Properties of the DWT and MODWT
1. Introduction
The use of quantitative modelling to enhance understanding of the agitationsedation (AS) system and the provision of an AS 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 agitationsedation (AS) models against empirical (recorded) dynamic AS infusion profiles. The DWT approach is shown to provide robust performance metrics of AS 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 agitationsedation deterministic models, and suggests new wavelet based diagnostics by which to assess the AS models.
Typically agitationsedation cycling in critically ill patients involves oscillations between states of agitation and oversedation, 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 oversedation and overuse of drugs. In clinical practice, however, a lack of understanding of the underlying dynamics of AS, 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 bolusfocused 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’ agitationsedation status over time (range [3,00125,261] 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 AS system. The resulting model can serve as a platform to develop and test semiautomated 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. AS 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 AS 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 AS 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  socalled 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 AS profiles to evaluate “closeness” or “discrimination” of simulated versus actual AS 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 timevarying 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
where Z is the set of all integers and the factor
where the twodimensional set of coefficients
Let
where
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
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 zerophase 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. Waveletbased estimators of correlation
The length
Let
where
Rationale for wavelets Wavelets allow time series data to be decomposed on a scale by scale basis, or to be discretized, into its socalled underlying subcomponents. Conventional time frequency domain methods results may be difficult to interpret, whereas the waveletcorrelation is able to display how the association between two time series change with wavelet scale. Transformation of the data (orthonormal) allows correlation, crosscorrelational 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 Wavelet coefficients There are The MODWT is a nondecimated variation of the DWT, which defines the jth level MODWT detail subcomponent of the time series as 
Cross Correlation and correlation The scale The wavelet cross correlation of scale 
MODWTMRA The orthonormal matrix, Specifically the level for a pre specified J_{0}, and are part of the MRA of 
where
As the usual crosscorrelation is used to determine lead or lag relationships between two series, the wavelet crosscorrelation provides a lead or lag relationship between
An unbiased estimator of the WCORR based on the MODWT in Equation (7) is
Applying the transformation tanh maps the confidence interval back to [−1
The quantity
3. Application to Agitation Sedation (AS) wavelet modelling
This section presents the application of a wavelets analysis of the agitationsedation (AS) data of 37 ICU patients’ bivariate time series, sourced from the research of Chase et al. (2004). An extensive description of AS modelling, as well as other references on AS control; along with details of the development and validation of the AS model are given in Chase et al. (2004). The model by Chase et al. (2004) serves as a platform to develop and test semiautomated 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 AS 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 AS 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 AS 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 AS 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 scalebyscale basis via a MODWTMRA (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 MODWTMRA (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 AS 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 timelag. 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 underassessment 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),

Poor trackers are identified via wavelet diagnostics as follows: Patients with a “Count of NS” greater or equal to 2 and a nonsignificant WCORR value at scale
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,
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 (kw) 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) (






1  2  4  8  16  32  64  128  AND  RAND  TI (SE)^{}  

S  S  S  S  S  NS  NS  S  2  0.51  0.62  87.0 (0.041) 
P2  NS  NS  NS  NS  NS  NS  NS  S  7  0.53  0.66  86.2 (0.037) 

S  S  S  S  S  S  N.S  S  1  0.70  0.83  88.8 (0.015) 
P4  NS  NS  NS  NS  NS  NS  NS  S  7  0.56  0.62  80.1 (0.095) 

S  S  NS  NS  NS  NS  NS  S  5  0.60  0.80  91.1 (0.016) 

S  S  S  S  S  NS  NS  S  2  0.70  0.84  87.0 (0.014) 
P7  S  S  NS  NS  NS  NS  NS  S  5  0.33  0.43  84.5 (0.068) 

S  S  S  S  S  NS  NS  S  2  0.45  0.59  87.4 (0.027) 
P9  NS  NS  NS  NS  NS  NS  NS  S  7  0.49  0.62  87.3 (0.024) 
P10  S  S  S  NS  NS  S  NS  S  3  0.27  0.34  83.4 (0.041) 
P11  NS  NS  NS  NS  NS  NS  NS  S  7  0.31  0.38  83.7 (0.080) 

S  S  S  S  S  S  NS  S  1  0.61  0.77  84.1 (0.033) 

S  S  S  S  S  NS  NS  S  2  0.37  0.45  86.1 (0.072) 

S  S  S  NS  S  NS  NS  S  3  0.48  0.56  93.1 (0.034) 

S  S  S  NS  NS  NS  NS  S  4  0.45  0.60  91.1 (0.011) 

S  S  S  S  S  NS  NS  S  2  0.44  0.57  87.9 (0.021) 

S  S  S  S  S  NS  NS  S  2  0.61  0.72  84.0 (0.037) 

S  S  S  S  NS  S  NS  S  2  0.55  0.68  94.6 (0.026) 

S  S  S  S  S  NS  NS  S  2  0.50  0.66  91.1 (0.014) 

S  S  S  NS  NS  NS  NS  S  4  0.53  0.65  87.3 (0.033) 
P21  NS  NS  NS  NS  NS  NS  NS  S  7  0.53  0.72  78.5 (0.095) 
P22  S  S  NS  NS  NS  NS  NS  S  5  0.35  0.45  85.2 (0.043) 

S  S  S  S  NS  NS  NS  S  3  0.72  0.85  84.8 (0.105) 

S  S  S  S  S  NS  NS  S  2  0.43  0.54  88.1 (0.023) 

S  S  S  S  S  NS  NS  S  2  0.50  0.66  92.4 (0.025) 

S  S  S  NS  NS  NS  NS  S  4  0.68  0.88  87.4 (0.031) 
P27  S  S  S  S  NS  S  S  S  1  0.39  0.49  74.9 (0.074) 
P28  S  S  S  S  NS  S  NS  S  2  0.34  0.44  89.2 (0.027) 
P29 













NS  NS  NS  NS  NS  NS  S  S  6  0.63  0.82  92.2 (0.021) 

S  S  NS  NS  NS  NS  NS  S  5  0.40  0.51  89.3 (0.030) 
P32  S  S  NS  NS  NS  S  NS  S  4  0.38  0.50  89.3 (0.022) 
P33 












P34  NS  NS  NS  NS  NS  S  NS  S  6  0.43  0.55  86.5 (0.034) 
P35 













S  S  NS  NS  NS  NS  NS  S  5  0.52  0.64  86.4 (0.095) 

S  S  S  S  S  NS  NS  S  2  0.53  0.59  79.9 (0.093) 
Poor Patient ID 




λ_{1}  λ_{2}  λ_{3}  λ_{4}  λ_{5}  λ_{6}  λ_{7}  λ_{8}  AND  RAND  TI(SE)  

0.005  0.005  0.005  0.005  0.008  0.186  0.034  0.649  0.53  0.66  86.2 (0.03) 

0.006  0.006  0.005  0.005  0.036  0.077  0.023  0.724  0.56  0.62  80.1 (0.09) 

0.081  0.081  0.081  0.081  0.037  0.066  0.034  0.785  0.33  0.43  84.5 (0.06) 

0.020  0.021  0.021  0.021  0.012  0.066  0.012  0.793  0.49  0.62  87.3 (0.02) 

0.051  0.051  0.051  0.051  0.052  0.125  0.040  0.591  0.27  0.34  83.4 (0.04) 

0.001  0.001  0.001  0.001  0.019  0.093  0.002  0.654  0.31  0.38  83.7 (0.08) 

0.016  0.017  0.017  0.017  0.026  0.148  0.031  0.779  0.53  0.72  78.5 (0.09) 

0.051  0.051  0.051  0.051  0.027  0.115  0.052  0.739  0.35  0.45  85.2 (0.04) 

0.108  0.108  0.108  0.108  0.088  0.227  0.201  0.660  0.39  0.49  74.9 (0.07) 

0.100  0.101  0.101  0.101  0.090  0.200  0.157  0.628  0.34  0.44  89.2 (0.02) 

0.616  0.616  0.616  0.616  0.582  0.497  0.294  0.627  0.38  0.45  77.6 (0.08) 

0.032  0.034  0.035  0.035  0.042  0.139  0.061  0.732  0.38  0.50  89.3 (0.02) 

0.046  0.046  0.046  0.046  0.049  0.127  0.012  0.691  0.28  0.36  88.7 (0.02) 

0.019  0.019  0.019  0.019  0.029  0.213  0.131  0.676  0.43  0.55  86.5 (0.03) 

0.172  0.172  0.172  0.172  0.139  0.155  0.166  0.576  0.38  0.46  85.9 (0.04) 
Poor Median 












(0.092 ,0.013) 
(0.094, 0.013) 
(0.094, 0.013) 
(094, 0.013) 
(0.069, 0.032) 
(0.195, 0.091) 
(0.147, 0.003) 
(0.636,0.736)  (0.343, 0.515) 
(0.444, 0.620) 
(81.333, 87.001) 
Good Pt ID  
λ_{1}  λ_{2}  λ_{3}  λ_{4}  λ_{5}  λ_{6}  λ_{7}  λ_{8}  AND  RAND  TI(SE)  

0.112  0.134  0.134  0.134  0.107  0.101  0.115  0.662  0.51  0.62  87.0 (0.04) 

0.503  0.504  0.504  0.504  0.439  0.221  0.242  0.675  0.70  0.83  88.8 (0.02) 

0.074  0.073  0.073  0.073  0.078  0.118  0.004  0.707  0.60  0.80  91.1 (0.02) 

0.202  0.202  0.202  0.202  0.155  0.090  0.167  0.666  0.70  0.84  87.0 (0.01) 

0.208  0.206  0.206  0.206  0.211  0.029  0.081  0.783  0.45  0.59  87.4 (0.03) 

0.359  0.359  0.359  0.359  0.316  0.264  0.179  0.645  0.61  0.77  84.1 (0.03) 

0.258  0.258  0.258  0.258  0.257  0.007  0.107  0.768  0.37  0.45  86.1 (0.07) 

0.378  0.379  0.380  0.380  0.301  0.074  0.003  0.785  0.48  0.56  93.1 (0.03) 

0.086  0.092  0.093  0.093  0.084  0.149  0.039  0.697  0.45  0.60  91.1 (0.01) 

0.168  0.169  0.169  0.169  0.173  0.067  0.035  0.758  0.44  0.57  87.9 (0.02) 

0.122  0.122  0.122  0.122  0.069  0.131  0.161  0.604  0.61  0.72  84.0 (0.04) 

0.134  0.134  0.134  0.134  0.130  0.243  0.211  0.628  0.55  0.68  94.6 (0.03) 

0.272  0.273  0.273  0.273  0.277  0.066  0.195  0.726  0.50  0.66  91.1 (0.01) 

0.057  0.057  0.057  0.057  0.057  0.106  0.051  0.613  0.53  0.65  87.3 (0.03) 

0.149  0.149  0.149  0.149  0.164  0.019  0.099  0.729  0.72  0.85  84.8 (0.11) 

0.231  0.232  0.232  0.232  0.232  0.007  0.157  0.793  0.43  0.54  88.1 (0.02) 

0.211  0.214  0.214  0.214  0.188  0.218  0.185  0.561  0.50  0.66  92.4 (0.03) 

0.122  0.123  0.123  0.123  0.073  0.057  0.011  0.579  0.68  0.88  87.4 (0.03) 

0.045  0.044  0.044  0.044  0.049  0.208  0.150  0.647  0.63  0.82  92.2 (0.02) 

0.040  0.040  0.040  0.040  0.047  0.108  0.024  0.701  0.40  0.51  89.3 (0.03) 

0.081  0.081  0.081  0.081  0.055  0.172  0.104  0.677  0.52  0.64  86.4 (0.10) 

0.272  0.273  0.273  0.273  0.273  0.066  0.195  0.726  0.53  0.59  79.9 (0.09) 
Good Median 












(0.122, 0.149) 
(0.134, 0.149) 
(0.134, 0.149) 
(0.134, 0.149) 
(0.108, 0.164) 
(0.149, 0.019) 
(0.015,0.036)  (0.065,0.726)  (0.479, 0.610)  (0.590, 0.771)  (86.984, 91.100) 
Kruskal Wallis P value (Poor vs Good) 
P=0.32  0.32  0.32  0.32  0.30  0.16  0.40  0.84 




Poor group median 
Good group median 
kw 
Modulus WCORR at 
0.046  0.159 

“Count of NS” 



AND  0.39  0.53 

RAND  0.50  0.66 

TI  85.20  87.7 

3.3. Using the Wavelet CrossCorrelation (WCCORR)
We can investigate possible lead or lag relationships between a given patient’s modelled (simulated) versus observed (recorded) AS profile by examining a plot of its MODWT based wavelet crosscorrelation (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
Patient 3 (good tracker)  

Level 1  Level 2  Level 3  Level 4  Level 5  Level 6  Level 7  Level 8 

0.528  0.539  0.554  0.573  0.544  0.398  0.485  0.382 

0.503^{+}  0.504^{+}  0.504^{+}  0.504^{+}  0.439^{+}  0.221^{+}  0.242^{+}  0.675^{+} 

0.477  0.467  0.451  0.428  0.321  0.027  0.036  0.845 
Patient 4 (poor tracker)  

Level 1  Level 2  Level 3  Level 4  Level 5  Level 6  Level 7  Level 8 

0.034  0.050  0.074  0.107  0.123  0.297  0.299  0.401 

0.006  0.006  0.005  0.006  0.036  0.077  0.023  0.724 

0.045  0.061  0.085  0.117  0.194  0.151  0.339  0.886 
Recall that Table 7 gives the values (

4. Conclusion
DWT and MODWTMRA 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 crosscorrelations (at all dyadic scales) between an ICU patient’s bivariate time series, namely their simulated and their recorded AS 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 agitationsedation (AS) 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 AS models. The DWT, wavelet correlation and crosscorrelation 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 AS modelling in ICU. Wavelet DWT analysis also demonstrates that the models of the AS 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 AS system and the provision of an AS simulation platform are critical tools in this area of patient critical care.
The DWT approach gives robust performance metrics of AS 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.
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