By Anne Fanatico
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Extra resources for Alternative poultry production systems and outdoor access
The true regression y(x) is given by the equation y(x) = sin(3x) sin(5x), where x is a uniformly distributed random number from the interval [0, π /2]. 2 The target value for training is d(x) = y(x) + n(x). 2 Heskes  used similar trigonometric functions for the true regression and the noise variance. V. Healy Fig. 1 Prediction bands for Example #1. Here, y(x) is the true regression. d(x) are the target data points. y∗ (x) is the estimate of the true regression. L and U are the true lower and upper prediction intervals and L∗ and U ∗ are the estimated prediction intervals obtained using σ ∗2 (x), the network estimate of the noise variance function The following procedure was adopted.
Improved performance in low noise regions of the input space is claimed, due to the form of weighted regression used. However, the use of weighted regression means the standard errors obtained must be interpreted with caution. This is because inference for NNs and other computational intelligence techniques rests on the assumption that an NLLS regression is performed, as discussed in Sect. 2. In weighted regression a penalty term is added to the least squares cost function and this may not be the case.
Bishop, Neural Networks for Pattern Recognition (Clarendon Press, Oxford, 1995) 4. M. S. Qazaz, Bayesian inference of noise levels in regression. Technical report, Neural Computing Research Group, Aston University (1995) 5. M. Dahl, S. Hylleberg, Flexible regression models and relative forecast performance. Int. J. Forecast. 20(2), 201–217 (2004) 6. B. J. Tibshirani, An Introduction to the Bootstrap (Chapman and Hall, New York, 1993) 7. F. Girosi, T. Poggio, Networks and the best approximation property.