Reconciling Fitts Law with Shannons Information Theory EMPG 2015 - PowerPoint PPT Presentation

Reconciling Fitts’ Law with Shannon’s Information Theory EMPG 2015 University of Padua, Sept 1-3, 2015 Julien Gori* Olivier Rioul** Yves Guiard*** *ENS Cachan **Telecom ParisTech ***CNRS LTCI Paris, France

Table of Contents Information Theory & Psychology Historical Perspective Channel Capacity Fitts’ Law What is Fitts’ Law ? Multiple formulas A Geometric Framework Partitioning the Space with Targets 3 New Derivations A Coherent Information Theoretic Model A Communication Channel A Capacity Formula 2/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Presentation outline Information Theory & Psychology Historical Perspective Channel Capacity Fitts’ Law What is Fitts’ Law ? Multiple formulas A Geometric Framework Partitioning the Space with Targets 3 New Derivations A Coherent Information Theoretic Model A Communication Channel A Capacity Formula 3/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Annus Mirabilis : 1948 Claude Shannon’s A Mathematical Theory of Communication Information Uncertainty Communication system Capacity 4/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Two Telling Quotes Information is quantifiable and measurable ! A tremendous impact on psychologists : We now call them experiments on the capacity of people to transmit information. ( G. A. Miller, 1956 , The Magical Number Seven, Plus or Minus Two ) Presented with a shiny new tool kit [information theory] and a somewhat esoteric new vocabulary to go with it, more than a few psychologists reacted with an excess of enthusiasm. ( F. Attneave, 1959 , Applications of Information Theory to Psychology ) 5/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

A Strong Reaction from Shannon and Colleagues The first paper has the generic title « Information Theory, Photosynthesis and Religion » ( [Elias, 1958] ) [. . .] the basic results of the subject are aimed in a very specific direction, a direction that is not necessarily relevant to such fields as psychology, economics, and other social sciences. ( [Shannon, 1956] ) 6/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

The Channel Capacity Maximum amount of information transmittable over noisy communication link (channel) 7/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

The Channel Capacity Maximum amount of information transmittable over noisy communication link (channel) A dditive W hite G aussian N oise channel y = s + n signal : s + noise : n 7/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

The Channel Capacity Maximum amount of information transmittable over noisy communication link (channel) A dditive W hite G aussian N oise channel y = s + n signal : s + noise : n Shannon’s famous Theorem 17 (1948) C = 1 1 + S = 1 � � 2 log 2 log ( 1 + SNR ) bits per channel use N N = E ( n 2 ) , S = E ( s 2 ) 7/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

The Channel Capacity Maximum amount of information transmittable over noisy communication link (channel) A dditive W hite G aussian N oise channel y = s + n signal : s + noise : n Shannon’s famous Theorem 17 (1948) C = 1 1 + S = 1 � � 2 log 2 log ( 1 + SNR ) bits per channel use N N = E ( n 2 ) , S = E ( s 2 ) Any achievable rate (=reliable communication) R ≤ C 7/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Whatever Happened to Information Theory in Psychology ? Information theory discredited in psychology One rarely sees Shannon’s information theory in contemporary psychology articles ( R. Luce, 2003 , Whatever Happened to Information Theory in Psychology ? ) There is one notable exception : Fitts’ Law , since 1954, and more generally the speed-accuracy trade-off for rapid aimed movement [Soukoreff and MacKenzie, 2009]. Part of ISO 9241-9. Used for device assessment and movement time prediction in HCI. 8/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Presentation outline Information Theory & Psychology Historical Perspective Channel Capacity Fitts’ Law What is Fitts’ Law ? Multiple formulas A Geometric Framework Partitioning the Space with Targets 3 New Derivations A Coherent Information Theoretic Model A Communication Channel A Capacity Formula 9/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

The Paradigm Aiming at a target of size W from a distance D Start D W 10/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

How Do D and W Affect Target Acquisition Time ? Fitts’ definition of an I ndex of D ifficulty (ID), by analogy with Shannon’s Theorem 17 : � 2 D � ID = log 2 ( bits ) W ( M ovement T ime) MT = a + b · ID through linear regression → Speed-accuracy trade-off a and b determined through experimentation. 11/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Other Formulations for ID Fitts’ original formulation, [Fitts, 1953] � 2 D � ID = log 2 W Welford’s formulation [Welford, 1960] � 0 . 5 + D � ID = log 2 W MacKenzie’s formulation [MacKenzie, 1989] � � 1 + D ID = log 2 W a ( D W ) b √ a + b A Many more formulations ! a + b log ( A W ) − a + b ( c + D ) log ( 2 A W ) 12/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

MacKenzie’s Formulation an analogy with Shannon’s capacity : � � � � 1 + D C = 1 1 + S ID = log 2 2 log 2 W N D , W target distance and size S , N powers of signal and noise is D W an amplitude SNR ? What is the communication model ? What are the input, output and noise ? What about the 1 2 factor ? 13/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Mackenzie’s Formulation (cont’d) Capacity for a system MacKenzie formulation → Communication model → Speed-accuracy trade-off y = s + n signal : s + noise : n C = 1 � 1 + S � � 1 + D � ID = log 2 2 log 2 N W Achievable rate (vanishing error probability) 14/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Presentation outline Information Theory & Psychology Historical Perspective Channel Capacity Fitts’ Law What is Fitts’ Law ? Multiple formulas A Geometric Framework Partitioning the Space with Targets 3 New Derivations A Coherent Information Theoretic Model A Communication Channel A Capacity Formula 15/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start D W W aiming at a target is equivalent to choosing one target among N 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory

Geometric Framework Idea : aiming = choosing ! stop start N targets D W W 16/34 Institut Mines-Télécom Reconciling Fitts’ Law with Shannon’s Information Theory