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Kalman extension of the genetic algorithm
Los Alamos National Laboratory, MS-F607, Los Alamos, NM 87545 LAUR-00-150, SPIE proc. vol. 4055, April, 2000 ABSTRACT
In typical GA application, the fitness assigned to a chromosome-represented individual in the context of a specified environment takes a deterministic calculable value. In many problems of interest, the fitness of an individual is stochastic, and the environment changes in unpredictable ways. These two factors contribute to an uncertainty that can be associated with the estimated fitness of the individual.
The Kalman formulation provides a mechanism for a useful treatment of uncertainty within the GA framework. It provides for tracking the best-estimated fitness, and for assessing the uncertainty of the best-estimated fitness. The stochastic uncertainty of an existing individual can be reduced by additional evaluation of the fitness. The process (environmental) noise causes an increase in uncertainty as the age of the last evaluation increases. In a Kalman-extended genetic algorithm, we want to know the value of re-evaluating an existing solution in the population relative to generating and evaluating new individuals. A scheme for efficient allocation of computational resources among existing individuals and new individuals is This Kalman-Darwin formulation is applied to the problem of maintaining an optimal network configuration of links between moving nodes, with time-dependent, stochastic blockages. The nodes, for example, could be environmental sensors with radio transmitter/relays located on vehicles, and the network is configured to provide communication paths from each sensor back to a central node with minimum message loss. As the sensors move, the optimal network changes, but information contained within the GA population of solutions allows new optima to be efficiently obtained. The performance of this approach is explored and results are presented.
1. INTRODUCTION
The Kalman Genetic Algorithm (KGA) is an extension of the genetic algorithm (GA)1-3 in which Kalman filtering4-6 is applied to the fitness values associated with the population of chromosomes. In the GA, a chromosome represents a trial solution to a problem. The chromosome has a well-defined measure of fitness within the context of the problem space or environment. In many interesting applications, the environment is dynamic. The fitness of a given chromosome will change in time, as changes occur in the environment. In addition, for environments of sufficient complexity, or for environments with stochastic elements, the fitness of a chromosome is unknowable, and can only be approximated. The Kalman formulation provides a natural mechanism for extending the domain of application of the genetic algorithm from static, prescribed problem spaces to dynamic, and complex or stochastic environments.
In the KGA, at any given time, each chromosome has a best-estimated fitness and an uncertainty associated with that estimate. The fitness and uncertainty are updated according to the Kalman formulation, as the environment changes with the * Correspondence: E-mail: stroud@lanl.gov; Telephone: 505-667-6654 passage of time, and when a chromosome is re-evaluated. Knowledge is acquired7 by re-evaluating an existing trial solution (thereby decreasing the uncertainty associated with its fitness estimate) or by generating and evaluating a new trial solution.
The principle behind the KGA is simple. The population, consisting of N individuals, each with a best-estimated fitness, can be characterized by the variance V of these N fitness values. When the uncertainty associated with all members of the population is smaller than V, the population is treated as sufficiently evaluated, and the algorithm will generate and evaluate a new trial solution. However, if any of individuals in the population have uncertainties greater than V, the algorithm will re- evaluate whichever individual has the highest uncertainty.
The literature contains applications in which GA is used to tune the parameters of a Kalman filter.8-11 There is also some research reported as a hybridization of Kalman filtering and GA, in which Kalman filtering and GA are used to optimize two separate parts of a problem.12,13 In previous work,14-17 the author used GA to periodically optimize controllers in the context of a dynamic environment. The heuristic of including the previous best solution in the next initial population was explored, but a theoretical formulation underlying the use of GA in noisy, dynamic environments was lacking. In a SciSearch® search of 17.5 million technical articles, no reference was found in which a Kalman filter is used as an integral part of the GA 2. MECHANICS OF THE KALMAN GENETIC ALGORITHM
The KGA keeps a population of individuals. An individual represents a trial solution to a problem. There is an upper limit, N, on the population size, and at any given time, there are actually n individuals in the population. The individual, i, has four elements: 1) a chromosome W , 2) a best estimated fitness, f , 3) an uncertainty, P , associated with that estimate, and 4) a timestamp, T , specifying the time that the uncertainty was last updated. The chromosomes can take a variety of forms.18 The requirements on the form of the chromosome are that it can be transformed into a possible solution of the problem, and that a set of genetic operators can be constructed for it. For this implementation, the chromosomes are simply fixed length sequences of bounded integers. A mechanism for transformation from the chromosomal form (the genotype) to the problem solution form (the phenotype) must be developed for each problem. As an aid to interpretation of the uncertainty, the underlying fitness can thought of as the best estimate of fitness, plus or minus the square root of the uncertainty.
When a real-time directed-search algorithm is used in a dynamic environment, the algorithm computational time is interwoven with the physical time. We consider the case in which an evaluation of the fitness of a trial solution takes a finite amount of time, τ . During a given period of time, there will thus be a finite number of trial evaluations that the algorithm A mechanism is required to evaluate the fitness of a solution. In environments with stochastic processes, the result of a fitness evaluation will be a real-number value that represents the true, underlying fitness value but has an uncertainty associated with it. If an ensemble of fitness evaluations were to be made on a trial solution, the variance of the ensemble of the resulting fitness values, R, is taken as the measure of the uncertainty in the fitness evaluation. This variance is analogous to the observation noise variance of the Kalman formulation, if a fitness evaluation is treated like an observation. R can be interpreted as the uncertainty with which the result of a fitness evaluation represents the true, underlying fitness of the trial solution. R designates the variance of the observation noise associated with a single evaluation of a trial solution.
Because the environment is dynamic, the underlying fitness of a given trial solution will change with time. In the Kalman formulation, this change is attributed to process noise. Like the observation noise, the process noise is characterized by a variance, in this case designated by Q. Q gives the variance in the change in the underlying fitness of trial solutions during a specified time interval, due to change in the environment. In this formulation, there is no fixed time interval to use to define Q, as is common in Kalman filtering applications. Instead, a “process noise rate”, given by q, is used. The process noise variance expected after the environment undergoes changes for a time interval dt, is Q = dt q. This presumes that the process noise variance increases linearly with time interval, although alternative formulations could be used as well. As with any Kalman application, q can be specified by knowledge about the environment, or it can be evaluated adaptively.
The KGA begins operating at time t = t . At this time, the population contains no trials yet (n = 0). The algorithm iterates over four main processes, as shown schematically in Fig. 1. The first process updates the uncertainties of the members of the population to the current time. If an uncertainty was last updated at time T , the associated uncertainty increases according to This is analogous to the Kalman equation that updates the covariance matrix for process noise. The time of the last update is then advanced to the current time, T = t.
Figure 1. The Kalman-extended genetic algorithm, shown schematically as a cycle over four major processes.
The second process characterizes some statistics of the population that will determine whether to generate a new trial, or re- evaluate an existing individual. The variance, V, of the best-estimated fitnesses of the individuals in the population is , of the most uncertain individual. These two statistics are used by the algorithm to determine which of the three elements of the third process to invoke.
The third process performs a fitness evaluation. There are three alternatives for this third process. If the population is not full (i.e. n<N), a new random chromosome is generated and evaluated. The second alternative for the evaluation process: if the population is full, and the uncertainty of the most uncertain individual exceeds the population’s fitness variance (i.e. P the most uncertain individual will be reevaluated. When a re-evaluation is performed (returning a value of g), the previous , is updated using the Kalman observation equation: In addition, a re-evaluation reduces the uncertainty P by multiplying by a factor of R/(P+R) as follows: The quantity P/(P+R) is analogous to the gain, K, of the Kalman formulation.
The third alternative for the evaluation process: if the population is full, and the uncertainty of the most uncertain individual is less than the population’s fitness variance, a new trial will be generated and evaluated. Traditional GA mechanics are used to generate new individuals. Two parents are selected based on their fitness, with more fit individuals being more likely to be selected. Genetic operators such as crossover and mutation are used to generate a new chromosome from those of the parents.
When a new chromosome is created and evaluated once, the best estimate of fitness f is the result of the evaluation, and the The fourth process incorporates the newly evaluated trial appropriately into the population. The best-estimated fitness and the uncertainty are combined to give an uncertainty-compensated fitness, F, for each individual, given by The algorithm thus searches for individuals with high fitness, but discounts the fitness if there is a high associated uncertainty. The population is sorted by this uncertainty-compensated fitness. If the uncertainty-compensated fitness of the new chromosome is better than that of the least fit individual in the population, the new individual is added to the population, and the least fit individual is discarded. If the population is not yet full, or if an existing individual was reevaluated, the population is simply re-sorted according to uncertainty-compensated fitness. At any time, the individual with the best uncertainty-compensated fitness is taken to be the best-adapted solution produced by the algorithm. These four processes take some time to perform, but the bulk of the time is presumably spent evaluating trials. When the cycle repeats, the time t is 3. DEVELOPMENTAL TEST-BED: NETWORK CONFIGURATION OPTIMIZATION
A test-bed has been constructed for developing and experimenting with the KGA in a simulated dynamic, noisy environment.
It has been implemented in Java, with a GUI that allows a user to easily construct scenarios and observe simulations of the The test problem is to configure a network of radio links connecting a set of nodes to a collection point. The task is to maintain a good configuration of links as the nodes move. This test application derives from a system of atmospheric aerosol sensors mounted on vehicles, which deliver their data via radio links to a collection point. Each sensor collects a stream of data, which it then transmits via a low-power radio link. A sensor node may transmit directly to the collection point or to another node. A node can receive and repeat data from other sensors. The network of links thus is an acyclic, directed graph (i.e. a tree) with the base being the collection point, and the sensors forming the other nodes. Each radio link forms an edge of In its pure, unconstrained form, this problem admits an algorithmic solution (via Dijkstra’s shortest-path tree algorithm).19-21 The existence of this solution does not render the application trivial, because slight variations in the problem definition, as would be encountered in practice, make this problem NP-complete.19 Examples of such variations include: a constraint on the number of links a given node may receive and retransmit; a bandwidth limitation on the transmission channels or a degradation in transmission fraction with increased message traffic; a maximum number of links in a path to the collection point; or the possibility of redundant paths, where a node may transmit to more than one receiver. Optimization of a complex network with routing and bandwidth constraints is not amenable to algorithmic approaches, but can be attacked with GA.22 The existence of an algorithmic solution for the pure form of the problem does allow the performance of the KGA to be assessed relative to the sequence of optimal solutions.
The nodes are treated as executing a random walk within a prescribed area. (The vehicles are not dedicated to the sensors, but are engaged in other activities.) The movement of the nodes makes the problem environment dynamic. As the nodes move, the transmission probability along the links changes, and the optimal network configuration also changes.
A prescribed area on the ground of arbitrary shape and size is represented on a 2D Cartesian cell grid. The cell size, and the number of cells spanning the two dimensions, are adjustable. A baseline scenario has been constructed, in which each cell represents a 160 by 160 meter square, and a grid of 125 by 125 cells covers a 20 by 20 km area. In this baseline scenario, the vehicle/sensor/nodes are restricted to a roughly 10 by 20 km elliptical region as shown in Fig 2.
Figure 2. Example scenario, showing a user-prescribed region, 25 randomly placed nodes, and a randomly generated link network. The collection point is shown as the open circle.
Let S designate the total number of nodes, exclusive of the collection point. The baseline scenario contains 25 sensor nodes.
These vehicle-mounted sensors can be initialized by assigning each sensor to a random cell within the prescribed area. Only one sensor can occupy a cell. Fig. 2 shows the locations of 25 randomly placed sensors.
Each node establishes a radio link to another node or to the collection point, to transmit its own data, and any data it receives from other nodes. Each node has a unique ID in the range of one to S. The receiver of a node is specified by an integer in the range of zero to S. A value of zero indicates that the link is transmitting directly to the collection point, while any other value specifies the ID of the receiving node. The network configuration is completely specified by giving the receiver for each node.19 This requires a total of S integers, each in the range 0 to S. A node can not have itself as a receiver. The number of distinct network configurations is then SS. For 25 sensors, there are 8.9(10)34 possible network configurations.
In this implementation, only valid spanning-trees configurations (including the collection point) are considered, i.e. all nodes and the collection point are included in the graph, and there are no cyclic paths in the graph. This reduces the size of the problem space. A simple algorithm is used to generate random trees, as follows. All nodes are initially unconnected. An unconnected node is selected at random. A receiver is selected at random from the set containing the collection point and all connected nodes. The selected unconnected node takes the randomly selected receiver to be its receiver, and then becomes a connected node. This process is repeated until all nodes are connected. A randomly generated network linking 25 sensors to a collection point is shown in Fig. 2. The 25 integer representation of this network is { 8 19 17 18 12 20 20 20 11 0 21 0 20 20 13 12 12 20 24 12 0 6 18 12 21}. Node 1 transmits to node 8, while nodes 10, 12 and 21 transmit to the collection point, etc.
The expected transmission fraction over a link depends on the length of the link. For short distances, the expected transmission is nearly perfect, because the signal is much higher than the noise. There is a scale distance, d, where the expected transmission fraction across a link is reduced to one half; for significantly longer transmission distances, the transmission fraction will be very small (extinction and the 1/r2 antenna gain relation degrades the signal, which becomes obscured in noise.) There is another scale distance, w, which characterizes the distance over which the transmission fraction drops from near one to near zero. A parameterized formulation is used for the transmission fraction from node i to node j: where d designates the distance from node i to node j. The transmission as a function of distance is shown in Fig. 3 for d = 5000 m and w = 100 m. If the path from a node to the collection point contains a sequence of links, the total transmission fraction from the node to the collection point is simply the product of the transmission fractions of each link in the path.
Fig. 3. The transmission fraction across a link, as a function of the length of the link, using the parameterized formulation with d=5000 m, The fitness of a network configuration can be characterized in a variety of ways, depending on what is important in any given application. For the test-bed application, the network configuration fitness is simply the average, over all nodes, of the transmission loss from each node to the collection point. The goal of maximizing transmission fraction is then equivalent to a goal of minimizing message transmission loss. For the random network shown in Fig. 2, the average message loss is 0.94594.
The shortest-path algorithm was used to generate the network that minimizes the transmission losses from each node to the collection point. The shortest-path tree algorithm minimizes the sum of the weights associated with all edges in the paths from any node to the root node. The algorithm can be used by equating the weight associated with an edge to the natural log of the inverse of the transmission fraction. Then minimizing the sum of the weights along a path is equivalent to maximizing the total transmission fraction along the path. The shortest-path network is shown in Fig 4, where the sensors have the same locations as in Fig. 2. For this network configuration, the average message loss fraction is 0.03185, which is representative of 25 sensors in a 10 by 20 km elliptical area, where the transmission range scale is 5 km.
Fig. 4. The shortest-path tree for 25 randomly located nodes.
The dynamic environment is implemented as follows. A simulation is run in which the time is advanced in steps of durationτ . During each time step, the fitness of one network configuration is evaluated, as prescribed by the KGA. In addition, during each time step, a sensor may move, causing a change in the environment. In order to explore sensitivity of the KGA to process noise, the expected change in fitness during a time step can be varied by controlling how often sensors are moved.
The process noise rate is then calculated adaptively, using a fixed-gain recursive filter and the observed fitness changes caused by the sensor movements. If, for example, sensors are moved (to one of the eight nearest-neighbor cell locations) at a rate of one movement per 500 time steps, the process noise variance is found to average approximately 8.51E-11 in a time step. During the 100 time steps which would be required to re-evaluate all members of a 100-member population, the process noise would be ~10E-8, corresponding to a typical fitness change of ~0.0001. This process noise level is about one part in 300 of typical near-optimal network configurations in the baseline scenario. In a simulation starting with the sensor configuration shown in fig. 2 and incurring 500 sensor movements, the fitness of the shortest-path tree network configuration varies over the range of 0.0295 to 0.0335. If the links in the initial shortest-path network configuration were left unchanged during these 500 sensor movements, the fitness would be significantly degraded.
The observation noise, R, is implemented as follows. The transmission fraction over a link depends not only on the length of the link, but also on stochastic factors such as the atmospheric conditions and line-of-sight obstructions. These stochastic factors contribute to the observation noise that gives variation in the actual fitness of a network configuration. In the test-bed, process noise is added to average transmission fraction of a network configuration whenever it is evaluated. An evaluation of the fitness of a given network configuration for a given set of node positions is artificially given an error, by adding (2r-1)sqrt(3R), where r is a random number between 0 and 1. This produces a distribution of observation noise with a An implementation of the genetic algorithm is used which takes a sequence of integers as its chromosome, where the integers are bounded in zero to a specified maximum value. In this case, the chromosome has S integers, each in the range of [0, S].
The population size is 100 individuals. The first 100 individuals are created as random tree configurations. When the algorithm determines that a new trial solution should be generated, two parents are selected. This selection is based on fitness rank, with the most fit individual being 2.5 times more likely to be selected than the median fit individual. Generation of new individuals uses two-point cross-over, followed by mutation and a repair step. The cross-over causes complete genes (i.e.
integers in 0 to S) to be taken from the parents into the child: there is no splitting of genes. The mutation step selects two genes at random, and resets them to new random values, corresponding to a receiver node selected at random from those connected to the collection point. The repair step ensures that the resulting network configuration is an acyclic spanning tree rooted on the collection point. Every gene that is part of a valid tree is retained – those that are not are replaced with alleles that correspond to a link to a connected node. If the algorithm determines that an existing individual should be re-evaluated, 4. RESULTS
A simulation was run in which the observation noise variance, R, of the fitness (the message loss fraction) was set to 1E-8.
Since the message loss fraction of good configurations is around 0.03, the standard deviation of the observation noise of 1E-4 is about one part in 300, which can be considered to be a low level of observation noise. For this case, the process noise variance is approximately 8.51E-11 per time step, which is achieved by moving the sensors at the rate of one movement per 500 time steps. As described above, for a population size of 100, this process noise standard deviation is below one part in 300, which can be considered to be a low level of process noise. Fig 5 shows the fractional error of the actual fitness (average message loss rate, with observation noise removed) of the best member (lowest uncertainty-compensated message loss fraction) of the population relative to the fitness of the shortest-path network configuration, i.e. (f-f )/f . Initially, when the KGA has evaluated only one random configuration (that shown in Fig. 2) the loss of the best configuration is 0.94594, while the loss with the optimal configuration is 0.03185, so the fractional error of the KGA relative to optimal has a high initial value of 2870%. As the KGA generates and evolves its population of trial solutions, the fractional error is seen to drop. After about 20,000 trial evaluations, the best solution in the KGA population is within 0.1% of the optimal solution. In this case, the KGA often attains the optimal solution.
During the 250,000 time steps of the simulated run, the KGA evaluated 100 random configurations, generated and evaluated 55,674 new configurations, and re-evaluated existing configurations 194,226 times. When the observation noise is low, the algorithm spends 22% of its resources on new trial solutions, and 78% on re-evaluating existing solutions. Note that once a population has converged somewhat on the optimal configuration, when a new trial is generated, it will have a larger uncertainty than the bulk of the population. If the fitness of the new trial is good enough to earn it a place in the population, the algorithm will then re-evaluate the new trial until its uncertainty drops below the population variance.
Several events can be seen in fig. 5 as spikes in the fractional error. These occur when the motion of the sensors causes the optimal network configuration to undergo a significant restructuring. The fractional error between the best KGA solution and the optimal solution may rise to a few percent on these occasions, but then drops back down as the KGA discovers the new configuration. Note that these recoveries require one or two thousand time steps, compared to the 20,000 time steps required to reach near-optimal solutions beginning from scratch.
Number of Evaluations
Fig. 5. The fractional error of the message loss fraction of the best KGA solution relative to that of the optimal network configuration, showing how closely the KGA tracks the optimal network configuration. There is one trial evaluation per time step. The process noise variance is approximately 10E-10 per timestep, and observation noise variance is 10E-8, for 25 sensor network starting from the locations shown in Fig. 2. The loss fraction of the optimal configuration ranges from 0.0295 to 0.0334.
A second case was run in which the process noise was increased by a factor of about 100. This was accomplished by moving the sensors at a rate of one sensor motion every 50 time steps, which results in a process noise variance that averaged 9.26E-9 per time step. This corresponds to a process noise standard deviation of about 0.001 in 100 time steps (one for each member of the population), which is about 3% of the fitness of near-optimal configurations. As shown in fig. 6, this increased process noise degrades the ability of the KGA to track the optimal network configuration. Where the optimal configuration has a message transmission fraction that varies from 0.0264 to 0.0357, with an average value of 0.02978 (excluding the first 20,000 time steps), the KGA has an average message loss fraction of 0.03093 during the same period. On average the KGA solution is 3.9% worse than the optimal configuration, excluding the first 20,000 time steps, at this process niose level of 3%. For this case, there is little change in the resource allocation between re-evaluation and new trial generation: the KGA called for new trial solutions in 21.1% of time steps.
A third case was run which was the same as the first, except that the observation noise variance was increased by a factor of 100, to a moderate level of 1.E-6. The process noise remained at the low level of 1E-10 per time step. The observation niose standard deviation (0.001) is about 3% of the fitness of near-optimal configurations for this third case. Fig. 7 shows how the KGA is able to track the optimal configuration at this noise level. After an initial period of 20,000 time steps, the KGA configuration is on average 1.92% worse than the optimal configuration. At this observation noise level, significantly more resources are allocated to re-evaluation of existing trials: new trial configurations are generated and evaluated in only 8.1% of Number of Evaluations
Fig. 6. The fractional error of the message loss fraction of the best KGA solution relative to that of the optimal network configuration. The case is identical to that shown in Fig 5, except that the process noise variance is increased by a factor of 100 to a value of approximately Number of Evaluations
Fig. 7. The fractional error of the message loss fraction of the best KGA solution relative to that of the optimal network configuration. The case is identical to that shown in Fig 5, except that the observation noise variance, R, is increased by a factor of 100 to a value of 5. CONCLUSIONS
The overall conclusion of this work is that the Kalman formulation can be integrated into the genetic algorithm formulation in a compelling way to create a Kalman-extended genetic algorithm that can perform ongoing directed searches in dynamic, noisy environments. The extension of a population of individuals, each with a fitness, by associating an uncertainty with the fitness of each individual, is conceptually clear, and easy to implement. The mechanics of updating the best-estimated fitness and the uncertainty after a re-evaluation follow directly from the Kalman formulation. The very simple implementation of the KGA, where a new trial is generated and evaluated whenever the uncertainty of the most uncertain individual is less than the variance of the population fitnesses, has been found to work, through experimentation in a test-bed.
The KGA can be viewed as a mechanism to allocate resources between the two complimentary activities of re-evaluating existing solutions for changing conditions, and generating new solutions. The allocation depends on the level of observation noise: the more uncertain the observations are, the more resources must be dedicated to re-evaluation of existing solutions.
Even when observation noise is low, however, a somewhat surprising result is that only 21 or 22% of effort should be expended on new solutions, while the remainder should go to re-evaluating existing solutions.
At low levels of process and observation noise (±0.3%), the KGA was shown to be able to maintain near-optimal solutions.
At moderate observation noise level (±3%), the KGA was still able to maintain configurations with fitness values within an average of 2% of the optimal configuration. At moderate process noise level (±3%), the KGA was able to maintain configurations with fitness values within an average of 4% of the optimal configuration.
In contrast to the traditional GA wherein a bigger population is better, in the KGA there is an optimal population size. If the population were too large, the KGA would waste time evaluating relatively poor individuals that have grown obsolete (i.e.
whose uncertainty has increased too much). There are indications that the optimal population size can be re-evaluated on an ongoing basis, using R, q τ , and V.
The formulation presented here was developed with simplicity as a criterion. Alternative forms may be more effective. For example, the uncertainty-compensated fitness, which is used to rank the individuals in the population, could be a more complex function of the best-estimated fitness, the uncertainty, and/or the square root of the uncertainty. The criteria for selecting which individual to evaluate next could depend on the best-estimated fitness in addition to the uncertainty. The population fitness variance could also be formulated differently.
When a new trial is generated and evaluated, its uncertainty is generally much higher than that of the existing individuals.
The new trial should therefore be re-evaluated several times to reduce its uncertainty to a level at which a valid comparison can be made to the existing individuals. Less fit trials can be discarded at higher level of uncertainty, improving the efficiency of the algorithm. This is an avenue for improvement of the formulation presented here, in which a new trial is compared with the population after only a single fitness evaluation.
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