Wavelet network

Zhang, Q., and Benveniste, A., Wavelet networks. IEEE Trans. Neural Networks 3, 889 (1992). 

Oussar, Y., Rivals, L, Personnaz, L., Dreyfus, G. Training wavelet networks for nonlinear dynamic input-output modeling. Neurocomputing 20, 173-188 (1998)... 

The output is calculated in two steps first, the input and output signals are delayed to different degrees. Second a nonlinear aetivation fimetion /( ) (here a static neural network) estimates the output. In (Nelles 2001) a sigmoid fimetion is proposed for the nonlinear activation function, which is used in this eontext. Other fimetions for nonlinear dynamie modeling e.g. Ham-merstein models, Wiener models, neural or wavelet network are also possible. 

Kruger, V., Bruns, S. and Sommer, G. (2000) Efficient head pose estimation with Gabor wavelet networks. In Proceedings of the IF British Machine Vision Conference, University of Bristol. UK, pp. 72-81. 

Calculates w(t) = f u t)) from the input data. w(t) is an input to the linear transfer function B/F. The input nonlinearity is a static (memoryless) function, where the value of the output a given time t depends only on the input value at time t. The input nonlinearity can be set as a sigmoid network, wavelet network, saturation, dead zone, piecewise linear function, one-dimensional polynomial, or a custom network. It is possible to remove the input nonlinearity. 

The space-frequency localization of wavelets has lead other researchers as well (Pati, 1992 Zhang and Benveniste, 1992) in considering their use in a NN scheme. In their schemes, however, the determination of the network involves the solution of complicated optimization problem where not only the coefficients but also the wavelet scales and positions in the input space are unknown. Such an approach evidently defies the on-line character of the learning problem and renders any structural adaptation procedure impractical. In that case, those networks suffer from all the deficiencies of NNs for which the network structure is a static decision. 

Bakshi, B., Koulouris, A., and Stepanopoulos, G., Learning at multiple resolutions Wavelets as basis functions in artificial nemal networks and inductive decision trees. In Wavelet Applications in Chemical Engineering (R. L. Motard and B. Joseph, eds.) Kluwer Academic Publishers, Dordrecht/Norwell, MA, p. 139 (1994). 

Pati, Y. C., Wavelets and time-frequency methods in linear systems and neural networks. 

Strang. G., Wavelets and dilation equations A brief introduction. SIAM Rev. 31, 614 (1989). Ungar, L. H., Powell, B. A., and Kamens, S. N., Adaptive Networks for fault diagnosis and process control. Comput. Chem. Eng. 14, 561 (1990). 

E.R. Collantes, R. Duta, W.J. Welsh, W.L. Zielinski and J. Brower, Reprocessing of HPLC trace impurity patterns by wavelet packets for pharmaceutical finger printing using artificial neural networks. Anal. Chem. 69 (1997) 1392-1397. 

Zhang et al.14 develop a neural network approach to bacterial classification using MALDI MS. The developed neural network is used to classify bacteria and to classify culturing time for each bacterium. To avoid the problem of overfitting a neural network to the large number of channels present in a raw MALDI spectrum, the authors first normalize and then reduce the dimensionality of the spectra by performing a wavelet transformation. 

Other approaches Hamming networks, pattern recognition, wavelets, and neural network learning systems are sometimes discussed but have not been commercially implemented. 

Since in many applications minor absorption changes have to be detected against strong, interfering background absorptions of the matrix, advanced chemometric data treatment, involving techniques such as wavelet analysis, principle component analysis (PCA), partial least square (PLS) methods and artificial neural networks (ANN), is a prerequisite. 

Also, we do not cover several typical chemometrics types of analyses, such as cluster analysis, experimental design, pattern recognition, classification, neural networks, wavelet transforms, qualimetrics etc. This explains our decision not to include the word chemometrics in the title. 

Konig R, Schramm G, Oswald M et al (2006) Discovering functional gene expression pattern in the metabolic network of Escherichia coli with wavelets transforms. BMC Bioinformatics 7 119... 

The evaluation of the measurements, the correlation between the medium components and the various ranges of the 2D-fluorescence spectrum was performed by Principal Component Analysis (PCA), Self Organized Map (SOM) and Discrete Wavelet Transformation (DWT), respectively. Back Propagation Network (BPN) was used for the estimation of the process variables [62]. By means of the SOM the courses of several process variables and the CPC concentration were determined. 

Piotrowski PL, Sumter BG, Mailing HV, et al. A toxicity evaluation and predictive system based on neural networks and wavelets. / Chem Inf Model. 2007 47 676-685. 

As to be expected there are a number of other methods and variants in addition to those described above. Recent examples include the utilization of Bayesian methods for class recognition, wavelet transforms, artificial neural networks (ANN) in biomedical NMR... 

Foy, B.R. and Theiler, J., Scene analysis and detection in thermal infrared remote sensing using independent component analysis. Independent Component Analyses, Wavelets, Unsupervised Smart Sensors, and Neural Networks 11, Proceedings of SPIE vol. 5439, 131-139(2004)... 

Nonlinear PCA To address the nonlinearity in the identity mapping of multivariate data, a nonlinear counterpart of the PCA can be used (see Section 3.6.1). As the versions of NLPCA make use of the neural network (NN) concept to address the nonlinearity, they suffer from the known overparameterization problem in the case of noise corrupted data. Data with small SNR will also give rise to extensive computations during the training of the network. Shao et al. [266] used wavelet filtering to pre-process the data followed by IT-net to detect the non-conforming trends in an industrial spray drier. 

Wavelets transforms are useful for compression of descriptors for searches in binary descriptor databases and as alternative representations of molecules for neural networks in classification tasks. 

The reduction of the descriptor size (i.e., the decrease in resolution) usually has a profound influence on the ability of the descriptor to characterize a molecule. Even though compressed, or filtered, wavelet transforms of descriptors have a reduced size, they preserve the similarity information well and in a much more efficient way. Figure 6.15 shows results from an experiment where a Kohonen neural network classifies the same data set (100 benzene derivatives plus 100 monocyclic cyclohexane derivatives) according to ring type. 

Tabaraki, R., Khayamian, T. and Ensafi, A.A. (2006) Wavelet neural network modeling in QSPR for prediction of solubility of 25 anthraquinone dyes at different temperatures and pressures in supercritical carbon dioxide./. Md. Graph. Model., 25, 46-54. 

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