3 Artificial neural networks

Neural networks have proven useful in a number of scientific disciplines for interpolating multidimensional data, and thus providing a nonlinear mapping between an input domain (in this case the DISPIs) and an output domain (the stellar parameters). For an overview of Artifical Neural Networks (ANNs) and their application in astronomy for stellar classification see, for example, Bailer-Jones (2002). The software used in this work is that of Bailer-Jones (1998).

A network consists of an input layer, one or two hidden layers and an output layer. Each layer is made up of several nodes. All the nodes in one layer are connected to all the nodes in the preceding and/or following layers. These connections have adaptable "weights", so that each node performs a weighted sum of all its inputs and passes this sum through a nonlinear transfer function. That weighted sum is then passed on to the next layer. Before the network can be used for parametrisation, it needs to be trained, meaning the weights have to be set to their appropriate values to perform the desired mapping. In this process, DISPIs together with known stellar parameters as target values are presented to the network. From these data, the optimum weights are determined by iteratively adjusting the weights between the layers to minimize an output error, i.e. the discrepancy between the targets and the network outputs. This is performed by a multidimensional numerical minimization, in this case with the conjugate gradients method. When this minimization converges, the weights are fixed and the network can be used in its "application" phase: now, only the DISPI input flux vector is presented and the network's outputs produce the stellar parameters of these DISPIs. Since we used only the central 51 effective pixels of the DISPIs (range 30 to 80, see Figs. 3 and 4), the input layer of the network was always made up of the same number of nodes, i.e. 51. We found that the performance was best when using two hidden layers, each containing 7 nodes. More nodes did not improve the result significantly but increased the training time considerably. With four output parameters this network then contains $51 \cdot 7 + 7 \cdot 7 + 7 \cdot 4 = 434$ weights (plus 18 bias weights).

Since we wanted to classify DISPIs solely based on their shapes, the absolute flux information was removed by area-normalizing each DISPI, i.e. each flux bin of a given DISPI was divided by the total number of counts in that DISPI. Given the non-uniform distribution of the training data over $T_{\rm eff}$, we classified DISPIs in terms of log  $T_{\rm eff}$ instead of $T_{\rm eff}$.

Note that, in our tests, we have not included distance information as it eventually might be done using DIVA parallaxes, since the present goal was to test the retrieval of stellar parameters from DISPIs only.

The parametrization errors given below are the average (over some set of DISPIs) errors for each parameter, i.e.

$$A = \frac{1}{N} \cdot \sum_{p=1}^N \left\vert C(p) - T(p)\right\vert$$ (1)

where p denotes the pth DISPI and Tis the target (or "true") value for this parameter. Since the network's function approximation can depend on the initial settings of the weights, it is sometimes recommended to use a "committee" of several networks with identical topologies but different initializations. The quantity C(p) is the classification output averaged over a committee of three networks.