Behind Fitts’ law: kinematic patterns in goal-directed movements

Half a century ago, Paul Fitts first discovered that the time necessary to complete a pointing movement (MT) linearly increases with the amount of information (ID) necessary to specify the target width (W) relative to the distance (D). The so-called Fitts’ law states that , with ID being a logarithmic function of the D/W ratio. With the rising importance of pointing in human–computer interaction, Fitts’ law is nowadays an important tool for the quantitative evaluation of user interface design. We show that changes in ID give rise to systematic changes in the kinematics patterns that determine MT, and provide evidence that the observed patterns result from the interplay between basic oscillatory motion and visual control processes. We also emphasize the generality and abstract nature of Fitts’ robust model of human psychomotor behavior, and suggest that some adaptations in the design of the (computer-mediated) coupling of perception and production of movement might improve the efficiency of the interaction.     

Fitts’ law 50 years later: applications and contributions from human–computer interaction

Since the advent of graphical user interfaces, electronic information has grown exponentially, whereas the size of screen displays has stayed almost the same. Multiscale interfaces were designed to address this mismatch, allowing users to adjust the scale at which they interact with information objects. The technology has progressed quickly and the theory has lagged behind. Multiscale interfaces pose a stimulating theoretical challenge: reformulating the classic target-acquisition problem from the physical world into an infinitely rescalable electronic world. We address this challenge by extending Fitts’ original pointing paradigm: we introduce the scale variable, thus defining a multiscale pointing paradigm. This article reports on our theoretical and empirical results. We show that target-acquisition performance in a zooming interface must obey Fitts’ law and, more specifically, that target-acquisition time must be proportional to the index of difficulty. Moreover, we complement Fitts’ law by accounting for the effect of view size on pointing performance, showing that performance bandwidth is proportional to view size, up to a ceiling effect. Our first empirical study shows that Fitts’ law does apply to a zoomable interface for indices of difficulty up to and beyond 30 bits, whereas classical Fitts’ law studies have been confined in the 2–10 bit range. Our second study demonstrates a strong interaction between view size and task difficulty for multiscale pointing, and shows a surprisingly low ceiling. We conclude with implications of these findings for the design of multiscale user interfaces.     

Speed–accuracy tradeoff in Fitts’ law tasks—on the equivalency of actual and nominal pointing precision

Pointing tasks in human–computer interaction obey certain speed–accuracy tradeoff rules. In general, the more accurate the task to be accomplished, the longer it takes and vice versa. Fitts’ law models the speed–accuracy tradeoff effect in pointing as imposed by the task parameters, through Fitts’ index of difficulty (I_d) based on the ratio of the nominal movement distance and the size of the target. Operating with different speed or accuracy biases, performers may utilize more or less area than the target specifies, introducing another subjective layer of speed–accuracy tradeoff relative to the task specification. A conventional approach to overcome the impact of the subjective layer of speed–accuracy tradeoff is to use the a posteriori “effective” pointing precision W_e in lieu of the nominal target width W. Such an approach has lacked a theoretical or empirical foundation. This study investigates the nature and the relationship of the two layers of speed–accuracy tradeoff by systematically controlling both I_d and the index of target utilization I_u in a set of four experiments. Their results show that the impacts of the two layers of speed–accuracy tradeoff are not fundamentally equivalent. The use of W_e could indeed compensate for the difference in target utilization, but not completely. More logical Fitts’ law parameter estimates can be obtained by the W_e adjustment, although its use also lowers the correlation between pointing time and the index of difficulty. The study also shows the complex interaction effect between I_d and I_u, suggesting that a simple and complete model accommodating both layers of speed–accuracy tradeoff may not exist.     

The performance of hand postures in front- and back-of-device interaction for mobile computing

Three studies of different mobile-device hand postures are presented. The first study measures the performance of postures in Fitts’ law tasks using one and two hands, thumbs and index fingers, horizontal and vertical movements, and front- and back-of-device interaction. Results indicate that the index finger performs well on both the front and the back of the device, and that thumb performance on the front of the device is generally worse. Fitts’ law models are created and serve as a basis for comparisons. The second study examines the orientation of shapes on the front and back of a mobile device. It shows that participants’ expectations of visual feedback for finger movements on the back of a device reverse the direction of their finger movements to favor a “transparent device” orientation. The third study examines letter-like gestures made on the front and back of a device. It confirms the performance of the index finger on the front of the device, while showing limitations in the ability for the index finger on the back to perform complex gestures. Taken together, these results provide an empirical foundation upon which new mobile interaction designs can be based. A set of design implications and recommendations are given based directly on the findings presented.     

Towards a standard for pointing device evaluation, perspectives on 27 years of Fitts’ law research in HCI

This paper makes seven recommendations to HCI researchers wishing to construct Fitts’ law models for either movement time prediction, or for the comparison of conditions in an experiment. These seven recommendations support (and in some cases supplement) the methods described in the recent ISO 9241-9 standard on the evaluation of pointing devices. In addition to improving the robustness of Fitts’ law models, these recommendations (if widely employed) will improve the comparability and consistency of forthcoming publications. Arguments to support these recommendations are presented, as are concise reviews of 24 published Fitts’ law models of the mouse, and 9 studies that used the new ISO standard.     

“Beating” Fitts’ law: virtual enhancements for pointing facilitation

We survey recent research into new techniques for artificially facilitating pointing at targets in graphical user interfaces. While pointing in the physical world is governed by Fitts’ law and constrained by physical laws, pointing in the virtual world does not necessarily have to abide by the same constraints, opening the possibility for “beating” Fitts’ law with the aid of the computer by artificially reducing the target distance, increasing the target width, or both. The survey suggests that while the techniques developed to date are promising, particularly when applied to the selection of single isolated targets, many of them do not scale well to the common situation in graphical user interfaces where multiple targets are located in close proximity.     

An interpolation problem associated with H-optimal design in systems with distributed input lags

A variety of H^∞ optimal design problems reduce to interpolation of compressed multiplication operators, f(s) → π_k(w(s)f(s)), where w(s) is a given rational function and the subspace K is of the form K=H² φ(s)H². Here we consider φ(s) = (1-e^α-5)/(s - α), which stands for a distributed delay in a system's input. The interpolation scheme we develop, adapts to a broader class of distributed lags, namely, those determined by transfer functions of the form B(e^−s)/b(s), where B(z) and b(s) are polynomials and b(s) = 0 implies B(e^−s) = 0.     

Book reviews

We present a new method to determine the near surface air temperature (T_a ) from satellite observations. The satellite observed parameters of total precipitable water (W), atmospheric boundary layer (～500 m) water vapour (W_b ), and sea surface temperature (SST) are used to derive T_a . A genetic algorithm (GA) is used to find the optimum relation between the input (W, W_b , SST) and output (T_a) parameters. The input data consist of 6 years (1988–1993) of instantaneous as well as monthly averages of W, W_b from the Special Sensor Microwave Imager (SSM/I), and SST data from the Advanced Very High Resolution Radiometer (AVHRR). T_a observations based on Comprehensive Ocean Atmospheric Data Set (COADS) are used to develop and evaluate the new methodology. The global mean root mean square (rms) error for instantaneous T_a estimates is 1.4°C and for monthly averages it decreases to 0.74°C. Slightly higher discrepancies between T_a derived from the new method and in situ data are found over the western boundary currents (such as the Kuroshio and Gulf Stream) during wintertime. These regions are characterized by continental cold air outbreak and seasonal current systems, particularly during wintertime. During these conditions weak coupling between SST and T_a may be one of the reasons for large error over these regions. Our method improves upon the air temperature estimates of earlier studies.     

Queueing analysis and optimal control of and systems

We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process (BMAP). The server serves customers in batches of maximum size ‘b’ with a minimum threshold size ‘a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N^* of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP). Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.     

Input Behavior When Using Two Fingers on a Multi-Touch Device

The purpose of this study is to analyze device input patterns using fingers in multi-touch mode and to derive Fitts’ law for multi-touch methods. Three specific input behaviors—dragging, rotating, and pinching—are investigated. Test participants performed two Fitts’ law experiments, a single-touch experiment using the thumb only and a multi-touch experiment using both the thumb and index finger. Within-subject factorial design was implemented with two levels of touch behavior (single-touch) and three levels of touch behavior (multi-touch), six levels of the index of difficulty (ID), four levels of direction, and two levels of repetition. The results indicate that a revised model of Fitts’ law is more appropriate for the multi-touch mode, achieving R² values above 0.895, and identifying the best nonlinear model among several regression models. The results of this study are used to determine the relationship between ID and movement time in terms of Fitts’ law and to investigate whether this relationship is sufficient to serve as a model to predict movement times. This study contributes to the prediction of movement times for products with multi-touch interfaces.     

On Parallel Integer Merging

The problem of merging two sorted arrays A = (a₁, a₂, ..., a_n1) and B = (b₁, b₂, ..., b_n2) is considered. For input elements that are drawn from a domain of integers [1...s] we present an algorithm that runs in O(log log log s) time using n/log log log s CREW PRAM processors (optimal speed-up) and O(ns^ε) space, where n = n₁ + n₂. For input elements that are drawn from a domain of integers [1...n] we present a second algorithm that runs in O(α(n)) time (where α(n) is the inverse of Ackermann′s function) using n/α(n) CREW PRAM processors and linear space. This second algorithm is non-uniform; however, it can be made uniform at a price of a certain loss of speed, or by using a CRCW PRAM.     

Some new paranormed sequence spaces

Maddox defined the sequence spaces ℓ_∞(p), c(p) and c₀(p) in [Proc. Camb. Philos. Soc. 64 (1968) 335, Quart. J. Math. Oxford (2) 18 (1967) 345]. In the present paper, the sequence spaces a₀^r(u,p) and a_c^r(u,p) of non-absolute type have been introduced and proved that the spaces a₀^r(u,p) and a_c^r(u,p) are linearly isomorphic to the spaces c₀(p) and c(p), respectively. Besides this, the α-, β- and γ-duals of the spaces a₀^r(u,p) and a_c^r(u,p) have been computed and their basis have been constructed. Finally, a basic theorem has been given and later some matrix mappings from a₀^r(u,p) to the some sequence spaces of Maddox and to some new sequence spaces have been characterized.     

A Criterion for Extractability of Mutual Information for a Triple of Strings

We say that the mutual information of a triple of binary strings a, b, c can be extracted if there exists a string d such that a, b, and c are independent given d, and d is simple conditional to each of the strings a, b, and c. It is proved that the mutual information between a, b, and c can be extracted if and only if the values of the conditional mutual informations I(a : b|c), I(a : c|b), and I(b : c|a) are negligible. The proof employs a non-Shannon-type information inequality (a generalization of the recently discovered Zhang–Yeung inequality).     

Modeling of deviation angle and performance losses in wet steam turbines using GMDH-type neural networks

In the present study group method of data handling (GMDH) type of artificial neural networks are used to model deviation angle (θ), total pressure loss coefficient (ω), and performance loss coefficient (ξ) in wet steam turbines. These parameters are modeled with respect to four input variables, i.e., stagnation pressure (P _z ), stagnation temperature (T _z ), back pressure (P _b), and inflow angle (β). The required input and output data to train the neural networks has been taken from numerical simulations. An AUSM–Van Leer hybrid scheme is used to solve two-phase transonic steam flow numerically. Based on results of the paper, GMDH-type neural networks can successfully model and predict deviation angle, total pressure loss coefficient, and performance loss coefficient in wet steam turbines. Absolute fraction of variance (R ²) and root-mean-squared error related to total pressure loss coefficient (ω) are equal to 0.992 and 0.002, respectively. Thus GMDH models have enough accuracy for turbomachinery applications.

     

Parallel Construction of (a, b)-Trees

We present an optimal parallel algorithm for the construction of (a, b)-trees-a generalization of 2-3 trees, 2-3-4 trees, and B-trees. We show the existence of a canonical form for (a, b)-trees, with a very regular structure, which allows us to obtain a scalable parallel algorithm for the construction of a minimum-height (a, b)-tree with N keys in O(N/p + log log N) time using p ≤ N/log log N processors on the EREW-PRAM model, and in O(N/p) time using p ≤ N processors on the CREW model. We show that the average memory utilization for the canonical form is at least 50% better than that for the worst-case and is also better than that for a random (a, b)-tree. A significant feature of the proposed parallel algorithm is that its time-complexity depends neither on a nor on b, and hence our general algorithm is superior to earlier algorithms for parallel construction of B-trees.     

Backstepping boundary control of Burgers’ equation with actuator dynamics

In this paper, we propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u_x(0,t), u_x(1,t), u_xx(0,t) and u_xx(1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H³ stable and well posed.     

Approximate regular expression pattern matching with concave gap penalties

Given a sequenceA of lengthM and a regular expressionR of lengthP, an approximate regular expression pattern-matching algorithm computes the score of the optimal alignment betweenA and one of the sequencesB exactly matched byR. An alignment between sequencesA=a₁a₂ ... a_M andB=b₁b₂... b_N is a list of ordered pairs, (i₁,j₁), (i₂j₂), ..., (i_t,j_tt) such that i_k < i_k+1 and j_k < j_k+1. In this case the alignmentaligns symbols a_ik and b_jk, and leaves blocks of unaligned symbols, orgaps, between them. A scoring schemeS associates costs for each aligned symbol pair and each gap. The alignment's score is the sum of the associated costs, and an optimal alignment is one of minimal score. There are a variety of schemes for scoring alignments. In a concave gap penalty scoring schemeS={, w}, a function (a, b) gives the score of each aligned pair of symbolsa andb, and aconcave function w(k) gives the score of a gap of lengthk. A function w is concave if and only if it has the property that, for allk > 1, w(k + 1) –w(k) w(k) –w(k –1). In this paper we present an O(MP(logM + log² P)) algorithm for approximate regular expression matching for an arbitrary and any concavew. This work was supported in part by the National Institute of Health under Grant RO1 LM04960.     

Unification Modulo <Emphasis Type="Italic">ACUI</Emphasis> Plus Distributivity Axioms

E-unification problems are central in automated deduction. In this work, we consider unification modulo theories that extend the well-known ACI or ACUI by adding a binary symbol * that distributes over the AC(U)I-symbol +. If this distributivity is one-sided (say, to the left), we get the theory denoted AC(U)ID_l; we show that AC(U)ID_l-unification is DEXPTIME-complete. If * is assumed two-sided distributive over +, we get the theory denoted AC(U)ID; we show unification modulo AC(U)ID to be NEXPTIME-decidable and DEXPTIME-hard. Both AC(U)ID_l and AC(U)ID seem to be of practical interest, for example, in the analysis of programs modeled in terms of process algebras. Our results, for the two theories considered, are obtained through two entirely different lines of reasoning. A consequence of our methods of proof is that, modulo the theory that adds to AC(U)ID the assumption that * is associative-commutative, or just associative, unification is undecidable.     

Cosmologies from nonlinear multidimensional gravity with acceleration and slowly varying <Emphasis Type="Italic">G</Emphasis>

We consider multidimensional gravity with a Lagrangian containing the Ricci tensor squared and the Kretschmann invariant. In a Kaluza-Klein approach with a single compact extra space of arbitrary dimension, with the aid of a slow-change approximation (as compared with the Planck scale), we build a class of spatially flat cosmological models in which both the observed scale factor a(τ) and the extradimensional one, b(τ), grow exponentially at large times, but b(τ) grows slowly enough to admit variations of the effective gravitational constant G within observational limits. Such models predict a drastic change in the physical laws of our Universe in the remote future due to further growth of the extra dimensions.     

Binding number and Hamiltonian (g,f)-factors in graphs II

A (g, f)-factor F of a graph G is called a Hamiltonian (g, f)-factor if F contains a Hamiltonian cycle. For a subset X of V(G), let N _G (X)= gcup _x∈X N _G (x). The binding number of G is defined by bind(G)=min{| N _G (X) |/| X|| ?≠X?V(G), N _G (X)≠V(G)}. Let G be a connected graph of order n, 3≤a≤b be integers, and b≥4. Let g, f be positive integer-valued functions defined on V(G), such that a≤g(x)≤f(x)≤b for every x∈V(G). Suppose n≥(a+b?4)²/(a?2) and f(V(G)) is even, we shall prove that if bind(G)>((a+b?4)(n?1))/((a?2)n?(5/2)(a+b?4)) and for any independent set X?V(G), N _G (X)≥((b?3)n+(2a+2b?9)| X|)/(a+b?5), then G has a Hamiltonian (g, f)-factor.