Complexity plots

In this paper, we present a novel visualization technique for assisting in observation and analysis of algorithmic\ud complexity. In comparison with conventional line graphs, this new technique is not sensitive to the units of\ud measurement, allowing multivariate data series of different physical qualities (e.g., time, space and energy) to be juxtaposed together conveniently and consistently. It supports multivariate visualization as well as uncertainty visualization. It enables users to focus on algorithm categorization by complexity classes, while reducing visual impact caused by constants and algorithmic components that are insignificant to complexity analysis. It provides an effective means for observing the algorithmic complexity of programs with a mixture of algorithms and blackbox software through visualization. Through two case studies, we demonstrate the effectiveness of complexity plots in complexity analysis in research, education and application

Measuring Visual Complexity of Cluster-Based Visualizations

Handling visual complexity is a challenging problem in visualization owing to the subjectiveness of its definition and the difficulty in devising generalizable quantitative metrics. In this paper we address this challenge by measuring the visual complexity of two common forms of cluster-based visualizations: scatter plots and parallel coordinatess. We conceptualize visual complexity as a form of visual uncertainty, which is a measure of the degree of difficulty for humans to interpret a visual representation correctly. We propose an algorithm for estimating visual complexity for the aforementioned visualizations using Allen's interval algebra. We first establish a set of primitive 2-cluster cases in scatter plots and another set for parallel coordinatess based on symmetric isomorphism. We confirm that both are the minimal sets and verify the correctness of their members computationally. We score the uncertainty of each primitive case based on its topological properties, including the existence of overlapping regions, splitting regions and meeting points or edges. We compare a few optional scoring schemes against a set of subjective scores by humans, and identify the one that is the most consistent with the subjective scores. Finally, we extend the 2-cluster measure to k-cluster measure as a general purpose estimator of visual complexity for these two forms of cluster-based visualization

Quantifying Model Complexity via Functional Decomposition for Better Post-Hoc Interpretability

Post-hoc model-agnostic interpretation methods such as partial dependence plots can be employed to interpret complex machine learning models. While these interpretation methods can be applied regardless of model complexity, they can produce misleading and verbose results if the model is too complex, especially w.r.t. feature interactions. To quantify the complexity of arbitrary machine learning models, we propose model-agnostic complexity measures based on functional decomposition: number of features used, interaction strength and main effect complexity. We show that post-hoc interpretation of models that minimize the three measures is more reliable and compact. Furthermore, we demonstrate the application of these measures in a multi-objective optimization approach which simultaneously minimizes loss and complexity

Nonlinear analysis of bivariate data with cross recurrence plots

We use the extension of the method of recurrence plots to cross recurrence plots (CRP) which enables a nonlinear analysis of bivariate data. To quantify CRPs, we develop further three measures of complexity mainly basing on diagonal structures in CRPs. The CRP analysis of prototypical model systems with nonlinear interactions demonstrates that this technique enables to find these nonlinear interrelations from bivariate time series, whereas linear correlation tests do not. Applying the CRP analysis to climatological data, we find a complex relationship between rainfall and El Nino data

Peeking Inside the Black Box: Visualizing Statistical Learning with Plots of Individual Conditional Expectation

This article presents Individual Conditional Expectation (ICE) plots, a tool for visualizing the model estimated by any supervised learning algorithm. Classical partial dependence plots (PDPs) help visualize the average partial relationship between the predicted response and one or more features. In the presence of substantial interaction effects, the partial response relationship can be heterogeneous. Thus, an average curve, such as the PDP, can obfuscate the complexity of the modeled relationship. Accordingly, ICE plots refine the partial dependence plot by graphing the functional relationship between the predicted response and the feature for individual observations. Specifically, ICE plots highlight the variation in the fitted values across the range of a covariate, suggesting where and to what extent heterogeneities might exist. In addition to providing a plotting suite for exploratory analysis, we include a visual test for additive structure in the data generating model. Through simulated examples and real data sets, we demonstrate how ICE plots can shed light on estimated models in ways PDPs cannot. Procedures outlined are available in the R package ICEbox.Comment: 22 pages, 14 figures, 2 algorithm

Evolution of complexity following a quantum quench in free field theory

Using a recent proposal of circuit complexity in quantum field theories introduced by Jefferson and Myers, we compute the time evolution of the complexity following a smooth mass quench characterized by a time scale $\delta t$ in a free scalar field theory. We show that the dynamics has two distinct phases, namely an early regime of approximately linear evolution followed by a saturation phase characterized by oscillations around a mean value. The behavior is similar to previous conjectures for the complexity growth in chaotic and holographic systems, although here we have found that the complexity may grow or decrease depending on whether the quench increases or decreases the mass, and also that the time scale for saturation of the complexity is of order $\delta t$ (not parametrically larger).Comment: V2: added references, new plots, and improved discussion of results on Section 5, V3: Few minor corrections. Published versio

Data complexity measured by principal graphs

How to measure the complexity of a finite set of vectors embedded in a multidimensional space? This is a non-trivial question which can be approached in many different ways. Here we suggest a set of data complexity measures using universal approximators, principal cubic complexes. Principal cubic complexes generalise the notion of principal manifolds for datasets with non-trivial topologies. The type of the principal cubic complex is determined by its dimension and a grammar of elementary graph transformations. The simplest grammar produces principal trees. We introduce three natural types of data complexity: 1) geometric (deviation of the data's approximator from some "idealized" configuration, such as deviation from harmonicity); 2) structural (how many elements of a principal graph are needed to approximate the data), and 3) construction complexity (how many applications of elementary graph transformations are needed to construct the principal object starting from the simplest one). We compute these measures for several simulated and real-life data distributions and show them in the "accuracy-complexity" plots, helping to optimize the accuracy/complexity ratio. We discuss various issues connected with measuring data complexity. Software for computing data complexity measures from principal cubic complexes is provided as well.Comment: Computers and Mathematics with Applications, in pres

Compositional End Members in Gale Crater, Mars

Geochemical data returned from the Mars Science Laboratory’s Curiosity rover over 1296 sols, has revealed a previously unforeseen martian geochemical complexity. Before Curiosity landed in Gale Crater, Martian SNC meteorite studies along with previous orbiter, rover and lander data showed Mars as being a predominantly basaltic planet with little magmatic differentiation. But through using ChemCam density contour plots to collate compositional data obtained by that instrument, we can identify 4 compositional end members in Gale sedimentary and igneous samples

Extended Recurrence Plot Analysis and its Application to ERP Data

We present new measures of complexity and their application to event related potential data. The new measures base on structures of recurrence plots and makes the identification of chaos-chaos transitions possible. The application of these measures to data from single-trials of the Oddball experiment can identify laminar states therein. This offers a new way of analyzing event-related activity on a single-trial basis.Comment: 21 pages, 8 figures; article for the workshop ''Analyzing and Modelling Event-Related Brain Potentials: Cognitive and Neural Approaches`` at November 29 - December 01, 2001 in Potsdam, German