1. Smoothing Serial Data

2. Serial Data Data collected over time Digitized Analog Signals
longitudinal
serial
Digitized Analog Signals
Data points are not independent
Serial Data

3. Smoothing Serial Data: Moving Average
3 point moving average
Serial Data

4. Simulated Signal
Signal simulated by sine wave data produced with sin() function in EXCEL
Serial Data

5. Simulated Noisy Signal
Original Sinusoid plus random error
Serial Data

6. 7-point Moving Average 7 adjacent points are averaged
Smoothed but still noisy
Serial Data

7. 21-point Moving Average
Smoother
Amplitude is reduced
Serial Data

8. Weighted Moving Averaging
Central points given more importance
Arbitrary weighting scheme
e.g
Serial Data

9. Weighted Moving Average
e.g. 5 data points
10, 9, 13, 12, 16
average = 12
with a weighting scheme of
weighted average:
= [(1x10)+(3x9)+(5x13)+(3x12)+(1x16)]/13
= 11.8

10. Mathematical Modeling of Serial Data

11. Signal Averaging of ECG
ECG + Noise
ECG
Identified QRS Peaks
Serial Data & Modeling

12. Mathematical Modeling of Serial Data
Signal Averaging
(a) (b) and (c) are QRS peak aligned ECG signal epochs
(d) is the result of averaging 100 such epochs
This works because noise tends to be random whereas signal has a consistent pattern
Mathematical Modeling of Serial Data

13. Smoothing Serial Data: Fitting Mathematical Equations
Often used to smooth noisy data
You can find an equation to fit most data
Can also be used for imputing (estimating) missing values
Serial Data

14. Mathematical Modeling of Serial Data

15. Mathematical Modeling of Serial Data
Modeling Serial Data
Differs from simple equation fitting in that the parameters of the equation must have meaning
Can be used to smooth
Can explain phenomena
Can be used to predict
Mathematical Modeling of Serial Data

16. Steps in Mathematical Modeling
Identification of the mechanism
Translation of that phenomenon into a mathematical equation
Testing the fit of the model to actual data
Modification of the model according to the results of the experimental evaluation
Mathematical Modeling of Serial Data

17. Criteria of Fit of the Model
Least Sum of Squares
Shape of the curve
Mathematical Modeling of Serial Data

18. Examination of Residuals
Residual = Actual Y - Predicted Y
Ideally there is no pattern to the residuals.
In this case there would be a horizontal normal distribution of residuals about a mean of zero.
However there is a clear pattern indicating the lack of fit of the model.
Mathematical Modeling of Serial Data

19. Ideal Characteristics of a Model
Simple
Fits the experimental data well
Has biologically meaningful parameters
Mathematical Modeling of Serial Data

20. Modeling Growth Data

21. Mathematical Modeling of Serial Data
Clinical Growth Charts
National Centre for Health Statistics (N.C.H.S.)1970’s revamped as Center for Disease Control C.D.C. charts, 2001
Most often used clinical norms for height and weight
Cross-sectional
Mathematical Modeling of Serial Data

22. Mathematical Modeling of Serial Data
Preece-Baines model I
where h is height at time t,
h1 is final height,
s0 and s1 are rate constants,
q is a time constant and
hq is height at t = q.
Mathematical Modeling of Serial Data

23. Smooth curves are the result of fitting Preece-Baines Model 1
to raw data
This was achieved using MS EXCEL rather than custom software

24. Examination of Residuals

26. Caribbean Growth Data
n =1697