Mean square error measurement noise

There are several ways of detecting peaks in such noisy signals. The Wiener-Hopf filter minimizes the expectation value of the noise power spectrum and may be used to optimally smooth the original noisy profile [19]. An alternative approach described by Hindeleh and Johnson employs knowledge of the peak shape. It synthesizes a simulated diffraction profile from peaks of known width and shape, for all possible peak amplitudes and positions, and selects that combination of peaks that minimizes the mean square error between the synthesized and measured profiles [20], This procedure is illustrated... 

Root mean square noise Measurement of spectrophotometric noise of NIR instruments by measuring a reference (usually a ceramic or spectralon) repeatedly and calculate the root mean square error. 

So far we have said nothing about the conditions needed before this reduction in mean squared error can be attained. The chief requirement is that the error e in the i-th measurement (i.e., the i-th reading of the spring balance or optical detector) be independent of the quantity being measured, for i = l,2,, n. In the weighing problem this implies that the objects to be weighed should be light in comparison with the mass of the balance, and in the optical problem that the noise in the detector be independent of the... 

Figure 6.7 The root mean square error of calibration (RMSEC), leave-one-out cross validation (RMSECV) and prediction (RMSEP) are plotted as a function of the signal-to-noise ratio (SNR). While the intrinsic SNR amounts to 3000, random noise was artificially added to mid-IR spectra of 247 serum samples (which decreases the SNR) and the concentration of glucose was recalculated by means of partial least squares (PLS) based on the noisy spectra. The open symbols refer to assessing the quality of quantification within the teaching set, while the filled symbols relate to an external validation set. The data show that the noise can be increased by more than an order of magnitude before the prediction accuracy of the independent external validation set (RMSEP) is affected. In addition, it can clearly be observed that the RMSEC is a poor measure of accuracy since it suggests delivering seemingly better results for lower SNRs, while in fact the calibration simply tends to fit the noise for low values of SNR (see section 6.7).

Maximum likelihood (ML) estimation can be performed if the statistics of the measurement noise Ej are known. This estimate is the value of the parameters for which the observation of the vector, yj, is the most probable. If we assume the probability density function (pdf) of to be normal, with zero mean and uniform variance, ML estimation reduces to ordinary least squares estimation. An estimate, 0, of the true yth individual parameters (pj can be obtained through optimization of some objective function, 0 (0 ). ModeP is assumed to be a natural choice if each measurement is assumed to be equally precise for all values of yj. This is usually the case in concentration-effect modeling. Considering the multiplicative log-normal error model, the observed concentration y is given by ... 

Suppose a given data set consisting of 20 measurements of the yield y, at different pressures (p) and temperatures (7 ) is available. Fit the model (5.1) to the data this results in a fit error SSEi, where SSE stands for sum of squared residuals (e ). If the same data set is fitted to model (5.2), then this results in a fit error SSE2. Then SSE2 < SSEi, because model (5.2) will always contain model (5.1). Model (5.1) is a constrained version of model (5.2), hence, there is a relationship between the fit errors of both models. This does not mean, however, that model (5.2) is better than model (5.1). If model (5.1) can describe the data well, the increased complexity of model (5.2) will only lead to overfitting, i.e., fitting of noise. 

In this simulated-data example, a twelve-story shear building is considered. It is assumed that this building has uniformly distributed floor mass and uniform stiffness across the height. The mass per floor is taken to be 100 metric tons, while the interstory stiffness is chosen to be k = 202.767 MN/m so that the first five modal frequencies are 0.900,2.686,4.429,6.103 and 7.680 Hz. The covariance matrix is diagonal with the variances corresponding to a 1.0% coefficient of variation of the measurement error of the squared modal frequencies and mode shapes for all modes, a reasonable value based on typical modal test results. For the simulated modal data, a sample of zero-mean Gaussian noise with covariance matrix was added to the exact modal frequencies and mode shapes. 

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