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Supplementary Material 1: Instructions for using program E-SURGE for implementing
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models of habitat dynamics in the presence of misclassification.
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It is possible to implement analysis of hidden markov chain models in program E-SURGE
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(Choquet and Nogue 2010), freely downloadable at http://www.cefe.cnrs.fr/biom/En/
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softwares.htm. Program E-SURGE works by decomposing dynamics into parameters associated
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with initial states, transitions between states, and classification probabilities that relate
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observations (events) to true states.
Occasion
t
t +1
Transition
True states
Events
Events
Observations
Transition
True states
Observations
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The basic structural form needed to run analyses is specified with the following “pattern"
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matrices in the “GEPAT" module in E-SURGE. Here, “*” entries denote the complement of the
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sum of positive row entries, and “-“entries denote zeroes.
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Note that E-SURGE was developed as software to analyze multi-state mark-recapture data in
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wildlife populations in order to estimate survival and transition parameters in the face of state
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uncertainty (Pradel 2005). Because of this, it includes an absorbing state called “Dead” in the
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structure of transition and events matrices and a state Not Observed in the event matrix. The
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three other states correspond to Forest (F), Managed (M) and Urban (U) for the pixel level
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analysis or to Dominant forest (F), Average forest (A) and Low forest (L) for the cell level
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analysis.
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The matrix notation used in E-SURGE is different than the matrix notation used in matrix
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models of demography: in E-SURGE, departure states (or FROM) are defined in rows and
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arrival states (or TO) in columns. In matrix models used in demography (the notation we used in
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this study), departure states (or FROM) are defined in columns and arrival states (or TO) in rows.
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We first describe the pattern matrices in GEPAT
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1-For the initial state vector, we have
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𝜋𝑡 = [𝜋
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2-For the transition parameter matrix, we have
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∗
𝜓
𝜓𝑡 = [
𝜓
−
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The fourth row and the fourth column correspond to the state Dead, an absorbing state from
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which transition elsewhere is not possible. In models of habitat dynamics, this might correspond
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to habitat that becomes completely unavailable (for example a parking lot). However, since these
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instances do not occur in our data, we constrained transitions to and from Dead to 0.
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3-For the event process (the habitat classification probabilities in our study), denoted as B:
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− ∗ 𝛽
− 𝛽 ∗
𝐵𝑡 = [
− 𝛽 𝛽
∗ − −
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The fourth row corresponds again to the state Dead, the first column to the state Not Observable.
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Thus corresponding classification parameters are constrained to 0 in our models.
𝜋
∗]
𝜓 𝜓 −
∗ 𝜓 −
]
𝜓 ∗ −
− − ∗
𝛽
𝛽
]
∗
−
2
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After these structures are specified in “GEPAT," the user must symbolically formulate
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design matrices to specify linear models for each parameter. This can be done using the
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“GEMACO" function in E-SURGE.
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We analyzed 3 types of models in this study:
1- Models where classification is not accounted for. This is equivalent to constraining all 𝛽
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parameters previously defined in GEPAT to 0 in GEMACO in order to set all
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classification probabilities to either 1 or 0 (no misclassification) ( in “GEPAT”, this
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matrix is slightly different from a diagonal matrix because of the presence of the Dead
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and Not Observable states).(Fig.1)
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Fig. 1: Schematic representation of type 1 models in E-surge : at each time step (or occasion), β
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parameters of the matrix B (so-called event probabilities in E-surge or classification probabilities
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in this study) are fixed to 0. Note that constraining the β (misclassification) parameters to 0
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results in the diagonal (correct classification) elements being 1.
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2- Models where information on true states at the level of a cell is obtained from computing
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expected true states based on pixel accuracy information. Dummy occasions are created
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at each time step in order to incorporate this additional information. Between dummy
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occasions and real occasions, all 𝜓 parameters of the transition matrix previously defined
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in GEPAT are fixed to 0 in GEMACO (resulting in an identity transition matrix) as well
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as all 𝛽 parameters of the event matrix fixed (Fig. 2)
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Fig. 2: Schematic representation of type 2 models in E-surge : for each time step (or occasion), a
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dummy occasion is created. Between dummy occasion and real occasion, both transition and
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event probabilities defined in GEPAT are fixed to 0.
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3- Lastly, we also analyzed a model where ground-truth data are available to determine true
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states, but only for a sample of habitat histories and only for the last occasion. Note that 𝛽
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parameters are estimable for the last time period. Application of these classification
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probabilities to data from previous time periods requires the assumption that these
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classification probabilities are constant over time.
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Fig. 3: Schematic representation of type 3 models in E-SURGE : in these models a dummy
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occasion is created only for the last time step, where ground-truth data are available for a sample
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of habitat histories (their habitat histories have a last column not equal to 0). Between the last
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occasion and the dummy occasion, both transition and event probabilities defined in GEPAT are
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fixed to 0. Note that when coding the habitat histories in E-SURGE, the group with no ground
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truth has a negative number in the last column (indicating no data for the last occasion)
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Reference
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Choquet, R., and E. Nogue. 2010. E-SURGE 1.7 User’s Manual. CEFE, Montpellier, France.
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