CHI Lab Reading Group 10 March 2015

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

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.

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]

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]

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

Deriving the AR Model

(XT ∙ X)-1

XT ∙ x

Autoregressive Model Autocorrelation Model

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

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

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

Transforming the AR Model

• Time Domain AR • Frequency Domain AR

Transforming the AR Model

Input Sequence

Error Term

Transfer Function

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:

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.

AR Model Example

• Input Signal (Pre-processed RR-interval signal)

AR Model Example

• Pole-zero plot • Transfer Function and Power Spectrum

AR Model Example

• Residual Noise