Properties of the DWT and MODWT
The use of quantitative modelling to enhance understanding of the agitation-sedation (A-S) system and the provision of an A-S simulation platform are key tools in this area of patient critical care. A suite of wavelet techniques and metrics based on the discrete wavelet transform (DWT) are developed in this chapter which are shown to successfully establish the validity of deterministic agitation-sedation (A-S) models against empirical (recorded) dynamic A-S infusion profiles. The DWT approach is shown to provide robust performance metrics of A-S control and also yield excellent visual assessment tools. This approach is generalisable to any study which investigates the similarity or closeness of bivariate time series of, say, a large number of units (patients, households etc) and of disparate lengths and of possibly extremely long length. This work demonstrates the value of the DWT for assessing ICU agitation-sedation deterministic models, and suggests new wavelet based diagnostics by which to assess the A-S models.
Typically agitation-sedation cycling in critically ill patients involves oscillations between states of agitation and over-sedation, which is detrimental to patient health, and increases hospital length of stay (Rudge et al., 2006a; 2006b; Chase et al., 2004; Rudge et al 2005). Agitation management via effective sedation management is an important and fundamental activity in the intensive care unit (ICU), where in the hospitalized adult agitation is defined as excessive verbal behaviour that interferes with patient care, and the patient’s medical therapies (Chase et al., 2004). The main goal of sedation is to control agitation, while also preventing over-sedation and over-use of drugs. In clinical practice, however, a lack of understanding of the underlying dynamics of A-S, combined with a lack of subjective assessment tools, makes effective and consistent clinical agitation management difficult (Chase et al., 2004; Rudge et al., 2005, 2006b). Early agitation management methods traditionally relied on subjective agitation assessment, and sedation assessment scales, combined with medical staff experience and intuition, to deliver appropriate sedation; and an appropriate sedation input response, from recorded at bedside agitation scales (Fraser & Riker, 2001b; Jaarsma et al., 2001; Ramsay et al., 1974; Ricker et al., 1999; Sessler et al., 2002). The clinical staff at the bedside, usually nurses, then select an appropriate infusion rate based upon their evaluation of these scales, experience, and intuition (Kress et al., 2002). This approach usually leads to the administration of largely continuous infusions which lack a bolus-focused approach, and commonly result in either over sedation, or insufficient sedation (Rudge et al., 2006b). Several recent studies have emphasised the cost and healthcare advantages of drug delivery protocols based on assessment scales of agitation and sedation. A minimal differential equation (DE) model to predict or simulate the patients’ agitation-sedation status over time (range [3,001-25,261] time points in minutes) was developed and validated statistically for the first time by Chase et al. (2004). This process is depicted in Figure 3 (see Chase et al., 2004). The goal of the research was to create a physiologically representative pharmacodynamic model that captured the fundamental dynamics of the A-S system. The resulting model can serve as a platform to develop and test semi-automated sedation management controllers that offer the potential of improved agitation management and thus, the clinically relevant outcomes of reduced length of stay in the ICU and reduced health care costs as a result. A-S models were later developed by Rudge et al. (2005, 2006a, 2006b). All these models used either kernel regression, tracking indices, kernel density estimation, a probability band or time within a band as metrics of similarity or closeness of the patient’s simulated and recorded A-S profiles. Lee et al. (2005) also developed a nonparametric regression approach with an Epanechnikov kernel (Wand & Jones 1995) to assess the validity of the deterministic A-S models.
The work in this chapter develops novel wavelet signatures and wavelet based statistics and threshold criterion (to assess closeness between pairs of time series). These are applied to the recorded and the simulated infusion rates obtained from the DE models of Chase et al. (2004) to test for commonality across patients, in terms of wavelet correlations. A major aim of this study is to test the feasibility of wavelet statistics to help distinguish between patients whose simulated profiles were “close” to their mean profile a majority of the time profile versus those for whom this was not the case - so-called good versus poor trackers. This research builds on initial work by Kang et al. (2005), which was a preliminary study to assess wavelet signatures for modelling ICU A-S profiles to evaluate “closeness” or “discrimination” of simulated versus actual A-S profiles with respect to wavelet scales. Another earlier application of some of our methods was the study by Kang et al. (2004) on historical, flowering records of 4 Eucalypt species, where it was established that wavelets add credibility to the use of phenological records to detect climate change (see also Hudson, 2010 , Hudson 2011, Hudson et al., 2010 c and Hudson et al., 2005). This early phenological study was recently expanded from 4 to 8 Eucalypt species by Hudson et al. (2011a, 2011b) (see also Hudson et al., 2010 a, 2010b and Hudson & Keatley, 2010 ).
2. Brief review of wavelets and associated mathematics
Section 2 gives a brief introduction of the basic ideas concerning wavelets. A wave is usually defined as an oscillating function that is localized in both time and frequency. A wavelet is a “small wave”, which has its energy concentrated in time to give a tool for the analysis of transient, nonstationary, or time-varying phenomena (Goupillaud et al., 1984; Morlet, 1983). Wavelets have the ability to allow simultaneous time and frequency analysis via a flexible mathematical foundation. Wavelets are well suited to the analysis of transient signals in particular. The localizing property of wavelets allows a wavelet expansion of a transient component on an orthogonal basis to be modelled using a small number of wavelet coefficients using a low pass filter (Barber et al., 2002). This allows application to a wide range of fields, such as signal processing, data compression, and image analysis (Mallat, 1998; Meyer, 2003; Kumar, 1993, 1994; Donoho, 1995; Chang et al., 2000a, 2000b). The wavelet decomposition of functions is analogous to Fourier decomposition methods (Ogden, 1997; Abramovich & Benjamini, 1995). The wavelet representation is presented first in terms of its simplest paradigm, the Haar wavelet (Haar, 1910). The Haar wavelet is used here to describe the concepts of multiresolution analysis (MRA). For more details about wavelets see, for example, Daubeches (1992), Chui (1992), Donoho & Johnstone (1994), Ogden (1997), Vidakovic (1999), Percival & Walden (2000), and Gencay et al. (2001).
2.1. The Discrete Wavelet Transform (DWT)
Wavelets may be formed from the mother wavelet function via dyadic dilation and integer translation by the following,
where Z is the set of all integers and the factor maintains a constant norm independent of scale
where the two-dimensional set of coefficients is called the discrete wavelet transform of A more specific form indicating how the’s are calculated, can be written using inner products (Swelden, 1996) as follows,
The structure of the
Let us now consider implementation of the DWT by using a pyramid algorithm (PA) (Mallat, 1989). Let
The Inverse DWT (IDWT) is achieved through upsampling the final level of wavelet and scaling coefficients, convolving them with their respective filters and adding up the two filtered vectors. Figure 2 gives a flow diagram for the reconstruction of from the second level wavelet and scaling coefficient vectors. Given a dyadic length time series, it was not necessary to implement the DWT down to level. A partial DWT (PDWT) may be performed instead that terminates at a level. The resulting vector of wavelet coefficients will then contain wavelets (Percival & Walden, 2000; Gencay et al., 2001). PDWT’s are more commonly used in practice than the full DWT because of the flexibility they offer in specifying a scale beyond which a wavelet analysis into individual large scales is no longer of real interest. A PDWT of level allows us to relax the restriction that
2.2. The Maximal Overlap Discrete Wavelet (MODWT)
The DWT is an alternative to the Fourier transform (FT) for time series analysis. The DWT provides wavelet coefficients that are local in both time and frequency. In this section the maximal overlap DWT (MODWT) which is a modified version of the discrete wavelet transform is discussed. Like the DWT, the MODWT is defined in terms of a computationally efficient pyramid algorithm (PA). The term MODWT comes from the relationship of the MODWT with estimators of the Allan variance (Allan, 1966). The MODWT gives up orthogonality in order to gain features the DWT does not possess. A consequence of this is that the wavelet and scaling coefficients must be rescaled in order to retain the variance preserving property of the DWT (Percival & Guttorp, 1994).
|Any sample size |
|Downsampling||Associated with zero phase filters|
|Does not hold||Holds and Invariant|
|Less efficient||Asymptotically Efficient|
The properties in Table 1 are important in distinguishing the MODWT from the DWT (Percival & Mofjeld, 1997; Percival & Walden, 2000; Gencay et al., 2001). The decomposition and reconstruction procedure and inverting of the MODWT is similar to the DWT. A key feature to an MRA using the MODWT is that the wavelet details and smooth are associated with zero-phase filters. Thus, interesting features in the wavelet details and smooth may be perfectly aligned with the original time series. This attribute is not available through the DWT since it subsamples the output of its filtering operations (Percival & Walden, 2000; Gencay et al., 2001).
2.3. Wavelet-based estimators of correlation
Let and be two time series of interest. The wavelet cross-covariance of for scale and lag τ is defined (Percival, 1995; Percival & Guttorp, 1994; Whitcher et al., 2000; Gencay et al., 2001) as follows,
where and are the scale
|Rationale for wavelets|
Wavelets allow time series data to be decomposed on a scale by scale basis, or to be discretized, into its so-called underlying subcomponents.
Conventional time frequency domain methods results may be difficult to interpret, whereas the wavelet-correlation is able to display how the association between two time series change with wavelet scale.
Transformation of the data (orthonormal) allows correlation, cross-correlational analyses of bivariate series to be performed - based on the derived wavelet coefficients.
DWT is often less computer intensive than other transformations (e.g. fast Fourier transform).
DWT offers easier analysis than the CWT as most time series are sampled as discrete values. DWT allows for the decorrelation of time series.
|Qualitative Description of the DWT and MODWT|
DWT transforms the original time series
There arewavelet coefficients for each scale λj ≡ 2j−1, j = 1, 2 . . , J0 where λj = 2j−1 is a so-called standardized scale, whereas λjΔt is a physical scale, where Δt is the sampling interval.
The MODWT is a non-decimated variation of the DWT, which defines the jth level MODWT detail subcomponent of the time series asand defines the jth level MODWT smoothed series,, which is related to the average (over scale N), and is normally interpretable as the trend.
|Cross- Correlation and correlation|
The wavelet cross correlation of
The orthonormal matrix,
Specifically the level
for a pre- specified J0, and are part of the MRA of
where is the wavelet variance with scale. and are the scale MODWT coefficients for
As the usual cross-correlation is used to determine lead or lag relationships between two series, the wavelet cross-correlation provides a lead or lag relationship between
An unbiased estimator of the WCORR based on the MODWT in Equation (7) is. The given estimated correlation coefficient, based on
3. Application to Agitation Sedation (A-S) wavelet modelling
This section presents the application of a wavelets analysis of the agitation-sedation (A-S) data of 37 ICU patients’ bivariate time series, sourced from the research of Chase et al. (2004). An extensive description of A-S modelling, as well as other references on A-S control; along with details of the development and validation of the A-S model are given in Chase et al. (2004). The model by Chase et al. (2004) serves as a platform to develop and test semi-automated sedation management controllers that offer the potential of improved agitation management and reduced length of stay in the ICU. Figure 3 presents a diagram of the feedback loop employing nursing staff feedback of subjectively assessed patient agitation through the infusion controller (Chase et al., 2004). We refer the reader also to the later works of Lee et al. (2005) and of Rudge et al. (2005, 2006a, 2006b) who developed further A-S models and metrics. Table 3 summarises the equations used, mathematical methods employed and the aims of the given study, along with the performance indicators derived for each of Chase et al. (2004), Rudge et al. (2006a, 2006b, 2005), and Lee et al. (2005). As such Table 3 and subsequently Table 8 show how the research presented in this chapter adds knowledge and insight into A-S modelling in the context of these earlier works.
3.1. Using the DWT and MODWT
The DWT, the maximal overlap (MODWT) and multiresolution analysis (MRA) were applied to all pairs of patient specific infusion profiles (recorded (R) and simulated (S)) for the 37 ICU patients. The aim of the analysis reported in section 3.1 – 3.3 is to investigate
whether wavelets based diagnostics can reliably assess how well the A-S model (simulation) captures the underlying dynamics of the true recorded infusion rates at different horizons via the DWT; and to compare these results with the diagnostics of Chase et al. (2004), Rudge et al. (2006a, 2006b, 2005) and Lee et al. (2005). For illustration of these concepts patient specific recorded (R) and simulated ( S) profiles (as the thick line, according to the equation of Chase et al. (2004)) are shown in Figure 4. It is noteworthy that simulation of the A-S states using the model of Rudge et al. (2005) showed that a reduction in both the magnitude of agitation and the severity of agitation sedation cycling is possible. Mean and peak agitation levels were reduced by 68.4% and 52.9%, respectively, on average, with some patients exhibiting in excess of a 90% reduction in mean agitation level through increased control gains. Implementation of automated feedback infusion controllers based on such models could thus offer simple and effective drug delivery, without significant increases in drug consumption and expenses.
The lag/lead relationship between the S and R infusion series was investigated on a scale-by-scale basis via a MODWT-MRA (using the LaDaub (8) filter); thereby each patient’s S and R series can be expressed as a new set of series, called details and smooth. Each of these series are associated with variations at a particular wavelet scale. The results of a MODWT-MRA (not detailed here due to space restrictions) reveal that thirteen patients (patients 3, 9, 11, 17, 20, 22, 26, 30, 33, 34, 35, 36, 37) (Figure 4) have recorded infusion series that lead their corresponding simulated infusion series. It is noteworthy that of these 13 patients, which exhibit such a lagged dependency, our DWT wavelet diagnostics (and those of Chase et al., (2004) and Rudge et al., (2006b)) identify the following as poor performers in common (Patients 9, 11, 17, 22, 33, 34 and 35). Overall it is thought that the simulated profile peaks later than the patient’s recorded infusion possibly due to the delay in distribution time for the drug. This result implies that, while performing well most of the time, the simulated rate is lagging behind the patient’s true infusion rate. These periods indicate times of the patient’s hospital length of stay in ICU, where the DE model may not capture the subject’s specific A-S dynamics (evidenced by the time lags). These periods may correspond to periods of marked distress or physiological alterations due to the patient’s state. A common reason for the departure of the simulated profile is this apparent time-lag. Particularly small departures indicate rapid increases (or decreases) in the recorded infusion rate, where the simulated infusion rate appears to lag behind. These differences may be a result of the medical staff’s over or under-assessment of the patient’s agitation status, this hypothesis is as yet not proven.
3.2. Wavelet correlation and other diagnostics
In section 3.2 an estimate of wavelet correlation (WCORR), from Equation (7), is derived per patient. This WCORR between the scale
Poor trackers are identified via wavelet diagnostics as follows: Patients with a “Count of NS” greater or equal to 2 and a non-significant WCORR value at scale (level 1) are said to be a poor tracker; as are patients with a “Count of NS” less or equal to 3 and a
Note that the threshold values delineating poor tracking for AND and RAND (according to Rudge et al. (2006b)) are not taken into account in classifying a patient as either a poor or good tracker (Table 4). In this chapter our criterion for tracking classification is based solely on the patient’s WCORR values at scales (
Figure 5 shows the estimated WCORR, and 95% CI for four patients, P2 and P4 (are poor trackers) and P8 and P14 (are good trackers). From Figure 5 WCORR is generally significant for wavelet levels 1, 2, 4, 8 and 16 for the good trackers (whether they are a positive or negative WCORR value). This is not the case for the poor trackers, who generally exhibit a non-significant WCORR at for
By using the data per patient (from Table 4), we can perform a Kruskal Wallis test to statistically compare the medians of the performance indicators between the wavelet based good and poor trackers. These results are summarized in Table 5 and Table 6. Specifically Table 6 gives the results of the Kruskal Wallis (k-w) tests for our wavelet based poor versus good tracker groups for measures of WCORR at scale (
Recall that 11 of the 15 DWT based poor trackers are also considered to be poor trackers by either or both Rudge et al.’s (2006b) and Chase et al.’s (2004) (non wavelet based) performance indicators. Indeed kappa tests of agreement show that our DWT WCORR criterion for poor tracking, as developed in this chapter, agrees significantly with that of the performance thresholds of Chase et al (2004) (
|Poor Patient ID|
|Good Pt ID|
|(0.015,0.036)||(0.065,-0.726)||(0.479, 0.610)||(0.590, 0.771)||(86.984, 91.100)|
(Poor vs Good)
|Modulus WCORR at||0.046||0.159|
|“Count of NS”|
3.3. Using the Wavelet Cross-Correlation (WCCORR)
We can investigate possible lead or lag relationships between a given patient’s modelled (simulated) versus observed (recorded) A-S profile by examining a plot of its MODWT based wavelet cross-correlation (WCCORR), according to Equation (8). Figure 8 shows this WCCORR plot for Patient 3 (P3: a good tracker) and Patient 4 (P4: a poor tracker). For Patient 3 the values are negative and statistically significant for all scales except 8 (Table 7), and there is also a large positive peak at a lag of 120 minutes for the first six wavelet scales (
|Patient 3 (good tracker)|
|Level 1||Level 2||Level 3||Level 4||Level 5||Level 6||Level 7||Level 8|
|Patient 4 (poor tracker)|
|Level 1||Level 2||Level 3||Level 4||Level 5||Level 6||Level 7||Level 8|
Recall that Table 7 gives the values (
DWT and MODWT-MRA decomposition and reconstruction are shown to provide clear and consistent, in regard to good or poor performance, “signatures” of, and values for the wavelet correlations and cross-correlations (at all dyadic scales) between an ICU patient’s bivariate time series, namely their simulated and their recorded A-S infusion profiles over time under sedation. A suite of wavelet techniques are advocated, based on the DWT, and applied successfully to assess whether an ICU patient’s mathematically simulated agitation-sedation (A-S) status reflects their true dynamic profile.
The wavelet correlation profiles of the good trackers are shown to be invariably significant at all scales (except at 32 and 64). Patients who exhibit poor tracking exhibit WCORR profiles which are invariably non significant at almost all wavelet scales,
Fifteen poor trackers are identified by the DWT based diagnostics derived in this chapter. Specifically it is found that the modulus of WCORR at
Other recent work by Kang et al. (
The work in this chapter provides a suite of new wavelet based diagnostics by which to achieve statistical model validation of the A-S models. The DWT, wavelet correlation and cross-correlation measures derived in this chapter are proved to be valid for assessing control, and mirror earlier validation measures; as do the more recent wavelet regression diagnostics (namely WTCI, ANWD, RANWD, and WPB 90%) of Kang et al.
Overall the various wavelet diagnostics strongly agree and confirm the value of A-S modelling in ICU. Wavelet DWT analysis also demonstrates that the models of the A-S studies of Chase et al., (2004), Rudge et al., (2005; 2006a; 2000b) and of Lee et al., (2005), are suitable for developing more advanced optimal infusion controllers. These offer significant clinical potential of improved agitation management and reduced length of stay in critical care. The use of quantitative modelling to enhance understanding of the A-S system and the provision of an A-S simulation platform are critical tools in this area of patient critical care.
The DWT approach gives robust performance metrics of A-S control and also yields excellent visual assessment tools - generalisable to any study which investigates the similarity or closeness between any bivariate time series of, say, a large number of units (patients, households etc) and of disparate lengths and possibly of extremely long length.