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## Correntropy as a novel measure for nonlinearity tests. Signal Processing 89(1 (2009)

Citations: | 14 - 0 self |

### Citations

12873 | Statistical Learning Theory
- Vapnik
- 1998
(Show Context)
Citation Context ...ace in which inner products can be computed efficiently without explicit knowledge of the mapping. The kernel functions employed in these techniques are positive-definite, satisfy Mercer’s conditions =-=[1]-=- and are denoted by κ ( xi , x j ) = φ( xi ), φ( x j ) (1) where φ is the mapping from the input to the feature space. A widely used Mercer kernel is the Gaussian kernel given by: ⎛ 2 ⎞ 1 ⎜ − xi − x j... |

2944 |
Numerical recipes in C : the art of scientific computing
- Press, Teukolsky, et al.
- 1992
(Show Context)
Citation Context ...scale.sB. Kolmogorov-Smirnov Goodness-of-Fit TestsThe two-sample Kolmogorov-Smirnov goodness-of-fit issa powerful tool that tests whether two time series aresrandom samples from the same distribution =-=[15]-=-. It is basedson the empirical cumulative distribution functions (ECDF)sobtained directly from the data samples. For the problem atshand, we want to compare the ECDFs calculated from thescorrentropy p... |

1539 | Nonlinear Component Analysis as a Kernel Eigenvalue Problem - Schölkopf, Smola, et al. - 1998 |

1493 |
Deterministic Nonperiodic Flow
- Lorenz
- 1963
(Show Context)
Citation Context ...mogorov-Smirnov test.sIn other words, the null hypothesis of a Gaussian linearssource was not rejected. Passing the Gaussian data throughshigher order linear filters did not alter the results.sLorenz =-=[16]-=- and Rössler [17] attractors are nonlinearschaotic systems which are widely used as reference inputssfor nonlinearity tests with surrogate methods [13]. Fig. 2spresents the time series of the first co... |

492 | Fisher discriminant analysis with kernels - Mika, Rätsch, et al. - 1999 |

452 | Kernel independent component analysis - Bach, Jordan - 2002 |

160 | Surrogate time series
- Schreiber, Schmitz
- 2000
(Show Context)
Citation Context ...d to the original datasunder a discriminating statistic with a confidence level insorder to reject the null hypothesis.sOne of the widely used surrogate generation methods issFourier based surrogates =-=[12]-=-, which require that thessurrogates have the same Fourier amplitudes as the data butswith random phases. The key point in these methods is thatsthe squared amplitude of the Fourier transform is asperi... |

127 |
Improved surrogate data for nonlinearity tests
- Schreiber, Schmitz
- 1996
(Show Context)
Citation Context ...tudes as the data butswith random phases. The key point in these methods is thatsthe squared amplitude of the Fourier transform is asperiodogram estimator of the (conventional) power spectralsdensity =-=[13]-=-. Hence, the original time series and itsssurrogates attained by this method share the same powersspectrum and thus the same autocorrelation functionsregardless the null hypothesis is true or false. H... |

120 |
Measurement and Analysis of Random Data
- Bendat, Piersol
- 1966
(Show Context)
Citation Context ...othesis. The Fourier Transform amplitudes of a randomsprocess are known to possess a chi squared distribution atseach frequency with the number of degrees of freedomsgiven by the length of the window =-=[14]-=-. The correntropysspectral density represents the distribution of total powersamongst frequencies. Thus, if normalized by the total power,sit is essentially the probability density function ofscorrent... |

105 | The Kernel-Adatron Algorithm: a Fast and Simple Learning Procedure for Support Vector Machines - Friess, Cristianini, et al. - 1998 |

73 |
Order within chaos
- Bergé, Pomeau, et al.
- 1984
(Show Context)
Citation Context ...avoided if the underlying signals are in fact linearsin nature, due to increased complexity of training nonlinearsmodels [7]. Conventional measures for testing nonlinearitysinclude Lyapunov exponents =-=[8]-=- and correlation dimensionss[9] which require embedding of signals into highersdimensional and extensive computations. Motivated by itssability to preserve nonlinear characteristics andscomputational ... |

40 | Generalized correlation function: Definition, properties, and application to blind equalization
- Santamaria, Pokharel, et al.
- 2006
(Show Context)
Citation Context ...cessing applications. Numerousskernel methods have emerged and have successfully beensapplied to problems of classification, pattern analysis andsdecomposition [1-5]. Recently introduced, correntropy =-=[6]-=- issa similarity measure which takes advantage of the power ofskernels to nonlinearly map the input space to a highersdimensional feature space in which inner products can bescomputed efficiently. The... |

32 |
Information Theoretic Learning: Renyi’s Entropy and its Applications to Adaptive System Training
- Erdogmus
- 2002
(Show Context)
Citation Context ...s.sAs the kernel size approaches infinity, the higher ordersmoment terms (m>2) in the correntropy definition convergesto zero. Thus, correntropy does not contain moresinformation than autocorrelation =-=[18]-=-. On the other hand,swhen too small a kernel size is chosen, most of the power issconcentrated in the zeroth lag, and information regardingsother lags are lost.sFigure 5 shows the effect of kernel siz... |

18 | A novel method for determining the nature of time series
- Gautama, Mandić, et al.
- 2004
(Show Context)
Citation Context ...racy inssystem modeling. In practical applications nonlinear filterssshould be avoided if the underlying signals are in fact linearsin nature, due to increased complexity of training nonlinearsmodels =-=[7]-=-. Conventional measures for testing nonlinearitysinclude Lyapunov exponents [8] and correlation dimensionss[9] which require embedding of signals into highersdimensional and extensive computations. Mo... |

2 |
Density Estimation for Statistics and Data Analaysis
- Silverman
- 1986
(Show Context)
Citation Context ...ernel sizesaffects the performance of the technique. The kernel sizesgenerally is determined empirically. In this study, thesbandwidth of the kernel is selected according to Silverman’ssrule of thumb =-=[10]-=-:s5/19.0 −=σ ANs(7) where A=min{standard deviation of data samples, datasinterquartile range/1.34} and N is the data length.sIII. SURROGATE METHODS FOR TESTS OF NONLINEARITYsMethods of surrogate data ... |

1 |
Surrogate Data Test on Time Series”, Nonlinear Deterministic Modeling and Forecasting of Economic and Financial Time Series
- Kugiumtzis
- 2000
(Show Context)
Citation Context ...atasinterquartile range/1.34} and N is the data length.sIII. SURROGATE METHODS FOR TESTS OF NONLINEARITYsMethods of surrogate data provide a statistically rigoroussframework for tests of nonlinearity =-=[11]-=-. The surrogate datasis generated to represent the null hypothesis that thesexamined time series is generated by a Gaussian linearsstochastic process. Properly designed surrogate data shouldsonly reta... |