Solar Particle Event Doses and Dose Rates for Interplanetary Crews: Predictions Using Artificial Intelligence and Bayesian Inference

For deep space missions, a major concern is the occurrence of large solar particle events (SPE) which can deliver doses to critical body organs in excess of 10 Gy at dose rates exceeding 1 Gy h over a period of several hours. Accurately predicting the likelihood of occurrence of these events before they begin has been an ongoing problem throughout the history of manned spaceflight. Recently, efforts to predict doses and dose rates from large SPEs using only dosimeter readings obtained early in the evolution of an event have been undertaken. Studies using artificial intelligence and Bayesian inference have been initiated. An innovative, artifiical neural network has been developed to predict total doses from SPEs early in an event, using only doses obtained at times early in that event. The network makes accurate predictions for large events, such as those of August and September 1989. The network is capable of updating its predictions as additional dose inputs are received. A second thrust has been to implement Bayesian inference techniques with Markov Chain Monte Carlo (MCMC) methods to calculate posterior parameter and dose rate distributions as well as predictive distributions for future observations and dose rates. In this work preliminary results obtained using these two methodologies are presented. INTRODUCTION In earlier work (Tehrani et al., 1999) it was demonstrated that an artificial neural network (ANN) could be trained and tested with dose data for simulated SPEs, and then used to predict total doses to interplanetary-mission crews from SPEs in a simulated real-time scenario. In most cases the network accurately predicted the total doses from large events, such as the August or September 1989 events, to within 4% very early in the event. The method uses an innovative network, called a Sliding Time Delay Neural network (STDNN), which is capable of updating its predictions during an event, as additional input dose data become available during the event. The STDNN, in its current form, cannot predict dose rates or the dose versus time profile for an event. Previous work (Zapp et al., 1999; Parsons and Townsend, 2000) has successfully modeled Solar Particle Event (SPE) dose and dose rate-time profiles for skin, the lens of the eye, and the blood forming organs using Weibull growth curve parameterizations utilizing least squares regression techniques. Also, Weibull, logistic, and Gompertz growth models have been used to fit SPE dose and dose rate-time profiles for skin and the lens of the eye utilizing Bayesian inference techniques as implemented by Markov Chain Monte Carlo (MCMC) methods (Neal and Townsend, 2000). Additionally, these earlier investigations used fits of dose data throughout the SPE to make model the parameter estimates. A more ambitious problem is that of predicting doses and dose rates from a subset of the data. This work reports preliminary results of efforts to predict both doses and dose rates from large SPEs using only dosimeter readings obtained early in the evolution of an event. In this paper, current methods under development using ANN and Bayesian inference to model SPE dose profiles are described and preliminary results reported. For the Bayesian studies, we use the deterministic, coupled neutron-proton space radiation computer code, BRYNTRN (Wilson et al., 1991), to transport SPE protons through a 1g/cm aluminum shield and to calculate the subsequent dose at the surface of the shield for the September 1989 and March 1991 SPEs. SLIDING TIME DELAY NEURAL NETWORK Multilayer Perceptrons (MLP) are feedforward neural networks in which the neurons are arranged in layers. Neurons from one layer are connected to neurons in adjacent layers with unidirectional links that are represented by weights, which act as signal multipliers. Layers between the input and output layers are called hidden layers The neurons in the input layer do not perform calculations, but feed signals to neurons in the first hidden layer, which feed the second hidden layer, and so forth until the output layer is reached. Although there is no theoretical limit on the number of hidden layers, in practice, one or two layers are used to reduce the chance of overfitting. In a feedforward MLP each neuron has one output and multiple inputs. Connections from higher layers to lower layers are not permitted, neither are connections between neurons in the same layer. Standard MLPs process a static mapping. In this work the dose profiles temporal in nature. The final dose may be estimated by using doses sampled from several different times. ANNs that do temporal processing are called Time Delay Neural Networks. This type of network was originally designed for speech recognition. These networks are MLPs in which the inputs are both past and present signals. By using present and past inputs, temporal behavior is learned. In the standard TDNN the time delays are constant. The network type developed for the work reported herein is the Sliding TimeDelay Network (STDNN). This is a variation of the TDNN. In the STDNN the time delays are variable and not fixed. The inputs are the values of the time dependent function f(t), with τ being the fractional size of an arbitrary time interval T, where T=nτ, and n is the number of input neurons. There are no feedback terms in an STDNN so training is accomplished using standard algorithms. Details of the network and how it is trained and used are found in Tehrani et al. (1999) and references therein. BAYESIAN INFERENCE In this work, inference techniques utilize Bayes’ Theorem to update probabilistic beliefs about model parameters after observing data. Bayes’ Theorem may be stated as where p(θ|D), the joint posterior distribution, is a probabilistic statement about θ, the parameter vector, after observing data, D; p(D|θ) is the likelihood of the data given the parameter(s); p(θ), the prior distribution, is a probabilistic statement of belief about θ before observing data; p(D) is the marginal distribution of the data. Marginal (individual) parameter posterior distributions require integration of the joint parameter posterior distribution over all other parameters. Parameter inference may be a useful end to itself, but it may also be viewed as an intermediate result in the process of predictive inference of future observables. The posterior predictive distribution for an observable y is given as where the posterior predictive distribution is shown to be an average of conditional predictions over the posterior distribution of θ. Because this work models SPE dose-time profiles with growth curves which use physically meaningful, asymptotic dose parameters, both parameter and predictive inference techniques produce useful results. The models in this work assumed normally distributed errors. Elicitation of parameter prior distributions for the Weibull model drew on previous work (Zapp et al., 1998) which allowed construction of uniform distributions for all growth curve parameters. A review of several SPE, other than those considered here, allowed construction of ∫ = θ θ θ d D p y p D y p ) | ( ) | ( ) | ( Weibull t D D ) ) ) ) ( exp( 1 ( γ α − − = ∞ Logistic rt C K D )) exp( 1 /( − + = Gompertz D t ) exp( βγ α − = ) ( ) ( ) | ( ) | ( D p p D p D p θ θ θ = uniform distributions for logistic and Gompertz growth curve parameters. The precision (inverse variance) parameter was taken to be distributed as a gamma distribution with shape parameter equal to 2.0 and scale parameter equal to zero, yielding a 1/(variance) prior distribution. MARKOV CHAIN MONTE CARLO METHODS Calculation of marginal and predictive distributions requires integration of potentially complex functions for which no analytical solution exists. MCMC methods create a Markov process whose stationary distribution is the joint posterior distribution, run the Markov process long enough so that the distribution of draws is close to the stationary distribution, also known as convergence, and then sample from the stationary distribution to approximate the posterior distribution of interest. For this investigation, the Bayesian Inference Using Gibbs Sampling (BUGS) software package (Spiegelhalter et al., 2000) is used for parameter and predictive inference and the Bayesian Output Analysis (BOA) software package (Smith, 2000) is used for monitoring convergence. Slice sampling methods (Neal, 1997) are used for sampling from distributions. The Raferty and Lewis convergence diagnostic (Raferty and Lewis, 1992) and the Heidelberger and Welch convergence diagnostic (Heidelberger and Welch, 1983) are used for this investigation. Both of these diagnostics attempt to uncover bias from a sample that is not representative of the underlying distribution, and both attempt to determine the number of samples to be drawn to produce estimates with small enough variance to meet some predefined accuracy. Model Assessment/Comparison After sampling from posterior distributions of interest, it is prudent to perform model assessment, checking how well a given model fits the data, and model comparison, checking different models to determine which best meets the comparison criteria. Model assessment is performed using posterior predictive checks, which examine the percentage of actual observations that fall within the predicted 50 and 95 percent confidence intervals. Model comparison checks are performed using the negative cross-validatory log-likelihood where y\i is the rest of the data excluding yi. This method, based on the “pseudo Bayes factor” (Gelfand and Dey, 1994), provides comparisons based on cross-validatory measures and is considered appropriate since the Weibull, logistic, and Gompertz models are only considered reasonable approximations of the true model. RESULTS Sliding Time Delay Neural Network Figures 1 and 2 display predictions of total dose (asymptotic dose) versus time since event onset for the August and September 1989 SPEs. Note that the STDNN predictions are within several percent after only a few hours of event data (4.5 hours for the September event and 2.5 hours for the August event).