Minimum Mean Square Error

It is quite a simple matter to generalize the simple prediction problem just discussed to the situation where we want to obtain the best (in the sense of minimum mean square error) linear estimate of one random variable fa given the value of another random variable fa. The quantity to be minimized is thus... 

C. Other Minimum Mean-Square-Error Variations 82... 

It is thus possible to convolve both spread function and data i(x) with s( — x). We may then use the relaxation methods as before. This time, however, we replace i(x) with s( — x) (g) i(x) and s(x) with s( — x) (x) s(x). Not only are we assured convergence, but we have also succeeded in band-limiting the data i(x) in such a way as to guarantee that all noise is removed from i(x) at frequencies where i(x) contains no information about o(x). Furthermore, Ichioka and Nakajima (1981) have shown that reblurring reduces noise in the sense of minimum mean-square error. 

We are permitted to specify the integrals for positive co only, because of the even property of the integrand. This simplication, in turn, stems from the real nature of all the x-space components of the integrand. Minimizing expression (9) is equivalent to asking that the physical solution conform to the Wiener inverse-filter estimate in the sense of minimum mean-square error after suitable weighting of the positive solution to ensure best conformance at frequencies of greatest certainty. 

Solutions for objects of finite extent are obtained through the familiar minimum mean-square-error criterion by selecting the Fourier components that minimize... 

Ephraim and Malah, 1984] Ephraim, Y. and Malah, D. (1984). Speech enhancement using a minimum mean-square error short-time spectral amplitude estimator. IEEE Trans. Acoust., Speech, Signal Processing, 32(6) 1109-1121. 

These two relations are the basis for other important developments of the Kalman filter equations. Concerning the problem considered above, the calculation of the minimum mean square error can be carried out either ... 

Consequently, the minimum mean square error estimate for y is y = Hm and the associated covariance of this is Cy= HC H + R. 

Now if z is also observed, then the minimum mean square error estimate of x for a given y and z is ... 

The minimum mean-squared error (MMSE) estimate xn of the original signal sample xn should be derived for each received sample rn. IID signals are assumed so that the sample index n is suppressed in the following. With help of the known key sequence sample k and known watermark letter d, the... 

Different numbers of hidden layer neurons and spread constants were tried in this study. The number of hidden layer neurons that gave the minimum mean square error (MSB) was determined. The spread that gave the minimum MSB was also found simply by adding some trial-and-error loops in the program codes. 

Minimum mean squared error predictor. For the joint dishibution in Exercise 7, eompute E y - IjvIx]]. Now, find the a and b which minimize the ftmction E y - a - bxf. Given the solutions, verify ih t E y - E y >c]f < E y-a-bxf. The result is fundamental in least squares theory. Verify that the a and h whieh you found satisfy (3-68) and (3-69). 

The Kalman Filter. As in the partially observable Markov decision process we presented above, in order to forecast ftiture demand the forecaster needs to estimate the actual state of the system. To this end, suppose that we would like to compute the minimum mean-square errors (MMSE) estimate of the state Xt, given the history of observations t-2, . , i, ... 

To summarize, we propose a so-called MMSE forecast adaptive base-stock policy. This policy employs the Kalman filter technique to calculate minimum mean square error (MMSE) forecasts of future demands at the beginning of each period. A fixed safety stock 7 set at the beginning of the planning horizon, is then added to the MMSE forecast to form the target level /3t for this period. Then, the following rule is applied if the current inventory position is lower than the target level, an order is placed to fill this gap otherwise, no order is placed. The advantage of our policy is that it is intuitive and easily implementable. But, not less importantly, it can be tailored for use in information-rich supply chains, for which the characterization of optimal policies is virtually impossible. 

Donoho, D.L. 1995. Denoising by soft thresholding. IEEE Trans. Information Theory 41(3) 613-627. Ephraim, Y. 1992. Statistical model based speech enhancement systems. Proc. IEEE 80(10) 1526-1555. Ephraim, Y. and Malah, D. 1984. Speech enhancement using a minimum mean-square error short time spectral amphtude estimator. IEEE Trans. Acoustics, Speech, and Signal Processing 32(6) 1109-1121. Ephraim, Y. and Van Trees, H.L. 1995. A signal subspace approach for speech enhancement. IEEE Trans, on Speech and Audio Processing 3 4) 25l-266. 

An unbiased, minimum variance estimator is also a minimum mean square error estimator. 

There are several criteria for selecting the best network. But, generally, minimum mean squared error (MSB) is used as the selection parameter. Mean squared error is measured as... 

The performances of different developed networks are presented in Table 5.5. It is seen from the table that increasing the number of neurons in the hidden layer does not always ensure that the mean squared error will decrease but it is clear that by increasing the number of neurons, the architecture gets complicated. The architecture 3-17-1 gives the minimum mean squared error and is selected as the best performing architecture. This network also provides the highest correlation coefficient and minimum mean absolute percentage error. 

To construct and train the network, pulse current intensity (I), pulse-on time (t,) and pulse-off time (tg) are used as the input parameters and corresponding fractal dimension (D) is used as the output. Only one hidden layer is used in the networks. The numbers of neurons are varied to select best network based on the minimum mean squared error (MSB). For training of the network, Levenberg-Marquardt algorithm is used. All the observations of the FCC design are used in the training of the networks. 

The MSB values for different architectures are presented in Table 5.21. From the table, it is seen that for mild steel, increase in the number of neurons in the hidden layer beyond four does not improve in the performance and thus the network with 3-4-1 architecture is selected based on minimum mean squared error. For mild steel, the maximum absolute percentage error is obtained as 2.42%, which implies that the ANN model outputs and experimental outputs are very close to each other. The comparative study of experimental and ANN model predicted fractal dimension is presented in Fignre 5.15. From this figure also, it is clear that the predicted and experimental... 

Figure 4.10 (a solid line 1) shows the displacement distance along the loading axis to the center of the equivalent elastic element reference values modulus cesium. The same graph (Fig. 4.10, a-2) shows the radial displacements of atoms from the distance to the center of mass of nanoparticles r, for constant Poisson s ratio. The graphs show that these dependencies are not identical. Therefore, changing the modulus of elasticity, it is necessary to achieve data fusion curves (Fig. 4.10b), which is the criterion for the minimum mean square error. The mean square error is a pronounced minimum (Fig. 4.11). We calculate the bulk modulus of nanoparticles of cesium from the system (the Eq. (7)).

Once the posterior PDF is known, the optimal estimate can be computed using different criteria, one of which is the minimum mean square error (MMSE) estimate. Although, the Bayesian solution is hard to compute analytically, the particle-based methods described herein in fact are approximations of this concept. For further details, the interested reader is pointed to the work of Ristic and Arulampalam (2004). 

Equations (3) and (4) form the basis for the optimal Bayesian solution. Once the posterior density Z ) is known, an optimal state estimate with respect to any criterion can also be computed. For example, the Minimum Mean Square Error (MMSE) estimate is the conditional mean of ... 

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