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The predictions are then averaged and output on the right side. All inputs and outputs are optional. For example, surface gravities, luminosities, and radii are not always available from observations e.
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In their absence, these quantities can be predicted instead of being supplied. In this case, those nodes can be moved over to the "prediction" side instead of being on the "observation" side. In addition to potentially unobserved inputs like stellar radii, other interesting model parameters can be predicted, such as core hydrogen mass fraction or surface helium abundance. The CART algorithm uses information theory to decide which rule is the best choice for inferring stellar parameters like age and mass from the supplied information Hastie et al.
At every stage, the rule that creates the largest decrease in mean squared error MSE is crafted.
We moreover use a variant on random forests known as extremely randomized trees Geurts et al. The process of constructing a random forest presents an opportunity for not only inferring stellar parameters from observations, but also for understanding the relationships that exist in the stellar models.
Each decision tree explicitly ranks the relative "importance" of each observable quantity for inferring stellar parameters, where importance is defined in terms of both the reduction in MSE after defining a decision rule based on that quantity and the number of models that use that rule.
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In machine learning, the variables that have been measured and are supplied as inputs to the algorithm are known as "features. The features that are used most often to construct decision rules are metallicity and temperature, which are each significantly more important features than the rest. Bellinger et al. Note that importance does not indicate indispensability: an appreciable fraction of decision rules being made based off one feature does not mean that another forest without that feature would not perform just as well.
That being said, these results indicate that the best area to improve measurements would be in metallicity determinations, because for stars being predicted using this random forest, less precise values here means exploring many more paths and hence arriving at less certain predictions.
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Figure 4. Relative importance of each observable feature in inferring fundamental stellar parameters as measured by a random forest regressor grown from a grid of evolutionary models. For example, the KOI data set see Section 3. We therefore must train random forests that predict those quantities instead of using them as features. We show the relative importance for the remaining features that were used to train these forests in Figure 5. Figure 5.
Box-and-whisker plots of relative importance for each feature in measuring fundamental stellar parameters for the hare-and-hound exercise data left , where luminosities are available; and the Kepler objects-of-interest right , where they are not. Octupole modes have not been measured in any of these stars, so and from evolutionary modelling are not supplied to these random forests. The boxes are sorted by median importance.
We choose random forests over any of the many other nonlinear regression routines e.
First, random forests perform constrained regression; that is, they only make predictions within the boundaries of the supplied training data see e. This is in contrast to other methods like neural networks, which ordinarily perform unconstrained regression and are therefore not prevented from predicting non-physical quantities such as negative masses or from violating conservation requirements.
Second, due to the decision rule process that is explained below, random forests are insensitive to the scale of the data. Unless care is taken, other regression methods will artificially weight some observable quantities like temperature as being more important than, say, luminosity, solely because temperatures are written using larger numbers e.
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Consequently, solutions obtained by other methods will change if they are run using features that are expressed using different units of measure. For example, other methods will produce different regressors if trained on luminosity values expressed in solar units verses values expressed in ergs, whereas random forests will not. Commonly, this problem is mitigated in other methods by means of variable standardization and through the use of Mahalabonis distances Mahalanobis However, these transformations are arbitrary, and handling variables naturally without rescaling is thus preferred.
Thirdly, random forests take only seconds to train, which can be a great benefit if different stars have different features available. For example, some stars have luminosity information available whereas others do not, so a different regressor must be trained for each. In the extreme case, if one wanted to make predictions for stars using all of their respectively observed frequencies, one would need to train a new regressor for each star using the subset of simulated frequencies that correspond to the ones observed for that star.
Ignoring the difficulties of surface-term corrections and mode identifications, such an approach would be well-handled by a random forest, suffering only a small hit to performance from its relatively small training cost. On the other hand, it would be infeasible to do this on a star-by-star basis with most other routines such as deep neural networks, because those methods can take days or even weeks to train.
Finally, as we saw in the previous section, random forests provide the opportunity to extract insight about the actual regression being performed by examining the importance of each feature in making predictions. There are three separate sources of uncertainty in predicting stellar parameters. The first is the systematic uncertainty in the physics used to model stars.
These uncertainties are unknown, however, and hence cannot be propagated. The second is the uncertainty belonging to the observations of the star. We account for the covariance between asteroseismic separations and ratios by recalculating them upon each perturbation. The final source is regression uncertainty. Fundamentally, each parameter can only be constrained to the extent that observations are able to bear information pertaining to that parameter. Even if observations were error-free, there still may exist a limit to which information gleaned from the surface may tell us about the physical qualities and evolutionary history of a star.
We quantify those limits via cross-validation: we train the random forest on only a subset of the simulated evolutionary tracks and make predictions on a held-out validation set. We randomly hold out a different subset of the tracks 25 times to serve as different validation sets and obtain averaged accuracy scores. We calculate accuracies using several scores. The first is the explained variance score :. This score tells us the extent to which the regressor has reduced the variance in the parameter it is predicting.
The value ranges from negative infinity, which would be obtained by a pathologically bad predictor; to one for a perfect predictor, which occurs if all of the values are predicted with zero error. The next score we consider is the residuals of each prediction, i. Naturally, we want this value to be as low as possible. We also consider the precision of the regression by taking the standard deviation of predictions across all of the decision trees in the forest.
Finally, we consider these scores together by calculating the distance of the residuals in units of precision, i. Figure 6 shows these accuracies as a function of the number of evolutionary tracks used in the training of the random forest. Since the residuals and standard deviations of each parameter are incomparable, we normalize them by dividing by the maximum value. We also consider the number of trees in the forest and the number of models per evolutionary track.
In this work, we use trees in each forest, which we have selected via cross-validation by choosing a number of trees that is greater than the point at which we saw that the explained variance was no longer increasing greatly; see Appendix D for an extended discussion. Figure 6. Evaluations of regression accuracy. Explained variance top left , accuracy per precision distance top right , normalized absolute error bottom left , and normalized uncertainty bottom right for each stellar parameter as a function of the number of evolutionary tracks used in training the random forest.
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