Unifying Time to Contact Estimation and Collision Avoidance across Species
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{"title"=>"Unifying Time to Contact Estimation and Collision Avoidance across Species", "type"=>"journal", "authors"=>[{"first_name"=>"Matthias S.", "last_name"=>"Keil", "scopus_author_id"=>"35401289100"}, {"first_name"=>"Joan", "last_name"=>"López-Moliner", "scopus_author_id"=>"6602219244"}], "year"=>2012, "source"=>"PLoS Computational Biology", "identifiers"=>{"issn"=>"1553734X", "pui"=>"365624242", "sgr"=>"84866075574", "doi"=>"10.1371/journal.pcbi.1002625", "scopus"=>"2-s2.0-84866075574", "isbn"=>"1553-7358 (Electronic)\\r1553-734X (Linking)", "pmid"=>"22915999"}, "id"=>"86319529-0e41-3b29-ac2f-0ec6b2b09593", "abstract"=>"The τ-function and the η-function are phenomenological models that are widely used in the context of timing interceptive actions and collision avoidance, respectively. Both models were previously considered to be unrelated to each other: τ is a decreasing function that provides an estimation of time-to-contact (ttc) in the early phase of an object approach; in contrast, g has a maximum before ttc. Furthermore, it is not clear how both functions could be implemented at the neuronal level in a biophysically plausible fashion. Here we propose a new framework--the corrected modified Tau function--capable of predicting both τ-type (\"τ(cm)\") and g-type (\"t(mod)\") responses. The outstanding property of our new framework is its resilience to noise. We show that t(mod) can be derived from a firing rate equation, and, as g, serves to describe the response curves of collision sensitive neurons. Furthermore, we show that tcm predicts the psychophysical performance of subjects determining ttc. Our new framework is thus validated successfully against published and novel experimental data. Within the framework, links between τ-type and η-type neurons are established. Therefore, it could possibly serve as a model for explaining the co-occurrence of such neurons in the brain.", "link"=>"http://www.mendeley.com/research/unifying-time-contact-estimation-collision-avoidance-across-species", "reader_count"=>33, "reader_count_by_academic_status"=>{"Professor > Associate Professor"=>3, "Researcher"=>6, "Student > Doctoral Student"=>3, "Student > Ph. D. Student"=>13, "Student > Master"=>4, "Student > Bachelor"=>2, "Lecturer"=>1, "Professor"=>1}, "reader_count_by_user_role"=>{"Professor > Associate Professor"=>3, "Researcher"=>6, "Student > Doctoral Student"=>3, "Student > Ph. D. Student"=>13, "Student > Master"=>4, "Student > Bachelor"=>2, "Lecturer"=>1, "Professor"=>1}, "reader_count_by_subject_area"=>{"Engineering"=>5, "Mathematics"=>1, "Medicine and Dentistry"=>2, "Agricultural and Biological Sciences"=>6, "Neuroscience"=>2, "Sports and Recreations"=>1, "Psychology"=>10, "Social Sciences"=>2, "Computer Science"=>4}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>5}, "Medicine and Dentistry"=>{"Medicine and Dentistry"=>2}, "Neuroscience"=>{"Neuroscience"=>2}, "Social Sciences"=>{"Social Sciences"=>2}, "Sports and Recreations"=>{"Sports and Recreations"=>1}, "Psychology"=>{"Psychology"=>10}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>6}, "Computer Science"=>{"Computer Science"=>4}, "Mathematics"=>{"Mathematics"=>1}}, "reader_count_by_country"=>{"United States"=>1, "Japan"=>1, "China"=>1, "Brazil"=>1, "France"=>1, "Germany"=>1}, "group_count"=>2}

Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/590784"], "description"=>"<p>The experimental data from <i>Gabbiani et al. </i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625-Gabbiani1\" target=\"_blank\">[26]</a> suggest a linear relationship between relative time of peak firing rate and the half-size to velocity ratio . The big shaded areas indicate one standard deviation from the mean value of . Notice the increase in with increasing . (<b>a</b>) Resampled <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi-1002625-g004\" target=\"_blank\">Figure 4a</a> from Reference <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625-Gabbiani1\" target=\"_blank\">[26]</a> (p. 1128). The locusts were stimulated by approaching dark squares with different sizes and velocities, such that various values of were covered. The circle symbol for each represents the mean of neuronal response curves across DCMD neurons. The result of a weighted least square regression fit reported by <i>Gabbiani et al.</i> had slope and intercept . With the manually resampled data points shown here, we obtained and , respectively. The light green shaded area indicates one standard deviation of slope. Additional statistical parameters of our weighted least square fit are shown above the figure. (<b>b</b>) An example of fitting a straight line to averaged random trials of the “noisified” <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.e175\" target=\"_blank\">equation (2)</a> with . “Noisifying” means that Gaussian noise with standard deviation was added to (according to <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.e393\" target=\"_blank\">equation 8</a>, page 1129 in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625-Gabbiani1\" target=\"_blank\">[26]</a>). The noise blurs the nonlinear character of the m-Tau function and makes it <i>appear</i> linear. The light red shaded area indicates one standard deviation of slope. Further simulation results are presented in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s002\" target=\"_blank\">Text S2</a></i>.</p>", "links"=>[], "tags"=>["m-tau", "nonlinearity"], "article_id"=>261282, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g003", "stats"=>{"downloads"=>1, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Masking_of_the_m_Tau_nonlinearity_by_noise_/261282", "title"=>"Masking of the m-Tau nonlinearity by noise.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:21:22"}
  • {"files"=>["https://ndownloader.figshare.com/files/590631"], "description"=>"<p>(<b>a</b>) The figure shows two m-Tau functions which are distinguished by (with values and , see legend). The horizontal bars denote their respective maxima for the default stimulus values (, , , ). The maxima shift to the left (circles) upon doubling the object radius to (“size effect”). They shift in the opposite direction (triangles) upon doubling both the approach velocity and the initial distance (“velocity effect”), such that remains unchanged (). The thin dotted lines (not identified in the legend) show the m-Tau functions with correspondingly doubled values. For the m-Tau function with , the two factors and are furthermore plotted, see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.e136\" target=\"_blank\">equation (1)</a>. The shift directions of the maxima are identical with the corresponding shifts observed with the -function, see <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s001\" target=\"_blank\">Text S1</a></i>. (<b>b</b>) Here it is shown how the maxima of seven m-Tau functions shift when the object diameter is halved or doubled with respect to its default value . Each point indicates (time of maximum) along with its corresponding amplitude . Circular symbols represent the default case with . All maxima lie on a line. With a smaller object diameter all maxima shift to the right (towards ), and an increase in object size causes a shift of all maxima to the left (away from ). All shifts proceed along the same straight line. Notice that some artifacts occur for the two leftmost points, because all maxima were computed numerically. The velocity effect is illustrated in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s001\" target=\"_blank\">Text S1</a></i>.</p>", "links"=>[], "tags"=>["modified", "tau"], "article_id"=>261129, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g001", "stats"=>{"downloads"=>2, "page_views"=>10, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_modified_Tau_function_8220_m_Tau_8221_/261129", "title"=>"The modified Tau function (“m-Tau”).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:18:49"}
  • {"files"=>["https://ndownloader.figshare.com/files/591118"], "description"=>"<p>(<b>a</b>) Psychophysical data points for “proportion of later responses” are shown for the presentation time and object diameters <i>big</i> (triangle symbols) and <i>small</i> (circle symbols), respectively. Each sigmoid curve represents a fit of a Gaussian cumulative density function (“GCDF” with mean and standard deviation ) to the data points of the respective object diameter. The GCDF-fits approximate the underlying psychometric functions, with the mean indicating the time point of subjective simultaneity. (<b>b</b>) The curves show how and depend on presentation time and object diameter. Each point represents the result of a GCDF-fit to the psychophysical data. If subjects responded correctly, the point of subjective simultaneity would coincide with ( is indicated by the dashed horizontal line).</p>", "links"=>[], "tags"=>["neuroscience", "mathematics"], "article_id"=>261610, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g007", "stats"=>{"downloads"=>1, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Psychometric_functions_/261610", "title"=>"Psychometric functions.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:26:50"}
  • {"files"=>["https://ndownloader.figshare.com/files/590890"], "description"=>"<p>The red square symbols denote data points , according to <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi-1002625-g003\" target=\"_blank\">Figure 3<i>a</i></a> from reference <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625-Gabbiani1\" target=\"_blank\">[26]</a>). In order to illustrate the nonlinear behavior of m-Tau, for each of these points an instance of m-Tau was created, such that the peaks of the -function and the m-Tau function coincide. The corresponding values of were computed with equation S7 in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s002\" target=\"_blank\">Text S2</a></i>, and are indicated in the figure. Along with the , the values of and are shown in small font size. The latter two values were obtained by “brute-force” fitting a straight line to the nonlinear m-Tau curves. We observe that: <i>(i)</i> the curvature of m-Tau (equation S6 in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s002\" target=\"_blank\">Text S2</a></i>) increases with decreasing values of . <i>(ii)</i> All “slopes” of the “brute-force” line-fit to m-Tau are smaller than suggested by the data from <i>Gabbiani et al.</i>, who reported (our fit of their re-sampled data is indicated by the green line and yielded ; see figure headline).</p>", "links"=>[], "tags"=>["nonlinear", "dependence", "m-tau"], "article_id"=>261390, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g004", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Illustration_of_nonlinear_dependence_of_m_Tau_maxima_/261390", "title"=>"Illustration of nonlinear dependence of m-Tau maxima.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:23:10"}
  • {"files"=>["https://ndownloader.figshare.com/files/309370", "https://ndownloader.figshare.com/files/309381", "https://ndownloader.figshare.com/files/309402", "https://ndownloader.figshare.com/files/309420", "https://ndownloader.figshare.com/files/309437", "https://ndownloader.figshare.com/files/309450", "https://ndownloader.figshare.com/files/309468", "https://ndownloader.figshare.com/files/309482"], "description"=>"<div><p>The -function and the -function are phenomenological models that are widely used in the context of timing interceptive actions and collision avoidance, respectively. Both models were previously considered to be unrelated to each other: is a decreasing function that provides an estimation of time-to-contact (<em>ttc</em>) in the early phase of an object approach; in contrast, has a maximum before <em>ttc</em>. Furthermore, it is not clear how both functions could be implemented at the neuronal level in a biophysically plausible fashion. Here we propose a new framework – the <em>corrected modified Tau</em> function – capable of predicting both -type (“”) and -type (“”) responses. The outstanding property of our new framework is its resilience to noise. We show that can be derived from a firing rate equation, and, as , serves to describe the response curves of collision sensitive neurons. Furthermore, we show that predicts the psychophysical performance of subjects determining <em>ttc</em>. Our new framework is thus validated successfully against published and novel experimental data. Within the framework, links between -type and -type neurons are established. Therefore, it could possibly serve as a model for explaining the co-occurrence of such neurons in the brain.</p> </div>", "links"=>[], "tags"=>["unifying", "estimation", "collision", "avoidance"], "article_id"=>121040, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1002625.s001", "https://dx.doi.org/10.1371/journal.pcbi.1002625.s002", "https://dx.doi.org/10.1371/journal.pcbi.1002625.s003", "https://dx.doi.org/10.1371/journal.pcbi.1002625.s004", "https://dx.doi.org/10.1371/journal.pcbi.1002625.s005", "https://dx.doi.org/10.1371/journal.pcbi.1002625.s006", "https://dx.doi.org/10.1371/journal.pcbi.1002625.s007", "https://dx.doi.org/10.1371/journal.pcbi.1002625.s008"], "stats"=>{"downloads"=>1, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Unifying_Time_to_Contact_Estimation_and_Collision_Avoidance_across_Species/121040", "title"=>"Unifying Time to Contact Estimation and Collision Avoidance across Species", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2012-08-16 00:17:20"}
  • {"files"=>["https://ndownloader.figshare.com/files/591050"], "description"=>"<p>The <i>corrected m-Tau</i> -function responds similar to , but with an improved noise suppression performance, as long as parameter values ( and ) are suitably chosen. More precisely, is constrained by the limit functions and . This means that <i>corrected m-Tau</i> can approach the former or the latter function for the corresponding (extreme) values of , but typically will perform somewhere between the two limit functions. For the simulations shown in this figure, uncorrelated normal-distributed noise was added to the angular variables and . Each curve represents a typical random trial, where noise was identical for all curves. The different shades of gray indicate different object diameters, as indicated in the legends. (<b>a</b>) “Normal” function, which is the limit function approached by for . Noise suppression is poor. Notice that the displayed range has been truncated so as to match it to the range of the figure on the right-hand side. (<b>b</b>) The function is the limit function that is approached for . It has an excellent noise suppression performance, owing to lowpass filtering of angular variables (, c.f. <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.e315\" target=\"_blank\">equation 4</a>). Further details are presented in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s003\" target=\"_blank\">Text S3</a></i>.</p>", "links"=>[], "tags"=>["functions"], "article_id"=>261553, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g006", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Limit_functions_of_the_corrected_m_Tau_function_/261553", "title"=>"Limit functions of the <i>corrected m-Tau</i> function.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:25:53"}
  • {"files"=>["https://ndownloader.figshare.com/files/591184"], "description"=>"<p>The proportion of later responses (i.e. subjects perceived <i>ttc</i> after ) are shown as a function of <i>ttc</i> for different presentation times : (<b>a</b>) , (<b>b</b>) , (<b>c</b>) , (<b>d</b>) , and (<b>e</b>) . Psychophysical results were pooled across subjects and are denoted by circles (<i>small</i> object diameter ) and triangles (<i>big</i> object diameter ), respectively. Predictions of the <i>corrected m-Tau</i> -model “” are represented by curves. In this figure, the prediction performance of was measured according to the root mean square error (“-score”). <i>Corrected m-Tau</i> -predictions with the three best performing parameter sets are juxtaposed (i.e. first three rows in Table S3 in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s005\" target=\"_blank\">Text S5</a></i> with smallest -score). Thinner and darker lines represent a better prediction performance. Furthermore, continuous curves are the -predictions for <i>small</i> (thus should match the circles), while dashed curves correspond to <i>big</i> (should match the triangles). Here, the same set of -parameters was used for both object diameters (“<i>combined</i> diameter”). The light-shaded areas correspond to the variability of simulated responses ( SD, see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#s4\" target=\"_blank\">Methods</a> Section): Yellowish shading for <i>small</i>, and bluish shading for <i>big</i>.</p>", "links"=>[], "tags"=>["predictions"], "article_id"=>261686, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g008", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Corrected_m_Tau_predictions_score_combined_diameter_/261686", "title"=>"<i>Corrected m-Tau</i> predictions ( score; combined diameter).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:28:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/590712"], "description"=>"<p>All symbols indicate the maxima in the neuronal recording data as a function of (with ). These data were manually resampled from previously published studies (see <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s004\" target=\"_blank\">Text S4</a></i> for further details). The line ends (lines start at the center of each symbol) denote where the fitted functions (thick gray bars) and (thin and red bars) have their respective maxima. Thus, the longer a bar, the bigger the difference between the predicted maxima and that of the neuronal data. The respective sum of absolute differences is indicated in the inset. The mean ( s.d., ) of absolute differences is (median : ) for the -function, and (median : ) for . The two continuous lines connect the data for a series of values from the same paper (light gray: reference <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625-Gabbiani1\" target=\"_blank\">[26]</a>; green: reference <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625-Gabbiani3\" target=\"_blank\">[39]</a>; first figure in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s004\" target=\"_blank\">Text S4</a></i>: all references.)</p>", "links"=>[], "tags"=>["experiments", "compared", "fitted"], "article_id"=>261208, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g002", "stats"=>{"downloads"=>1, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_from_experiments_symbols_compared_to_fitted_bars_/261208", "title"=>"from experiments (symbols) compared to fitted (bars).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:20:08"}
  • {"files"=>["https://ndownloader.figshare.com/files/590971"], "description"=>"<p>For compiling this figure, a value of was first selected. Then, noisified curves () were generated and averaged, assuming a noise level of in equation S10 in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s002\" target=\"_blank\">Text S2</a></i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625-Gabbiani1\" target=\"_blank\">[26]</a>. A pair of intercept and slope values ( and , respectively) were obtained from a weighted linear regression fit to the average curve (weights variance). Now, was parsed from to in steps of (totaling values). For each value of , the weighted linear regression fit to the averaged -curves was repeated times. The small grey circles represent the mean value of these intercept-slope pairs. Statistical parameters for each fit were also recorded, and the corresponding figures are included in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s002\" target=\"_blank\">Text S2</a></i>. The main axis of the ellipse are in the direction of the eigenvectors of the covariance matrix. The matrix was computed from all intercept-slope pairs (i.e. samples for each ). The lengths of the eigenvectors were scaled with the square root of their associated eigenvalues. The area enclosed by the ellipse thus corresponds to one standard deviation (legend: and ). (Note that the ellipse shown in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi-1002625-g004\" target=\"_blank\">Figure 4b</a> from <i>Gabbiani et al.</i> denotes instead a confidence region for intercept and slope). The noise-free correlation is indicated by the straight line. Notice that the abscissa values are defined up to an arbitrary additive constant.</p>", "links"=>[], "tags"=>["neuroscience", "mathematics"], "article_id"=>261467, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g005", "stats"=>{"downloads"=>1, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Simulation_of_Figure_4b_from_Reference_26_p_1128_/261467", "title"=>"Simulation of <b>Figure 4b</b> from Reference [<b>26</b>] (p. 1128).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:24:27"}
  • {"files"=>["https://ndownloader.figshare.com/files/591328"], "description"=>"<p>In order to predict psychophysical performance with the <i>corrected m-Tau</i> -model, its parameters were optimized. Prediction performance was measured with a score measure, either the root mean square error (, shown here), or an outlier-insensitive robust error (; shown in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s005\" target=\"_blank\">Text S5</a></i>). The -parameter set with which the best prediction was achieved was assigned rank one, the second best rank two, and so on. Thus, rank one corresponds to the parameter set with the smallest score measure. The figure shows the median value of the noise probability <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.e638\" target=\"_blank\">equation (9)</a> of: (<b>a</b>) angular size , and (<b>b</b>) angular velocity , as a function of rank. Abscissa values of , , etc. signify that the median value across the first , first , etc. values of and , respectively, was computed, according to “-ranking”. Shaded areas indicate of the corresponding robust estimation of standard deviation . The continuous curves were computed with the -values optimized for the <i>small</i> object diameter (listed in Table S1 in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s005\" target=\"_blank\">Text S5</a></i>), and broken curves denote corresponding values for the <i>big</i> diameter (Table S2 in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s005\" target=\"_blank\">Text S5</a></i>). The curves shown here suggest that the <i>small</i> object diameter is associated with a higher noise level. This conclusion is valid for until rank (curves become indistinguishable beyond that value), and for until rank ten: For ranks bigger than ten, reveals a certain dependence on the score measure and the averaging procedure (not visible in this plot, but see corresponding figures in <i><a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002625#pcbi.1002625.s005\" target=\"_blank\">Text S5</a></i>).</p>", "links"=>[], "tags"=>["probabilities"], "article_id"=>261820, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Matthias S. Keil", "Joan López-Moliner"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002625.g009", "stats"=>{"downloads"=>1, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Median_value_of_noise_probabilities_as_a_function_of_rank_/261820", "title"=>"Median value of noise probabilities as a function of -rank.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-08-16 00:30:20"}

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Relative Metric

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