Measuring Entrainment: Some Methods and Issues
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Measuring Entrainment:Some Methods and Issues
J. Devin McAuleyJ. Devin McAuley
Center for Neuroscience, Mind & BehaviorCenter for Neuroscience, Mind & Behavior
Department of PsychologyDepartment of Psychology
Bowling Green State UniversityBowling Green State University
Email: Email: mcauley@bgnet.bgsu.edumcauley@bgnet.bgsu.edu
Entrainment Network III, Milton Keynes & Cambridge, UK, December 9 th – 12th 2005
J. Devin McAuley 2
Outline of Talk
• A Few Examples of EntrainmentA Few Examples of Entrainment
• Entrainment Involves Circular DataEntrainment Involves Circular Data
• Statistics for Circular DataStatistics for Circular Data
• What Can I Do With Circular Statistics?What Can I Do With Circular Statistics?
• What What Can’tCan’t I Do? I Do?
J. Devin McAuley 3
A Simple Example
Target TTarget T
(A) Stimulus Sequence(A) Stimulus Sequence
Produced Interval (P)Produced Interval (P)
(B) Tapping Sequence(B) Tapping Sequence
...
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A More Complex Example …
J. Devin McAuley 5
A Mystery Example …
J. Devin McAuley 6
Entrainment Involves Circular Data
• A simple way to describe any rhythmic behavior is using a circle.
• Each point on the circle represents a position in relative time (a phase angle).
• The start point is arbitrary.
J. Devin McAuley 7
Polar versus Rectangular Coordinates
r
(x, y)
180 0, 360
270
90
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(1,0)
(1, 0)
(0, 1)
(-1, 0)
0
90
180
270
(x, y)
R =
1
x = cos
y = sin
J. Devin McAuley 9
A Simple Example
Target TTarget T
(A) Stimulus Sequence(A) Stimulus Sequence
Produced Interval (P)Produced Interval (P)
(B) Tapping Sequence(B) Tapping Sequence
...
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A Tale of Two Oscillators
r
r
Driven Oscillator Driving Oscillator
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Case 1: Perfect Synchrony
= 0 = 0
Driven Oscillator Driving Oscillator
Each Produced Tap
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Case 2: Taps Lag Tones
= 0
= 45
Driven Oscillator Driving Oscillator
Each Produced Tap
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Case 3: Taps Ahead of Tones
= 0
= 315
Driven Oscillator Driving Oscillator
Each Produced Tap
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Case 4: Entrainment
= 0
→ ,
as n ↑
Driven Oscillator Driving Oscillator
Each Produced Tap
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Why won’t linear statistics work?
• With circular data there is a cross-over problem.
• For example, measured in degrees, the linear mean of 359 and 1 is 180, not 0
• This problem arises no matter what the start point is, and is independent of unit of measurement.
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Statistics for Circular Data
• Descriptive Statistics– Mean Direction, – Mean Resultant Length, R– Circular Variance, V
• Inferential Statistical Tests
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-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
90
0180
270
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(1,0)
(1, 0)
(0, 1)
(-1, 0)
0
90
180
270
(x, y)
R =
1
x = cos
y = sin
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Calculating a Mean
(x1, y1)(x2, y2)
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Calculating a Mean
X = x1 + x2
Y = y1 + y2
(X, Y)
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Mean Direction,
0,0
0,02tan
0tan
0,02
0,0tan
1
1
1
YXifundefined
YXifYX
XifYX
YXif
YXifYX
n
jjY
1
sin
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Mean Resultant Length, R
nRR
n
jjC
1
cos
n
jjC
1
cos
n
jjC
1
cos
n
jjY
1
sin
22 YXR (Pythagorean Theorem)
R R
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Circular Variance, V
V = 1 – R
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-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
90
0180
270
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-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
90
0180
270
= 50
R = 0.34
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90
0180
270
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
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90
0180
270
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
= 344
R = 0.88
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Statistics for Circular Data
• Descriptive Statistics– Mean Direction, – Mean Resultant Length, R– Circular Variance, V
• Inferential Statistical Tests
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Logic of Hypothesis Testing
• State Null & Alternative Hypotheses
• Determine Critical Value– for pre-selected alpha level (e.g., = 0.05)
• Calculate Test Statistic
• If Test Statistic > Critical Value– then Reject Null (e.g., p < 0.05)– otherwise Retain Null
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Inferential Statistics
• Test for uniformity
• Test for unspecified mean direction
• Test for specified mean direction
J. Devin McAuley 31
Logic of Hypothesis Testing
• State Null & Alternative Hypotheses
• Determine Critical Value– for pre-selected alpha level (e.g., = 0.05)
• Calculate Test Statistic
• If Test Statistic > Critical Value– then Reject Null (e.g., p < 0.05)– otherwise Retain Null
J. Devin McAuley 32
What can I do with circular stats?(not an exhaustive list)
• Descriptive statistics– Mean direction and length
– Variance, Standard Deviation
– Skewness, Kurtosis
• Inferential statistics– Uniformity, symmetry
– Unspecified and specified mean direction
– Comparison of two or more samples
– Confidence intervals
J. Devin McAuley 33
What can’t I do with circular stats?
• Circular statistics do not address sequential dependencies.
J. Devin McAuley 34
Stability Across Lifespan
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
4-5 6-7 8-9 10-12 18-38 39-59 60-74 75+Age Range (years)
Mea
n R
(p
has
e co
up
lin
g)
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