[Depmix-commits] r420 - papers/jss
noreply at r-forge.r-project.org
noreply at r-forge.r-project.org
Wed Jun 30 17:06:55 CEST 2010
Author: ingmarvisser
Date: 2010-06-30 17:06:55 +0200 (Wed, 30 Jun 2010)
New Revision: 420
Modified:
papers/jss/dpx4.sh
papers/jss/dpx4Rev.Rnw
papers/jss/dpx4Rev.bib
Log:
Minor updates to the jss paper.
Modified: papers/jss/dpx4.sh
===================================================================
--- papers/jss/dpx4.sh 2010-06-30 13:30:54 UTC (rev 419)
+++ papers/jss/dpx4.sh 2010-06-30 15:06:55 UTC (rev 420)
@@ -4,6 +4,8 @@
R --vanilla < dpx4Sweave.R ;
+R --vanilla < dpx4SweaveNoEval.R ;
+
R --vanilla < dpx4Stangle.R ;
Modified: papers/jss/dpx4Rev.Rnw
===================================================================
--- papers/jss/dpx4Rev.Rnw 2010-06-30 13:30:54 UTC (rev 419)
+++ papers/jss/dpx4Rev.Rnw 2010-06-30 15:06:55 UTC (rev 420)
@@ -17,12 +17,12 @@
\author{Ingmar Visser\\University of Amsterdam \And
Maarten Speekenbrink\\University College London}
-\title{\pkg{depmixS4} : An \proglang{R}-package for hidden Markov models}
+\title{\pkg{depmixS4} : An \proglang{R}-package for Hidden Markov Models}
%% for pretty printing and a nice hypersummary also set:
\Plainauthor{Ingmar Visser, Maarten Speekenbrink} %% comma-separated
-\Plaintitle{depmixS4: An R-package for hidden Markov models} %% without formatting
+\Plaintitle{depmixS4: An R-package for Hidden Markov Models} %% without formatting
\Shorttitle{depmixS4: Hidden Markov Models} %% a short title (if necessary)
@@ -31,16 +31,18 @@
\pkg{depmixS4} implements a general framework for defining and
estimating dependent mixture models in the \proglang{R}
- programming language \citep{R2009}. This includes standard Markov
+ programming language \citep{R2010}. This includes standard Markov
models, latent/hidden Markov models, and latent class and finite
mixture distribution models. The models can be fitted on mixed
multivariate data with distributions from the \code{glm} family,
- the (logistic) multinomial, or the multivariate normal distribution.
- Other distributions can be added easily, and an example is
- provided with the exgaus distribution. Parameters are estimated by
- the EM algorithm or, when (linear) constraints are imposed on the
- parameters, by direct numerical optimization with the
- \pkg{Rsolnp} or \pkg{Rdonlp2} routines.}
+ the (logistic) multinomial, or the multivariate normal
+ distribution. Other distributions can be added easily, and an
+ example is provided with the {\em exgaus} distribution.
+ Parameters are estimated by the EM algorithm or, when (linear)
+ constraints are imposed on the parameters, by direct numerical
+ optimization with the \pkg{Rsolnp} or \pkg{Rdonlp2} routines.
+
+}
\Keywords{hidden Markov model, dependent mixture model, mixture
model, constraints}
@@ -88,22 +90,24 @@
\SweaveOpts{echo=TRUE}
\usepackage{a4wide}
-%\usepackage{Sweave}
+%% \usepackage{Sweave}
\begin{document}
+
%set width of figures produced by Sweave
\setkeys{Gin}{width=0.9\textwidth}
\maketitle
+<<echo=FALSE>>=
+.Options$digits<-3
+library(depmixS4)
+@
+
%% include your article here, just as usual
%% Note that you should use the \pkg{}, \proglang{} and \code{} commands.
-\begin{center}
-\bf{DRAFT: DO NOT QUOTE WITHOUT CONTACTING AUTHOR}
-\end{center}
-
\section{Introduction}
%% Note: If there is markup in \(sub)section, then it has to be escape as above.
@@ -119,7 +123,9 @@
latent Markov models are usually referred to as hidden Markov models.
See for example \citet{Fruhwirth2006} for an overview of hidden Markov
models with extensions. Further examples of applications can be found
-in e.g.\ \citet[][chapter~1]{Cappe2005}.
+in e.g.\ \citet[][chapter~1]{Cappe2005}. A more gentle introduction
+into hidden Markov models with applications is the book by
+\citet{Zucchini2009}.
The \pkg{depmixS4} package was motivated by the fact that while Markov
models are used commonly in the social sciences, no comprehensive
@@ -181,8 +187,6 @@
\begin{figure}[htbp]
\begin{center}
<<fig=TRUE,echo=FALSE,height=5,width=7,eps=FALSE>>=
-.Options$digits<-3
-library(depmixS4)
data(speed)
plot(as.ts(speed[1:168,]),main="Speed-accuracy trade-off")
@
@@ -215,12 +219,6 @@
distributions, the initial mixture probabilities and transition
probabilities can all depend on covariates $\vc{z}_t$.
-%transition probability functions $a_{ij}$ and the initial state
-%probability functions $\greekv{\pi}$ may depend on covariates as well
-%as the response distributions $\vc{b_{j}^{k}}$. See for example
-%\citet{Fruhwirth2006} for an overview of hidden Markov models with
-%extensions.
-
In a dependent mixture model, the joint likelihood of observations
$\vc{O}_{1:T}$ and latent states $\vc{S}_{1:T} = (S_1,\ldots,S_T)$,
given model parameters $\greekv{\theta}$ and covariates $\vc{z}_{1:T}
@@ -259,6 +257,7 @@
state probability functions $\greekv{\pi}$ may depend on covariates as
well as the response distributions $b_{j}^{k}$.
+
\subsection{Likelihood}
To obtain maximum likelihood estimates of the model parameters, we
@@ -357,7 +356,6 @@
the expected values $\gamma_t(j)$ are used as prior weights of the
observations $O^k_t$.
-
The EM algorithm however has some drawbacks. First, it can be slow to
converge towards the end of optimization. Second, applying
constraints to parameters can be problematic; in particular, EM can
@@ -370,15 +368,17 @@
linear (in)equality constraints (and optionally also non-linear
constraints).
+
\section[Using depmixS4]{Using \pkg{depmixS4}}
Two steps are involved in using \pkg{depmixS4} which are illustrated
below with examples:
\begin{enumerate}
- \item model specification with function \code{depmix} (or with \code{mix}
- for latent class and finite mixture models, see example below on adding
- covariates to prior probabilities)
+ \item model specification with function \code{depmix} (or with
+ \code{mix} for latent class and finite mixture models, see example
+ below on adding covariates to prior probabilities)
+
\item model fitting with function \code{fit}
\end{enumerate}
We have separated the stages of model specification and model fitting
@@ -520,10 +520,10 @@
distribution for the response). The resulting model indicates two
well-separated states, one with slow and the second with fast
responses. The transition probabilities indicate rather stable
-states, i.e., the probability of remaining in either of the states is around 0.9.
-The initial state probability estimates indicate that state 1 is the
-starting state for the process, with a negligible probability of starting in
-state 2.
+states, i.e., the probability of remaining in either of the states is
+around 0.9. The initial state probability estimates indicate that
+state 1 is the starting state for the process, with a negligible
+probability of starting in state 2.
\subsection{Covariates on transition parameters}
@@ -543,7 +543,7 @@
repeated measurement data ($T=2$) using logistic regression on the
transition parameters; they rely on Bayesian methods of estimation.
Covariates on the transition probabilities can be specified using a
-one-sided formula as in the following example:
+one-sided formula as in the following example:
<<>>=
set.seed(1)
mod <- depmix(rt~1, data=speed, nstates=2, family=gaussian(),
@@ -614,9 +614,10 @@
instead of \code{conpat}, with zeroes for fixed parameters and other
values (ones e.g.) for non-fixed parameters. Fitting the models
subject to these constraints is handled by the optimization routine
-\code{solnp} or, optionally, by \code{donlp2}. To be able to construct the \code{conpat} and/or
-\code{fixed} vectors one needs the correct ordering of parameters
-which is briefly discussed next before proceeding with an example.
+\code{solnp} or, optionally, by \code{donlp2}. To be able to
+construct the \code{conpat} and/or \code{fixed} vectors one needs the
+correct ordering of parameters which is briefly discussed next before
+proceeding with an example.
\paragraph{Parameter numbering} When using the \code{conpat} and
\code{fixed} arguments, complete parameter vectors should be supplied,
@@ -673,19 +674,20 @@
@
Using \code{summary} on the fitted model shows that the regression
-coefficients are now estimated at the same value of 12.66. Setting
-elements 13 and 14 of \code{conpat} to zero resulted in fixing those parameters
-at their starting values of 0.5. This means that state 1 can now be interpreted
-as a 'guessing' state which is associated with comparatively fast responses.
-Similarly for elements 1 and 2, resulting in fixed initial probabilities. The
-function \code{llratio} computes the likelihood ratio
-$\chi^2$-statistic and the associated $p$-value with appropriate
-degrees of freedom for testing the tenability of constraints
-\citep{Dannemann2007}. Note that these arguments (i.e. \code{conpat}
-and \code{conrows}) provide the possibility for arbitrary constraints,
-also between, e.g., a multinomial regression coefficient for the
-transition matrix and the mean of a Gaussian response model. Whether
-such constraints make sense is hence the responsibility of the user.
+coefficients are now estimated at the same value of 12.66. Setting
+elements 13 and 14 of \code{conpat} to zero resulted in fixing those
+parameters at their starting values of 0.5. This means that state 1
+can now be interpreted as a 'guessing' state which is associated with
+comparatively fast responses. Similarly for elements 1 and 2,
+resulting in fixed initial probabilities. The function \code{llratio}
+computes the likelihood ratio $\chi^2$-statistic and the associated
+$p$-value with appropriate degrees of freedom for testing the
+tenability of constraints \citep{Dannemann2007}. Note that these
+arguments (i.e. \code{conpat} and \code{conrows}) provide the
+possibility for arbitrary constraints, also between, e.g., a
+multinomial regression coefficient for the transition matrix and the
+mean of a Gaussian response model. Whether such constraints make
+sense is hence the responsibility of the user.
\subsection{Adding covariates on the prior probabilities}
@@ -694,8 +696,8 @@
included another data set with \pkg{depmixS4}. The \code{balance}
data consists of 4 binary items (correct-incorrect) on a balance scale
task \citep{Siegler1981}. The data form a subset of the data
-published in \citet{Jansen2002}. Before specifying specifying a model
-for these data, we briefly describe them.
+published in \citet{Jansen2002}. Before specifying specifying a model
+for these data, we briefly describe them.
The balance scale taks is a famous task for testing cognitive
strategies developed by Jean Piaget \citep[see][]{Siegler1981}.
@@ -1036,27 +1038,28 @@
We are currently working on implementing the gradients for response
and transition models with two goals in mind. First, to speed up
-(constrained) parameter optimization using \pkg{Rdonlp2}. Second,
-analytic gradients are useful in computing the Hessian of the
-estimated parameters so as to arrive at standard errors for those. We
-are also planning to implement goodness-of-fit statistics
-\citep{Titman2008}.
+(constrained) parameter optimization using \pkg{Rdonlp2} or
+\pkg{Rsolnp}. Second, analytic gradients are useful in computing the
+Hessian of the estimated parameters so as to arrive at standard errors
+for those. We are also planning to implement goodness-of-fit
+statistics \citep{Titman2008}.
\section*{Acknowledgements}
Ingmar Visser was supported by an EC Framework 6 grant, project 516542
-(NEST). Maarten Speekenbrink was supported by ESRC grant RES-062-23-1511
-and the ESRC Centre for Economic Learning and Social Evolution (ELSE).
-Han van der Maas provided the speed-accuracy data \cite{Dutilh2010} and thereby
-neccessitated implementing models with time-dependent covariates.
-Brenda Jansen provided the balance scale data set \citep{Jansen2002}
-which was the perfect opportunity to test the covariates on the prior
-model parameters. The examples in the help files use both of these
-data sets.
+(NEST). Maarten Speekenbrink was supported by ESRC grant
+RES-062-23-1511 and the ESRC Centre for Economic Learning and Social
+Evolution (ELSE). Han van der Maas provided the speed-accuracy data
+\citep{Dutilh2010} and thereby neccessitated implementing models with
+time-dependent covariates. Brenda Jansen provided the balance scale
+data set \citep{Jansen2002} which was the perfect opportunity to test
+the covariates on the prior model parameters. The examples in the
+help files use both of these data sets.
\bibliography{dpx4Rev}
+
\end{document}
Modified: papers/jss/dpx4Rev.bib
===================================================================
--- papers/jss/dpx4Rev.bib 2010-06-30 13:30:54 UTC (rev 419)
+++ papers/jss/dpx4Rev.bib 2010-06-30 15:06:55 UTC (rev 420)
@@ -2,21 +2,54 @@
%% http://bibdesk.sourceforge.net/
-%% Created for Ingmar Visser at 2010-03-10 17:01:34 +0100
+%% Created for Ingmar Visser at 2010-06-30 16:36:57 +0200
%% Saved with string encoding Unicode (UTF-8)
+ at book{Zucchini2009,
+ Address = {Boca Raton},
+ Author = {Walter Zucchini and Iain MacDonald},
+ Date-Added = {2010-06-30 14:53:53 +0200},
+ Date-Modified = {2010-06-30 14:53:53 +0200},
+ Number = {110},
+ Publisher = {CRC Press},
+ Series = {Monographs on statistics and applied probability},
+ Title = {Hidden Markov Models for Time Series: An Introduction Using R},
+ Year = {2009}}
+
+ at manual{Ghalanos2010,
+ Author = {Alexios Ghalanos and Stefan Theussl},
+ Date-Added = {2010-06-30 14:45:04 +0200},
+ Date-Modified = {2010-06-30 14:45:31 +0200},
+ Note = {R package version 1.0-2},
+ Title = {Rsolnp: General Non-linear Optimization Using Augmented Lagrange Multiplier Method},
+ Url = {https://r-forge.r-project.org/projects/rino/},
+ Year = {2010},
+ Bdsk-Url-1 = {https://r-forge.r-project.org/projects/rino/}}
+
+ at manual{R2010,
+ Address = {Vienna, Austria},
+ Author = {{R Development Core Team}},
+ Date-Added = {2010-06-30 14:42:17 +0200},
+ Date-Modified = {2010-06-30 14:42:17 +0200},
+ Note = {{ISBN} 3-900051-07-0},
+ Organization = {R Foundation for Statistical Computing},
+ Title = {R: A Language and Environment for Statistical Computing},
+ Url = {http://www.R-project.org},
+ Year = {2010},
+ Bdsk-Url-1 = {http://www.R-project.org}}
+
@book{Salzberg1998,
Address = {Amsterdam},
Booktitle = {Computational methods in molecular biology},
Date-Added = {2010-03-10 16:58:42 +0100},
- Date-Modified = {2010-03-10 16:58:42 +0100},
+ Date-Modified = {2010-06-30 16:34:58 +0200},
Editor = {S. L. Salzberg and D. B. Searls and S. Kasif},
Publisher = {Elsevier},
- Title = {Computational methods in molecular biology},
+ Title = {Computational Methods in Molecular Biology},
Year = 1998}
@phdthesis{Ye1987,
@@ -29,9 +62,9 @@
Address = {San Francisco},
Author = {Thomas D. Wickens},
Date-Added = {2010-03-05 15:47:54 +0100},
- Date-Modified = {2010-03-05 15:47:54 +0100},
+ Date-Modified = {2010-06-30 16:36:50 +0200},
Publisher = {W. H. Freeman and Company},
- Title = {Models for Behavior: Stochastic processes in psychology},
+ Title = {Models for Behavior: Stochastic Processes in Psychology},
Year = 1982}
@manual{Vermunt2003,
@@ -70,27 +103,27 @@
Address = {Voorburg: The Netherlands},
Author = {Frank {Van de Pol} and Rolf Langeheine and W. De Jong},
Date-Added = {2010-03-05 15:46:41 +0100},
- Date-Modified = {2010-03-05 15:46:41 +0100},
+ Date-Modified = {2010-06-30 16:33:16 +0200},
Publisher = {Statistics Netherlands},
- Title = {PANMARK 3. Panel analysis using {M}arkov chains. A latent class analysis program [User manual].},
+ Title = {PANMARK 3. Panel Analysis Using {M}arkov Chains. A Latent Class Analysis Program [User manual].},
Year = 1996}
@article{Titman2008,
Author = {Andrew C. Titman and Linda D. Sharples},
Date-Added = {2010-03-05 15:46:23 +0100},
- Date-Modified = {2010-03-05 15:46:23 +0100},
+ Date-Modified = {2010-06-30 16:36:30 +0200},
Journal = {STATISTICS IN MEDICINE},
Pages = {2177--2195},
- Title = {A general goodness-of-fit test for Markov and hidden Markov models},
+ Title = {A General Goodness-of-fit Test for Markov and Hidden Markov Models},
Volume = {27},
Year = {2008}}
@manual{Tamura2009,
Author = {Ryuichi Tamura},
Date-Added = {2010-03-05 15:46:10 +0100},
- Date-Modified = {2010-03-05 15:46:10 +0100},
+ Date-Modified = {2010-06-30 16:36:07 +0200},
Note = {R package version 0.4},
- Title = {Rdonlp2: an R extension library to use {Peter Spelluci's DONLP2} from {R}.},
+ Title = {Rdonlp2: An R Extension Library to Use {Peter Spelluci's DONLP2} from {R}.},
Url = {http://arumat.net/Rdonlp2/},
Year = {2009},
Bdsk-Url-1 = {http://arumat.net/Rdonlp2/}}
@@ -128,52 +161,40 @@
@book{Siegler1981,
Author = {Robert S. Siegler},
Date-Added = {2010-03-05 15:44:59 +0100},
- Date-Modified = {2010-03-05 15:44:59 +0100},
+ Date-Modified = {2010-06-30 16:35:42 +0200},
Number = {46},
Publisher = {SRCD},
Series = {Monographs of the Society for Research in Child Development},
- Title = {Developmental sequences within and between concepts},
+ Title = {Developmental Sequences Within and Between Concepts},
Year = {1981}}
@article{Ratcliff1978,
Author = {Roger Ratcliff},
Date-Added = {2010-03-05 15:44:46 +0100},
- Date-Modified = {2010-03-05 15:44:46 +0100},
+ Date-Modified = {2010-06-30 16:34:44 +0200},
Journal = {Psychological Review},
Pages = {59-108},
- Title = {A theory of memory retrieval},
+ Title = {A Theory of Memory Retrieval},
Volume = {85},
Year = {1978}}
@article{Rainer2000,
Author = {Gregor Rainer and Earl K. Miller},
Date-Added = {2010-03-05 15:44:29 +0100},
- Date-Modified = {2010-03-05 15:44:29 +0100},
+ Date-Modified = {2010-06-30 16:34:29 +0200},
Journal = {Neurocomputing},
Pages = {961--966},
- Title = {Neural ensemble states in prefrontal cortex identified using a hidden {M}arkov model with a modified EM algorithm},
+ Title = {Neural Ensemble States in Prefrontal Cortex Identified Using a Hidden {M}arkov Model with a Modified EM Algorithm},
Volume = {32--33},
Year = 2000}
- at manual{R2009,
- Address = {Vienna, Austria},
- Author = {{R Development Core Team}},
- Date-Added = {2010-03-05 15:44:13 +0100},
- Date-Modified = {2010-03-05 15:44:13 +0100},
- Note = {{ISBN} 3-900051-07-0},
- Organization = {R Foundation for Statistical Computing},
- Title = {R: A Language and Environment for Statistical Computing},
- Url = {http://www.R-project.org},
- Year = {2009},
- Bdsk-Url-1 = {http://www.R-project.org}}
-
@article{Lystig2002,
Author = {Theodore C. Lystig and James P. Hughes},
Date-Added = {2010-03-05 15:43:58 +0100},
- Date-Modified = {2010-03-05 15:43:58 +0100},
+ Date-Modified = {2010-06-30 16:32:48 +0200},
Journal = {Journal of Computational and Graphical Statistics},
Summary = {Authors introduce an interpretation of the forward variables in terms of conditional probabilities and a new scaling regime that allows exact computation of the score function and the observed information matrix.},
- Title = {Exact computation of the observed information matrix for hidden Markov models},
+ Title = {Exact Computation of the Observed Information Matrix for Hidden Markov Models},
Year = {2002}}
@article{Leroux1992,
@@ -205,34 +226,34 @@
Booktitle = {Computational methods in molecular biology},
Chapter = 4,
Date-Added = {2010-03-05 15:43:21 +0100},
- Date-Modified = {2010-03-10 17:01:34 +0100},
+ Date-Modified = {2010-06-30 16:31:57 +0200},
Editor = {S. L. Salzberg and D. B. Searls and S. Kasif},
Keywords = {hidden markov model, application},
Pages = {45--63},
Publisher = {Elsevier},
- Title = {An introduction to hidden {M}arkov models for biological sequences},
+ Title = {An Introduction to Hidden {M}arkov Models for Biological Sequences},
Year = {1998}}
@article{Kim1994,
Author = {Chang-Jin Kim},
Date-Added = {2010-03-05 15:43:13 +0100},
- Date-Modified = {2010-03-05 15:43:13 +0100},
+ Date-Modified = {2010-06-30 16:31:31 +0200},
Journal = {Journal of Econometrics},
Keywords = {markov models},
Pages = {1--22},
- Title = {Dynamic linear models with {M}arkov-switching},
+ Title = {Dynamic Linear Models with {M}arkov-switching},
Volume = {60},
Year = 1994}
@article{Jansen2002,
Author = {Brenda R. J. Jansen and Han L. J. van der Maas},
Date-Added = {2010-03-05 15:43:02 +0100},
- Date-Modified = {2010-03-05 15:43:02 +0100},
+ Date-Modified = {2010-06-30 16:31:21 +0200},
Journal = {Journal of Experimental Child Psychology},
Keywords = {balance scale},
Number = {4},
Pages = {383-416},
- Title = {The development of children's rule use on the balance scale task},
+ Title = {The Development of Children's Rule Use on the Balance Scale Task},
Volume = {81},
Year = {2002}}
@@ -248,12 +269,6 @@
Volume = {12},
Year = 1994}
- at manual{Ghalanos2010,
- Author = {Alexios Ghalanos and Stefan Theussl},
- Note = {R package version 1.0-1.},
- Title = {Rsolnp: General Non-linear Optimization Using Augmented Lagrange Multiplier Method},
- Year = {2010}}
-
@book{Fruhwirth2006,
Author = {Sylvia Fr\"uhwirth-Schnatter},
Date-Added = {2010-03-05 15:42:33 +0100},
@@ -273,7 +288,7 @@
Title = {Likelihood Ratio Testing for Hidden Markov Models Under Non-standard Conditions},
Volume = {25},
Year = {2007},
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@article{Chung2007,
Author = {Hwan Chung and Theodore Walls and Yousung Park},
@@ -287,7 +302,7 @@
Url = {http://ideas.repec.org/a/spr/psycho/v72y2007i3p413-435.html},
Volume = {72},
Year = 2007,
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@book{Cappe2005,
Address = {New York},
@@ -302,22 +317,22 @@
@article{Baum1966,
Author = {L. E. Baum and T. Petrie},
Date-Added = {2010-03-05 15:41:40 +0100},
- Date-Modified = {2010-03-05 15:41:40 +0100},
+ Date-Modified = {2010-06-30 16:30:30 +0200},
Journal = {Annals of Mathematical Statistics},
Keywords = {hidden markov model},
Pages = {1554--40},
- Title = {Statistical inference for probabilistic functions of finite state {M}arkov Chains},
+ Title = {Statistical Inference for Probabilistic Functions of Finite State {M}arkov Chains},
Volume = 67,
Year = 1966}
@article{Rabiner1989,
Author = {Lawrence R. Rabiner},
Date-Added = {2010-03-05 15:40:56 +0100},
- Date-Modified = {2010-03-05 15:40:56 +0100},
+ Date-Modified = {2010-06-30 16:33:50 +0200},
Journal = {Proceedings of IEEE},
Number = 2,
Pages = {267--295},
- Title = {A tutorial on hidden {M}arkov models and selected applications in speech recognition},
+ Title = {A Tutorial on Hidden {M}arkov Models and Selected Applications in Speech Recognition},
Volume = {77},
Year = 1989}
@@ -335,11 +350,11 @@
@article{Schmittmann2006,
Author = {Verena D. Schmittmann and Ingmar Visser and Maartje E. J. Raijmakers},
Date-Added = {2010-03-05 15:39:23 +0100},
- Date-Modified = {2010-03-05 15:39:23 +0100},
+ Date-Modified = {2010-06-30 16:35:25 +0200},
Journal = {Neuropsychologia},
Number = 11,
Pages = {2079--2091},
- Title = {Multiple learning modes in the development of rule-based category-learning task performance},
+ Title = {Multiple Learning Modes in the Development of Rule-based Category-learning Task Performance},
Volume = 44,
Year = 2006,
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