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--- avalanche_protection_in_davos_switzerland [2019/05/16 12:55] – stritiha
+++ avalanche_protection_in_davos_switzerland [2023/04/21 15:30] (current) – external edit 127.0.0.1
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 Remote sensing products (e.g. land cover classifications or LiDAR-based measurements of vegetation cover) represent a proxy of the actual state of the ecosystem. We make the uncertainty in the measurements or classifications explicit by creating separate nodes representing the observed value (Y) and the actual state (X) of the variable. The observation is caused by the actual state, not vice-versa, and defining the structure of the network based on this causality helps to define conditional probabilities.
-We used this principle to account for uncertainties in the land cover classification. Classification errors are commonly expressed in confusion matrices, which contain counts of predicted classes for objects where the true class is known (in our case, obtained from 110 ground truth locations), with rows representing the classes in reality c, and columns representing the classes predicted by the classification (c’). Based on these counts, we can calculate either backward probabilities P(X = c | Y = c’) (e.g. the probability that a patch classified as forest is a forest in reality); or the forward probabilities P(Y = c’ | X = c) (that a forest patch will be classified as forest). The backward probabilities depend on the prior distribution of land cover – if we sample ground truth locations in a densely forested landscape, it is likely that many of the patches classified as forest will in fact be forested, leading to a higher backward probability than if we sample in a sparsely vegetated area. On the other hand, forward probabilities are inherent to the error process in the remote sensing data and the classification algorithm (Cripps et al. 2009), and are therefore consistent over the whole area. Therefore, we define the classification node Y as the child of the actual class X, and the rows of its CPT then correspond to the forward probabilities P(Y | X).
+We used this principle to account for uncertainties in the land cover classification. Classification errors are commonly expressed in confusion matrices, which contain counts of predicted classes for objects where the true class is known (in our case, obtained from 110 ground truth locations), with rows representing the classes in reality c, and columns representing the classes predicted by the classification (c’). Based on these counts, we can calculate either backward probabilities P(X = c | Y = c’) (e.g. the probability that a patch classified as forest is a forest in reality); or the forward probabilities P(Y = c’ | X = c) (that a forest patch will be classified as forest). The backward probabilities depend on the prior distribution of land cover – if we sample ground truth locations in a densely forested landscape, it is likely that many of the patches classified as forest will in fact be forested, leading to a higher backward probability than if we sample in a sparsely vegetated area. On the other hand, forward probabilities are inherent to the error process in the remote sensing data and the classification algorithm ([[http://www.mucm.ac.uk/Pages/Downloads/Technical%20Reports/08-03.pdf|Cripps et al. 2009]]), and are therefore consistent over the whole area. Therefore, we define the classification node Y as the child of the actual class X, and the rows of its CPT then correspond to the forward probabilities P(Y | X).
 {{:confusion_cpt.png?600|}}
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   p(Crown_cover_Lidar | Crown_cover) = NormalDist(Crown_cover_Lidar, Crown_cover, 0.12*Crown_cover)
-{{:measurement_cpt_distribution.png?200 |}}
+{{:measurement_cpt_distribution.png?180 |}}
 // Distribution of actual crown cover, given a measurement of crown cover. The CPT of Crown cover (Lidar) is defined as a normal distribution around the actual crown cover.//
 === 2.2 Learning from process-based models ===
-The process-based avalanche model RAMMS (Christen et al. 2010) simulates avalanche flows and also snow detrainment in forests during avalanches. In order to quantify the CPT of the node “Detrainment”, we simulated five known avalanche events in RAMMS with varying input parameters (e.g. different snow heights and parameters of snow erodibility, to account for uncertainty in the model). Then, we used the outputs of the simulations to “learn” the CPT of “Detrainment”, using the Expectation Maximisation algorithm in Netica.
+The process-based avalanche model RAMMS ([[https://doi.org/10.1016/j.coldregions.2010.04.005|Christen et al. 2010]]) simulates avalanche flows and also snow detrainment in forests during avalanches. In order to quantify the CPT of the node “Detrainment”, we simulated five known avalanche events in RAMMS with varying input parameters (e.g. different snow heights and parameters of snow erodibility, to account for uncertainty in the model). Then, we used the outputs of the simulations to “learn” the CPT of “Detrainment”, using the Expectation Maximisation algorithm in Netica.
 === 2.3 Incorporating empirical models===
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 However, this procedure results in a very large CPT for the node (a line for each combination of parameters and predictor variables). Since the parameter nodes will not be modified with evidence, we can reduce the CPT by using the function “Absorb nodes”, which removes the nodes from the network, but retains the associated information in the reduced CPT.
-{{:empirical_cpt.png?800|}}
+{{:empirical_cpt.png?700|}}
 //The parameters of an empirical model can be included explicitly as nodes in the network, to account for model uncertainty when calculating the CPT. Then, these nodes can be "absorbed" to reduce the size of the CPT.//
 === 2.4 Expert knowledge: linking quantitative variables to qualitative categories ===
-Expert knowledge is often related to qualitative categories rather than quantitative variables. For example, it may be easier for an expert to estimate the avalanche protection capacity of forests that are either “open”, “scattered”, or “dense”, rather than based on a percentage of crown cover. Linking such categories to numerical values is associated with a type of linguistic uncertainty (vagueness), where the delineation between categories is not sharp (Regan et al. 2002). Linguistic uncertainty is commonly addressed using fuzzy logic (Zadeh 1965), where membership functions m(y) define the level of membership (between 0 and 1) in a specific class for values of y. For example, we define trapezoidal membership functions of crown cover (Y) for the classes of forest density (X) (see Figure 6.1-5, a). The thresholds between classes have been defined by experts, whereas the slopes of the membership functions are defined based on the standard deviation of measured crown cover at locations where the forest density was classified in the field (method adapted from (Petrou et al. 2013)). At the expert-defined threshold of Y = 70 % crown cover, the probability of the forest being classified as “dense” is 0.5, while a forest with 100 % crown cover will certainly be classified as “dense” (P(X = dense) = 1). We use the membership function to define the probability of the class (X) given an observation y, P(X|Y=y), which is proportional to P(Y|X) * P(X).
+Expert knowledge is often related to qualitative categories rather than quantitative variables. For example, it may be easier for an expert to estimate the avalanche protection capacity of forests that are either “open”, “scattered”, or “dense”, rather than based on a percentage of crown cover. Linking such categories to numerical values is associated with a type of linguistic uncertainty (vagueness), where the delineation between categories is not sharp ([[https://doi.org/10.1890/1051-0761(2002)012[0618:ATATOU]2.0.CO;2|Regan et al. 2002]]). Linguistic uncertainty is commonly addressed using fuzzy logic (Zadeh 1965), where membership functions m(y) define the level of membership (between 0 and 1) in a specific class for values of y. For example, we define trapezoidal membership functions of crown cover (Y) for the classes of forest density (X) (see figure below). The thresholds between classes have been defined by experts, whereas the slopes of the membership functions are defined based on the standard deviation of measured crown cover at locations where the forest density was classified in the field (method adapted from ([[https://doi.org/10.1016/j.patrec.2013.11.002|Petrou et al. 2013]])). At the expert-defined threshold of Y = 70 % crown cover, the probability of the forest being classified as “dense” is 0.5, while a forest with 100 % crown cover will certainly be classified as “dense” (P(X = dense) = 1). We use the membership function to define the probability of the class (X) given an observation y, P(X|Y=y), which is proportional to P(Y|X) * P(X).
 {{:fuzzy_cpt.png|}}
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 === 2.5 Expert knowledge: estimating distributions ===
-For nodes where no data was available (e.g. “Potential detrainment”), we used expert knowledge to quantify the CPT.  To avoid overconfidence, we used the “four-point estimation method” (Speirs-Bridge et al. 2010), where we asked the expert to estimate the lowest and highest value they would expect, the most likely value, and their confidence that the true value is within this range (Metcalf and Wallace 2013). For example, for a dense evergreen forest on rough terrain, the expert estimated the lowest possible detrainment factor to be 24 Pa, the highest 96 Pa, and the best estimate at 48 Pa, with a confidence of 80%. This gives us the quantiles and mode of the distribution, to which we fitted a simple asymmetric triangular distribution (see figure below).
+For nodes where no data was available (e.g. “Potential detrainment”), we used expert knowledge to quantify the CPT.  To avoid overconfidence, we used the “four-point estimation method” ([[https://doi.org/10.1111/j.1539-6924.2009.01337.x|Speirs-Bridge et al. 2010]]), where we asked the expert to estimate the lowest and highest value they would expect, the most likely value, and their confidence that the true value is within this range ([[https://doi.org/10.1016/j.biocon.2013.03.005|Metcalf and Wallace 2013]]). For example, for a dense evergreen forest on rough terrain, the expert estimated the lowest possible detrainment factor to be 24 Pa, the highest 96 Pa, and the best estimate at 48 Pa, with a confidence of 80%. This gives us the quantiles and mode of the distribution, to which we fitted a simple asymmetric triangular distribution (see figure below).
-{{:expert_cpt.png?600|}}
+{{:expert_cpt.png?500|}}
 //Expert-based distribution of potential detrainment for a dense evergreen forest on rough terrain.//
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 The model and the resulting maps of avalanche protection provision and demand, as well as the underlying ecosystem functions, were presented and discussed with local experts. In addition, we performed a sensitivity analysis of the model, using the Netica function “Sensitivity to findings” to calculate the reduction of entropy (uncertainty) on the target nodes in response to findings on other nodes in the network. The entropy reduction (also called mutual information) gives us an indication of which variables in the system have the highest influence on the ecosystem service.
-We also performed a stepwise sensitivity analysis to visualize the flow of information in the network. For each node X, we calculated the proportion of its entropy that can be reduced by a finding on each of its parents. These relative mutual information values were used as weights for links between nodes in a Sankey diagram of the network (Figure 6.1-8)  For each node, the thickness of incoming (from the left) links show how much the entropy on the node can be reduced by findings on preceding nodes. Mutual information is not additive, i.e. if both parent nodes can reduce the entropy of a child by 50%, this does not mean that findings on both parents will result in complete certainty on the child node. Nonetheless, plotting the MI gives an indication of the main sources of uncertainty in the model. When the value of MI for all the parents of a node is rather low, this means that the node will have a wide probability distribution even when the states of its parents are known, implying high uncertainty in the corresponding links. If such a node has a large influence on the outcome of the network, this indicates a knowledge gap.
+We also performed a stepwise sensitivity analysis to visualize the flow of information in the network. For each node X, we calculated the proportion of its entropy that can be reduced by a finding on each of its parents. These relative mutual information values were used as weights for links between nodes in a Sankey diagram of the network (see figure below). For each node, the thickness of incoming (from the left) links show how much the entropy on the node can be reduced by findings on preceding nodes. Mutual information is not additive, i.e. if both parent nodes can reduce the entropy of a child by 50%, this does not mean that findings on both parents will result in complete certainty on the child node. Nonetheless, plotting the MI gives an indication of the main sources of uncertainty in the model. When the value of MI for all the parents of a node is rather low, this means that the node will have a wide probability distribution even when the states of its parents are known, implying high uncertainty in the corresponding links. If such a node has a large influence on the outcome of the network, this indicates a knowledge gap.
 {{:avalanche_sensitivity.png?800|}}