Parsimonious models

These, such as the black box that was the receptor at the turn of the century, usually are simple input/output functions with no mechanistic description (i.e., the drug interacts with the receptor and a response ensues). Another type, termed the Parsimonious model, is also simple but has a greater number of estimatable parameters. These do not completely characterize the experimental situation completely but do offer insights into mechanism. Models can be more complex as well. For example, complex models with a large number of estimatable parameters can be used to simulate behavior under a variety of conditions (simulation models). Similarly, complex models for which the number of independently verifiable parameters is low (termed heuristic models) can still be used to describe complex behaviors not apparent by simple inspection of the system. 

Because variables in models are often highly correlated, when experimental data are collected, the xrx matrix in Equation 2.9 can be badly conditioned (see Appendix A), and thus the estimates of the values of the coefficients in a model can have considerable associated uncertainty. The method of factorial experimental design forces the data to be orthogonal and avoids this problem. This method allows you to determine the relative importance of each input variable and thus to develop a parsimonious model, one that includes only the most important variables and effects. Factorial experiments also represent efficient experimentation. You systematically plan and conduct experiments in which all of the variables are changed simultaneously rather than one at a time, thus reducing the number of experiments needed. 

A one standard error rule is described in Hastie et al. (Hastie et al. 2001). It is assumed that several values for the measure of the prediction error at each considered model complexity are available (this can be achieved, e.g., by CV or by bootstrap, Sections 4.2.5 and 4.2.6). Mean and standard error (standard deviation of the means, s) for each model complexity are computed, and the most parsimonious model whose mean prediction error is no more than one standard error above the minimum mean prediction error is chosen. Figure 4.4 (right) illustrates this procedure. The points are the mean prediction errors and the arrows indicate mean plus/minus one standard error. 

The most parsimonious model for upregulation that acconunodates current information suggests that nicotine binding to inunature subnnits enhances receptor assembly by provoking quaternary structure rearrangements (that might resnlt in altered subunit stoichiometry), leading to accelerated maturation of nAChRs and a net... 

PLS is very efficient in using only the information in X that is related to the analyte. Hence it will lead to more parsimonious models than PCR, that is, in general, PLS will require fewer factors than PCR to obtain a model with a similar predictive performance. 

The simultaneous use of information from X and Y makes PLS more complex than PCR. However, it can allow PLS to develop better regression vectors, i.e., more harmonious with respect to the bias/variance trade-off. Some authors also report that PLS can sometime provide acceptable solutions for low-precision data where PCR cannot. Other authors have reported that PLS has a greater tendency to overfit noisy Y data compared to PCR. It is often reported in the literature that PLS is preferred because it uses fewer factors than PCR and, hence, forms a more parsimonious model. This is not the case, and the literature [38, 39,43,45, 53] should be consulted. 

Simplest Base Model Is Backbone. The simplest base model that characterizes the underlying patterns in the data should form the backbone for developing a population model. The principle of simplicity stipulates that models with the minimum number of parameters should be used. This is called the parsimony principle. The model with the smallest number of parameters that describe the data well is the most parsimonious model. 

There are many suggested choices for a it, some of them providing more parsimonious models and other more complex models. The most known FPE criteria are shown in Table Rl. 

In multi-scale studies, the larger-scale study can be based on a simpler (parsimonious) model, while in the smaller-scale study a more detailed model is required. 

How can parsimonious models be constructed There are several possible approaches, however in this chapter a combination of data compression and variable selection will be used. Data compression achieves parsimony through the reduction of the redundancy in the data representation. However, compression without involving information about the dependent variables will not be optimal. It is therefore suggested that variable selection should be performed on the compressed variables and not on the original variables which is the usual strategy. Variable selection has been applied with success in fields such as analytical chemistry [1-4], quantitative structure-activity relationships (QSAR) [5-8] and analytical biotechnology [9-11]. 

In this chapter, compression is achieved by assuming that the data profiles can be approximated by a linear combination of smooth basis functions. The bases used originate from the fast wavelet transform. The idea that data sets are really functions rather than discrete vectors is the main focus of functional data analysis [12-15] which forms the foundation for the generation of parsimonious models. 

There is a whole family of different wavelet methods available depending on the signal properties and the type of information that is to be extracted. However, this chapter will only focus on the fast wavelet transform (FWT) which is based on Mallat s algorithm [39.40]. It should be mentioned that the described methods to achieve parsimonious models are not dependent on one particular type of wavelet transform. Other types of wavelet transforms can be used. FWT is not always optimal for all types of problems and other techniques such as wavelet packets [41], continuous transforms [42,43] and biorthogonal transforms [37] should be considered. Some of the properties of the FWT that makes it an attractive transform are ... 

In this section, several different strategies for creating parsimonious models are described. 

The interesting regions are zoomed to avoid large PRESS values to dominate the plot. Note the rank ridge at the left part of each plot. By inspecting the SEC surfaces, possible candidates for parsimonious models can be made, see Table 1. 

Models with high prediction errors all have one thing in common They are not selecting scales 5, 6 or 7. What is the most parsimonious model from this analysis Sorting the scale combinations with respect to the total number of wavelet coefficients, it was found that the smallest model is using only scale 5 (2 = 32 variables, A = 5 PLS factors, RMSEP = 5.6%). 

So what does one need to do First, the three methods obviously may not provide the researcher with the same resultant model. To pick the best model requires experience in the field of study. In this case, using the backward elimination method, in which all x, variables begin in the model and those less significant than F-to-Leave are removed, the media concentration was rejected. In some respects, the model was attractive in that and s were more favorable. Yet, a smaller, more parsimonious model usually is more useful across studies than a larger, more complex one. The fact that the interaction term was left in the model when X2 was rejected makes the interaction of Xj x X2 a moot point. Hence, the model to select seems to be the one detected by both stepwise and forward selection, y = ho + 2 2, as presented in Table 10.5. 

For model selection, the prediction rule with the smallest CV error averaged over the V evaluation sets is labeled as the best model. Nonetheless, in practice a 1-SErule is used, which suggests selection of the most parsimonious model whose error is no more than one standard error above that of the best model s. 

C.R. Nelson and A.F. Siegel, Parsimonious Modelling of Yield Curves, Journal of Business 60 (1987). 

One consideration for all statistical adjustment techniques in drug safety evaluation studies is the danger of the statistical adjustment itself introducing bias (Greenland and Morgenstern 2001). Statistical control can sometimes either increase bias or decrease precision without affecting bias and can thereby produce less reliable effect estimates (Schisterman et al. 2009). For example, bias can also be induced if an analysis improperly stratifies on a collider variable (Cole et al. 2010), that is, a variable that is itself directly influenced by two other variables. As a result, care is necessary in any evaluation study to develop a parsimonious model that achieves an appropriate balance between bias and variance. 

The principle of parsimony (de Noord, 1994 Flury and Riedwyl, 1988 Seasholtz and Kowalski, 1993) states that if a simple model (that is, one with relatively few parameters or variables) fits the data then it should be preferred to a model that involves redundant parameters. A parsimonious model is likely to be better at prediction of new data and to be more robust against the effects of noise (de Noord, 1994). Despite this, the use of variable selection is still rare in chromatography and spectroscopy (Brereton and Elbergali, 1994). Note that the terms variable selection and variable reduction are used by different researchers to mean essentially the same thing. 

Standard parsimony model (all characters treated as unordered). The fossil was linked iteratively to all possible branches of the backbone tree and the number of required parsimony changes (steps) was calculated for each resulting tree. 

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