Fahrmeir, L. and Tutz, G. (1994) Multivariate Statistical Modelling Based on Generalized Linear Models. Springer Verlag, Berlin.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of this issue can be found in Lin et al. 3] 3.4 Generalized Linear Model (GLM) A GLM for binary responses is built up from a linear model for the input data, and the model output f (x) w x is in turn input to the link function. For Bernoulli distributions, the natural link function [1] is the logistic transfer function s( f (x) 1 e ) 1 . The overall output of the GLM s( f (x) computes p(x) the probability of the input x belonging to class 1. Training of the linear model was done by iteratively re weighted least squares (IRLS) 4 Training and Evaluation One of ....

Fahrmeir, L. and Tutz, G. Multivariate Statistical Modelling Based on Generalized Linear Models. Springer Verlag, 2nd edn., 2001.

.... ( j The covariance matrix is dynamically scaled until a reasonable acceptance rate in the MH algorithm is observed. Thus, to determine the multivariate Normal proposal PDF, we need to nd the posterior mode, or alternatively minimize ( To this end, we employ the Newton Raphson algorithm [27], and make use of automatic di erentiation [28] to calculate the rst and second order partial derivatives of ( This can be done to the same degree of accuracy as the function evaluation itself. We use automatic di erentiation implemented in a C class library which combines an array language ....

L. Fahrmeir and G. Tutz, Multivariate Statistical Modelling Based on Generalized Linear Models. (Springer, New York, 1994).

....(z) zj0; 1) Z z 1 N (xj0; 1) dx; 2) where N (vjm; C) denotes a Gaussian density with mean m and (co)variance C. The re scaled probit (3 z 2 j0; 1) is plotted in Fig. 1, together with the logistic function, showing that (apart from a scale factor) they are almost indistinguishable [11]. Of course, both the logistic and probit functions can be re scaled (horizontally) but this scale is implicitly absorbed by . 5 0 5 0 0.2 0.4 0.6 0.8 1 z logistic(z) probit 3 z 2 ( Figure 1. The logistic and (re scaled) probit link functions. To extend the probit (or the ....

L. Fahrmeir and G. Tutz. Multivariate Statistical Modelling Based on Generalized Linear Models. Springer Verlag, New York, 1994.

....sparse table, concerning a credit scoring data set, where we compare, using a hierarchical prior, the extended MCS and the RJMCMC algorithm. For the latter, we also derive the posterior distribution of the odds ratios of interest, and compare the results with non bayesian analysis of this data set (Fahrmeir and Hamerle, 1994). ....

Fahrmeir and Hamerle (1994) Multivariate statistical modelling based on generalized linear models. Springer, New York.

....continuous) were treated as factors as the purpose of the analysis was to find the best of the selected factors levels. Analysis of deviance was used to judge the significance of terms, with insignificant terms being dropped from the models. Information on generalised linear models can be found in [64, 47, 16]. The mean time to solution on the real and generated data sets was compared with the predicted values using two sided z tests (or t tests when there were less than 30 correct runs) Each test used the null hypothesis that the actual mean time was equal to the predicted time. 2.3 Results 2.3.1 ....

Ludwig Fahrmeir and Gerhard Tutz. Multivariate Statistical Modelling Based on Generalized Linear Models. Springer-Verlag, 1994.

....state space model using a Schaefer surplus production model as a basic example. This approach extends to other assessment methodologies, including delaydifference and age structured models. Introduction State space models are among the most powerful tools for dynamic modeling and forecasting (Fahrmeir and Tutz 1994). They have started to enjoy an increasing popularity in fisheries stock assessment (Sullivan 1992, Pella 1993, Gudmundsson 1994, Schnute 1994, Freeman and Kirkwood 1995, Reed and Simons 1996, Kinas 1996, Meyer and Millar , Millar and Meyer 1998a) as they can realistically account for both ....

Fahrmeir, L., and Tutz, G. 1994. Multivariate Statistical Modelling Based on Generalized Linear Models. Springer, New York.

....that the derivation of this estimator is based on yet another adhoc cost function instead of a sound statistical paradigm. Furthermore, we point out major aws in its derivation. We suggest a Bayesian approach instead, by integrating the problem into the framework of nonlinear state space modeling [17,18]. This alleviates both problems i) and ii) by incorporating the known serial correlation as prior information in a complete probability model for the observations and the unknown states. We even consider the more realistic generalization where the underlying dynamic evolution is not assumed ....

....a proper statistical paradigm requires treating the system states as stochastic instead of as deterministic. We therefore consider the more realistic case that the system dynamics are subject to random disturbances. This casts the problem into the general framework of a Bayesian state space model [17,18], one of the most powerful tools for dynamic modeling and forecasting. State space models relate time series observations to unobserved states by a stochastic observation model. The states are assumed to follow a stochastic transition over time, given by the state equations. The state equations, ....

L. Fahrmeir and G. Tutz, Multivariate Statistical Modelling Based on Generalized Linear Models. (Springer, New York, 1994).

....di erent areas of currently active research: econonometrics, sheries, and physics. 1 Introduction The state space approach is one of the most powerful tools for dynamic modeling and forecasting of time series and longitudinal data. Excellent overviews are given in West and Harrison (1997) and Fahrmeir and Tutz (1994). A state space model consists of observation and state equations. The observation equations specify the conditional distributions of the observations y t at time t as a function of unknown states t . But unlike a static model, the state of nature, t , changes over time according to a ....

FAHRMEIR, L. and TUTZ, G. (1994): Multivariate Statistical Modelling Based on Generalized Linear Models. Springer, New York.

....6 sin 12 t t u it ; i = 1; 2; t = 1; 366; and in contrast to Kitagawa (1987) the state variables f t g here follow the stationary AR(1) process with a common and bounded variance 2 . Note again that we take the probit link instead of the logistic link used in Kitagawa (1987) and Fahrmeir and Tutz (1994). Our experience with the prior choice in the simulation study led us to favor informative prior for 2 . So we used the uniform prior on the truncation interval [ 05; 1] A total of 10,000 iterations of this MCMC algorithm adopted to the binomial model were run with every 10th iteration ....

Fahrmeir, L. and Tutz, G. (1994) Multivariate Statistical Modelling Based on Generalized Linear Models. Springer Verlag, Berlin.

.... since cloglog( yjx) log( Gamma log(1 Gamma (yjx) x 0 fi 55 Likelihood function P (Y = y) y Gamma1 Y i=1 (1 Gamma (ijx) yjx) P (Y y) y Y i=1 (1 Gamma (ijx) ffl Easy estimation etc with software for Generalized Linear Models ffl Markov structure in time (Fahrmeir Tutz, 1994) 56 Choice of Covariates Two aspects: 1. Time dependent covariates 2. Spatial dependence, covariates depend on neighbour trees One tree ffl Diameter dbh ffl Basal area ffl Position x and y Several trees ffl Competition index Take spatial structure into account (Rathbun Cressie, 1994) 57 ....

Fahrmeir, L. & Tutz, G. (1994), Multivariate Statistical Modelling Based on Generalized Linear Models, Springer Series in Statistics, Springer-Verlag.

....waldalpha0=1,0, 1,2,0 [waldnamesalpha] namealpha0=equi distant 2.3 Specifying start values You can specify start values for the parameters. The syntax is given in section 4. 2.4 Sequential logit link For binary data the sequential logit link is now implemented, too. For details see e.g. Fahrmeir and Tutz (1994). 3 New options for the GEE module For the Independence Estimating Equation (IEE) and the method of Prentice in the GEE module several methods for estimating the covariance matrix of the parameters are now available. For the IEE method you can choose between the usual implementation for the ....

Fahrmeir, L. and Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models, Springer, New York.

.... the local estimating equation 0 = n X i n i X r ir;0 Z T ir hfZ ir fi(t 0 )g j T Var(e y ir ) Gamma1 h e y ir Gamma ir f fi(t 0 )g i : 5) The solution of (5) does not necessarily provide a valid estimate since it is not guaranteed that fi 0k (t 0 ) fi 0k 1 (t 0 ) Fahrmeir Tutz (1994) suggest the simple reparameterization 01 : fi 01 ; 0k : log(fi 0k Gamma fi 0k Gamma1 ) for k = 2; q to overcome this point. For simplicity of notation however we neglect this reparameterization in the sequel. For notational simplicity we abbreviate the component Z T ir hfZ ir ....

....integrated mean squared error of the estimates and hence automatically takes the correlation among the observations into account. The cumulative model (4) allows for a further interpretation. The ordinal response y ir can be seen as coarser version of a latent score variable u ir , say, see e.g. Fahrmeir Tutz (1994). By setting u ir = Gammax ir fi x (t ir ) ir with ir latent and distributed according to distribution F ( Delta) in (3) we can interpret the main effects fi 0k (t ir ) as thresholds. This means we get y ir = k if fi 0k Gamma1 (t ir ) u ir fi 0k (t ir ) for k = 1; q and fi 00 (t ....

Fahrmeir, L. and Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. New York: Springer Verlag.

....two approaches. The linear predictor is extended to j ti = z T ti fi t w T ti b i ; so both time dependent as well as unit specific parameters are allowed. To include multivariate models such as cumulative or sequential models for multicategorical responses (a recent survey is given in Fahrmeir Tutz, 1994), a more general form j ti = Z ti fi t W ti b i is considered. Here j ti is a vector of dimension q. Since dynamic and mixed models are combined this framework is called a dynamic generalized linear mixed model (DGLMM) Note that dynamic models (D = 0) as well as mixed models (fi t j fi) are ....

....and pointwise one posterior standard deviation confidence bands (dotted lines) Figure 2 (bottom) True b i s ( Theta) and posterior mean estimates Sigma one posterior standard deviation (fi) The units are ordered respective to the mean estimates. 4. 2 Business test data Fahrmeir (1992a) Fahrmeir Nase (1994) and Knorr Held (1995) analyzed data from the IFO business test applying a dynamic cumulative model. This monthly data is based on a questionnaire, answered by n = 55 firms of a specific industrial branch for the years 1980 to 1990. The response variable short range production plans is given in ....

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Fahrmeir, L. & Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. New York: Springer--Verlag.

....; T (2:4) with transition matrix F t 2 IR p;p , initial state ff 0 N (a 0 ; Q 0 ) We summarize the hyperparameters a 0 ; Q 0 ; Q t in the vector . Let be fixed and known for the moment. The exponential family state space model (2.1) 2.2) 2. 4) covers many well known time series models, cf. Fahrmeir and Tutz (1994) chapter 8. In this framework we want to estimate the unobservable states ff t via penalized likelihood estimation which could be motivated by posterior mode smoothing outlined in Fahrmeir and Wagenpfeil (1994) With ff = ff 0 0 ; ff 0 1 ; ff 0 T ) 0 , the penalized likelihood ....

....space model f( p(y ) The aim is to maximize the approximative likelihood f( in (3.4) with respect to . Therefore we give explicit formulae for [detfV ( g] 1=2 and pfa( y g. Repeated application of Bayes theorem, using (2.1) 2.2) 2. 4) and further independence assumptions, cf. Fahrmeir and Tutz (1994) chapter 8, yields ln p fa( y g = ln n (2) Gammam=2 o ln(det Q 0 ) Gamma1=2 T X t=1 ln(det Q t ) Gamma1=2 PLfa( g (3.5) with the penalized log likelihood PL( Delta) from (2.6) and the densities from (2.1) 2.4) PLfa( g = T X t=1 l t fa t ( g Gamma 1 2 fa 0 ( ....

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Fahrmeir, L. and Tutz, G. (1994): Multivariate Statistical Modelling Based On Generalized Linear Models, Springer-Verlag, New York.

....responses (McCullagh, 1980) and multivariate extensions. Such models exploit, in a parsimonious way, the ordered scale of the outcomes. An important example for a cumulative model is the well known proportional odds model. This and other ordinal response models have been discussed in detail by Fahrmeir and Tutz (1994, ch. 3) First, we give a short review of models for correlated ordinal outcomes; these models are related to the approach used in this paper. The methods are similar in that they all use odds ratios to describe the association between responses. Dale (1986) proposed a model for bivariate ....

....unobserved. What we observe, Y , is a categorization of U into q 1 intervals. By this mechanism the density of U is divided into slices determined by the thresholds fi 1 ; fi q . The explanatory part, 4 fi q 1 z q 1 : fi m z m , shifts the location of the underlying response U (Fahrmeir and Tutz, 1994, ch.3) To further explain the model, we have to give some definitions. Letting z i = z i1 ; z im ) 0 and fi = fi 1 ; fi m ) 0 , we can define a predictor j (r) i = z 0 i fi. As usual, the response Y i is represented as a vector y i = y (1) i ; y (r) i ; ....

FAHRMEIR, L., TUTZ, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. New York: Springer-Verlag.

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Fahrmeir, L. & Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. New York: Springer--Verlag.

....the whole brain. As well the lag and the shape of the activation should be allowed to differ between the voxels and therefore, pixelwise parameters d i and i for the stimulus transformations are introduced. Thus, the final model can be formulated as follows: y it = a it z it b it ffl it ; [6] with the error distributions and priors of the parameters a and b as defined in Eq[5] Estimation This section will give a brief description of the method of estimation in the presented model. Because most algorithms used for the above model have already been described very well and detailed in ....

....on the data. The residuals are plotted in the last column. The second row displays an approach with the nonparametrically modelled baseline (Eq. 3] but still a fixed effect for the pixelwise estimate of the reference function. In the last row the effects of the fully dynamic model (Eq. [6]) are plotted. Here, for illustration, the estimated dynamic effect of the stimulus was multiplied with the reference function. As it is indicated through the residual plots, in both models with a single fixed parameter for the influence of the stimulus, an adequate fit to the data is not given. ....

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L. Fahrmeir, G. Tutz. "Multivariate Statistical Modelling Based on Generalized Linear Models". Springer Verlag New York, New York, 1994.

....e ects and interactions which leads to varying coecient models as introduced by Hastie Tibshirani (1993) This allows to model smooth interaction between factorial and continuous regressors. Background material to cumulative and sequential models is found for instance in Agresti (1990) Fahrmeir Tutz (1994), Greenland (1994) or Barnhart Sampson (1994) Simono (1996) discusses the smoothing of sparse ordinal data, which applies if the number of categories are large and correspondingly the cell frequencies are small, a topic not focussed in this paper (see also Hall Titterington, 1987) ....

....(19) is equal to the likelihood of a logit model for the random sample y ir ; x i ; z i for i = 1; n, r = 1; q and d ir = 1. The corresponding model for y ir equals P (y ir = 1jx i ; z i ; d ir = 1) Ff 0r z 1i z 0 (x i ) z 2i z (x i )g and restriction 01 0 (compare Fahrmeir Tutz, 1994, ch 9) Estimation is carried out by local and pro le likelihood as in the previous section. Let Z 1;i and Z 2;i be de ned as in the previous section and let Z 1;ir and Z 2;ir denote the r th row of Z 1;i and Z 2;i , 13 respectively. Considering as xed, the local likelihood for estimating ....

Fahrmeir, L. and Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. New York: Springer Verlag.

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Fahrmeir, L. and Tutz, G. (1994) Multivariate Statistical Modelling Based on Generalized Linear Models. Springer Verlag, Berlin.

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Fahrmeir, L. and Tutz, G. (1994) Multivariate Statistical Modelling Based on Generalized Linear Models. Springer Verlag, Berlin.

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L. Fahrmeir and G. Tutz. Multivariate Statistical Modelling Based on Generalized Linear Models. New York, Springer-Verlag, 2nd edition, 2001.

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Fahrmeir, L., and Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. Springer Verlag, New York.

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Fahrmeir, L. and Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. Springer.

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Fahrmeir, L. and Tutz, G. (1994). Multivariate Statistical Modelling Based in Generalized Linear Models. Springer.

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Fahrmeir, L. and Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. Springer.

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