Distinguishing noise from chaos: objective versus subjective criteria using Horizontal Visibility Graph

A recently proposed methodology called the Horizontal Visibility Graph (HVG) [Luque {\it et al.}, Phys. Rev. E., 80, 046103 (2009)] that constitutes a geometrical simplification of the well known Visibility Graph algorithm [Lacasa {\it et al.\/}, Proc. Natl. Sci. U.S.A. 105, 4972 (2008)], has been used to study the distinction between deterministic and stochastic components in time series [L. Lacasa and R. Toral, Phys. Rev. E., 82, 036120 (2010)]. Specifically, the authors propose that the node degree distribution of these processes follows an exponential functional of the form $P(\kappa)\sim \exp(-\lambda~\kappa)$, in which $\kappa$ is the node degree and $\lambda$ is a positive parameter able to distinguish between deterministic (chaotic) and stochastic (uncorrelated and correlated) dynamics. In this work, we investigate the characteristics of the node degree distributions constructed by using HVG, for time series corresponding to $28$ chaotic maps and $3$ different stochastic processes. We thoroughly study the methodology proposed by Lacasa and Toral finding several cases for which their hypothesis is not valid. We propose a methodology that uses the HVG together with Information Theory quantifiers. An extensive and careful analysis of the node degree distributions obtained by applying HVG allow us to conclude that the Fisher-Shannon information plane is a remarkable tool able to graphically represent the different nature, deterministic or stochastic, of the systems under study.Comment: Submitted to PLOS On

Uncovering Molecular Biomarkers That Correlate Cognitive Decline with the Changes of Hippocampus' Gene Expression Profiles in Alzheimer's Disease

Background: Alzheimer’s disease (AD) is characterized by a neurodegenerative progression that alters cognition. On a phenotypical level, cognition is evaluated by means of the MiniMental State Examination (MMSE) and the post-morten examination of Neurofibrillary Tangle count (NFT) helps to confirm an AD diagnostic. The MMSE evaluates different aspects of cognition including orientation, short-term memory (retention and recall), attention and language. As there is a normal cognitive decline with aging, and death is the final state on which NFT can be counted, the identification of brain gene expression biomarkers from these phenotypical measures has been elusive. Methodology/Principal Findings: We have reanalysed a microarray dataset contributed in 2004 by Blalock et al. of 31 samples corresponding to hippocampus gene expression from 22 AD subjects of varying degree of severity and 9 controls. Instead of only relying on correlations of gene expression with the associated MMSE and NFT measures, and by using modern bioinformatics methods based on information theory and combinatorial optimization, we uncovered a 1,372-probe gene expression signature that presents a high-consensus with established markers of progression in AD. The signature reveals alterations in calcium, insulin, phosphatidylinositol and wnt-signalling. Among the most correlated gene probes with AD severity we found those linked to synaptic function, neurofilament bundle assembly and neuronal plasticity. Conclusions/Significance: A transcription factors analysis of 1,372-probe signature reveals significant associations with the EGR/KROX family of proteins, MAZ, and E2F1. The gene homologous of EGR1, zif268, Egr-1 or Zenk, together with other members of the EGR family, are consolidating a key role in the neuronal plasticity in the brain. These results indicate a degree of commonality between putative genes involved in AD and prion-induced neurodegenerative processes that warrants further investigation

Identification of a 5-Protein Biomarker Molecular Signature for Predicting Alzheimer's Disease

Background: Alzheimer’s disease (AD) is a progressive brain disease with a huge cost to human lives. The impact of the disease is also a growing concern for the governments of developing countries, in particular due to the increasingly high number of elderly citizens at risk. Alzheimer’s is the most common form of dementia, a common term for memory loss and other cognitive impairments. There is no current cure for AD, but there are drug and non-drug based approaches for its treatment. In general the drug-treatments are directed at slowing the progression of symptoms. They have proved to be effective in a large group of patients but success is directly correlated with identifying the disease carriers at its early stages. This justifies the need for timely and accurate forms of diagnosis via molecular means. We report here a 5-protein biomarker molecular signature that achieves, on average, a 96% total accuracy in predicting clinical AD. The signature is composed of the abundances of IL-1α, IL-3, EGF, TNF-α and G-CSF. Methodology/Principal Findings: Our results are based on a recent molecular dataset that has attracted worldwide attention. Our paper illustrates that improved results can be obtained with the abundance of only five proteins. Our methodology consisted of the application of an integrative data analysis method. This four step process included: a) abundance quantization, b) feature selection, c) literature analysis, d) selection of a classifier algorithm which is independent of the feature selection process. These steps were performed without using any sample of the test datasets. For the first two steps, we used the application of Fayyad and Irani’s discretization algorithm for selection and quantization, which in turn creates an instance of the (alpha-beta)-k-Feature Set problem; a numerical solution of this problem led to the selection of only 10 proteins. Conclusions/Significance: the previous study has provided an extremely useful dataset for the identification of A biomarkers. However, our subsequent analysis also revealed several important facts worth reporting: 1. A 5-protein signature (which is a subset of the 18-protein signature of Ray et al.) has the same overall performance (when using the same classifier). 2. Using more than 20 different classifiers available in the widely-used Weka software package, our 5- protein signature has, on average, a smaller prediction error indicating the independence of the classifier and the robustness of this set of biomarkers (i.e. 96% accuracy when predicting AD against non-demented control). 3. Using very simple classifiers, like Simple Logistic or Logistic Model Trees, we have achieved the following results on 92 samples: 100 percent success to predict Alzheimer’s Disease and 92 percent to predict Non Demented Control on the AD dataset

Causality and the Entropy-Complexity Plane: Robustness and Missing Ordinal Patterns

We deal here with the issue of determinism versus randomness in time series. One wishes to identify their relative weights in a given time series. Two different tools have been advanced in the literature to such effect, namely, i) the "causal" entropy-complexity plane [Rosso et al. Phys. Rev. Lett. 99 (2007) 154102] and ii) the estimation of the decay rate of missing ordinal patterns [Amig\'o et al. Europhys. Lett. 79 (2007) 50001, and Carpi et al. Physica A 389 (2010) 2020-2029]. In this work we extend the use of these techniques to address the analysis of deterministic finite time series contaminated with additive noises of different degree of correlation. The chaotic series studied here was via the logistic map (r = 4) to which we added correlated noise (colored noise with f-k Power Spectrum, 0 {\leq} k {\leq} 2) of varying amplitudes. In such a fashion important insights pertaining to the deterministic component of the original time series can be gained. We find that in the entropy-complexity plane this goal can be achieved without additional computations.Comment: submitted to Physica

Report of the results of the 24 classifiers when using the 10-Protein biomarker.

<p>Report of the results of the 24 classifiers when using the 10-Protein biomarker.</p

Classification and prediction of clinical Alzheimer's diagnosis in subjects with Alzheimer's disease.

<p>(a) An undirected graph, where each node corresponds a different protein belonging to the 10-protein signature we identified; each edge indicates the existence of a direct relation as obtained by searching the PubMed database, (using the Pathway Studio software). (b) Identification of the maximum clique of the graph, uncovering a robust 6-protein signature; each node on the clique has a direct relation with each other. Simple Logistic was used to classify and predict Alzheimer's (AD) and non-Alzheimer's class, in the training set (c), the blinded test set ‘AD’ (d). All the results are shown in a confusion matrix, for the training set a 10-fold cross-validation was applied 10 times, in both cases Simple Logistic was used with the default parameters of Weka package. All the p-values were calculated using the Fisher exact test.</p

Histograms of the number of errors of the random forest classifier using 20 randomly selected signatures with 18 proteins.

<p>The arrow indicates the results under the same conditions of the 18-protein signature proposed by Ray <i>et al</i>.</p

Number of errors from the 6-genes randomly selected signatures on the “AD” validation test set.

<p>The Random forest algorithm was used as classifier, for each signature 10 runs with different seeds were done. We used the WEKA software implementation, and the algorithm was allowed to generate 150 trees. The best and worst signatures are highlighted in bold text. This result shows what it is expected, that a 6-signature, when the biomarkers are randomly chosen, is performing significantly worse than the panel of 18 biomarkers selected by Ray <i>et. al.</i> Now the best result (81.5%) is worse than the average result of a random 18-signature (86%).</p