System and method for analyzing structure of probability tree

Abstract

A method of performing probability tree analysis. The method includes identifying a plurality of nodes in a probability tree, wherein each of the plurality of nodes has a probability vector. At least one node structure value is calculated for each of the plurality of nodes. The at least one node structure value quantifies an entropy of a sub-tree extending from a corresponding one of the plurality of nodes. At least one node structure value is assigned to a corresponding one of the plurality of nodes. An analyzed probability tree is output that includes at least one node structure value assigned to a corresponding one of the plurality of nodes.

Description

Technical Field

[0001] The present disclosure relates to automated decision making, and in particular to systems and methods for analyzing the structure of probability trees. Background Art

[0002] Artificial intelligence (AI) can be used in autonomous or semi-autonomous vehicles to determine the actions to be taken by the vehicle. AI algorithms are applied to calculate and determine the driving behavior of semi-autonomous and fully autonomous vehicles by generating probability trees. Probabilistic trees (rooted graphs with probabilities on each vertex) are common data structures used in many automated systems (e.g., systems or machines that make automated decisions, such as automated vehicles). An example of such a tree is the Monte Carlo Tree Search (MCTS), which generates a probabilistic tree structure as an output for sequential decision making. To reach a decision, the algorithm builds a probabilistic tree data structure where the root node is the current state and each vertex is a different action that can be taken from the root node. Summary of the invention

[0003] A method for performing probability tree analysis is disclosed herein. The method includes identifying a plurality of nodes in a probability tree, wherein each of the plurality of nodes has a probability vector. At least one node structure value is calculated for each of the plurality of nodes. The at least one node structure value quantifies the entropy of a subtree extending from a corresponding one of the plurality of nodes. The at least one node structure value is assigned to a corresponding one of the plurality of nodes. An analyzed probability tree is output, which includes the at least one node structure value assigned to the corresponding one of the plurality of nodes.

[0004] Another aspect of the disclosure may be wherein the entropy defines a span of a subtree extending from a corresponding one of the plurality of nodes.

[0005] Another aspect of the present disclosure may be wherein the subtree includes at least one branch extending therefrom, and the probability vector includes a number of values ​​matching the number of branches in the at least one branch extending, wherein the values ​​in the probability vector correspond to the size of the subtree extending therefrom.

[0006] Another aspect of the present disclosure may be that the at least one node structure value includes a standardized node structure value and a non-standardized node structure value.

[0007] Another aspect of the disclosure may be wherein the normalized node structure value provides a value for comparison between different layers of the probability tree.

[0008] Another aspect of the present disclosure may be wherein the probability tree is generated from a Monte Carlo tree search.

[0009] Another aspect of the present disclosure may include generating an updated probability tree based on at least one node structure value in the analyzed probability tree.

[0010] Another aspect of the present disclosure may include generating an updated probability tree based on the analyzed probability tree. Identifying a plurality of nodes in the updated probability tree. Calculating at least one node structure value for each of the plurality of nodes in the updated probability tree. Assigning the at least one node structure value to a corresponding one of the plurality of nodes in the updated probability tree. Outputting the updated probability tree includes at least one node structure value assigned to a corresponding one of the plurality of nodes.

[0011] Another aspect of the disclosure may be wherein the analyzed probability tree advances the updated probability tree by at least one time step.

[0012] Another aspect of the disclosure may be wherein the probability tree is generated prior to identifying a plurality of nodes in the probability tree.

[0013] Another aspect of the disclosure may be wherein identifying a plurality of nodes and calculating at least one node structure value for each of the plurality of nodes occurs during generating the probabilistic tree using a construction algorithm.

[0014] Another aspect of the disclosure may be wherein updating the probability tree includes at least one node corresponding to a node in the probability tree having a reduced span extending therefrom.

[0015] Another aspect of the present disclosure may include determining a degree of similarity between a probability tree and an updated probability tree by comparing at least one node structure value of at least one corresponding node between the probability tree and the updated probability tree.

[0016] Another aspect of the present disclosure may include increasing the number of branches spanning from at least one of the plurality of nodes in the updated probability tree if at least one node structure value of the analyzed probability tree is below a predetermined value.

[0017] Another aspect of the present disclosure may include limiting the number of branches spanning from at least one of the plurality of nodes in the updated probability tree if at least one node structure value of the analyzed probability tree is greater than a predetermined structure value.

[0018] Another aspect of the present disclosure may include providing explanations regarding possible decisions considered from a probabilistic tree based on at least one node structure value.

[0019] Disclosed herein is a non-transitory computer-readable medium containing programming instructions, which, when executed by a processor, are operable to perform a method. The method includes identifying a plurality of nodes in a probability tree, wherein each of the plurality of nodes includes a probability vector. Calculating at least one node structure value for each of the plurality of nodes, wherein the at least one node structure value quantifies the entropy of a subtree extending from a corresponding one of the plurality of nodes. Assigning the at least one node structure value to a corresponding one of the plurality of nodes. Outputting an analyzed probability tree, which includes the at least one node structure value assigned to the corresponding one of the plurality of nodes.

[0020] An automation system is disclosed herein. The automation system includes a plurality of sensors and a controller in communication with the plurality of sensors. The controller is configured to identify a plurality of nodes in a probability tree, wherein each of the plurality of nodes includes a probability vector. The controller is further configured to calculate at least one node structure value for each of the plurality of nodes, wherein the at least one node structure value quantifies the entropy of a subtree extending from a corresponding one of the plurality of nodes. The controller is further configured to assign the at least one node structure value to a corresponding one of the plurality of nodes, and output an analyzed probability tree including the at least one node structure value assigned to the corresponding one of the plurality of nodes.

[0021] This article discloses the following solutions: Solution 1. A method for performing probability tree analysis, the method comprising: identifying a plurality of nodes in a probability tree, wherein each of the plurality of nodes comprises a probability vector; calculating at least one node structure value for each of the plurality of nodes, wherein the at least one node structure value quantifies entropy of a subtree extending from a corresponding one of the plurality of nodes; assigning the at least one node structure value to a corresponding one of the plurality of nodes; and An analyzed probability tree is output, which includes the at least one node structure value assigned to a corresponding one of the plurality of nodes.

[0022] Solution 2. The method according to Solution 1, wherein the entropy defines a span of a subtree extending from a corresponding one of the multiple nodes.

[0023] Option 3. A method according to Option 1, wherein the subtree includes at least one branch extending from it, and the probability vector includes a number of values ​​matching the number of branches in at least one branch extending from it, wherein the values ​​in the probability vector correspond to the size of the subtree extending from it.

[0024] Option 4. A method according to Option 1, wherein the at least one node structure value includes a standardized node structure value and a non-standardized node structure value.

[0025] Option 5. A method according to Option 4, wherein the normalized node structure value provides a value for comparison between different layers of the probability tree.

[0026] Option 6. A method according to Option 1, wherein the probability tree is generated from a Monte Carlo tree search.

[0027] Solution 7. The method according to Solution 1 includes generating an updated probability tree based on at least one node structure value in the analyzed probability tree.

[0028] Solution 8. The method according to Solution 7, comprising: generating an updated probability tree based on the analyzed probability tree; Identify and update multiple nodes in the probability tree; calculating at least one node structure value for each of a plurality of nodes in the updated probability tree; assigning at least one node structure value to a corresponding one of a plurality of nodes in an updated probability tree; and outputting an updated probability tree including the at least one node structure value assigned to the corresponding one of the plurality of nodes.

[0029] Option 9. The method according to Option 8, wherein the analyzed probability tree advances the updated probability tree by at least one time step.

[0030] Option 10. The method according to Option 1, wherein the probability tree is generated before identifying multiple nodes in the probability tree.

[0031] Option 11. A method according to Option 1, wherein identifying multiple nodes and calculating at least one node structure value for each of the multiple nodes occurs during generation of a probability tree using a construction algorithm.

[0032] Embodiment 12. The method of embodiment 8, wherein the updated probability tree includes at least one node corresponding to a node in the probability tree having a reduced span extending therefrom.

[0033] Solution 13. The method according to Solution 8 comprises determining the degree of similarity between the probability tree and the updated probability tree by comparing at least one node structure value of at least one corresponding node between the probability tree and the updated probability tree.

[0034] Solution 14. The method according to Solution 8 includes increasing the number of branches spanning from at least one of the multiple nodes in the updated probability tree when at least one node structure value of the analyzed probability tree is below a predetermined value.

[0035] Solution 15. The method according to Solution 8 includes limiting the number of branches spanning from at least one of the multiple nodes in the updated probability tree when at least one node structure value of the analyzed probability tree is greater than a predetermined structure value.

[0036] Option 16. The method of Option 1, comprising providing explanations regarding possible decisions considered from the probability tree based on at least one node structure value.

[0037] Embodiment 17. A non-transitory computer readable medium containing programming instructions, which, when executed by a processor, are operable to perform a method comprising: identifying a plurality of nodes in a probability tree, wherein each of the plurality of nodes comprises a probability vector; calculating at least one node structure value for each of the plurality of nodes, wherein the at least one node structure value quantifies entropy of a subtree extending from a corresponding one of the plurality of nodes; assigning the at least one node structure value to a corresponding one of the plurality of nodes; and An analyzed probability tree is output, which includes the at least one node structure value assigned to a corresponding one of the plurality of nodes.

[0038] Embodiment 18. A computer-readable medium according to Embodiment 17, wherein the entropy defines a span of a subtree extending from a corresponding one of the plurality of nodes.

[0039] Embodiment 19. The computer-readable medium according to embodiment 17, wherein the method comprises: generating an updated probability tree based on the analyzed probability tree; Identify and update multiple nodes in the probability tree; calculating at least one node structure value for each of a plurality of nodes in the updated probability tree; assigning at least one node structure value to a corresponding one of a plurality of nodes in an updated probability tree; and outputting an updated probability tree including the at least one node structure value assigned to the corresponding one of the plurality of nodes.

[0040] Solution 20. An automated system comprising: Multiple sensors; a controller in communication with the plurality of sensors and configured to: identifying a plurality of nodes in a probability tree, wherein each of the plurality of nodes comprises a probability vector; calculating at least one node structure value for each of the plurality of nodes, wherein the at least one node structure value quantifies entropy of a subtree extending from a corresponding one of the plurality of nodes; assigning the at least one node structure value to a corresponding one of the plurality of nodes; and An analyzed probability tree is output, which includes the at least one node structure value assigned to a corresponding one of the plurality of nodes. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 A vehicle is shown that may operate in an autonomous mode or an automated mode according to an exemplary embodiment.

[0042] Figure 2 A behavioral model is shown that may be used by a controller to select actions to be performed by a vehicle in an illustrative embodiment.

[0043] Figure 3 A flow chart of a method for performing a probabilistic tree analysis in an illustrative embodiment is shown.

[0044] Some embodiments of the present disclosure will now be described by way of example only and with reference to the accompanying drawings. The same reference number represents the same element or the same type of element in all drawings. DETAILED DESCRIPTION

[0045] According to an exemplary embodiment, Figure 1 A vehicle 100 is shown that can operate in an autonomous mode or an automated mode. The vehicle 100 can be a fully autonomous vehicle or a semi-autonomous vehicle. The vehicle 100 includes a driving system 102 that controls the autonomous operation of the vehicle. The driving system 102 includes: a sensor system 104 for obtaining information about the surroundings or environment of the vehicle 100; a controller 106 for calculating possible actions of the autonomous vehicle based on the information obtained, and for implementing one or more of the possible actions; and a human-machine interface 108 for communicating with an occupant of the vehicle (e.g., a driver or a passenger). The sensor system 104 can include devices such as cameras, Lidar, radar, GPS, etc.

[0046] The controller 106 may include processing circuitry that may include an application specific integrated circuit (ASIC), an electronic circuit, a processor (shared, dedicated, or grouped) and a memory that executes one or more software or firmware programs, a combinatorial logic circuit, and / or other suitable components that provide the described functionality. According to one or more embodiments detailed herein, the controller 106 may include a non-transitory computer-readable medium storing instructions that, when processed by one or more processors of the controller 106, implement a method of analyzing the structure of at least one probability tree and providing a node structure value for at least one node in the probability tree. The controller 106 may operate a program that executes artificial intelligence.

[0047] The human-machine interface 108 may include one or more interfaces, such as a speaker, a display, a tactile device, etc. Each interface is suitable for a given message format. For example, a speaker can be used to present an auditory message. A display can be used to present a written message, a graphic, or an animation. A tactile device can be used to present a tactile message felt by the passenger, such as a vibration. In another embodiment, the human-machine interface 108 can be a mobile device that the passenger can use inside or outside the vehicle.

[0048] Figure 2 A behavior model 200 is shown that can be used by the controller 106 to select actions to be performed at the vehicle 100 in an illustrative embodiment. In the example shown, the behavior model 200 includes a probability tree having a plurality of nodes. An edge connects any two nodes of the probability tree. Each node represents a state of the vehicle 100, and each edge represents an action that takes the vehicle from one state to another state (a subsequent state or a future state). The top node (or root node) of the probability tree represents the current state of the vehicle 100. The subsequent possible states are shown below the top node. Each action or edge has an associated probability. State X in the probability tree i represents random variables that receive values ​​from possible actions (e.g., [right, left, speed up, slow down, none]). This defines a discrete-time random process X0,X1,X2,X3,X4,X5,X6…,X n , where n is the depth of the tree.

[0049] The sequence of actions through the behavioral model 200 is called a trajectory. When using the behavioral model 200, the controller 106 starts at the root node (i.e., the current state of the vehicle 100). The controller 106 selects an action to take by executing an optimization algorithm that considers all possible futures in the probability tree evaluated from the root node. Once the action is executed, the vehicle now moves to the resulting state. The controller 106 identifies the resulting state as the new root node and expands the new tree so that the optimization algorithm can be executed from the new root node. The controller 106 can operate an artificial intelligence planning system to optimize the sequential decision making process to achieve automated behavior. In one embodiment, the sequential decision making process can be implemented as in a Monte Carlo Tree Search (MCTS) algorithm.

[0050] From the current state X0 (first state 202), the controller 106 can calculate multiple trajectories through the probability tree to reach the desired end state or subsequent state, and can calculate the probability of each trajectory. For illustrative purposes, a first expected trajectory 204 and a second expected trajectory 206 are shown. The first expected trajectory sequentially passes through states 204a, 204b, 204c, 204d, 204e, and 204f. For illustrative purposes, the first expected trajectory 204 can be the trajectory with the highest expected value based on the decision process.

[0051] Figure 3 For example, in Figure 2 Flowchart of method 300 for analyzing at least one probability tree on the behavior model 200 shown. Method 300 can be applied to existing probability trees afterwards, or applied based on iteration during the construction of probability trees. Analysis of probability trees provides node structure values, which describe the structure of branches or subtrees spanned from a given node in the probability tree. The subtree includes at least one branch extending from it, and the probability vector includes the number of values ​​that match the number of branches in at least one branch extending from it. In addition, the value in the probability vector corresponds to the size of the subtree of the child node. This allows the analysis performed by method 300 to be related to the structure of the probability tree. Therefore, the node structure value indicates the structure of the probability tree spanned from each node, rather than going down along the probability tree and depicting its structure. Larger node structure values ​​reflect more uncertainty in the probability tree. This corresponds to a wider tree (e.g., more branches in different layers of the subtree). The node structure value can be standardized to reflect the uncertainty of each node so that a smaller subtree can be compared with a larger subtree.

[0052] Method 300 includes identifying multiple nodes in a probability tree (block 302). Although the present disclosure describes the method 300 used in conjunction with the driving system 102, the method 300 can be used with any automated system that generates or outputs a probability tree. The node can come from a previously constructed probability tree, or the node can come from a probability tree being constructed by a construction algorithm. The method 300 then calculates at least one node structure value for each node (block 304). The node structure value provides an entropy value for a given node to describe the structure of the probability tree spanned from the node. For nodes that do not include branches or subtrees, the node structure value is zero. However, the present disclosure is not limited to real-time, offline analysis, or behavioral models that consider the time of calculating entropy metrics.

[0053] EQ.1-5 below show example methods for calculating the node structure value or entropy of a given node. EQ.1 below provides a measure of entropy T i The naming format is represented by the possible backward paths to the node at depth i, where T0 is empty. EQ.2 provides a measure of entropy T i The naming format is represented by possible forward paths to the node at depth i. T i =X0,X1,…,X i-1 EQ.1 T i =X i , X i+1 , …X n EQ.2

[0054] EQ.3 shows an equation that can be recursively implemented on a probability tree to determine entropy and assign at least one node structure value to a corresponding one of a plurality of nodes in a probability tree (e.g., an existing probability tree). The function H() in EQ.3 refers to the entropy function, Pr(X i =x j |T i =t) is when T i When X has value t (i.e., a specific trajectory) i Get the value x j The conditional probability of . In EQ.3 below, the variable Represents the action space of the probability tree. The action space represents the various possible actions considered and from which the desired outcome can be selected.

[0055] Although the output values ​​based on the calculations in EQ.3 above provide node structure values ​​for a given node in the probability tree, in order to compare values ​​between trees in different layers, the values ​​are normalized, as shown in EQ.4 below. Therefore, EQ.3 will provide non-normalized node structure values, and EQ.4 will provide normalized values ​​for a given node in the probability tree. EQ.4 represents the average entropy at each time step as

[0056] The variable l in EQ.4 above is the depth of the subtree extracted from the probability tree. The lower bound is exact if the entropy at a leaf or branch end is assumed to be zero. If the process continues and leaves are truncated, this uses l LB Create an upper limit on H(\bar{X}). A lower limit l LB Described in EQ.5 below.

[0057] variable can be replaced with the actual number of branches that have been expanded in the probability tree (and make the upper limit clearer). When there is a very large action space, the actual number of branches can be quite large, where most actions may not be valid or cannot be explored further.

[0058] Alternatively, the method 300 can be incorporated into or piggybacked onto a standard construction algorithm for a probability tree, such as the MCTS algorithm, to determine the entropy of a node. When the method 300 is incorporated into a standard construction algorithm, the root node is initially identified and corresponds to Figure 2. When a probability tree is formed by adding new nodes spanning from the root node, at least one node structure value is calculated for each node in the probability tree. The at least one node structure value may include both a standardized value and a non-standardized value, depending on the desired use of the node structure value. This allows for intermediate node structure values ​​(standardized and non-standardized) to be available at any point during the construction of the probability tree, which are iteratively updated as the probability tree grows. Nodes that are identified during the construction process that do not include branches or subtrees will also have a single node structure value of zero.

[0059] In one example, whether the analysis is performed on an existing tree or during the construction of a probability tree, the node structure values ​​of two identical probability trees will not change. However, as will be discussed in more detail below, the node structure values ​​can be used to guide the construction of the probability tree, such that the analysis incorporated into the construction of the probability tree can result in the growth or reduction of the probability tree in certain areas, depending on the node structure values.

[0060] In one example embodiment, incorporating method 300 into a standard construction algorithm occurs during the back propagation step. In the standard construction algorithm, back propagation updates the number and value of visits from the far end of the leaf or branch up the probability tree to the root node. Using method 300, the node structure value or entropy of the tree is updated by the encapsulated method and the calculation of the depth of each node to facilitate the normalization (lower limit) of the entropy (see block 304).

[0061] Using the node structure values ​​calculated using the method as described above for an existing probability tree or a probability tree during construction, the method 300 assigns at least one normalized or non-normalized node structure value to a corresponding one of the plurality of nodes at block 306. The method 300 may then output an analyzed probability tree at block 308, which includes at least one node structure value assigned to a corresponding one of the plurality of nodes.

[0062] The analyzed probability tree from block 308 may then be provided back to the driving system 102 or an automated system for further analysis or development of an updated probability tree based on the analyzed probability tree, as will be discussed in greater detail below.

[0063] Determining a node structure value or entropy of a probability tree is a feature that generates an enhanced probability tree construction algorithm and guides the search by the driving system 102. The node structure value can also be used as a metric to determine whether the resulting search by the system 102 has fully explored the action space. This metric can also be used for meaningful reduction of probability trees for storage or simplification purposes. In addition, the node structure value can be used for extensibility for both developers and drivers / passengers, regarding a given probability tree or as a metric for comparison between consecutive probability trees based on time to evaluate the occurrence of meaningful changes.

[0064] A probability tree is created at each time step of the planning system, and may include creating several probability trees per second. Storing the probability trees for backup or post-hoc verification is limited by the amount of storage available. The number of nodes in the probability tree also limits the analysis that can be performed on the probability tree, and transmitting the probability tree to a backend is also expensive. Therefore, using node structure values ​​to limit or reduce the number of nodes in the probability tree can improve the amount of analysis that is performed, such as by allowing the probability tree to be transmitted to a backend, such as a remote computing location, for analysis.

[0065] A probability tree is created at each time step of the planning system. The standard approach in probability tree based decision making is to not consider the last probability tree, but rather the previous probability tree to check if the decision is consistent. This is done by comparing probability trees from a series of time steps. However, probability trees are not identical because their construction is probabilistic, but evaluating the level of similarity between probability trees is still informative in the decision making process of system 102. Node structure values ​​can also provide a way to make comparisons by comparing node structure values ​​between corresponding nodes in different probability trees. If the node structure values ​​are within a predetermined range, the probability trees are considered similar. If the node structure values ​​are not within the predetermined range, the probability trees are not considered similar.

[0066] The driving system 102 generates a probability tree to determine the actions that the vehicle 100 will perform. Explaining these decisions to the driver / passenger is done to increase confidence in the decisions made by the system 102. Engineers who develop these systems can also benefit from explanations that describe the reasoning process of the original system. Algorithms used to provide action explanations can benefit from having structural metrics, such as node structure values, which indicate which options the planning algorithm considered and to what extent these options were considered (the reasoning process of the planning algorithm). Explanations of decisions by the automated system can be provided to the user of the vehicle 100 through the interface 108.

[0067] Although the probability tree is constructed based on an algorithm that expands the tree from the current state, the construction of the probability tree is limited by computing power and time. The node structure value can be used to check the construction process at the intermediate point to evaluate whether the algorithm has fully explored in a particular direction, and may indicate that the process can be stopped, thereby saving calculations. Using this example application of the node structure value, the analyzed probability tree with the node structure value is provided as feedback to an automated system such as the driving system 102 for building the probability tree or updating the next iteration of the probability tree. The feedback obtained from reviewing the analyzed probability tree with the node structure value can indicate to the automated system where it may be helpful to expand the probability tree and where expansion may not be necessary.

[0068] Some automated systems consider the value of taking an action and tend to reconsider actions that have a stronger preference or have shown promise multiple times. The node structure value can be viewed as an additional independent parameter to balance the search in directions that are not fully considered. One feature of incorporating node structure values ​​into probabilistic trees (particularly implementations that occur during the construction of probabilistic trees) is that the node structure values ​​are incorporated into each intermediate tree, so these values ​​are available throughout the tree construction process to guide the construction of the tree.

[0069] Although various embodiments have been described, the description is intended to be exemplary rather than restrictive. It will be apparent to those of ordinary skill in the art that more embodiments and implementations within the scope of the embodiments are possible. Unless specifically limited, any feature of any embodiment may be used in combination with or replaced with any other feature or element in any other embodiment. Therefore, the embodiments are not limited except in accordance with the appended claims and their equivalents. In addition, various modifications and variations may be made within the scope of the appended claims.

Claims

1. A method for performing probability tree analysis, the method include: identifying a plurality of nodes in a probability tree, wherein each of the plurality of nodes comprises a probability vector; calculating at least one node structure value for each of the plurality of nodes, wherein the at least one node structure value quantifies entropy of a subtree extending from a corresponding one of the plurality of nodes; assigning the at least one node structure value to a corresponding one of the plurality of nodes; and An analyzed probability tree is output, which includes the at least one node structure value assigned to a corresponding one of the plurality of nodes.

2. The method according to claim 1, in, The entropy defines a span of a subtree extending from a corresponding one of the plurality of nodes.

3. The method according to claim 1, in, The subtree includes at least one branch extending therefrom, and the probability vector includes a number of values ​​matching the number of branches in the at least one branch extending therefrom, wherein the values ​​in the probability vector correspond to the size of the subtree extending therefrom.

4. The method according to claim 1, in, The at least one node structure value includes a standardized node structure value and a non-standardized node structure value.

5. The method according to claim 4, in, The normalized node structure values ​​provide values ​​for comparison between different layers of the probability tree.

6. The method according to claim 1, in, The probability tree is generated from a Monte Carlo tree search.

7. The method of claim 1, comprising generating an updated probability tree based on at least one node structure value in the analyzed probability tree.

8. The method according to claim 7, include: generating an updated probability tree based on the analyzed probability tree; Identify and update multiple nodes in the probability tree; calculating at least one node structure value for each of a plurality of nodes in the updated probability tree; assigning at least one node structure value to a corresponding one of a plurality of nodes in the updated probability tree; and An updated probability tree is output, which includes at least one node structure value assigned to a corresponding one of the plurality of nodes.

9. The method according to claim 8, in, The analyzed probability tree advances the updated probability tree by at least one time step.

10. The method according to claim 1, in, The probability tree is generated before identifying a plurality of nodes in the probability tree.

Images