Slide 1

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 2

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 3

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 4

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 5

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 6

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 7

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 8

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 9

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 10

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 11

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 12

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 13

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 14

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 15

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 16

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 17

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 18

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 19

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 20

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 21

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).


Slide 22

Outline
 Abstract
 Introduction
 Methodology

 Results

Abstract
 Most of the currently accepted approaches to

compute heart rate and assess heart rate
variability operate on interpolated, continuousvalued heart rate signals, thereby ignoring the
underlying discrete structure of human heart beats.
 To overcome this limitation, we model the

stochastic structure of heart beat intervals as a
history-dependent, inverse Gaussian process and
derive from it an explicit probability density
describing heart rate and heart rate variability.

 We estimate the parameters of the inverse

Gaussian model by local maximum likelihood and
assess model goodness-of-fit using Q-Q plot

analyses (Quantile-Quantile Normal Plots-常態分
位數圖)


goodness-of-fit (適合度):
此種檢定是看我們之實際值(或觀測值)是否服從某一理
論之分配。這種實際值（或觀測值）與理論值之間之配
合程度之檢定問題稱之為適合度檢定。

 We apply our model in an analysis of human

heart beat intervals from a tilt-table
experiment.

Introduction
 In the last 40 years, heart rate (HR) and heart

rate variability (HRV) have been established as
important quantitative indices of
cardiovascular control by the autonomic nervous
system, as well as effective diagnostic tools
and predictors of mortality for diseases
related to cardiovascular function and
regulation

 HR is the number of R-wave events (heart beats)

per unit time on the electrocardiogram (ECG).
 HRV is defined as the variation in the R-R

intervals, i.e., in the times between the Rwave events.
 Neither HR nor HRV can be observed directly

from the ECG, but both must be estimated from

the sequence of R-R intervals

 There are several methodological limitations to

current methods used to estimate HRV.
 In research studies, current time domain,

frequency domain, dynamical systems, and
entropy methods for HRV analysis generally
require several minutes or more of ECG
measurements in order to produce meaningful
analyses, and these data often must be
collected under stationary conditions.

 In addition, most of these methods must convert R-

R interval data into evenly spaced, continuousvalued measurements for analysis by first
interpolating the HR series estimate computed from
either the local averages model or the reciprocal
model
 While all of these methods give important

characterizations of human heart beat dynamics,

none provides a goodness-of-fit assessment to
measure how well the R-R interval data are
described by a particular model.

 In response to these shortcomings, we present a

new statistical framework that models the Rwave events as a discrete event defined by an
inverse Gaussian parametric probability
function
 As such, our approach is able to avoid the need

for conversion to continuous-valued signals and
is also able to formally assess model goodnessof-fit through well established techniques for
comparing discrete events models.

Methdology
 A. Heart Rate Probability Model

• In an observation interval (0,T]
• 0 event times detected from an ECG.
• Hun is the history of the R-R intervals up to Un
• θ is a set of p model parameters

 The history term represents the influence of recent

parasympathetic and sympathetic inputs to the SA node
on the R-R interval length by modeling the mean as a
linear function of the previous R-R intervals.

 The Mean and Standard deviation of the R-R interval

probability model in (1) are , respectively:

 高斯分佈
 早在18世紀就有數學家和天文學家開始探討這樣的一

條曲線。德國天文家兼數學家高斯（Carl Friedrich
Gauss，1777-1855）利用常態分佈研究天文學觀察中
誤差的分佈情形，因此常態分佈又稱高斯分佈。
 另一位著名的數學和統計學家Karl Pearson（1857-

1936）將高斯分佈稱為常態分佈。

90
80
70

人 數

60
50
40
30
20
10
0
150

155

160

165

170

身 高

175

180

185

190

 這條曲線的數學函數為



Y  f X ; ,

2



1

 2

e

1  X  
 

2  

2

 其中p = 3.1416，e是自然對數之底2.7183，X介在正

負無限大，m是平均數，s是標準差。一旦確定平均數
和標準差後，帶入公式算得f(X)。

 要決定常態分佈的形狀，就必須知道平均數m和變異數s2

（或者標準差s）。常態分佈取決於兩個參數
（parameter）：m和s2。
 只要設定這兩個參數，就可以畫出那條常態分佈曲線。只

要m或s2不同，曲線就不同。
 這也就是為何在上述公式裡，表明 其中分號後面代表的就

是決定這個函數的參數。假如變數X服從常態分佈，平均數
為m，變異數為s2，則寫成：X ~ N(m, s2)，其中~表示服
從，N表示常態分佈。

 B. Model Goodness-of-Fit
 Because the R-R interval model in (1) defines

an explicit discrete event model, we can use a
quantile-quantile (Q-Q) analysis to evaluate
model goodness-of-fit

Result

Fig. 2. Autoregressive spectral estimation of the supine (top panel) and the
tilt (bottom panel) segments of the interpolated reciprocal R-R intervals in
Fig. 1A (dotted line) and our HR estimates in Fig. 1C (solid line).