Integral moments (2008)(en)(17s)
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Boros, G. Cambridge, England: Cambridge University Press, Borwein, J. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Experimentation in Mathematics: Computational Paths to Discovery. Glasser, M. Heidelberg, Germany: Springer-Verlag, Hildebrand, F. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. Mitchell, C. In "Media Clips" Ed. Cibes and J. Teacher , , Dec. Moll, V. Oloa, O. Press, W. Cambridge, England: Cambridge University Press, pp.
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[DOWNLOAD] Integral moments ()(en)(17s) by Garrett P.. Book file PDF easily for everyone and every device. You can download and read online Integral . Garrett: Integral moments [Edinburgh, 04 Aug ]Integral moments Paul Garrettgarrett@giuliettasprint.konfer.eu giuliettasprint.konfer.eu
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Introduction
Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. See methods for estimation of click mislocalization from model parameters. Two lines of evidence suggest that our estimate of the level of sensory noise is reliable.
First, ref. Second, we found click mislocalization levels were constant across a wide range of click rates in ref. Rats discount evidence.
Error bars show standard deviation. Error bars are omitted for clarity.
Psychophysical reverse correlation The computation of the reverse correlation curves was very similar to methods previously reported 12 , 13 , Figure 4 Open in figure viewer PowerPoint. Within this period, because the fastest leading edge is truncated, the variance of the truncated plumes is no longer infinite but increases almost linearly with time which can be confused with the Gaussian model. Although the analytical methods have intrinsic limitations, they can provide the most reliable and accurate solutions. A fourth integral proposed by a challenge is also trivially computable in modern versions of the Wolfram Language ,.
Only the reverse correlation curve for the right choice are shown for clarity. Each curve was fit with an exponential function example red. The fit parameters are used in d. Example in c show with red dot. The discounting term of Eq. To gain insight, consider that if evidence is very reliable, accurate decisions can be made by only using a few clicks from a small time window. However, if evidence is unreliable, a longer time window must be used to average out unreliable clicks.
This intuition is confirmed by plotting the discounting function for a variety of evidence reliability values Fig. Decreasing reliability weakens the evidence discounting term creating longer integration timescales. The optimal inference equation attempts to predict the hidden state. As the hidden state dynamically transitions, we expect the inference process to track, albeit imperfectly, the dynamic transitions. From the perspective of a subject this dynamic tracking leads to changes of mind in the upcoming choice.
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- Rats adopt the optimal timescale for evidence integration in a dynamic environment.
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Through the optimal inference process we can predict the timing of changes of mind by looking for times when the sign of the inference process changes sign a. We simulated the optimal inference agent on a large dataset of trials assuming either no sensory noise black , or average rat sensory noise pink.
For both agents we computed the distribution of when changes of mind happen relative to changes in the hidden state of the trial. As expected, hidden state changes trigger changes of mind with a temporal delay that increases with sensory noise. There are many possible linear approximations with different slopes.
A linear approximation using the slope of sinh at the origin will fail to capture the strong discounting farther from the origin. The best approximation is the one that achieves the highest accuracy at predicting the underlying state.
We found the best linear approximation numerically. We find this to be true across a wide range of noise values Fig. While the optimal linear discounting strength at each noise level changes Fig. For the average level of sensory noise, we find the linear agent to have The linear model was optimized on a training set of trials, and both models were evaluated on a test set of held-out trials and achieved It is important to note that a linear approximation in general will not always be close in accuracy to the full nonlinear theory 11 , but for our specific click rate parameters it is an accurate approximation.
Given that a linear discounting function matches the accuracy of the nonlinear model, we analyze rat evidence discounting behavior by looking for the appropriate discounting rate or equivalently the appropriate integration timescale. Specifically, we compare the rat behavior to this linear discounting equation:.
We did not examine whether rats demonstrate nonlinear evidence discounting because the linear approximation matches the accuracy of the nonlinear theory. Here we use reverse correlation to find the integration timescale used by the rats. We normalized the reverse correlation curve to have an area under the curve equal to one. This step lets the curves be interpreted in units of effective weight at each time point. A flat reverse correlation curve indicates even weighting of evidence across all time points.
Previous studies in a static environment find rats with flat reverse correlation curves 12 , 13 , The stimulus earlier in the trial is weighted less than the stimulus at the end of the trial indicating evidence discounting. The curves were generated from a synthetic dataset of 20, trials. The weaker the discounting rate, the flatter the reverse correlation curves. To quantify the discounting timescale from the reverse correlation curves, an exponential function e bt was fit to each curve. We then computed the reverse correlation curves for both the rats and the optimal linear agent Fig.
We then fit an exponential function to each of the reverse correlation curves. Rat behavior was compared with two optimal agents. The first optimal linear agent assumes no sensory noise; while the second agent uses the optimal timescale given the average level of sensory noise across rats reported in ref.
Integration formulas to evaluate functions of random variables
When the average level of sensory noise is taken into account, the rats match the optimal timescale. The reverse correlation analysis shows that rats are close to the optimal timescale given the average level of sensory noise in a separate cohort of rats. Rats discount evidence with the optimal timescale. The optimal linear inference agent was simulated on the same trials the rat performed. Shaded area shows one standard deviation. Rat integration timescales plotted with optimal agents with no sensory noise black , or with sensory noise pink.
The variability in optimal discounting rates is a result of measuring the reverse correlation curves on a different set of trials each rat actually performed. Error bars omitted for clarity. In order to examine individual variations in noise level and integration timescales, we fit a behavioral accumulation of evidence model from the literature to each rat 12 , 13 , This model generates a moment-by-moment estimate of a latent accumulation variable.
The dynamical equations for the model are given by:. In addition to the click integration and linear discounting that was present in our normative theory, this model also parameterizes many possible sources of noise. In addition to the sensory noise, each click is also filtered through an adaptation process, C. See ref.
Moment of inertia
The only modification to the model from previous studies is the removal of the sticky bounds B , which are especially detrimental to subject performance given the dynamic nature of the task. This model is a powerful tool for the description of behavior on this task because of its flexibility at characterizing many different behavioral strategies 12 , 13 , We parameterized the model with linear discounting, rather than nonlinear discounting in the full theory for three reasons.
First, the linear discounting model has been fit to rat behavior in static environments, allowing a direct comparison to previous results. Second, the linear model has an analytical solution that greatly facilitates analysis. Third, the linear model has comparable accuracy to the nonlinear model with less parameters, simplifying the fitting procedure and providing a more parsimonious description of rat behavior.