Probabilistic networks for assessing the course of emotions

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Probabilistic networks for assessing the course
of emotions

• Frank Rijmen

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overview

• Motivating example ecological momentary
  assessment of emotions among anorectic patients
• Panel data characteristics
• A latent markov model and statement of the
  problem
• Probabilistic graphical models
• Application a hierarchical latent markov model

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overview

• Multinomial regression
• Incorporating covariates
• Model testing
• Note on software
• Conclusion

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ecological momentary assessment of emotions among
anorectic patients

• Daily diary method
• Participants report their momentary experiences
  contingent to
• Event
• Signal
• s
• moment-to-moment covariation
• Ecological validity

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Current study

• 32 female anorectic patients from inpatient
  eating disorders unit
• Emotion questionnaire (e.g. At this moment, I
  feel angry) (responses were dichotomized)
• 9 times a day (stratified random administration)
• 7 days
• 24 missing data (ignorable missingness assumed)

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• 6 pairs of emotions (Diener et al., 1995)
• anger and irritation (anger)
• shame and guilt (shame)
• anxiety and tension (fear)
• sadness and loneliness (sadness)
• happiness and joy (joy)
• love and appreciation (love)
• Research questions
• Co-occurrence of emotions
• Evolution over time

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Panel data characteristics

• Panel data
• Multivariate (multiple emotions)
• Repeated (97 measurement occasions)
• Sources of dependency
• Between-person variability
• Within-person variability
• Between days
• Serial correlation within a day
• (co-occurrence of emotions within and across
  measurement occasions)

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• Modeling the dependency structure
• Latent variables for between-person and
  -day variability
• observation- or parameter-driven process for the
  serial correlation within a day
• Approach we follow
• Discrete response and latent variables
• -gt latent class models
• Transitions between states (classes) modelled as
  a Markov process (parameter-driven process)
• Known as latent transition or markov models

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Exemplary model latent markov model

• 2 main assumptions
• at a given measurement occasion t, the responses
  Yit1 ,,YitJ are independent, given latent state
  Zit
• latent state at measurement occasion t depends on
  past latent states through latent state at
  measurement occasion t-1 only

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• Marginal probability of a response pattern yi
• where
• summation is over ST terms

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4 state model, ignoring dependencies between days
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Transition probabilities
State at timepoint t
State at timepoint t-1
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• State probabilities over time

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Statement of the problem

• Estimation (ML framework) involves computation of
  marginal probabilities
• Brute force summation ST terms (49262144)
• More efficient algorithms exist -gt probabilistic
  graphical models

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Probabilistic graphical models

• Principle
• Given statistical model for set of (discrete)
  random variables
• Translate statistical model into graph
• Nodes correspond to random variables
• Conditional dependency relations are represented
  by edges
• -gt visualization
• Apply transformations to the graph
• Transformed graph can be used to factorize joint
  probability distribution
• -gt efficient computation

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Construction of directed acyclic graph for latent
markov model
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• d-separation (Pearl, 86) in the DAG implies
  conditional independence relations in statistical
  model
• Zi2 d-separates Zi1 and Zi3 Zi1 and Zi3 are
  conditionally independent, given Zi2
• Zi1 d-separates Yi11Yi1J Yi11Yi1J are
  conditionally independent, given Zi1
• Zi1 -gt Zi2 lt- Zi3 no d-separation given Zi2 ,
  Zi1 and Zi3 are conditionally dependent in at
  least one distribution compatible with the DAG

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Applying transformations

• Moralization
• marry all non-adjacent parents
• Drop directions

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• Triangulation
• Add edges so that chordless cycles contain no
  more than three nodes
• Chordless none other than successive pairs of
  nodes in the cycle are adjacent

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• Junction tree
• Tree of cliques (clique maximal subset of nodes
  that are all interconnected)
• Has running intersection property each node that
  appears in any two cliques also appears in all
  cliques on the undirected path between these two
  cliques
• Separators intersection of the linked cliques

Zi1
Zi1
Zi1
Zi2
Zi2
Zi2
Zi3
Zi3
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Factorizing the joint probability function

• Given the junction tree representation, the joint
  probability function can be factorized as the
  product of clique marginals over separator
  marginals
• Leads to efficient computational schemes
• Can be done automatically by applying algorithms
  from graph theory

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EM-algorithm (Lauritzen, 1995)

• E-step
• efficient computation of posterior probabilities
  of hidden variables based on junction tree
  factorization (Jensen, Lauritzen, Olesen, 1990)
• M-step
• as usual
• For the latent markov model, this is the
  forward-backward algorithm (the snake bites its
  tail)

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Application a hierarchical latent markov model

• Latent classes on signal- and day-level
• Graphical representation
• 2 classes for signal- and day-level

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Transition probabilities
State at timepoint t
SIGNALS
Day-state 1
Day-state 2
State at timepoint t-1
DAYS
State at timepoint t
State at timepoint t-1
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• State probabilities over time

SIGNALS
DAYS
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Multinomial regression

• Hitherto no restrictions on conditional
  probabilities
• explosion of number of parameters with
• increasing number of parents
• increasing number of states of hidden variables
• Restrictions can be incorporated by modeling the
  conditional probabilities under a multinomial
  logistic regression model
• E.g. incorporating only main effects of parents

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• Response probabilities were restricted
• Main effects of items
• Interaction between day-state and positive vs.
  negative emotions
• Interaction between signal-state and positive vs.
  negative emotions
• gt states only differ wrt shift common to all
• Positive emotions
• Negative emotions
• Shifts for day- and signal-states are additive

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• Saturated model for transition probabilities
• Reduction from 57 to 32 parameters
• Restricted model is rejected (LR 352.4, df 32,
  plt.001)

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Incorporation of covariates

• Covariates can be incorporated as additional
  predictors in the multinomial regression models
• we now factorize Pr(xcovariates) instead of
  Pr(x) according to the junction tree
• Alternatively, incorporate covariates as nodes in
  the graph and factorize Pr(x, covariates)
• More complex graphs
• Not possible when dealing with continuous
  covariates for discrete (hidden) outcome
  variables
• Must when missing data for covariates

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• Before fixed time intervals were assumed between
  two signals lt-gt stratified random administration
• Test time interval as covariate for transition
  probabilities between latent states at
  signal-level
• No effect (LR 7, df 4, p.14 LR 12.4, df
  8, p .13)

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Model testing

• Model testing
• Comparing nested models likelihood-ratio test
• Not applicable when nested model on the boundary
  of the parameter space of the unrestricted model
  (e.g. testing 3-state vs. 4 state-model)
• Global goodness-of-fit
• Sparse data, reference distribution of LR-test
  unknown
• AIC, BIC (but are also deviance-based ???)
• Alternative model diagnostics
• Comparison of expected to observed log-odds
  ratios (averaged)

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• Underestimation of
• dependencies between positive emotions
• dependencies between negative emotions
• Worst for item pairs belonging to same emotional
  category
• Latent markov model better in explaining
  dependencies within a day
• Hierarchical latent markov model better in
  explaining dependencies over days
• Not a surprise

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• Note on software
• BNT matlab toolbox (Murphy, 2004) for operations
  on graph (construction junction tree)
• Set of matlab functions for EM-algorithm
• Multinomial logistic nodes
• Incorporation of covariates
• Nodes can have same CPTs or different CPTs but
  sharing same set of parameters (e.g. when
  different values on covariates)
• parameter restrictions
• Continuous latent variables (gaussian quadrature
  or npml)
• M-step Fischer scoring
• Information matrix numerical differentiation of
  score function in MLEs
• For basic models design matrix is constructed
  automatically
• For more advanced models design matrix is
  defined by the user

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Conclusion

• Graphical models
• Visualization
• Computational efficiency
• Conditional independence relations are derived
  automatically, regardless of numerical
  specifications
• Can also be applied to continuous variables and
  mixed sets of discrete and continuous variables
  (but)
• Posterior configurations, trajectories