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Physical Review B 97 Fluctuation and Noise Letters 17 , Nonlinear Dynamics 91 :3, Advances in Nonlinear Geosciences, Thermodynamics of Computation. Unconventional Computing, International Journal of Food Properties 21 :1, Journal of Applied Physics , Nonlinearity 30 :7, Physical Review X 7 Nonlinear Dynamics 87 :3, Journal of Statistical Mechanics: Theory and Experiment , Analog Integrated Circuits and Signal Processing 89 :1, The European Physical Journal B 89 Advances in Difference Equations Frontiers in Applied Mathematics and Statistics 1.
Chinese Physics Letters 32 :9, Current Opinion in Biotechnology 34 , Neurocomputing , Communications in Nonlinear Science and Numerical Simulation 22 , Science China Earth Sciences 58 :3, Food Chemistry , Stochastic Systems. Multiple Time Scale Dynamics, Encyclopedia of Complexity and Systems Science, Annals of Clinical and Translational Neurology 1 :7, The European Physical Journal B 87 Physics Letters A , Nonlinear Dynamics 76 :1, Neural Computing and Applications 24 , Electrochimica Acta , Mathematical Problems in Engineering , Exit Problems for Diffusion Processes and Applications.
Stochastic Processes and Applications, Chinese Physics B 22 , Physical Review E 88 Applied Mechanics and Materials , Annual Review of Condensed Matter Physics 4 :1, Measuring Out-of-Equilibrium Fluctuations. Nonequilibrium Statistical Physics of Small Systems, International Journal of Genomics , Neural Networks 37 , Chinese Physics B 21 , The European Physical Journal B 85 Physical Review E 85 Computational Complexity, Climate Dynamics 37 , Physical Review E 84 Physical Review E 83 International Journal of Modern Physics B 25 , Nano Communication Networks 2 :1, Some evidences from Hurst autocorrelation analysis.
Advances in Space Research 47 :4, Journal of Physics A: Mathematical and Theoretical 43 , Physical Review E 82 Paleoceanography 25 Annals of the New York Academy of Sciences :1, Progress of Theoretical Physics :1, No other differences can be noted at higher pedestal contrasts. In sum, fitting the rescaled data of all noise conditions at once leads to a perhaps reasonable overall quality of fit, but the occurrence of systematic trends in the deviance residuals implies that the null hypothesis that contrast discrimination is invariant to the presence of noise must be rejected.
Though these analyzes were run on the pooled deviance residuals, data of all observers displayed the trends described above. To analyze the differences between noise conditions in more detail, we fitted the gain-control model to the data of each noise condition separately. Modelling results II—Separate fits to the data of each noise condition. The most parsimonious modification of the gain-control model is to allow the response exponent and gain-control exponent to vary over noise conditions while the response gain and semisaturation contrast are frozen to the estimates of the fit to all data.
Indeed, the exponents in the generalized Naka—Rushton equation determine the depth of the dip. Freezing the response gain parameter a scaling parameter guarantees that all models operate on the same scale and are thus easily comparable to each other and to the fit to all data at once. Freezing the semi-saturation contrast forces the dip to have the same location for all conditions, i. An example of these fits can be seen in the upper row of Figure 4 for observer E. Figure 4. As can be seen from these plots, allowing the exponents to vary over conditions indeed leads to different estimates of the depth of the dip for different noise conditions.
It is further noticeable that, in the absence of a pedestal, psychometric functions are estimated to be steeper without noise than in the presence of weak noise for each observer see Figures 4d — 4g , as was also borne out by our data see Figure 1a. The psychometric functions derived from the model fit are plotted in the bottom row of Figure 4. Table 2 lists the parameter estimates and normalized deviance for all observers. Comparing the normalized deviance values of Table 2 with those of Table 1 reveals that quality of fit improved for each noise condition and all observers.
It can be seen that normalized deviance is noticeably high for observer E. Fitting this condition with an expanded 6-free parameter version of the gain-control model i. Most likely, this data set is over-dispersed, i. Parameter estimates and normalized deviance for the fit of the gain-control model to the separate noise conditions.
As for the model fit to all conditions at once, we analyzed the pooled deviance residuals of all observers by means of a linear regression analysis relating deviance residual to the logarithm of pedestal contrast. The raw data of this analysis are plotted in Figure 5 , the summary of this analysis in Figure 6. For the decreasing part of the dipper, all differences between noise conditions have vanished. As we hypothesized, allowing the response- and gain-control exponent to vary over noise conditions is sufficient to capture the systematic differences between conditions at low pedestal contrasts.
At higher contrasts, virtually nothing has changed, so there is still a small but systematic misfit for the weak-noise condition. Figure 5. Deviance residuals of the separate fits as a function of pedestal contrast for all observers. Different colors indicate different noise conditions; different symbols indicate different observers. The thick line represents the average deviance residual. Figure 6. Results of a standard linear regression procedure, relating the logarithm of pedestal contrast to the deviance residuals of the separate fits, split by noise condition.
Compared to the simultaneous fit to all data, allowing the exponents to vary over noise conditions leads to an improvement in quality of fit and the disappearance of systematic trends in the deviance residuals at low pedestal contrasts. We now assess whether the improvement brought about by more free parameters is sufficiently large as assessed by methods of model selection. Model selection: Simultaneous vs. As explained in the paragraph on model selection, there are different ways to assess which of the two modelling approaches has the highest predictive accuracy.
We first consider the outcome of the AIC procedure in detail. First, and most general, we may consider the overall predictive accuracy for all observers and all conditions, analog to pooling of data across observers and conditions. We must thus compare the AIC of a free parameter model i. For the fit to all conditions at once, AIC is The reduction in AIC thus equals We may thus conclude that, considered across noise conditions and observers, the response gain and semisaturation contrast may be frozen, but it is better not to freeze the exponents of the Naka—Rushton equation. Second, we can also do this analysis for each noise condition across observers and for each observer across noise conditions.
And finally, we can do this analysis for each condition within each observer. Results of these analyses are summarized in Table 3. Each cell denotes an analysis at the third, most detailed level. Each marginal total denotes an analysis at the second, intermediate level of detail. Significant reductions in AIC are marked by means of stars in Table 3. It will be noted that the second model is favored over the first for three of four observers. For observer L. We suspect that this may at least in part result from the relative lack of data that reduces statistical power—she completed only 4, trials—because her deviance residuals and parameter estimates are not inconsistent with other observers.
Reduction in AIC for each observer and each noise condition. Whenever the theoretical degrees of freedom were not a natural number e. Table 4 summarizes the reduction in BIC. Positive numbers indicate that the separate fits should be selected, negative numbers that the simultaneous fit should be selected. At the most general level, i.
At the level of individual observers, the same trend is obvious.
At the level of noise conditions, the no-noise and weak-noise condition clearly benefit from the additional free parameters in the separate fits. This is not the case for the moderate noise condition, for which BIC selects the simultaneous fit. Reduction in BIC for each observer and each noise condition. Table 5 summarizes the average CV index i.
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As indicated by the positive differences in test error, the separate fits provide better predictions for unseen data for each observer. Average test error for each observer across noise conditions for the simultaneous and separate fits and their difference. For Model II, the presence of stars indicates that the average D Test was outside the relevant confidence interval for at least one noise condition. In sum, despite differences in the aspects of model complexity captured by AIC, BIC, and CV, all support the same conclusion: When pooled data and observers are considered, the separate fits are always selected over the joint fit.
This conclusion also holds at the level of individual observers. At the level of the noise conditions, the improved predictive accuracy is a result of the better fit to both the no-noise and weak-noise condition. Given our comprehensive model selection, we are now in a position to use the parameter estimates of the cross-validation to assess differences between noise conditions.
Because the response gain and semisaturation contrast were frozen, we only need to consider the exponents of the Naka—Rushton function. We first compare parameter estimates for the no-noise and weak-noise condition. For all observers, both exponents are estimated to be reduced in the presence of weak noise. Furthermore, the difference between the exponents is always higher for the weak-noise condition than for the no-noise condition.
It is this difference, together with the absolute value of the response exponent, which determines the strength of the pedestal effect: The smaller the difference and the larger the response exponent, the bigger the pedestal effect is.
In other words, the depth of the dipper function is reduced in the presence of weak noise for all observers. It is interesting to note that results are not as systematic for the moderate-noise condition. For some observers, exponents are estimated to be reduced relative to the no-noise condition e.
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A similar variability over observers is present in the differences between both exponents: For some this difference has increased in the presence of moderate noise e. This variation over observers is not inconsistent with data sets that have been published earlier: The dipper function of some observers seems to be invariant to the presence of strong noise, while this is not the case for others e. Figure 7.
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In all these box plots, the central horizontal line indicates the second quartile i. Whiskers indicate one and a half times the interquartile range.
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Outliers have been omitted for clarity. We thus find three stable differences between the no noise and weak-noise condition, namely, a reduction in both the response- and gain-control exponent and an increased difference between these exponents in the presence of weak noise. This indicates that the pedestal effect was reduced for all observers in the presence of weak noise. For the moderate-noise condition, results vary over observers.
In this region, the contrast response behaves as an accelerating nonlinearity. It is this acceleration that leads to the response expansion that underlies the pedestal effect. The larger the log—log steepness at low contrasts, the stronger the pedestal effect. Comparing the no-noise condition to the weak-noise condition illustrates that the log—log steepness, and thus the pedestal effect, is reduced for all observers in the presence of weak noise.