Zero-mean Gaussian process

For a given set of model parameters, the response x is a zero-mean Gaussian process and the (/, / ) element of its power spectral density matrix function S is given by [249] ... 

If the process X(t) is a stationary, zero mean Gaussian process. Thus, the condition for statlonarlty is converted Into a statistical statement in eq. (30). This allows the derivation of a measure C for nonstatlonarityi, ... 

Model errors are caused by travel-time prediction errors due to unknown velocity heterogeneities in the Earth and may introduce location bias. After phase identification errors are accounted for, model errors represent the most significant contribution to the error budget, especially at local and regional distances. Representing model errors as zero mean, Gaussian processes commonly lead to erroneous formal uncertainty estimates, as the distribution of model errors is not zero mean and is often multimodal. Because of their systematic nature (a velocity model will always produce the same travel-time prediction error along the same ray path), model errors can only be reduced by improved travel-time predictions that account for the 3D velocity structure of the Earth. 

An alternative approach has been proposed by Cacciola (2010) the author s contribution allows a straightforward evaluation of a non-separable power-spectral density function compatible with a target response spectrum. In the model proposed by Cacciola (2010), it is assumed that the nonstationary spectrum-compatible evolutionary ground motion process is given by the superposition of two independent contributions the first one is a fully nonstationary known counterpart which accounts for the time variability of both intensity and frequency content the second one is a corrective term represented by a quasi-stationary zero-mean Gaussian process that adjusts the nonstationary signal in order to make it spectrum compatible. Therefore the grotmd motion can be split in two contributions ... 

Unlike SPC techniques, standard feedback control methods such as PID-control, do exert control upon a process, in an effort to minimize y, — yk. Control in Statistical Process Control is as such not regulatory control, but a semantic means of relating SPC to quality control—a means that often leads to the hybrid term SQC. Ogunnaike and Ray [14, Sec. 28.4] offer advice on when to use SPC and when to use standard feedback control methods When the sampling interval is much greater than the process response time, when zero-mean Gaussian measurement noise dominates process disturbances, and when the cost of regulatory control action is considerable, SPC is preferred. 

The prediction error is modeled as a discrete zero-mean Gaussian white noise vector process , with e R °, and it satisfies the following correlation structure ... 

The flat spectrum indicates equal power of the signal at all frequencies. It is well known that for given model parameters the response x is a zero-mean Gaussian random process. The... 

Since y is a zero-mean Gaussian vector process, both yj (Gaussian random vectors. Furthermore, in the limit when At 0+, and for a non-zero and non-Nyquist frequency tok, it can be shown that the covariance matrix of the... 

The modal forcing f is a linear combination of the components of g so it is also a zero-mean Gaussian vector process. It is stationary with the spectral density matrix function ... 

It is assumed that only the first three x-directional and y-directional modes are measured but not any of the torsional modes. This is done deliberately to simulate a common situation where some of the modes are not excited sufficiently to be able to observe. In the identification process, it is unknown that there are some missing modes. The six measured modes correspond to the 1st (3.432 Hz), 2nd (3.837 Hz), 4th (10.10 Hz), 5th (11.29 Hz), 7th (18.08 Hz) and 9th (21.31 Hz) modes. Sensors are placed on the +y and —y faces of the 1st, 2nd, 5 th and 6th floors, and the -x face of all floors to measure the modal frequencies and mode shape components. The covariance matrix Te is diagonal with 0.5% COV for the modal data. For the simulated modal data, a sample of zero-mean Gaussian noise with covariance matrix Ze was added to the exact modal frequencies and mode shapes. Initial values for all stiffness parameters are taken to be 100 MN/m, which overestimates the values by 100% and 150% for the x and y faces, respectively. 

As noted before, the Brownian force n t) may be modeled as a white noise stochastic process. White noise is a zero mean Gaussian random process with a constant power spectrum given in (72). Thus,... 

Here 0 is a p x 1 vector of unknown system parameters, Xk and / are the r x 1 state and externally applied force vectors in the time discretized form, (ftk is a nonlinear state transition vector, and Wk is the x 1 process noise which represents the error in arriving at mathematical model for the vibrating system, modeled as a sequence of zero-mean Gaussian random variables with known covariance, i.e.,... 

Assuming the ground acceleration process as zero-mean Gaussian nonstationary non-separable... 

Let us assume now that the forcing term is a mono-correlated zero-mean Gaussian random process vector given by the relationship ... 

In order to take into account of the spatial variability of earthquake-induced ground motions, let us consider an n-DoF structural system subjected to an N support motion. It follows that the stochastic forcing vector process has to be modeled as a multi-correlated zero-mean Gaussian random vector process ... 

This relationship shows that the stationary counterpart of the multi-correlated stochastic process is a vector process, N(( ) (of order AT), with orthogonal increments. Furthermore, Gnn(Hermitian matrix function which describes the one-sided PSD function matrix of the so-called embedded stationary counterpart vector process, N(m). After some algebra it can be proved that the autocorrelation function matrix of the zero-mean Gaussian nonstationary random vector process F(t) can be obtained as... 

Let us consider an oscillator, whose differential equation governing the motion is written in canonical form in Eq. 23, forced by the uniformly modulate Gaussian zero-mean nonstationary process F t), defined as... 

In Figs. 7 and 8 the mean frequencies, v (t), and the normalized time-varying central frequencies, fflc,Ma(0/fflo. defined in Eq. 21, of the response of the two oscillators before analyzed for the Hsu and Bernard (1978) and Spanos and Solomos (1983) modulating functions of the nmistationary zero-mean Gaussian input process, are depicted. 

If the detector noise originates from stationary, Gaussian random processes and the instrument is calibrated with zero-offset, this matched Alter output is a zero-mean Gaussian random variable with variance equal to the power signal-to-noise ratio of the known signal ... 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

For normal statistics, the mean and the variance are completely sufficient to characterize the process all the other moments are zero. For standard normal or Gaussian statistics (i.e., normal statistics with zero mean), the variance p,2... 

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