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Dimensional regression relation
The multinomial logit model is defined only with the complete parametrization ηj = αj + xtδj . For a given explanatory vector x of dimension p in the most general form, this is the maximal possible parametrization with (J − 1)(1 + p) parameters. We could define the multinomial logit model with the proportional parametrization ηj = αj + xtδ. Taking into account x, this is the minimal possible parametrization, with J − 1 + p parameters. With this in mind, Anderson proposed a large set of parametrizations to cover the range between minimal and maximal parametrizations. Thus he used model (1.14) with a particular parametrization for the J − 1 slopes δj . First, Anderson proposed to relax slightly the proportional assumption (δ1 = . . . = δJ−1 = δ) using the parametrization δj = φj δ, (1.15). for j = 1, . . . , J − 1, where φj are scalars. To avoid identifiability problems, the number of parameters must be less than the number of parameters of the maximal model (J − 1)2 + p ≤ (J − 1)(1 + p).
This is the case for J > 2 and p > 1. Otherwise, if J = 2 or p = 1, one scalar must be fixed (φ1 = 1 such as in Anderson (1984)). He referred to model (1.14) with parametrization (1.15) as the one-dimensional stereotype model. He then proposed the two-dimensional stereotype model using parametrization δj = φj δ + ψj γ, for j = 1, . . . , J −1. Finally, he defined the d-dimensional stereotype model using parametriza-tion d δj = φk,j γk, k=1 for j = 1, . . . , J − 1, where φk,j ∈ R and γk ∈ Rp. The number of different slopes γk is such that d ≤ J − 1 for identifiability issues. Therefore, the maximal number of identifiable scale parameters φk,j is J − 1 − d. Anderson proposed the maximal dimension d = min(J − 1, p). Finally, using the flexibility of slope parametrization in two ways (scale parameters φk,j and slope parameters γk), Anderson proposed a multinomial logit model with any number of parameters between the minimal and the maximal number of parameters (i.e. between (J − 1 + p) and (J − 1)(1 + p)).
Ordered regression relation
Anderson proposed to order the parameters (φj )j=1,…,J of the one-dimensional stereotype model to obtain an ordered regression relation 1 =φ1>φ2>…>φJ =0.
It should be noted that Anderson did not use this ordering as an a priori constraint. If theˆ estimated parameters φj are ordered, then the order is considered as a consequence of the estimation. As noticed by Tutz (2012), the stereotype model cannot be considered to be an ordinal model, even though it is used for ordinal response variables in applications.
Indistinguishability procedure
Anderson (1984) proposed a testing procedure – useful for ordinal data – to identify successive categories that can be clearly distinguished by the explanatory variables x. These categories are said to be indistinguishable with respect to x when the explanatory variables x do not have significantly different effects on them. He proposed to aggregate the corresponding successive slope parameters δj and use a deviance test. More precisely, he proposed an iterative proce-dure to find the best splitting point among categories 1, . . . , J with respect to x. The minimal number of splitting point is zero, corresponding to the simple model without explanatory vari-ables (null hypothesis H0), and the maximal number of splitting points is J − 1, corresponding to the classical multinomial logit model with J − 1 different slopes.
The first step is to find the best split into two groups of categories. The hypothesis H(2;r) is then introduced H(2;r) : δ1 = . . . = δr; δr+1 = . . . = δJ = 0, for r = 1, . . . , J − 1. Comparing the corresponding log-likelihood values l(2;r) yields the best splitting point r∗ such that l2 = l(2;r∗) = maxr l(2;r). The hypothesis H(2;r∗) is then tested against H0, using the deviance statistic 2(l2 − l0) which follows a χ2p distribution under H0. Finally, if the splitting point r∗ is accepted, the procedure must be restarted in parallel for the two groups {1, . . . , r∗} and {r∗ + 1, . . . , J} in order to obtain the best partition into three groups.
This is a dichotomous partitioning procedure with at most J(J −1)/2 different parametriza-tions to test. It should be noted that this procedure is simplified for the one-dimensional stereotype model since the equality between slopes δ1 = . . . = δr becomes equality between scalar parameters φ1 = . . . = φr. In practice, only this particular case of the procedure is used.
Partitioned conditional generalized linear models
The main idea is to recursively partition the J categories then specify a conditional GLM at each step. This is why this type of models is called partitioned conditional GLMs. Such models have already been proposed, e.g. the nested logit model (McFadden et al., 1978), the two-step model (Tutz, 1989) and the partitioned conditional model for partially-ordered sets (POS-PCM) (Zhang and Ip, 2012). In a first level, the entire set of categories is partitioned into L groups (or “nests”) as follows L {1, . . . , J} = Nl, l=1 and in a second level, the groups are partitioned into categories. Thus, a basic event is decomposed as follows {Y = j} ⇔ {Y ∈ Nl} ∩ {Y = j|Y ∈ Nl}.
Partitioning step The groups Nl are considered as categories and a categorical regression model is used to describe the probabilities P (Y ∈ Nl|x) for l = 1, . . . , L.
Conditioning step For each group Nl, the response variable Y is conditioned on Nl and a GLM is used to describe the conditional probabilities P (Y = j|Y ∈ Nl; x) for j ∈ Nl.
More generally, this partitioning/conditioning process may be reiterated for Nl. However, these models have been formally defined and used with only two or three levels. The first aim of these models is to aggregate the categories that are influenced by the same explanatory variable.This is very similar to the principle of distinguishability defined by Anderson. The nested logit model were introduced in this way. The categories are aggregated according to the explanatory variable. On the other hand, the two-step model takes account of the order assumption among the categories, aggregating only successive categories. A more recent aim was to use the partial ordering among the categories to define an adapted partitioned condi-tional model. For any partially-ordered structure, Zhang and Ip (2012) proposed a particular recursive partition which “conserves” the order relation.
Partitioned conditional model for partially-ordered set
In categorical data analysis, the case of nominal and ordinal data has been investigated in depth while the case of partially-ordered data has been comparatively neglected. Zhang and Ip (2012) introduced the partitioned conditional model for partially-ordered set (POS-PCM). The main idea was to recursively partition the J categories in order to lead the partial order back to degenerate cases: total order and no order. Thus, the odds proportional logit model was used for the total order case and the multinomial logit model was used for the no order case.
Zhang and Ip introduced the partially-ordered set theory into the GLM framework. A partial ordered set (poset) (P, ) is summarized by a Hasse diagram. The order relation j k is represented by an edge between the two nodes (categories) and node k is above node j. A chain in a poset (P, ) is a totally ordered subset C of P , whereas an antichain is a set A of pairwise incomparable elements. Zhang and Ip described an algorithm of category partitioning which can be used to show the following proposition.
Proposition 1. (Zhang and Ip, 2012) A finite poset (with one component) can always be partitioned into antichains that are totally weakly ordered.
Note that the weakly order must be introduced because, in this approach, sets of categories are compared. Two subsets N1 and N2 in P are weakly ordered if at least one element in N2 is dominated by elements in N1 and no element in N2 dominates any element in N1. One says that N1 weakly dominates N2.
Specification of generalized linear models for categorical data
Consider the situation of regression analysis, with the multivariate response variable Y and the vector of Q explanatory variables X = (X1, . . . , XQ) in a general form (i.e. categorical variables being represented by indicator vectors). The dimension of X is thus denoted by p with p ≥ Q. We are interested in the conditional distribution of Y |X, observing the values (yi, xi)i=1,…,n taken by the pair (Y, X). All the response variables Yi are assumed to be con-ditionally independent of each other, given {Xi = xi}. The variables Yi follow the conditional truncated multinomial distribution Yi|Xi = xi ∼ T M(π(xi)),
with at least J = 2. In the following we will misuse some notations for convenience. For example, we will often forget the conditioning on X and the individual subscript i. Moreover, the response variable will sometimes be considered as a univariate categorical variable Y ∈ {1, . . . , J } in order to use the univariate notation {Y = j} instead of the multivariate notation {Y1 = 0, . . . , Yj−1 = 0, Yj = 1, Yj+1 = 0, . . . , YJ−1 = 0}.
Table of contents :
1 Background and literature review
1.1 Fracture mechanics issues
1.2 Bridge between fracture mechanics and fatigue crack propagation .
1.2.1 Fatigue crack propagation behavior
1.2.2 Eects of crack closure and R-ratio on fatigue crack propagation
1.2.3 Plastic work eect ahead of a long propagating fatigue crack .
1.2.4 Heat production mechanisms during fatigue crack propagation
1.3 Conclusion of Chapter I
2 Characterization and assessment of the heat sources and their re- sulting temperature elds
2.1 Theoretical concepts behind the heat sources
2.1.1 Fundamental concepts of thermodynamics
2.1.2 Application of the thermodynamic approach in the presence of a long propagating fatigue crack
2.2 Methodology for quantifying the heat sources
2.3 Material and geometry of the specimens
2.4 Quantication of the heat sources and computation of the associated temperature elds
2.4.1 The thermoelastic source
2.4.2 The intrinsic dissipation due to microplasticity
2.4.3 The cyclic plasticity dissipated into heat in the RCPZ
2.5 Conclusion of Chapter II
3 Thermomechanical analysis – Eects of the heat sources on the SIF 71
3.1 Techniques to solve the SIF solution in crack problems involving thermal stresses
3.2 Methodology and assumptions for computing the thermal corrections of the SIF
3.3 Thermomechanical problem
3.4 Computing the thermal corrections of the SIF
3.4.1 Consequence of the thermoelastic source on the SIF
3.4.2 Consequence on the SIF of the intrinsic dissipation due to microplasticity
3.4.3 The eect on the SIF of cyclic plasticity dissipated into heat in the RCPZ
3.4.4 Comparison of the three thermal eects on the SIF through the applied fatigue crack growth tests
3.5 Consequences of the heat sources on the fatigue crack parameters
3.6 Consequence of the cyclic plasticity dissipated into heat in the RCPZ, on the fatigue crack growth rate
3.7 Conclusion of Chapter III
4 Exploring the consequences of the loading frequency and the ma- terial behavior on the cyclic plasticity dissipated into heat in the RCPZ
4.1 Eect of the loading frequency on the cyclic plasticity dissipated into heat in the RCPZ
4.2 Another approach to estimate the cyclic plasticity dissipated into heat in the RCPZ
4.2.1 Brief review of the cyclic behavior of materials
4.2.2 Cyclic tension/compression tests to determine the non-linear kinematic hardening coecients of C40 steel
4.2.3 Computing the cyclic plastic dissipation in the RCPZ
4.3 Conclusion of Chapter IV
Conclusions and Prospects
References
