Fuzzy Wavelet Neural Network Using a Correntropy Criterion for Nonlinear System Identification

Recent researches have demonstrated that the Fuzzy Wavelet Neural Networks (FWNNs) are an efficient tool to identify nonlinear systems. In these structures, features related to fuzzy logic, wavelet functions, and neural networks are combined in an architecture similar to the Adaptive Neurofuzzy Inference Systems (ANFIS). In practical applications, the experimental data set used in the identification task often contains unknown noise and outliers, which decrease the FWNN model reliability. In order to reduce the negative effects of these erroneous measurements, this work proposes the direct use of a similarity measure based on information theory in the FWNN learning procedure. The Mean Squared Error (MSE) cost function is replaced by the Maximum Correntropy Criterion (MCC) in the traditional error backpropagation (BP) algorithm. The input-output maps of a real nonlinear system studied in this work are identified from an experimental data set corrupted by different outliers rates and additive white Gaussian noise. The results demonstrate the advantages of the proposed cost function using the MCC as compared to the MSE. This work also investigates the influence of the kernel size on the performance of the MCC in the BP algorithm, since it is the only free parameter of correntropy

Expression profile of the target genes SvAP3/PI and SvPEPC.

<p>The plot corresponding to the expression profile of the MADS-box gene SvAP3/PI (magenta bars) and the enzyme SvPEPC (green bars) responsible for the initial carbon fixation on C4 photosynthesis is shown in (A). The developmental dataset was normalized using the Si034613 and Si035045 genes. It can be observed in (B) that the expression profile of SvPEPC on the leaf gradient was normalized using the gene pair that included Si021373 and Si034613. This result is compared with the transcriptome data (light blue line) for the same gene (personal communication with Todd Mockler). The small plots correspond to the same described target genes normalized with the poorly ranked reference genes: Si018607 and Si025395 for the developmental set and Si000245 and Si018607 for leaf gradient. The reference samples are indicated with an asterix.</p

Candidate genes and their primer sequences that were selected for evaluation of expression stability using qPCR analysis on <i>S</i>. <i>viridis</i> tissues, and the sequences of two genes of interest.

<p>¹Score and identity values assessed through alignment between S. italica sequences and S. viridis sequences (TM personal communication). NA indicates non-available data due to missing sequence of S. viridis</p><p>Candidate genes and their primer sequences that were selected for evaluation of expression stability using qPCR analysis on <i>S</i>. <i>viridis</i> tissues, and the sequences of two genes of interest.</p

RNA integrity profile.

<p>The assessment of the best methodology for RNA extraction for each representative tissue: seedling (A); shoot (B); root (C), inflorescence (D) and axis (E). Electropherograms were obtained using an Agilent 2100 Bioanalyzer. RNA quality is expressed as the RNA integrity number (RIN).</p

geNorm results for expression stability values (M) and ranking of the candidate reference genes.

<p>Average expression stability values (M) of the reference genes measured during geNorm stepwise exclusion of the least stable reference genes. Both the developmental (A) and leaf gradient (B) datasets are shown; lower values of average expression stability, M, indicate more stable expression.</p

Validating Internal Control Genes for the Accurate Normalization of qPCR Expression Analysis of the Novel Model Plant <i>Setaria viridis</i>

<div><p>Employing reference genes to normalize the data generated with quantitative PCR (qPCR) can increase the accuracy and reliability of this method. Previous results have shown that no single housekeeping gene can be universally applied to all experiments. Thus, the identification of a suitable reference gene represents a critical step of any qPCR analysis. <i>Setaria viridis</i> has recently been proposed as a model system for the study of Panicoid grasses, a crop family of major agronomic importance. Therefore, this paper aims to identify suitable <i>S</i>. <i>viridis</i> reference genes that can enhance the analysis of gene expression in this novel model plant. The first aim of this study was the identification of a suitable RNA extraction method that could retrieve a high quality and yield of RNA. After this, two distinct algorithms were used to assess the gene expression of fifteen different candidate genes in eighteen different samples, which were divided into two major datasets, the developmental and the leaf gradient. The best-ranked pair of reference genes from the developmental dataset included genes that encoded a phosphoglucomutase and a folylpolyglutamate synthase; genes that encoded a cullin and the same phosphoglucomutase as above were the most stable genes in the leaf gradient dataset. Additionally, the expression pattern of two target genes, a SvAP3/PI MADS-box transcription factor and the carbon-fixation enzyme PEPC, were assessed to illustrate the reliability of the chosen reference genes. This study has shown that novel reference genes may perform better than traditional housekeeping genes, a phenomenon which has been previously reported. These results illustrate the importance of carefully validating reference gene candidates for each experimental set before employing them as universal standards. Additionally, the robustness of the expression of the target genes may increase the utility of <i>S</i>. <i>viridis</i> as a model for Panicoid grasses.</p></div

Expression profile of the target genes SvAP3/PI and SvPEPC.

<p>The plot corresponding to the expression profile of the MADS-box gene SvAP3/PI (magenta bars) and the enzyme SvPEPC (green bars) responsible for the initial carbon fixation on C4 photosynthesis is shown in (A). The developmental dataset was normalized using the Si034613 and Si035045 genes. It can be observed in (B) that the expression profile of SvPEPC on the leaf gradient was normalized using the gene pair that included Si021373 and Si034613. This result is compared with the transcriptome data (light blue line) for the same gene (personal communication with Todd Mockler). The small plots correspond to the same described target genes normalized with the poorly ranked reference genes: Si018607 and Si025395 for the developmental set and Si000245 and Si018607 for leaf gradient. The reference samples are indicated with an asterix.</p

Optimal number of control genes for an accurate normalization.

<p>The geNorm pairwise variation <i>V</i>_{<i>n</i>(<i>n</i>+1)} was analyzed between the normalization factors (NF) for both developmental (A) and leaf gradient (B) datasets. All values are below the cutoff of 0.15.</p

Candidate genes ranked by geNorm algorithm according to their average pairwise variation compared with all other genes.

<p>Stability values are listed from the most stable to the least stable.</p><p>Candidate genes ranked by geNorm algorithm according to their average pairwise variation compared with all other genes.</p