Conditional PDFs

In this section, the general expression is given for the conditional probability densities involving Np previous points p(y 0 y -Mp, y -Wp-n, , yn-i, C) in Equation (4.34), with n Np . First, the random vector has the zero mean and covariance matrix 

Since the measured response has zero mean, the optimal estimator y of y conditional on Nn-Np,n-i is given by (refer to Appendix C or Brockwell and Davis [35] for the proof)  

Therefore, the conditional probability density p(yn yn-Np yn-Np+i y -i. O follows an Ao-variate Gaussian distribution with mean y and covariance matrix 

Therefore, the reduced-order likelihood function and the conditional PDFs in the approximated expansion are available and the procedure can be summarized as follows  

Use Equation (4.30) with the likelihood function p(T 0,C) being computed through the approximation in Equation (4.34). 

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This conditional pdf can be seen as the likelihood function of the unknown value p(x) ... 

The probability of making a correct decision to clean is l-a(x), and has been mapped on Figure 3b Most of the time it appears that this probability is less than 50 percent except in the central zone next to the pollution source. Besides changing the threshold value 500, one way to improve this probability is to take more samples (increase the number of data N) which would decrease the variance and skewness of the conditional pdf fx(z (N)). 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

The first term on the right-hand side of this expression can be calculated from the conditional PDF /1, [Pg.86]

In terms of these PDFs, the conditional PDF of the composition variables given the mixture fraction is defined by... 

Mathematically, the conditional expected value (Z f) can be found by integration starting from the conditional PDF /zif(z x, t) ... 

More precisely, the Fourier coefficients in (4.27) can be replaced by random variables with the following properties k. U = 0 and (U ) = 0 for all k such that k > kc. An energy-conserving scheme would also require that the expected value of the residual kinetic energy be the same for all choices of the random variable. The LES velocity PDF is a conditional PDF that can be defined in die usual manner by starting from die joint PDF for the discrete Fourier coefficients U. ... 

Note that hv operates on the random field U(r, f) and (for fixed parameters V, x, and t) produces a real number. Thus, unlike the LES velocity PDF described above, the FDF is in fact a random variable (i.e., its value is different for each realization of the random field) defined on the ensemble of all realizations of the turbulent flow. In contrast, the LES velocity PDF is a true conditional PDF defined on the sub-ensemble of all realizations of the turbulent flow that have the same filtered velocity field. Hence, the filtering function enters into the definition of /u u(V U ) only through the specification of the members of the sub-ensemble. 

The marginal PDF of the age, /a(a), will depend on the fluid dynamics and flow geometry.104 On the other hand, the conditional PDF of

[Pg.214]

Note that even if /a (a ) were known for a particular reactor, the functional form of / a(V Iq9 will be highly dependent on the fluid dynamics and reactor geometry. Indeed, even if one were only interested in at the reactor outlet where fA ( ) = E(a), the conditional PDF will be difficult to model since it will be highly dependent on the entire flow structure inside the reactor. 

In this equation, Y should be replaced by die conditional expected value of Y given = . However, based on the structure of the flamelet, it can be assumed that the conditional PDF is a delta function centered at Y( , r). 

For simple chemistry, we have seen in Section 5.5 that limiting cases of general interest exist that can be described by a single reaction-progress variable, in addition to the mixture fraction.131 For these flows, the chemical source term can be closed by assuming a form for the joint PDF of the reaction-progress variable Y and the mixture fraction . In general, it is easiest to decompose the joint PDF into the product of the conditional PDF of Y and the mixture-fraction PDF 132... 

One could also do the same with the mixture-fraction-vector PDF. However, it is much more difficult to find acceptable forms for the conditional PDF. 

Note finally that, for any given value of the mixture fraction (i.e., f f), the multienvironment presumed PDF model discussed in this section will predict a unique value of 4>. In this sense, the multi-environment presumed PDF model provides a simple description of the conditional means (0 f) at Ve discrete values of f. An obvious extension of the method would thus be to develop a multi-environment conditional PDF to model the conditional joint composition PDF / (-i/d x, / ). We look at models based on this idea below. 

In a multi-environment conditional PDF model, it is assumed that the composition vector can be partitioned (as described in Section 5.3) into a reaction-progress vector y>rp and a mixture-fraction vector . The presumed conditional PDF for the reaction-progress vector then has the form 155... 

The multi-environment conditional PDF model thus offers a simple description of the effect of fluctuations about the conditional expected values on the chemical source term. 

For this reason, mixing in the conditional PDF model is orthogonal to mixing in the unconditional model. Thus, y in the conditional model need not be the same as in the unconditional model, where its value controls tiie mixture-fraction-variance decay rate. 

Despite these difficulties, the multi-environment conditional PDF model is still useful for describing simple non-isothermal reacting systems (such as the one-step reaction discussed in Section 5.5) that cannot be easily treated with the unconditional model. For the non-isothermal, one-step reaction, the reaction-progress variable Y in the (unreacted) feed stream is null, and the system is essentially non-reactive unless an ignition source is provided. Letting Foo(f) (see (5.179), p. 183) denote the fully reacted conditional progress variable, we can define a two-environment model based on the E-model 159... 

As compared with the other closures discussed in this chapter, computation studies based on the presumed conditional PDF are relatively rare in the literature. This is most likely because of the difficulties of deriving and solving conditional moment equations such as (5.399). Nevertheless, for chemical systems that can exhibit multiple reaction branches for the same value of the mixture fraction,162 these methods may offer an attractive alternative to more complex models (such as transported PDF methods). Further research to extend multi-environment conditional PDF models to inhomogeneous flows should thus be pursued. 

In Section 4.2, the LES composition PDF was introduced to describe the effect of residual composition fluctuations on the chemical source term. As noted there, the LES composition PDF is a conditional PDF for the composition vector given that the filtered velocity and filtered compositions are equal to U and 0, respectively. The LES composition PDF is denoted by U, 0 x, /), and a closure model is required to describe it. 

The key step here is to use the conditional PDF to eliminate the velocity dependence. However, this generates a new unclosed term. Note that we assume the mean velocity field to be solenoidal in the last line. 

Again, the key step here is to use the conditional PDF to eliminate the velocity dependence. 

This conditional PDF is governed by the corresponding Fokker-Planck equation (Gardiner 1990) ... 

Due to the simple codebook structure in SCS and DM, the sample-wise embedding and extraction procedure, and the IID original data, s can be considered a realization of an IID stochastic process s with the PDF ps (s). For performance evaluation of the considered watermarking schemes, the conditional PDFs ps (s d, k) for all d A V are required. Conditioning on the key k is necessary since otherwise the key hides any structure of the watermarked data. For simplicity, k = 0 is assumed for the presented... 

We introduce a simple model for the conditional PDFs pr (r k Ky) of the received pilot elements in order to motivate the afterwards described estimation of r0nS(H. The model is motivated by the observation that each PDF pr (r k Ky) shows local maxima with a distance of A,.. Let pr (r) denote the PDF of the received signal samples rn. It can be assumed that pr (r) reflects more or less the host signal PDF (pr (r) px (a )) if the embedding distortion and attack distortion is small relative to the host... 

Fig. 18 depicts an example for the given model. The local maxima of the conditional PDFs pr (r k G IQ,) with a relative distance of Ar = 10 are clearly visible. 

Fig. 18. Total and conditional PDFs of the received pilot sequence. Lbink = 3 different ranges for the key are distinguished. The example is for a Gaussian distribution of rn, and for the parameters Ar — 10 and r0ffset — 0.

The parameters /o and 0 of the model given in (30) have to be computed from the given conditional PDFs pr (r k IQ,) and the given unconditional PDF pr (r). Fourier analysis is appropriate for this task since /o and 0 are the frequency and a constant phase contribution of the cosine term in (30). 

In the latter case, the spectrum A(f) would have another peak at / = -/o which increases the required sampling interval for the numerical computation of the conditional PDFs. 

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