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CHI Lab Reading Group 10 March 2015
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Intro: Autocorrelation Function
• Average of the product of a data sample x[n] with a version of itself advanced by lag
• Autocorrelation model defined by:
Autocorrelation Value of x
Sample delay
Number of Data Points
N-k
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Autoregressive Model and DFT
AR Model • Advantage: Provides smoother and
more easily interpretable power spectrum
• Advantage: More exact determination of frequency components
• Advantage: Can be run on smaller time windows
• Disadvantage: Complex model identification
DFT • Advantage: Simple to run
• Disadvantage: Harder to interpret
• AR model is an alternative to DFT in calculation of a power spectrum density function of a time series.
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Deriving the AR Model
• Predicts current values of a time series based on past values
• AR model is a set of autocorrelation functions
• AR model defined by:
Current Value
Model Order
Previous Values
Weighting Coefficients One-Step
Prediction Error
*Assume Model Order = 3 Points: x[n] x[n-1] x[n-2] x[n-3]
a1*x[n-1]
a2*x[n-2]
a3*x[n-3]
ε[n] +
+
+
=
x[n]
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Deriving the AR Model
• Unknowns • ε[n]
• a1 , a2 , a3 … aM
x = (N-M) x 1 matrix X = (N-M) x M matrix a = M x 1 matrix ε = (N-M) x 1 matrix Thus, if: N = 25 M = 3
[22x1] = [22x3] x [3x1] + [22x1]
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Deriving the AR Model
ε and a optimization • (aopt) occurs when ε is
orthogonal to each explanatory vector xi, i = 1…M
• To do this, solve each xi
Tε=0 • Can simplify this to XTε=0
where as in the last slide X = all xi vectors
• This is a direct least squares solution, called the covariance method
Remember: a, ε Optimization:
XT ∙ ε = 0
XT ∙ (x – X ∙ aopt) = 0
XT ∙ x – XT ∙ X ∙ aopt = 0
XT ∙ x = XT ∙ X ∙ aopt
XT ∙ x ∙ (XT ∙ X)-1 = XT ∙ X ∙ (XT ∙ X)−1 ∙aopt
XT ∙ x ∙ (XT ∙ X)-1 = I∙aopt
XT ∙ x ∙ (XT ∙ X)-1 = aopt
x = X ∙ a + ε
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Deriving the AR Model
(XT ∙ X)-1
XT ∙ x
Autoregressive Model Autocorrelation Model
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Deriving the AR Model
XT ∙ x ∙ (XT ∙ X)-1 = R-1 ∙ r = aopt
• Yields Yule-Walker Equation
• Yule-Walker can be solved through Levinson-Durbin algorithm • Recursive
• Yields solutions for all lower M values, makes M selection easier
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AR Model Optimization
• Methods for selecting M • Final Prediction Eror • Criterion Autoregressive Transfer
Function • Rissanen Minimum Description
Length • Akaike’s Information Criterion
(AIC): • The higher M is the better the
model fits the measurments • Often why M for cardiovascular
signals is large than predicted by AIC
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Transforming the AR Model
• Convolution • Used to determine output
sequence of a filter
• Method for converting AR in time domain to frequency domain
• Autocorrelation
Impulse Response Filter
Output Sequence
Input Sequence
Convolution
*If h[n] = x[n] the only difference between autocorrelation and convolution is the time reversal in convolution
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Transforming the AR Model
• Time Domain AR • Frequency Domain AR
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Transforming the AR Model
Input Sequence
Error Term
Transfer Function
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Transforming the AR Model
• AR Filter All Pole Structure • Different frequency components in time domain estimated from poles of H(z),
the roots of the X(z) polynomial
• H(z) peaks whenever the frequency point on the circle passes a pole, the nearer the unit circle it is, the sharper the peak.
• Thus, using the poles, H(z) represented as:
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Autoregressive Model and DFT
AR Model • Advantage: Provides smoother and
more easily interpretable power spectrum
• Advantage: More exact determination of frequency components
• Advantage: Can be run on smaller time windows
• Disadvantage: Complex model identification
DFT • Advantage: Simple to run
• Disadvantage: Harder to interpret
• AR model is an alternative to DFT in calculation of a power spectrum density function of a time series.
![Page 15: CHI Lab Reading Group 10 March 2015 - University of Oxforddavidc/pubs/ht2015_db.pdf · Sample delay Number of Data Points N-k . Autoregressive Model and DFT ... where as in the last](https://reader034.vdocument.in/reader034/viewer/2022051919/600c549ca70f8c490634dfcf/html5/thumbnails/15.jpg)
AR Model Example
• Input Signal (Pre-processed RR-interval signal)
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AR Model Example
• Pole-zero plot • Transfer Function and Power Spectrum
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AR Model Example
• Residual Noise