Properties of the DWT and MODWT

## 1. Introduction

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

where Z is the set of all integers and the factor *j*. The entire set of wavelets *j* and translation index *k*. Higher *j* corresponds to lower frequency wavelets, and higher *k* corresponds to a rightward shift. The wavelet transform is usually considered to be a continuous wavelet transform (CWT) (Vidakovic, 1999; Percival & Walden, 2000; Gencay et al., 2001) since it is applied to a function *f* ( ) defined over the entire real axis. However, we only have a finite number *N* of sampled values, as is usually the case for real data applications. This approach leads to the discrete wavelet transform (DWT). To some degree of approximation, we can regard the DWT as being formed by taking slices through a corresponding CWT (Mc Coy et al., 1995). Any wavelet in

where the two-dimensional set of coefficients

Let *X* _{t }be a dyadic length column vector containing a sequence*N*=2^{ J }observations of a real-valued time series. The length *N* vector of discrete wavelet coefficients *W* is obtained via *W*= *X*, where is an *N* × *N* orthonormal matrix defining the DWT. The vector of wavelet coefficients may be organised into *J +* 1 vectors,

where *W* _{j }is a length *N/* *V* _{J }is a length *N/*2^{ J }vector scaling coefficients associated with averages on a scale of length 2^{ J }= 2λ_{ J. }Wavelet coefficients thus tell us about variations in adjacent averages (Percival & Walden, 2000).

The structure of the *N* × *N* dimensional matrix _{1}, …, _{J }and V_{ J }(scaling coefficient matrix) via

Let us now consider implementation of the DWT by using a pyramid algorithm (PA) (Mallat, 1989). Let *h* = (*h* _{0 },..., *h* _{L-1 }) be the vector of wavelet (high-pass) filter coefficients and *g* = (*g* _{0 },..., g_{ L-1 }) be the vector of scaling (low-pass) filter coefficients (Daubechies, 1992). Graphical representation of the DWT as applied to a dyadic length vector *X* _{t }is given by Figure 1 and Figure 2. These depict the analysis of _{1}, W_{2} and V_{2} using the pyramid algorithm (PA). The synthesis of _{1}, W_{2} and V_{2} use the inverse of the PA (Figure 2).

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 *N* satisfy, *J* and replace this restriction with the condition that *N* can be an integer multiple 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).

Property | DWT | MODWT |

Data | N=2 J | Any sample size N |

Detail and Smooth Coefficients of MRA | Downsampling | Associated with zero phase filters |

Circularly shifting | Does not hold | Holds and Invariant |

Wavelet Variance | 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

The length *N* vector of discrete wavelet coefficients *W* is obtained via *W* = W*X* and *N* × *N* orthonormal matrix defining the DWT, *W* _{j }is a length *N/* *V* _{J }is a length *N/*2^{ J }vector scaling coefficients associated with averages on a scale of length 2^{ J }= 2λ_{ J }in Section 2.1. Due to orthonormality, *X* = W^{T} *W,* and the squared wavelet coefficients of *W* the norm,

Let

where *λ* _{j }MODWT coefficients for *X* _{t }to *Y* _{t }, *X* _{t }or *Y* _{t }denoted by *X* _{t }, *Y* _{t }) at scale *λ* _{j }=2^{ j-1} is then defined as follows,

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 X into its DWT coefficients W = wY, where w is a N × N orthonormal matrix− the components, Wj, of W are associated with coefficients for each scale λj = 2 j−1Wavelet coefficients Wj inform on the variations in adjacent averages over λ_{j.}There are _{j} ≡ 2^{j−1}, j = 1, 2 . . , J_{0} where λ_{j} = 2^{j−1} is a so-called standardized scale, whereas λ_{j}Δt 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 as |

Cross- Correlation and correlation The scale λj MODWT coefficients may be used to investigate the correlation and cross-correlation of two time series, Xt and Yt .The wavelet cross correlation of Xt, Yt at scale λj=2 j-1 for a time lag τ, is defined asscale λj MODWT coefficients. Correlation is the cross-correlation at lag τ = 0. | MODWT-MRA The orthonormal matrix, N × N, leads to scale-based decompositions. Given the MODWT coefficients, Y can be constructed as an additive decomposition, known as a multiresolution analysis (MRA).Specifically the level J0 MODWT-based MRA is given byj= 1, 2, . . ., J0 )for a pre- specified J _{0}, and are part of the MRA of Y that can be attributed to variations on a scale of λj. . |

where *X* _{t }and *Y* _{t }, respectively (Percival & Walden, 2000; Gencay et al., 2001). As with the usual correlation coefficient (between two random variables), the range of *j*. The typical cross-correlation statistic is purely a function of the cross-covariance and standard deviations. Thereby the MODWT estimator of the wavelet cross-correlation (WCCORR) of the two processes, which are at variance by an integer lag τ, is defined as

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 *X* _{t }and *Y* _{t }on a scale-by-scale basis. In terms of confidence intervals (CIs) for the WCORR and WCCORR parameters, a nonlinear transformation is required to produce reasonable CIs for the correlation coefficient (Gencay et al., 2001). We use the Fisher’s z-transformation (Dépué 2003) which is defined as follows,

An unbiased estimator of the WCORR based on the MODWT in Equation (7) is*N* independent Gaussian observations, has the following limiting distribution (Percival & Walden, 2000; Gencay et al., 2001).

Applying the transformation tanh maps the confidence interval back to [−1*,* 1] to produce an approximate 95% CI for *ρ* as follows (Whitcher et al. 2000; Gencay et al. 2001; Hudson et al., 2010 b)

The quantity

## 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), *λ* _{j }wavelet coefficients of each patient’s bivariate (S, R) time series is used to assess how the simulated (S) and recorded (R) infusion series correlate. Graphical assessment tools and derived wavelet-based metrics are then suggested and proven valid for ICU A-S management. The wavelets based performance indices developed in this chapter pertain to the modulus of the wavelet correlation at wavelet scale (*λ* _{j }) (level 1) and also to summary statistics founded on measures of the wavelet correlations and wavelet cross-correlations over scales, as defined below (Table 6). For the *i*th patient (*i*=1, 2, …, 37) count the number of its wavelet scales (out of a maximum of 8) for which the WCORR value at scale *j*=1, 2, …, 8) as either significant (S) or not significant (NS) in Table 4. Patient specific AND (average normalized density), RAND (relative average normalized density), and TI (tracking index) values, all derived by Rudge et al (2006b) and Chase et al. (2004), are also shown in Table 4. Definitions for the AND, RAND and TI performance indicators or diagnostics are detailed in Table 3.

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 *significant negative* WCORR value at scale *significant negative* WCORR value at scale_{1} and

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 *j*=1, 2, …, 8), their significance or otherwise, and on their “Count of NS” wavelet correlations (Table 4). As mentioned above Table 4 also indicates patients which are considered to track poorly according to both Rudge et al’s (2006b) and Chase et al.’s (2004) diagnostics, the latter are based on completely different mathematical methodologies as summarised in Table 3. It is noteworthy that 11 of our 15 wavelet based poor trackers are also considered to be poor trackers by either or both of Rudge et al.’s (2006b) and Chase et al.’s (2004) performance indices.

Figure 5 shows the estimated WCORR, *j*< 5. Figures 6 and 7 display each patient’s multivariate profile of AND, RAND and “Count of NS” (divided by 10 for axis scaling purposes), for increasing

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 *j*=1, 2, …, 8), “Count of NS”, *j*=1, 2,…, 8) is 5.0 for the poor trackers, significantly higher than the median of 2.0 for the good tracking group (P = 0.001) (Table 6). It is noteworthy also that the patient specific WCORR profiles are good visual “signatures” of the patient’s tracking status (good or poor tracking) (see Figure 5).

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) (*kappa* = 0.2127, P=0.01) and with that of Rudge et al. (2006b) (*kappa* = 0.5856, P=0.001). It is noteworthy also that the threshold criterion for poor tracking for RAND by both Rudge et al.’s (2006b) and Chase et al.’s (2004) performance indicators is corroborated by our poor trackers, identified by the wavelet metrics derived in this chapter, shown to have a median RAND of 0.50 (Table 6), which is exactly the threshold used by Rudge et al. (2006b) and Chase et al. (2004).

Patient ID | Wavelet scale λ j (j= 1, 2, …, 8) | Count of NS* | Rudge et al. (2006b) diagnostics | Chase et al. (2004) diagnostics | ||||||||

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

P1 | 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) |

P3 | 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) |

P5 | S | S | NS | NS | NS | NS | NS | S | 5 | 0.60 | 0.80 | 91.1 (0.016) |

P6 | 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) |

P8 | 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) |

P12 | S | S | S | S | S | S | NS | S | 1 | 0.61 | 0.77 | 84.1 (0.033) |

P13 | S | S | S | S | S | NS | NS | S | 2 | 0.37 | 0.45 | 86.1 (0.072) |

P14 | S | S | S | NS | S | NS | NS | S | 3 | 0.48 | 0.56 | 93.1 (0.034) |

P15 | S | S | S | NS | NS | NS | NS | S | 4 | 0.45 | 0.60 | 91.1 (0.011) |

P16 | S | S | S | S | S | NS | NS | S | 2 | 0.44 | 0.57 | 87.9 (0.021) |

P17 | S | S | S | S | S | NS | NS | S | 2 | 0.61 | 0.72 | 84.0 (0.037) |

P18 | S | S | S | S | NS | S | NS | S | 2 | 0.55 | 0.68 | 94.6 (0.026) |

P19 | S | S | S | S | S | NS | NS | S | 2 | 0.50 | 0.66 | 91.1 (0.014) |

P20 | 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) |

P23 | S | S | S | S | NS | NS | NS | S | 3 | 0.72 | 0.85 | 84.8 (0.105) |

P24 | S | S | S | S | S | NS | NS | S | 2 | 0.43 | 0.54 | 88.1 (0.023) |

P25 | S | S | S | S | S | NS | NS | S | 2 | 0.50 | 0.66 | 92.4 (0.025) |

P26 | 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 | S | S | S | S | S | S | NS | S | 1 | 0.38 | 0.45 | 77.6 (0.083) |

P30 | NS | NS | NS | NS | NS | NS | S | S | 6 | 0.63 | 0.82 | 92.2 (0.021) |

P31 | 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 | S | S | S | NS | NS | S | NS | S | 3 | 0.28 | 0.36 | 88.7 (0.020) |

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

P35 | S | S | S | S | S | S | NS | S | 1 | 0.38 | 0.46 | 85.9 (0.044) |

P36 | S | S | NS | NS | NS | NS | NS | S | 5 | 0.52 | 0.64 | 86.4 (0.095) |

P37 | S | S | S | S | S | NS | NS | S | 2 | 0.53 | 0.59 | 79.9 (0.093) |

Poor Patient ID | Wavelet Correlations (scale λ j ) | Rudge et al. (2006b) diagnostics | Chase et al. (2004) diagnostics | ||||||||

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

P2 | 0.005 | 0.005 | 0.005 | 0.005 | -0.008 | -0.186 | -0.034 | 0.649 | 0.53 | 0.66 | 86.2 (0.03) |

P4 | 0.006 | 0.006 | 0.005 | 0.005 | 0.036 | -0.077 | 0.023 | 0.724 | 0.56 | 0.62 | 80.1 (0.09) |

P7 | -0.081 | -0.081 | -0.081 | -0.081 | -0.037 | -0.066 | 0.034 | 0.785 | 0.33 | 0.43 | 84.5 (0.06) |

P9 | -0.020 | -0.021 | -0.021 | -0.021 | 0.012 | -0.066 | 0.012 | 0.793 | 0.49 | 0.62 | 87.3 (0.02) |

P10 | 0.051 | 0.051 | 0.051 | 0.051 | 0.052 | -0.125 | -0.040 | 0.591 | 0.27 | 0.34 | 83.4 (0.04) |

P11 | -0.001 | -0.001 | -0.001 | -0.001 | 0.019 | 0.093 | 0.002 | 0.654 | 0.31 | 0.38 | 83.7 (0.08) |

P21 | 0.016 | 0.017 | 0.017 | 0.017 | 0.026 | -0.148 | -0.031 | 0.779 | 0.53 | 0.72 | 78.5 (0.09) |

P22 | -0.051 | -0.051 | -0.051 | -0.051 | -0.027 | -0.115 | -0.052 | 0.739 | 0.35 | 0.45 | 85.2 (0.04) |

P27 | -0.108 | -0.108 | -0.108 | -0.108 | -0.088 | -0.227 | -0.201 | 0.660 | 0.39 | 0.49 | 74.9 (0.07) |

P28 | -0.100 | -0.101 | -0.101 | -0.101 | -0.090 | -0.200 | -0.157 | 0.628 | 0.34 | 0.44 | 89.2 (0.02) |

P29 | -0.616 | -0.616 | -0.616 | -0.616 | -0.582 | -0.497 | -0.294 | 0.627 | 0.38 | 0.45 | 77.6 (0.08) |

P32 | 0.032 | 0.034 | 0.035 | 0.035 | 0.042 | -0.139 | -0.061 | 0.732 | 0.38 | 0.50 | 89.3 (0.02) |

P33 | 0.046 | 0.046 | 0.046 | 0.046 | 0.049 | -0.127 | -0.012 | 0.691 | 0.28 | 0.36 | 88.7 (0.02) |

P34 | -0.019 | -0.019 | -0.019 | -0.019 | -0.029 | -0.213 | -0.131 | 0.676 | 0.43 | 0.55 | 86.5 (0.03) |

P35 | -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 | -0.019 | -0.019 | -0.019 | -0.019 | -0.008 | -0.139 | 0.040 | 0.676 | 0.390 | 0.500 | 85.2 |

(95%CI) | (-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) | |

P1 | -0.112 | -0.134 | -0.134 | -0.134 | -0.107 | -0.101 | -0.115 | 0.662 | 0.51 | 0.62 | 87.0 (0.04) |

P3 | -0.503 | -0.504 | -0.504 | -0.504 | -0.439 | -0.221 | -0.242 | 0.675 | 0.70 | 0.83 | 88.8 (0.02) |

P5 | 0.074 | 0.073 | 0.073 | 0.073 | 0.078 | -0.118 | -0.004 | 0.707 | 0.60 | 0.80 | 91.1 (0.02) |

P6 | -0.202 | -0.202 | -0.202 | -0.202 | -0.155 | -0.090 | -0.167 | 0.666 | 0.70 | 0.84 | 87.0 (0.01) |

P8 | 0.208 | 0.206 | 0.206 | 0.206 | 0.211 | -0.029 | 0.081 | 0.783 | 0.45 | 0.59 | 87.4 (0.03) |

P12 | -0.359 | -0.359 | -0.359 | -0.359 | -0.316 | -0.264 | -0.179 | 0.645 | 0.61 | 0.77 | 84.1 (0.03) |

P13 | 0.258 | 0.258 | 0.258 | 0.258 | 0.257 | 0.007 | 0.107 | 0.768 | 0.37 | 0.45 | 86.1 (0.07) |

P14 | -0.378 | -0.379 | -0.380 | -0.380 | -0.301 | -0.074 | -0.003 | 0.785 | 0.48 | 0.56 | 93.1 (0.03) |

P15 | 0.086 | 0.092 | 0.093 | 0.093 | 0.084 | -0.149 | -0.039 | -0.697 | 0.45 | 0.60 | 91.1 (0.01) |

P16 | 0.168 | 0.169 | 0.169 | 0.169 | 0.173 | -0.067 | 0.035 | 0.758 | 0.44 | 0.57 | 87.9 (0.02) |

P17 | -0.122 | -0.122 | -0.122 | -0.122 | -0.069 | 0.131 | -0.161 | 0.604 | 0.61 | 0.72 | 84.0 (0.04) |

P18 | -0.134 | -0.134 | -0.134 | -0.134 | -0.130 | -0.243 | -0.211 | 0.628 | 0.55 | 0.68 | 94.6 (0.03) |

P19 | 0.272 | 0.273 | 0.273 | 0.273 | 0.277 | 0.066 | 0.195 | 0.726 | 0.50 | 0.66 | 91.1 (0.01) |

P20 | 0.057 | 0.057 | 0.057 | 0.057 | 0.057 | -0.106 | -0.051 | 0.613 | 0.53 | 0.65 | 87.3 (0.03) |

P23 | 0.149 | 0.149 | 0.149 | 0.149 | 0.164 | -0.019 | 0.099 | 0.729 | 0.72 | 0.85 | 84.8 (0.11) |

P24 | 0.231 | 0.232 | 0.232 | 0.232 | 0.232 | -0.007 | 0.157 | 0.793 | 0.43 | 0.54 | 88.1 (0.02) |

P25 | -0.211 | -0.214 | -0.214 | -0.214 | -0.188 | -0.218 | -0.185 | 0.561 | 0.50 | 0.66 | 92.4 (0.03) |

P26 | -0.122 | -0.123 | -0.123 | -0.123 | -0.073 | 0.057 | -0.011 | 0.579 | 0.68 | 0.88 | 87.4 (0.03) |

P30 | -0.045 | -0.044 | -0.044 | -0.044 | -0.049 | -0.208 | -0.150 | 0.647 | 0.63 | 0.82 | 92.2 (0.02) |

P31 | 0.040 | 0.040 | 0.040 | 0.040 | 0.047 | -0.108 | 0.024 | 0.701 | 0.40 | 0.51 | 89.3 (0.03) |

P36 | 0.081 | 0.081 | 0.081 | 0.081 | 0.055 | -0.172 | -0.104 | 0.677 | 0.52 | 0.64 | 86.4 (0.10) |

P37 | 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 | 0.049 | 0.049 | 0.0485 | 0.0485 | 0.051 | -0.096 | -0.025 | 0.676 | 0.525 | 0.655 | 87.7 |

(95%CI) | (-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 | 0.004 | 0.003 | 0.005 |

Variable | Poor group median | Good group median | k-w # : P value |

Modulus WCORR at | 0.046 | 0.159 | 0.001 |

“Count of NS” | 5 | 2 | 0.001 |

AND | 0.39 | 0.53 | 0.004 |

RAND | 0.50 | 0.66 | 0.003 |

TI | 85.20 | 87.7 | 0.005 |

### 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 *j*=1, 2, …, 6) (Figure 8). At scale

Patient 3 (good tracker) | ||||||||

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

L. CI | -0.528 | -0.539 | -0.554 | -0.573 | -0.544 | -0.398 | -0.485 | 0.382 |

ρ ˜ | -0.503^{+} | -0.504^{+} | -0.504^{+} | -0.504^{+} | -0.439^{+} | -0.221^{+} | -0.242^{+} | 0.675^{+} |

U. CI++ | -0.477 | -0.467 | -0.451 | -0.428 | -0.321 | -0.027 | 0.036 | 0.845 |

Patient 4 (poor tracker) | ||||||||

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

L. CI | -0.034 | -0.050 | -0.074 | -0.107 | -0.123 | -0.297 | -0.299 | 0.401 |

ρ ˜ | 0.006 | 0.006 | 0.005 | 0.006 | 0.036 | -0.077 | 0.023 | 0.724 |

U. CI++ | 0.045 | 0.061 | 0.085 | 0.117 | 0.194 | 0.151 | 0.339 | 0.886 |

Recall that Table 7 gives the values (*signature* over wavelet scales) of

## 4. Conclusion

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, *particularly* at *λ* _{1 }. Moreover, *cross*-correlation (WCCORR) signatures also show a common pattern for the good trackers, which is distinctly different to the pattern associated with poor tracking. It is also shown that the lead or lag relationship between a patient’s simulated and recorded agitation-sedation infusion series can be investigated on a scale by scale basis via the implementation of the MODWT-MRA. The MRA is shown to successfully indicate local features of interest in the simulated and recorded series, with the smooth MRA series offering a good visual summary of the overall long-term trends in a patient’s A-S status.

Fifteen poor trackers are identified by the DWT based diagnostics derived in this chapter. Specifically it is found that the modulus of WCORR at *λ* _{1 }, |*λ* _{1 }|, is invariably higher (and significantly so) for the good compared to the poor trackers. The profile of (“Count of NS”) across all patients is higher for the poor trackers; and the values of RAND, AND and |*λ* _{j }| invariably higher for the good trackers. Specifically the median absolute value of the wavelet correlation at *λ* _{1 }, and median value of AND, RAND and TI are highly significantly lower for the poor trackers (15 of 37 patients). The median of the number of non significant wavelet correlations (the “Count of NS” variable) is 5.0 for the poor trackers, significantly higher than the median of 2.0 for the good tracking group (P = 0.001). It is noteworthy that 11 of the 15 DWT based poor trackers are also found to be poor trackers by either or both of Rudge et al.’s (2006b) and Chase et al.’s (2004) (non wavelet based) performance indicators, showing significant agreement between the wavelet DWT and the earlier kernel based methods - with a kappa test of agreement between the WCORR criterion for poor tracking and Chase et al (2004) of 0.2127 (P=0.01); and with Rudge et al. (2006b) of 0.5856 (P=0.001).

Other recent work by Kang et al. (*In Prep)* has used Bayesian densities and wavelet shrinkage methods to create a novel wavelet probability band (WPB). A 90% value for the WPB implies that for at least 90% of the time, the estimated mean value of the patient’s recorded infusion rate lies within the band. A 90% WPB was constructed by Kang et al. *(In Prep*) for each of the 37 patient profiles, and the time and duration of any deviations from the wavelet probability band recorded for each patient. Likewise wavelet analogues of the AND and RAND diagnostics of Rudge et al. (2006a; 2006b), namely the average normalized wavelet density (ANWD) and the relative average normalized wavelet density (RANWD) have been derived by Kang et al. *(In Prep*); as has a Wavelet Time Coverage Index (WTCI) – all by using Bayesian wavelet thresholding. The resultant WTCI and 90% WPB provide very strong support for the DWT wavelet diagnostics derived in this chapter. Indeed of the 15 DWT based poor trackers identified in this chapter, 10 also exhibit a low WPB (WPB 90% < 70%) and a low wavelet density based RANWD measure (RANWD*In Prep)*. Statistically speaking the wavelet probability band and wavelet density diagnostics mirror the poor versus good classification of this chapter’s WCORR DWT based diagnostics (kappa = 0.7701, P = 0.0001). The main reason for the reduced total time within the WPB (and for the non significant WCORRs) for this minority group of 10 (or 15) poor trackers (of a total of 37 patients), is the consistently poor performance of the DE model throughout their *total* length of the A-S simulation.

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. *(In Prep*). Wavelets are shown to visually and quantitatively discriminate patients for whom the A-S model captures their fundamental A-S dynamics, versus those, for whom this is not so. Wavelet WCORR and WCCORR signatures thus form a possibly alternative and appropriate feedback mechanism for comparison of improved sedation administration controllers and gain. The wavelets based visual tools and quantitative measures thus contribute to the task of improving and of refining A-S models. This chapter thus demonstrates that wavelets provide a new diagnostic tool by which to assess the agitation-sedation of ICU patients, and show that it is possible to evaluate A-S models via wavelet diagnostics for accurate evaluation of A-S management, where the latter represents a trade-off between the benefits of low patient agitation versus the cost of high infusion rates and increased total dose requirements (Rudge et al., 2003; 2005; 2006a; 2006b). Wavelets are thus suitable for clinical implementation in ICU agitation and sedation control.

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.