Smoothing Serial Data

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

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

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

Simulated Noisy Signal Original Sinusoid plus random error Serial Data

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

21-point Moving Average Smoother Amplitude is reduced Serial Data

Weighted Moving Averaging Central points given more importance Arbitrary weighting scheme e.g 1 3 5 3 1 Serial Data

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

Mathematical Modeling of Serial Data

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

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

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

Mathematical Modeling of Serial Data

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

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

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

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

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

Modeling Growth Data

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

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

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

Examination of Residuals

Caribbean Growth Data n =1697