Transcript FUNCTIONAL DATA ANALYSIS: TECHNIQUES FOR EXPLORING

Functional Data Analysis:
Techniques for Exploring Temporal
Processes in Music
Bradley W. Vines
McGill University
Collaborators
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Daniel Levitin (McGill University)
Carol Krumhansl (Cornell University)
Jim Ramsay (McGill University)
Regina Nuzzo (McGill University)
Stephen McAdams (IRCAM)
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Talk Outline
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What is Functional Data Analysis?
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Steps of a typical FDA
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Demonstrate some of the major FDA tools
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Smoothing
Registration
General Linear Modeling (significance testing)
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An Example of Functional
Data
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Solo clarinet performances
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3 Treatment Groups
 Auditory
only
 Visual only
 Auditory + Visual
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Continuous Tension Judgments
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An example of functional data
QuickTime™ and a
Video decompressor
are needed to see this picture.
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What is Functional Data Analysis?
(Ramsay & Silverman, 1997)
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For data drawn from continuous processes
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Growth curves, market value, movement, ERP’s
Model data as functions of time
Temporal dynamics in music
(Vines, Nuzzo, & Levitin, under review)
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continuous measurements of emotion
expressive timing profiles
physiological measurements
movement tracking
Software tools available in Matlab and in S-Plus
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Modeling data as functions of time
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Basis functions
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Element functions that can be added together
to approximate the data.
W1*F1(t) + W2*F2(t) + W3*F3(t)…
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A least squares algorithm is used to
determine the weighting coefficients.
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Two basis types
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Fourier
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B-spline
 Polynomial
functions
 Knots
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Visualizing B-spline Bases
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Visualizing B-spline Bases
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Steps in a typical FDA
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Representing the data in Matlab: Matrices
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Each row: a sample point in time
Each column: an observation (participant/performer)
Third dimension for multivariate observations
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As in Schubert’s multi-dimensional continuous
interface
Valence
 Arousal
(Schubert, 1999)
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Steps in a typical FDA
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Modeling the data with functions
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Two major considerations:
Order of the B-spline bases
 The number of basis functions
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Steps in a typical FDA
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Modeling the data with functions
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Two major considerations:
Order of the B-spline bases
 The number of basis functions
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The order of B-spline bases
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Determines how many derivatives will be smooth.
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Steps in a typical FDA
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The number of basis functions
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Affects the quality of fit to the data
The more B-splines, the smaller the error
Tradeoff:
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Modeling data accurately
Excluding unimportant noise in the data
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Original Data
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Modeled Data
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Modeled Data
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Major FDA Tools
Controlling Unwanted Variability
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Curvature (high frequency noise)
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Amplitude
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Smoothing
Scaling
Phase
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Registration
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Nine Tension Judgments
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Nine Tension Judgments
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Nine Tension Judgments
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Time Warping
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General Linear Modeling
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Functional regression
Functional significance test (F-test)
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The effect of adding video
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Functional Linear Model
Y(t) = U(t) + B1(t) [if video is added]
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Results
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Significance Testing
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Analogous components to traditional F-testing:
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MSE(t) = SSE(t) / df(error)
-with df(error) = N participants - P parameters
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MSR(t) = [SSY(t) – SSE(t)]/df(model)
-with df(model) = P parameters - 1
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FRATIO(t) = MSR(t)/MSE(t)
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Significance Testing
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Other FDA techniques that
are available
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Analysis of covariance
Functional correlation analysis
Canonical correlation analysis
Principal Components Analysis
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FUNCTIONAL DATA ANALYSIS:
TECHNIQUES FOR EXPLORING TEMPORAL PROCESSES
IN MUSIC
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Prof. James Ramsay’s ftp site:
http://www.psych.mcgill.ca/faculty/ramsay/fda.html
[email protected]
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Smoothing
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The smoothing parameter, lambda,
controls the curvature of a function.
Trade off between perfect fit to the original
data and a best linear approximation for
the data. Penalizes variance
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Smoothing
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Examples of curves before and after
smoothing (try to find a good singly
participant who is nice and dynamic for all
of this, or a mean curve, I suppose)
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Principal Components Analysis
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Traditional statistics:
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Identifying major modes of variation
Reducing the number of dimensions in the data
Determine which variables are related
Functional analogue:
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Reveals major modes of variation
Can reveal trends in phase and in magnitude
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Principal Components Analysis
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Monthly temperature data
(available on the ftp website)
 Weather stations across Canada
 Exploring trends in the data and
grouping weather stations
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Monthly Weather Data
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Eigenvalues, VARIMAX PCA
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VARIMAX Principal Components
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VARIMAX Principal Components
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VARIMAX Principal Components
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VARIMAX Principal Components
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Component Scores
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