Using generalized additive models to decompose time series and waveforms, and dissect heart–lung interaction physiology

Figures

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  Fig. 1. Splines fitted to simulated data (n = 70). The data-generating function is Y = sin(X) with added normally distributed noise. a Vertical dashed lines show the position of the 8 knots. *In the cyclic spline there are effectively 7 knots, since the first and last line represent a single knot, joining the ends of the spline. b Comparison of a penalised and an unpenalised spline fitted to the same data. The unpenalised spline with 20 knots is clearly too wiggly and overfits the data. Penalising the spline on wiggliness reduces the risk of overfitting, but keeps the model flexible in case the data demand it

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  Fig. 2. How a generalized additive model (GAM) can be fitted to a series of pulse pressure measurements (derived from the arterial waveform). a and b For each beat, systolic and diastolic pressure are detected, and pulse pressure (PP) is calculated. A GAM with two smooths c and d is fitted to the PP time series (b). c This first smooth represents the variation in PP explained by the beats’ position in the respiratory cycle. Coloured points (beats) correspond between panels b and c. d The second smooth represents the trend in PP over time with the model constant (α) added. The sum of these two smooths (b and c) gives the model prediction. Residuals of the model (ε) are the vertical distance from the smooth to the points in panel c (i.e. the scatters are partial residuals). Dashed curves represent 95% confidence intervals

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  Fig. 3. This patient has a heart-rate-to-respiratory rate ratio just beyond 2:1 (52:24). a From the pulse pressure (PP) plot, it is difficult to assess pulse pressure variability (PPV), and it seems to be changing. b When PP is modelled as a smooth function of each beat’s position in the respiratory cycle, a tight relationship between respiration and PP is revealed. Dashed curves represent 95% confidence intervals

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  Fig. 4. Generalized additive model of central venous pressure (CVP). Variation in CVP is explained by the effects of the cardiac cycle and the respiratory cycle. In this model there is no interaction between the two effects. Grey shades in b, c and d represent 95% confidence intervals (often too narrow to be visible)

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  Fig. 5. How a generalized additive model (GAM) can be fitted to a CVP waveform. a Each sample from a 125 Hz CVP waveform is represented with three predictor variables: position in cardiac cycle, position in respiratory cycle and time (seconds since sample start). A GAM is fitted giving the smooth functions b to e (the model constant (α) is added to the smooth function in d. f Model fit including residuals that are markedly reduced compared to the model without an interaction term, visualised in Fig. 4. Grey shades in panel b, c and e represent 95% confidence intervals (often too narrow to be visible)

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  Fig. 6. Comparison of sections of a generalized additive model (GAM). The model is fitted to two one-minute sections of central venous pressure (CVP) recordings. One section before a 250 ml fluid infusion and one after. A grey shade in panel c represents 95% confidence intervals (often too narrow to be visible). The constant terms (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta }_{s}$$\end{document}βs) are included in all predictions (b, c and d)

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  Fig. 7. Quantile generalized additive models (QGAM) robustly fit medical signals with non-normal errors. The models correspond to the model shown in Fig. 5