x101mm round which gave 750 m/s, and the 39 kg FF S firing a 20x110mm round at 830 m/s at a slightly slower 470 rounds per minute. The original guns became known as the FF F from this point on.

The FF F was licensed by the Japanese and produced as the Type 99-1, along with the FF L as the Type 99–2. Hispano-Suiza built the FF S as the HS.7, and slightly improved HS.9. This design was later abandoned by Hispano-Suiza. In the 1930s, Marc Birkigt designed an entirely new gas-operated cannon with a locked bolt, the HS.404, which became one of the best 20 mm weapons of the war. Ikaria Werke in Germany started production of the FF F with a slightly more powerful 20x80mm round as the MG FF, but later introduced a new mine shell round which was lighter and improved velocity. The resulting MG FF/M was a common weapon until about 1944. Starting in late 1940 these were replaced by the very different Mauser MG 151/15 and Mauser MG 151/20 autocannon, respectively of 15mm (early) and 20mm calibre (primary production).

title: Oerlikon FF


text: User talk:86.128.40.234

title: User talk:86.128.40.234


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Polestar (disambiguation)

Polestar is the official Volvo Cars performance company and brand.

Pole star, polestar or polar star may refer to:

 * Pole star, a bright star that is approximately aligned with the axis of rotation of a planet
 * Polaris, North Pole star of the Earth
 * Polestar, a monument in Letterkenny
 * Polestar, the company of Philip Segal
 * Polestar, developer of the Japan-exclusive game Magical Pop'n
 * Polestar Book Publishers, a literary book publishing division owned by Raincoast
 * Polestar Racing Group, an American motorsport team currently competing in the Atlantic Championship
 * Polestar Racing, a Swedish motorsport team currently competing in the STCC and V8 SuperCars Championship
 * Polestar Preschool, an area in the SNES RPG EarthBound
 * Polyarnaya Zvezda (disambiguation) (Russian-language titles)
 * Polar Star (Decembrist journal)
 * Polar Star (Herzen)
 * Operation Polar Star, World War II Soviet military operation

Literature and arts

 * Polestar an the alias of Cosmic Boy, a DC Comics superhero in the Legion of Super-Heroes
 * Polestar (comics), a Spider-Man villain
 * Polar Star (novel), a 1989 novel by Martin Cruz Smith
 * "Polar Star" (song), a song by F.T. Island

Ships

 * MS Polstjerna, a former Norwegian sealing ship
 * MV Polar Star, a former cruise ship by Polar Star Expeditions
 * NLV Pole Star, a lighthouse tender vessel
 * RV Polarstern, a German research icebreaker
 * USCGC Polar Star (WAGB-10), an icebreaker

title: Polestar (disambiguation)


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Samuel L. Popkin

Samuel L. Popkin (born June 9, 1942) is an American political scientist who teaches at the University of California, San Diego. Popkin has played a role in the development of rational choice theory within political science. He is also noted for his work as a pollster.

Early life
Popkin was born on June 9, 1942. He received his Ph.D. from the Massachusetts Institute of Technology in 1969.

Career
His most recent book is The Candidate: What it Takes to Win (and Hold) the White House. Earlier he wrote  The Reasoning Voter: Communication and Persuasion in Presidential Campaigns and co-authored Issues and Strategies: The Computer Simulation of Presidential Campaigns. He also co-edited Chief of Staff: Twenty-Five Years of Managing the Presidency.

When he was assistant professor of Government, in 1972, he was jailed for a week, for his refusal to answer questions before a grand jury investigating the Pentagon Papers case, during a hearing before the Boston's Federal District Court. The Faculty Council later passed a resolution condemning the government's interrogation of scholars on the grounds that "an unlimited right of grand juries to ask any question and to expose a witness to citations for contempt could easily threaten scholarly research."

Popkin is equally well known for his work on peasant society, with particular reference to East and Southeast Asia, including The Rational Peasant: The Political Economy of Rural Society in Vietnam.

Popkin has also been a consulting analyst in presidential campaigns, serving as consultant on Bill Clinton's presidential campaign on polling and strategy, to the CBS News election units from 1983 to 1990 on survey design and analysis, and more recently to the Gore campaign. He has also served as consultant to political parties in Canada and Europe and to the Departments of State and Defense. His current research focuses on presidential campaigns and the relationship of public opinion to foreign policy.

Personal life
Popkin is married to Susan Shirk, research professor and chair of the 21st Century China Center at the Graduate School of Global Policy and Strategy at the University of California, San Diego.

Selected publications

 * The Reasoning Voter: Communication and Persuasion in Presidential Campaigns
 * The Rational Peasant: The Political Economy of Rural Society in Vietnam
 * Candidates, Issues and Strategies
 * Chief of Staff: Twenty-Five Years of Managing the Presidency, co-editor with Samuel Kernell

title: Samuel L. Popkin


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Ramón Castro (catcher)

Ramón Abraham Castro (born March 1, 1976) is a Puerto Rican former professional baseball catcher, who played in Major League Baseball (MLB) for the Florida Marlins, New York Mets and Chicago White Sox.

Early career
On June 2, 1994, Castro was selected by the Houston Astros in the first round (17th overall pick) of the Major League Baseball draft, becoming the first player to be drafted out of Puerto Rico in the first round. He remained the highest-selected player drafted directly from a Puerto Rico high school until 2012, when Carlos Correa became the first overall pick.

Florida Marlins
He was traded to the Florida Marlins in 1998 for relief pitcher Jay Powell.

Castro made his major league debut on August 27, 1999 in a game against the Houston Astros. He spent 1999 and 2000 competing with Mike Redmond for playing time and, when Charles Johnson arrived in Florida, Castro spent most of the 2001 season in the minor leagues. In 2002, he was back in the majors but as a third option and, in 2003, he played behind Iván Rodríguez as the Marlins won the 2003 World Series. However, Castro did not appear in the series. In 2004 Rodríguez was gone, giving Castro his first opportunity as a starter but a .135 batting average and an early-June season-ending injury finished his time in Florida. Paul Lo Duca finished out the Marlins' catching duties for 2004. The Marlins released Castro in October 2004.

Castro is known as a good defensive catcher, but, despite showing occasional home run power has never displayed sufficient hitting skills to win a job as a starter, along with struggling with injuries throughout his career. While never hitting for a superior average, however, his hits are often well-timed.

New York Mets
In December 2004, Castro signed with the New York Mets. In 2005, he was backup catcher to Mike Piazza, with 41 RBIs in 209 at-bats, a career-best .244 batting average, and playing solid defense.

When Piazza became a free agent after the season, it looked like Castro might again get a chance to be a starter, but the Mets traded for Paul Lo Duca who previously had replaced Castro in Florida. Injuries further reduced his role in 2006, limiting him to 126 at-bats.

When Lo Duca left after the 2007 season, many fans called for Castro to start, but the job instead went to newcomer Brian Schneider. It was most likely due to his injury-riddled career, which became evident once more when he strained his hamstring late in spring training that year, missing the beginning of the season.

Chicago White Sox
On May 29, 2009, Ramon was traded to the Chicago White Sox for pitcher Lance Broadway after Omir Santos won the backup job with the return of Brian Schneider from the DL.

On July 23, 2009, Castro became the 17th backstop in Major League Baseball history to catch a perfect game, receiving Mark Buehrle's second no-hitter. It was his first time catching for Buehrle, who stated afterwards that he did not shake off Castro at any time during the game.

On January 12, 2010, Castro re-signed with the Chicago White Sox on a 1-year $800,000 contract with a club option for 2011. He played in 37 games in 2010 and 23 games in 2011.

Los Angeles Dodgers
After not playing in 2012, Castro signed a minor league contract with the Los Angeles Dodgers in January 2013. He was released on March 18, 2013.

Long Island Ducks
Castro signed with the Long Island Ducks of the Atlantic League of Professional Baseball.

Telemarket Rimini
Castro signed a contract to play with Telemarket Rimini of Italian Baseball League during the 2013 season.

Personal life
Ramon Castro was raised in Sabana Puerto Rico. Since he was little he loved playing baseball. In high school he was the first Puerto Rican to be drafted by the Houston Astros. Recently he married Brenda Castro. They have two daughters: Mairym (born December 21, 1995), and Nathalie (born July 15, 1999), Miosoty (born January 23, 2001) and he has another daughter by a different mother, Isabella (born December 22, 2001) They are currently living in Miramar, Florida.

title: Ramón Castro (catcher)


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Wikipedia:Peer review/History of music/archive1

History of music
It is a pretty good article but was rejected from being a featured article. -- Tony Jin | (talk) 00:39, Apr 19, 2005 (UTC)
 * And I see not much has been done to adress the FAC objections. First, adress them, then come back here. I could simply post my FAC objections again. What's the point? --Piotr Konieczny aka Prokonsul Piotrus Talk 10:14, 19 Apr 2005 (UTC)
 * That does seem pointless. I have written to Tony asking him to respond to Piotrus here, so that this listing can receive some attention, or, alternatively, be removed as an inactive request (see Peer review/Request removal policy). --Bishonen | talk 14:10, 24 Apr 2005 (UTC)