Model for the Data

The modeling of the acquired data is a fundamental step because it determines the choice of the reconstruction technique and the quality of the parametric map obtained. This phase is delicate because it depends on both methodologic factors (sequence used) and the properties of the tissue being studied. 

For a given sequence, Bloch equations give the relationship between the explanatory variables, x, and the true response, i]. The / -dimensional vector, 0, corresponds to the unknown parameters that have to be estimated x stands for the m-dimensional vector of experimental factors, i.e., the sequence parameters, that have an effect on the response. These factors may be scalar (m — 1), as previously described in the TVmapping protocol, or vector (m 1) e.g., the direction of diffusion gradients in a diffusion tensor experiment.2 The model ](x 0) is generally non-linear and depends on the considered sequence. Non-linearity is due to the dependence of at least one first derivative 5 (x 0)/50, on the value of at least one parameter, 6t. The model integrates intrinsic parameters of the tissue (e.g., relaxation times, apparent diffusion coefficient), and also experimental nuclear magnetic resonance (NMR) factors which are not sufficiently controlled and so are unknown. 

This is the case in 7, mapping using an inversion recovery sequence and collecting n images at different inversion delays (jc = 77). In this case, 

Where 8 and d2 are, respectively, the intrinsic 7j and an experimental factor accounting for a spatially dependent imperfect spin inversion. In some cases, generic parametric models (e.g., polynomial series) can be used to identify complicated responses, such as the free induction decay (FID) signal dephasing induced by the presence of magnetic field inhomogeneities.3 

The validation is an essential step for checking the validity of the parametric model in the case of practical data. It is usually carried out by performing measurements on phantoms in the range of x judged of interest by the experimenter.4 6 This approach is often tedious, especially when m 1, because the space of the experimental factors has to be scanned. In addition, the test objects often display characteristics that are remote from those of biologic tissues. For this reason, phantoms with more complex behavior have been proposed.7,8 

---

Setting up the probability model for the data and parameters of the system under smdy. This entails defining prior distributions for all relevant parameters and a likelihood function for the data given the parameters. 

Another interesting comparison is between the coefficients for the functions representing the best-fitting models for the data and the coefficients for the functions that result from performing the linearity test as described in the previous chapter [4], We have not looked at these before since they are not directly involved in the linearity test. Now, however, we consider them for their pedagogic interest. These coefficients, for the case of testing a quadratic nonlinearity of the data from Figure 67-1, are listed in Table 67-2. 

Distributions are derived from data to give a mathematical description or a model for the data. You will find much more detailed information on this topic in textbooks on statistics. 

In a more general setting the recipe [91] can be considered as an implementation of another suggestion by Gunnarsson and Lundqvist [99] and von Barth [100] known also at a pretty early stage of the development of the DFT technique of employing different functionals to describe different spin or symmetry states. In other words the simplified model for the data fit Eq.(lO) changes to ... 

Table 13.2 Predictions from the CoMFA Model for the Data from Kuiper et al. (1997 1998)... 

Box and Meyer also derived a useful result (which is applied in some of the subsequent methods in this chapter) that relates dispersion effects to location effects in regular 2k p designs. We present the result first for 2k designs and then explain how to extend it to fractional factorial designs. First, fit a fully saturated regression model, which includes all main effects and all possible interactions. Let /3, denote the estimated regression coefficient associated with contrast i in the saturated model. Based on the results, determine a location model for the data that is, decide which of the are needed to describe real location effects. We now compute the Box-Meyer statistic associated with contrast j from the coefficients 0, that are not in the location model. Let i o u denote the contrast obtained by elementwise multiplication of the columns of +1 s and—1 s for contrasts i and u. The n regression coefficients from the saturated model can be decomposed into n/2 pairs such that for each pair, the associated contrasts satisfy i o u = j that is, contrast i o u is identical to contrast j . Then Box and Meyer proved that equivalent expressions for the sums of squares SS(j+) and SS(j-) in their dispersion statistic are... 

The Bayesian approach to subset selection is outlined in Sections 2 to 4. Section 2 gives the mathematical ingredients of the analysis a probability model for the data, prior distributions for the parameters (J3, a, 5) of the model, and the resultant posterior distribution. 

Similarly as with the diffuse layer model, for the data sets presented in Figs. 5.58 and 5.59 better fit can be obtained taking K (reaction (5.23)) somewhat higher than the PZC, and for the data sets presented in Figs. 5.57 and 5.60 better fit can be obtained taking K (reaction (5.23)) somewhat lower than the PZC. With the data sets... 

Multiple imputations are generated by assuming a particular imputation model. Therefore, the success or failure of MI depends on the propriety of the assumed imputation model. Assumptions required in MI are (a) a model for the data values, (b) a prior distribution for parameters of the data model, and (c) the nonresponse mechanism. However, with nonparametric methods of MI, minimal distributional assumptions are required (see Section 9.6.6). 

One hundred bootstrap samples are generated and the appropriate structural model that best describes the data from each sample is determined. This is done to ensure that the model that best describes the bootstrap data is not different from the basic structural model used for developing the population PK model for the data before bootstrapping. With the right structural model POSTHOC individual Bayesian estimates are generated and the data subjected to GAM. 

The resulting GEMANOVA model for the data in Table 9.2 can be written as equation 9.6, sinee the effect of the 02-level is insignificant in the interval between 40-80% O2 (Bro and Jakobsen 2002). The interaction term Day Temp Light c02 describes deviations from the a -value on day 0 in a very simple way, and interpretation of the model parameters can be performed from Fig. 9.2. 

The first step of nearly every PK data analysis procedure is to calculate the natural logarithm of each of the measured plasma concentration values. The values of In(C ) are then plotted versus the time (t) of sample collection. If the plot shows a series of points falling near a straight line, then the data can be well represented by the one-compartment bolus IV model. Early or late data points that do not fall on the line created by other points can be indicative of plasma sample analysis problems, or could indicate that the one-compartment bolus rV model is not the best PK model for the data, as indicated in Figure 10.22. 

If the model chosen accurately reflects the true underlying model for the data, then the chosen model will represent any sample function of the source and not just those in the training set. The sample densities of the training data only represents samples that are in the training set. 

The regression analysis fi amework, shown in Fig. 3.1, is an iterative procedure that seeks to determine the best model for the data. Before the procedure can be started, three things must be determined ... 

Consider the previously examined Edmonton temperature data series detailed in Sect. A5.1. For the purposes of this example, consider the problem of estimating a model for the mean summer temperature. The autocorrelation and the partial autocorrelation plots have already been shown and analysed previously (see Figs. 5.2 and 5.3). Using the results from there, obtain an initial model for the data. 

A wavelet model for the data consisted of the following regressors i, y/ 2> y, 3, and yt 4, plus for each of the inputs the terms between rtk+X and 11 + 5. The total time delay was assumed to be one for the process between the i and hi and two between the and hi. Note that the time delay used here must include the one-sample time delay introduced by sampling a system, that is, the total time delay equals The estimated model parameters will not be included here as they are... 

Multiple regression analysis. This is suitable for data modeling and expresses data as a simple equation. The process begins with experimentation to produce a vector of measured data known as the dependent variables. Then a limited number of factors are considered to be significant for the determination of data values, and these independent variables are used to prepare a model for the data. Finally, coefficients, as shown below, are calculated by least-squares analysis to represent the significance or weighting of the independent variables. The result is a calculation of regression coefficients to prepare a mathematical model that is suitable for preditions,... 

---