WO2013113111A1

WO2013113111A1 - System and method for prediction of self-similar signals

Info

Publication number: WO2013113111A1
Application number: PCT/CA2013/050062
Authority: WO
Inventors: Chi-Guhn LEE; Jue Wang
Original assignee: University of Toronto
Current assignee: University of Toronto
Priority date: 2012-01-30
Filing date: 2013-01-29
Publication date: 2013-08-08
Anticipated expiration: 2014-07-30

Description

SYSTEM AND METHOD FOR PREDICTION OF SELF-SFMILAR SIGNALS CROSS REFERENCE [0001] This application claims priority from United States Patent Application No.

61/592,352 filed January 30, 2012, which is hereby incorporated herein by reference. TECHNICAL FIELD [0002] The following relates generally to signal prediction and more specifically to prediction of self-similar signals. BACKGROUND [0003] Physiological signals are of great importance in the diagnosis and prognosis of critical illness. Advances in information technology have enabled the acquisition and storage of a large amount of physiological signal data. [0004] Various models have been proposed for analyzing physiological signal data and extracting information from them, including time series models. Several conventional time series models have been applied to physiological signals, such as autoregressive and moving average model (ARMA) and Hidden Markov model (HMM). Another model is fractional Brownian motion (fBM). However, these models have not been widely used in practice, one reason being that physiological signals often exhibit much more complex dynamics than what conventional models may handle. [0005] An examination of a mean arterial pressure (MAP) signal, an example of which is shown in Fig. 1, reveals its fractal property: parts of the signal resemble the whole. More specifically, the statistical properties of the signal remains the same when measured at different time scales, also known as statistical self-similarity. A fractal pattern can be understood as a recurring pattern in a chaotic environment. [0006] A number of physiological signals including heart rate variability, arterial blood pressure, respiratory rate, gait, as well as spatial patterns of physiological structures such as blood vessels, dendrites in neurons and airways in the lung, have been found to exhibit the statistical self-similarity property. The statistical self-similarity property of some physiological signals has been shown to relate to certain disease and aging. However, there are challenges to modeling a statistically self-similar (SSS) signal. [0007] Self-similar signals are also found in many natural and social phenomena, including turbulence, natural temperature variations, noise in electrical components, traffic in Ethernet networks and financial time series. [0008] A (MAP) signal is a representative self-similar signal. Fig. 2 shows a comparison of simulated signals using an autoregressive (AR) model (Fig. 2(a)), a HMM model (Fig. 2(b)) of a MAP signal and a real observed MAP signal (Fig. 2(c)) which illustrates disadvantages of the models. The AR model displays a uniform variation with no fundamental shifts. The HMM model may represent a non-homogeneous signal with piece-wise stationarity. However, a real MAP signal, which is characterized by irregular fluctuations with random bursts, is like neither of the modeled signals. [0009] Despite their prevalence, SSS processes are generally considered difficult to model. Classical models such as ARMA model typically only model processes with exponentially decreasing autocorrelation function, while the autocorrelation function of a SSS process decreases hyperbolically. [0010] It is an object of the following to obviate or mitigate at least one of the above disadvantages. SUMMARY [0011] In one aspect, a method for predicting a substantially self-similar signal is provided, the method comprising: (a) obtaining an observed substantially self-similar signal; (b) discretizing the observed substantially self-similar signal into a plurality of intervals; (c) performing a mathematical transformation to generate a statistical value for each of the plurality of intervals at a plurality of time scales; (d) generating, by one or more processors, at least one additional statistical value from an across-scale relationship of at least a subset of the statistical values; and (e) predicting the substantially self-similar signal from the at least one additional statistical value. [0012] In another aspect, a system for predicting a substantially self-similar signal is provided, the system comprising a signal prediction engine operable to: (a) obtain an observed substantially self-similar signal; (b) discretize the observed substantially self-similar signal into a plurality of intervals; (c) perform a mathematical transformation to generate a statistical value for each of the plurality of intervals at a plurality of time scales; (d) generate at least one additional statistical value from an across-scale relationship of at least a subset of the statistical values; and (e) predict the substantially self-similar signal from the at least one additional statistical value. DESCRIPTION OF THE DRAWINGS [0013] The features will become more apparent in the following detailed description in which reference is made to the appended drawings wherein: [0014] FIG. 1 is a representative MAP signal; [0015] FIG. 2 is a comparison of simulated signals using an autoregressive (AR) model (a), a HMM model (b) and a real observed MAP signal (c); [0016] FIG. 3 is a block diagram illustrating a system for predicting self-similar signals; [0017] FIG. 4 is a flow chart illustrating a method for predicting self-similar signals; [0018] FIG. 5 is a hierarchical illustration of the relationships between the discrete signal X„ and wavelet coefficients; [0019] FIG. 6 is a plot of the autocorrelation of an original MAP signal and its corresponding Haar wavelet coefficients; [0020] FIG. 7 is an illustration of partial correlation of wavelet coefficients for MAP signal; [0021] FIG. 8 is an illustration of an across-scale prediction scheme; [0022] FIG. 9 is an illustration of a down-sampling algorithm with moving implementation; [0023] FIG. 10 is a comparison of an across-scale predicted mean and within-scale predicted mean; [0024] FIG. 11 is a receiver operating characteristic (ROC) curve for arterial blood pressure prediction; [0025] FIG. 12 is an illustration of a desired prediction time and corresponding time intervals of the input signal; and [0026] FIG. 13 is a ROC curve for heart rate variability prediction. DETAILED DESCRIPTION [0027] Embodiments will now be described with reference to the figures. It will be appreciated that for simplicity and clarity of illustration, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements. In addition, numerous specific details are set forth in order to provide a thorough understanding of the embodiments described herein. However, it will be understood by those of ordinary skill in the art that the embodiments described herein may be practiced without these specific details. In other instances, well-known methods, procedures and components have not been described in detail so as not to obscure the embodiments described herein. Also, the description is not to be considered as limiting the scope of the embodiments described herein. [0028] It will also be appreciated that any module, engine, unit, application, component, server, computer, terminal or device exemplified herein that executes instructions may include or otherwise have access to computer readable media such as storage media, computer storage media, or data storage devices (removable and/or non-removable) such as, for example, magnetic disks, optical disks, or tape. Computer storage media may include volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information, such as computer readable instructions, data structures, program modules, or other data. Examples of computer storage media include RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by an application, module, or both. Any such computer storage media may be part of the device or accessible or connectable thereto. Any application, module or engine herein described may be implemented using computer readable/executable instructions that may be stored or otherwise held by such computer readable media. [0029] A system and method for predicting a self-similar signal are provided. The prediction is made based on historical data of the self-similar signal (hereinafter also referred to as simply a "signal").

[0030] Referring now to Fig. 3, a system (300) for predicting self-similar signals comprises a signal prediction engine (302), a signal input (304) and a signal output (306). The signal input (302) is operable to obtain an input signal (308) from a signal source (310). The signal input (304) provides the input signal (308) to the signal prediction engine (302), which generates an output signal (312). The signal prediction engine provides the output signal (312) to the signal output (306), which may provide the output signal (312) to any signal storage (314) or signal di spl ay (316) medi a . [0031] The signal prediction engine (302) may comprise a processor (318) and a memory (320). The memory (320) may have stored thereon computer instruction which, when executed by the processor (318), provide the functionality described herein. The memory (320) may, for example, comprise any one or more of a magnetic disk, RAM, ROM, Flash, EPROM, EEPROM, or other storage media. The processor (318), when providing the functionality described herein, may temporarily or permanently store the input signal (308) and/or output signal (312) on the memory (320), which may be used for further analysis. [0032] It will be appreciated that the input signal need not be "strictly" self-similar. That is, the signal X„ need not strictly satisfy the scale invariance property wherein the statistical properties of an SSS process remain unchanged when the scale is enlarged or reduced. Pursuant to the scale invariance property, a stochastic process X(t), t G E is said to be SSS with parameter H if and only if : X (at ) = ^H .r (†) . a > O. f e l where ^— denotes equivalent in finite joint distributions. While in an idealized model, this property holds on all scales, in the real world upper bound a_u and lower bound are imposed over which such scale-invariant behavior applies. It is understood that the input signal need not be strictly statistically self-similar and may have such an upper and lower bound so as to be substantially self-similar. In the following description, the terms SSS and self-similar signal shall be understood to include a substantially self-similar signal. [0033] It has been found that it is possible to predict the long-term trend of a signal by enlarging its small-scale picture. In other words, short-term variations in a signal may be used to help predict its long-term variations. Furthermore, long-term variations may be used to estimate short-term variations. Counter intuitively, it has been found that it is possible to reverse the undesirable attributes of SSS behaviour which hamper the use of classic time series model by using the SSS property for signal prediction. [0034] The signal prediction engine may apply mathematical transformation to the input signal to model the output signal. Exemplary transformations comprise wavelet analysis, Fourier transform, statistical moments and autocorrelation. Statistical moments may comprise mean, variance, skewness and kurtosis. It shall be understood that while the following describes the application of wavelet analysis for the prediction of self-similar signals, the application of other mathematical transformations would be apparent to a person of skill in the art. One particular advantage of the wavelet analysis is that wavelet transformation may approximately whiten a SSS process. [0035] Referring now to Fig. 4, in one aspect, the signal prediction engine (302) obtains an observed self-similar signal X„ from the signal input. The signal prediction engine applies an across-scale prediction model (404) to generate the predicted signal using the obtained signal X„. The signal prediction engine discretizes the signal (406) or a subset thereof into a set of n intervals. The number and time duration of intervals selected for discretization may determine the time period for which the output signal can be predicted. [0036] The signal prediction engine generates a statistical value, which for a wavelet transformation may be a wavelet coefficient (408), for each of the n intervals, for example by applying a discrete wavelet transform (DWT) to each of the n intervals. The DWT may be applied recursively (410) based on the desired prediction time period for the signal to generate (or predict) wavelet coefficients across multiple time scales. Using the generated wavelet coefficients, for the observed intervals, at least one wavelet coefficient may be predicted (410). The at least one predicted wavelet coefficient may be transformed to a predicted signal (412) which may be output (414) as the output signal. [0037] The wavelet coefficients may be expressed as d_{j+ J:}„ and may be generated from the original obtained signal X„ as follows. [0038] The signal prediction engine applies a wavelet transform to the input signal. The wavelet transform may, for example, comprise a Haar wavelet transform. The Haar wavelet transform is based on a piecewise constant function (also referred to as the mother wavelet) expressed as:

[0039] By dilating and translating the mother wavelet function, an orthonormal dyadic wavelet basis of L² (R) may be generated:

where j, ) G TL. Any finite energy signal X_t may be decomposed over this Haar wavelet basis:

where dj _n = {X_t, i j_in) = / X_tipj _n(t) dt are the Haar wavelet coefficients.

[0040] Referring now to Fig. 5, the discrete wavelet transform (DWT) computes the wavelet coefficients of a discrete signal X_n recursively:

in which are the scaling coefficients, and particularly s_0,„ = X_n-

[0041] From these recursive formulae, it has been found that the wavelet coefficients may also be expressed in a form related to the original signal as follows: d_{J+ n} = 2 ^(x^ - X^) where

the aggregated process defined in the previous section. The above equation is herein referred to as the "wavelet equation".

[0042] The wavelet equation reveals a powerful relationship obtainable from X„. Referring again to Fig. 5, the wavelet equation illustrates that the wavelet coefficient is actually the scaled difference between the average X„ in interval AB (i.e.,

^{tne avera}g^e -¾ ^m interval (i.e., x ^). [0043] Thus, supposing the current time is at time point B in Fig. 5, then the signal prediction engine will already have enough information to compute

case, the signal prediction engine may predict the wavelet coefficient d_j+ j_:„ to indirectly predict average X„ in the window BC. In other words, the signal prediction engine may predict the wavelet coefficient at one level (scale) higher in order to transform it into the average X„ in the next 2¹ period through the wavelet. As such, the prediction of original variables along the time axis may be transformed into the prediction of wavelet coefficients across different scales. In relation to the depiction shown in Fig. 5, a higher scale may be referred to as a coarser scale relative to a lower scale which may be referred to as a finer scale. [0044] The signal prediction engine may select the appropriate coarsest scale used for the prediction based on the particular application and the particular requirements for prediction duration. The signal prediction engine may further select the discrete time period for each interval based on the particular application and the particular requirements for prediction duration. For example, in certain clinical decision-making, people may be concerned with the average value of a physiological variable in a specified time window, which may be represented \>y X& [0045] Referring now to Fig. 12, in a particular example, a desired prediction time (shown as prediction window EF) for certain diagnoses may be 10 minutes. In such an example, the signal prediction engine may discretize the input signal into a plurality of 10 minute intervals (shown as windows AB, BC and CD). It will be appreciated that the signal prediction engine may be configured suitably for different applications that require signal prediction in fractions of a second to many years. [0046] The signal prediction engine further applies a covariance function to predict the wavelet coefficient of a different scale. [0047] Referring now to Fig. 6, suppose that process {X(t), t E E} is statistically self- similar with Hurst parameter H. r(u) and f(co) denote its covariance function and spectrum respectively. The Fourier transform Ψ(ω) of the wavelet function ip(t) is:

[0048] Thus, Ψ_{/ η}(ω) =

d the covariance of an arbitrary pair of wavelet coefficients may be expressed by: jj η)ψ_{] η} (ν)ψ_{/ η}, (u + v)dudv \ f( )2^(]+ Ψ(2^] ω) (2^f ω) where Ψ (ω) is the complex conjugate of Ψ(ω). For the wavelet coefficients located at the same scale (i.e. J =j '), the covariance becomes

which is a function of (n - n ') only. [0049] Thus the wavelet coefficients are covariance stationary at each scale. Fig. 6 shows the within-scale autocorrelation function of the original MAP signal versus its Haar wavelet coefficients. It is clear that the autocorrelation structures of wavelet coefficients are similar across different scales. Furthermore, the discrete wavelet transform (DWT) significantly reduces the magnitude of within-scale autocorrelation. Such a "decorrelation" property allows the wavelet coefficients to be modeled by classical time series model. [0050] However, the across-scale decorrelation for wavelet coefficients is less powerful since

= 2^jl2X(2^Ju)i//(u -n)du

y(H+l/2) where j,n G TL. The second equality follows from a change of variable t = ίιι and the third

P

equality is the result of X(2^Ju) = 2^JH X(u) . [0051] In other words, because the statistical property of X(t) is scale-invariant, then the wavelet transform on one scale should have similar statistical property with the wavelet transform on another scale. [0052] Referring back to Fig. 5, for example, the finer scale wavelet coefficient d_j:2„.j covers X_n' S on smaller interval AB, while the coarser scale wavelet coefficient d_j+j_:„ covers X_n>s on a larger interval AC. If the X^ sequence on interval AB is statistically self-similar to the sequence on interval AC, then it may be expected that a correlation exists between the corresponding wavelet coefficients d_j:2n-i and d_j+ „. [0053] This across-scale correlation enables the signal prediction engine to predict the parent coefficient d_j+ „ of the coarser scale from the descendant coefficient d_ji2n-i of the finer scale. [0054] As opposed to other wavelet coefficient covariance models (such as the multiscale autoregressive model, for example), for SSS signals the wavelet coefficients at several scales apart are still directly related. [0055] Referring now to Fig. 7, empirical evidence from MAP signal shows estimates of partial correlation coefficient P(pi,Dj)\D__{{i }]} between wavelet coefficients Z¾ and D_j conditional on the remaining wavelet coefficients D.^_jj = {Dk, k≠ [0056] Pearson correlation cannot tell whether two variables are directly dependent or indirectly dependent through a third "latent" variable. However, partial correlation may give a more accurate picture by excluding the influence of the "latent" variables. For example, since D13 is correlated with both D15 and D10, then D15 may be dependent on D10 through D13. Similarly, D9 and D10 may be highly correlated due to their joint correlation with the parent coefficient DJ3. [0057] Furthermore, the correlation between D15 and D10 may be significantly conditional on D13 and other wavelet coefficients. Moreover, wavelet coefficients at several scales apart, e.g. D15 and D4, may still be correlated conditional on the intermediate scales. This is consistent with the across-scale correlation herein discovered from the property of self-similarity. [0058] To exploit the herein discovered across-scale correlation, the signal prediction engine may apply a modified vector autoregressive (VAR) model to model both within-scale and cross- scale correlations of wavelet coefficients. Traditional VAR models are flexible time series models that may capture complex dynamic interrelations among multiple time series. They may also infer the structural relation such as causality among those multiple time series. However, a problem arises when applying traditional VAR models to wavelet coefficients as traditional VAR model use the same autoregressive order for all streams, but the number of wavelet coefficients are not the same in different scales, as can be seen in Fig. 7(a) wherein scale J has double amount of coefficients compared with scale j+1. [0059] To overcome this problem, the signal prediction engine may either "pad" the coarse scales with more wavelet coefficients (which may be referred to as a redundant wavelet transform) or down-sample the fine scales in order to match the number of wavelet coefficients per scale. Both methods may synchronize the wavelet coefficients before feeding them into the VAR model. [0060] In the "padding" implementation, the signal prediction engine may apply a redundant wavelet transform to the selected set of wavelet coefficients. The dyadic wavelet transform specified by the wavelet equation is a decimated algorithm, with the wavelet coefficient d_j:„ being updated every 2¹ periods, resulting in unequal number of coefficients in different scales. The signal prediction engine may apply a non-decimated or redundant Haar wavelet transform to generate the same number of coefficients by updating d_j:„ for all scales at every period, as follows:

[0061] In this implementation, the indexing in redundant wavelet transform is different from the original index. [0062] The signal prediction engine further implements down sampling. The use of redundant wavelet transform traditionally imposes serial correlation to the wavelet coefficients. 1 The serial correlation results from the fact that two adjacent coarser scale (parent) wavelet

2 coefficients will have an equal finer scale (child) wavelet coefficient, since two adjacent child

3 coefficients will be redundant. This serial correlation causes an over-parameterization issue,

4 which may reduce the accuracy of the predicted signal.

5 [0063] To alleviate the over-parameterization issue, the signal prediction engine may

6 implement a down-sampling algorithm that selects wavelet coefficients that are not strongly

7 correlated.

8 [0064] Referring now to Fig. 8, the synchronized wavelet coefficients vector may be denoted

9 as: i Q D_n = (d_{J+l n}, d_{j n} ,...,d_{0 n})

11 [0065] At time t=n, the signal prediction engine has enough information to compute

12 {-¾}, k = 1,2,... ,n. Then from the wavelet equation, the signal prediction engine may predict the

13 average X„ in the next 2¹ period of time by first predicting the coarse level wavelet coefficient

14 d_{j+l n+2}j.

15 [0066] Referring now to Fig. 9, in order to predict d_{j+1 n+2}j, the signal prediction engine

16 may down-sample X„ at every 2¹ periods to get wavelet coefficients vector D_n_₂j_k, k = 0,1, ....

17 The signal prediction engine may achieve parsimony while maintaining completeness using the

18 down-sampling algorithm. Notably, the down-sampling algorithm applied by the signal

19 prediction engine differs from the lagging approach known in the prior art, with one significant 0 difference being that the lagging approach attempts to predict the finest scale, so the prediction 1 window is restricted to be just one period, but the signal prediction engine is operable to predict 2 the target coarse scale. Therefore, the signal prediction engine is operable to generate a signal 3 prediction for a relatively large prediction window. 4 [0067] The signal prediction engine may model the down-sampled wavelet coefficients using 5 the VAR model with order p: g D_n " = " +A_X ¹D n- ₇2_j ^J + ... + A PD n- _?2,^J p + _t ¹ 7 where error vector e_t follows multivariate normal distribution N(0,∑). [0068] The signal prediction engine may select the autoregressive order p by choosing the integer that minimizes the Akaike information criteria (AIC):

AIC(p) = ln \∑(p) \ ₊ ^p(j ₊ 2)²

T

where T is the sequence length and ^t=1 is the variance-covariance matrix of residuals. [0069] Once the autoregressive order is determined, the signal prediction engine may estimate the VAR model, for example by implementing the Yule-Walker method or maximum likelihood method. Since the down-sampling approach is updated every 2¹ periods of time, the signal prediction engine may apply a moving implementation, as shown in Fig. 9, to fully exploit the coefficients at all scales. In the moving implementation, when a new observation X_n (or do_,n) becomes available, a new wavelet coefficients vector D„ may be computed and a new prediction of D_n+2i may be generated. [0070] A multi-step ahead prediction may also be generated by treating the predicted value as historical data for processing by the signal prediction engine. [0071] Additionally, the across-scale correlation of wavelet coefficients may be generated using alternate processes, including, for example, hidden Markov tree, linear regression and Markov-switching Multifractal model. Hidden Markov tree (HMT), for example, is similar to traditional hidden Markov chain models: both are models with hidden states, parameterized by transition probability matrix and emission distributions. A difference between them is that HMT models the correlation of wavelet coefficients across adjacent scales, while the hidden Markov chain models the serial correlation along the same scale. [0072] Linear regression model may also enable the quantification of dependences among wavelet coefficients across different scales. The wavelet coefficients in the target scale can be selected as the response variable in the linear regression, with wavelet coefficients from finer scales as the design variables. One advantage of linear regression is its simplicity of

implementation and efficiency of computation. [0073] Markov-switching Multifractal model (MSM) models the volatility of a signal by multiplying together a finite number of random first-order Markov components. MSM enables a multifrequency stochastic volatility model with a closed-form likelihood. The MSM model captures long-memory features, intermediate frequency volatility dynamics, and thick tails in distributions all with the single regime-switching approach. [0074] Furthermore, in across-scale prediction, the signal prediction engine may apply parameter estimation. Parameters may be estimated from historical data using statistical estimation methods. Two exemplary implementations of parameter estimation include maximum likelihood estimation and Bayesian estimation. In maximum likelihood estimation, an optimal value of a parameter in a given model may be generated by maximizing the likelihood, whereas in Bayesian estimation, a posterior distribution may be generated using Bayes' theorem from a given prior on the parameters and the historical data. For models with many parameters, such as the across-scale prediction model, for example, the signal prediction engine may provide estimation by implementing simulation techniques, such as Markov chain Monte Carlo, for example, which applies simulation to sample the posterior distribution and provide approximate estimations. [0075] In a further embodiment, the signal prediction engine may be applied to estimate an observed coarse scale signal at a finer scale. In this embodiment, the accuracy of estimation may be improved using the information from different (coarser or finer) time scales. [0076] In an illustrative example, the signal prediction engine may be used to predict an arterial blood pressure signal. The arterial blood pressure signal may be obtained by the signal input from a device present in an intensive care unit (ICU). [0077] In an example implementation, a total of 120 patients (61.5 ± 14.7 years old, Male/Female=l .29) were randomly selected from the MFMIC II data base. Among them, 46 patients had experienced acute hypotensive episode (defined as the mean arterial pressure is below 60 mmHg for at least 90% time in a 30 min window) during the monitoring period. The first 180 min monitoring record was chosen as the training set. The target scale J = 4 was selected, which corresponds to the prediction window with length 2^4"1 = 8 min. [0078] The across-scale VAR model uses the lagged values from finer scales d_j:„ = 1,2,3 to predict d^„, while the within-scale AR model only uses the lagged values on the target scale <¾, i, d_4tn_2, . . . Table 1 illustrates the result from one patient as an example. Table 1

Mean Squared Error (In sample

p= l * p=2

across- scale V^'ARipj 20. 1 3 20.0 L 20.07

within-s ale A Ri ) 21 .78 2 1.7 L 21 .7?

GC-lcsl P- value 1.48e-9 7..i9e-9 4.460-8 [0079] For this result, the Granger causality test is significant for different lags from p=l to p=3. This result suggests that the across-scale VAR model produces significantly lower mean squared error (MSE) compared with within-scale AR model. The different f-values of Granger causality tests from all k patients are combined using Fisher's method with statistic

*² = -2∑ g(p,)

i=l

where p_t is the p-value for patient /^'. X has a chi-square distribution with 2k degrees of freedom. This test illustrates a significant result (P < 0.0001), which suggests MAP variations at finer scales may help predict the variations at coarser scale. [0080] Table 2 shows the out-of-sample prediction MSE from across-scale VAR(l) model and within-scale AR(1) model. Table 2

Out- of- sample MSE Correlation

across-scale V^'ARi l j 53.9B 0.41

within-scale ARi U 62.6B -0.08 [0081] Table 2 shows that, for this example, the across-scale prediction is better in terms of mean squared error (with 14% reduction in MSE) than the within-scale prediction. However, it is also important to note that MSE is not the only criterion, and not necessarily the best criterion, to assess prediction performance. Pearson correlation, for example, is an alternative criterion measuring the consistency of relative trends between two time series. In Table 2, the across-scale model achieves a higher Pearson correlation than the within-scale model. The high correlation indicates that when the cross-scale model predicts an increase, the corresponding value of actual signal also increases. It is the relative change or trend that is common for predicted and actual signals, although the values may be far from equal. [0082] A comparison of predicted signals and the actual signal is shown in Fig. 10. Although both predictions generate smaller ranges compared with the actual signal, the across-scale prediction has a more consistent trend with the actual signal, especially in the major decreases and increases. Such property is highly valuable in clinical practice, where the relative change is sometimes more important than the exact values. This is because every patient has her/his own baseline values, but typically when a critical illness develops the common phenomenon is a change deviating from the baseline value. [0083] In the intensive care units, a sudden decrease of MAP is usually more dangerous than a sudden increase. This is because the decrease of MAP may indicate insufficient circulation which requires immediate treatment. Therefore we should be more interested in predicting the drop than the rise. To evaluate the prediction performance on this negative change, we use the receiver operating characteristic curve (ROC). We set the objective to predict the episode of major decrease in 8-min average MAP exceeding the amplitude of 10 mmHg. Such episode will translate to a wavelet coefficient d_{4 n}≥ 20 in the actual signal following the wavelet equation, since:

[0084] A good prediction should give a relatively large positive wavelet coefficient before the major decrease episode, although the exact value of the predicted wavelet coefficient does not have to be close to 20. The ROC curve may be used to quantify the prediction performance. The ROC curve summarizes the true positive rate (TPR) and false positive rate (FPR) for different critical thresholds on the predicted signal. It is suitable for the present example to measure whether a predicted large wavelet coefficient also corresponds to a large actual one, although the exact values may not be matched. [0085] The averaged ROC curve for across-scale prediction based on the 120 patients is shown in Fig. 11. The area under the curve (AUC) for different prediction windows are compared between across-scale model and within-scale model in Table 3. Table 3

[0086] The prediction performance is significantly better in across-scale model for short prediction window. [0087] The signal prediction engine may be implemented as a component of suitable devices applied in various fields. [0088] In a particular implementation example, the signal prediction engine may be provided as a component of a monitoring device. For example, a blood pressure monitoring device may comprise typical blood pressure monitoring components suitable for a typical blood pressuring monitoring device. The blood pressure monitoring device may further comprise a signal prediction engine operable to obtain an input signal that is also provided to the typical blood pressure monitoring components. The signal prediction engine may be linked to a display of the blood pressure monitoring device or another signal output to provide the predicted signal. The blood pressure monitoring device may further be configured to display an alert if the predicted signal meets preconfigured criteria, such as that corresponding to a predicted warning condition or fatal condition. [0089] It will be appreciated that the signal prediction engine could be implemented in a standalone device, or in a device comprising the signal prediction engine, the signal input and the signal output. [0090] Additionally, a plurality of signal prediction engines may be provided wherein each one of the signal prediction engines is configured to provide a predicted signal of a different desired time period. Predicted signals of a shorter time period may provide a more accurate predicted signal while predicted signals of a longer time period may provide more advanced warning of a condition. The signal prediction engines may be configured to operate in parallel, with the signal input providing the input signal to each signal prediction engine. One or more signal output may be provided. [0091] It will be also appreciated that the monitoring device could be any other device for monitoring or diagnosing any condition on the basis of a physiological signal including heart rate variability, arterial blood pressure, respiratory rate, insulin uptake data for diabetics, gait, as well as spatial patterns of physiological structures such as blood vessels, dendrites in neurons and airways in the lung; or could be any other device for monitoring or diagnosing any other condition of a non-physiological signal including natural and social phenomena, including natural geometric objects, turbulence, natural temperature variations, noise in electrical components, traffic in Ethernet network, financial time series (e.g., market fluctuations), sound, digital images and molecular motion. [0092] Several illustrative examples of such signals are now described. These signals are generally considered to exhibit fractal geometry. Fractal geometry has become a common tool to describe objects or phenomena in which a scale invariance of some sort exists. The variety of scientific applications is large as such structures exist from physics to astrophysics, from biology to chemistry but also in market fluctuations analysis. [0093] Mainstream capital market theory, for example, is based on efficient market assumptions, even though the markets themselves exhibit characteristics that are symptomatic of nonlinear dynamic systems. Financial markets are fractal in this way, as the general observer would be unable to view an unlabeled price chart and determine whether it is an hourly, monthly, or even 5 minute chart of trading activity. Fractal market analysis is used to discover and characterize the order hidden within seemingly random financial markets, and determines the probability of future events. In an example implementation, the signal prediction engine is operable to generate a projection of market activity given historical market activity. One benefit of knowing these fractal patterns and projections is they identify specific critical balance points where the potential energy may resolve in one direction or another. [0094] In another example implementation, the signal prediction engine is operable to generate a predicted heartbeat time-series. Careful analysis of heartbeat time-series could give cardiologists diagnostic tools in the battle against heart disease. Similar analyses of brain waves and stride length in walking could give researchers insights into such conditions as epilepsy and Parkinsons disease. [0095] The heart is part of a large feedback system whose dynamics are nonlinear, non- stationary, and multi-scale. It is generally accepted that heart rate variability (HRV) is self- similar. Referring now to Fig. 13, a study similar to that described above for MAP was conducted for HRV. The ROC curves are generated by varying the threshold and computing the corresponding true positive rate and false positive rate of the prediction. Fig. 13 illustrates the calculated ROC curve for HRV when the prediction window is 6 min and 8 min respectively. From Fig. 13, the accuracy of short-term HRV prediction in this example is shown to be similar to that of arterial blood pressure in the example above. [0096] In yet another example implementation, the signal prediction engine is operable to generate a DNA model given a DNA barcode. It has been determined that very short stretches of DNA, such as a DNA barcode which usually represents less than one one-millionth of the genome, can enable identification of most animal species. DNA barcoding is effective since patterns seen in very short DNA sequences usually reflect patterns seen in longer sequences. In this way, DNA barcodes demonstrate self-similarity. In other words, for unsequenced genomes, the DNA barcodes can provide a quick preview of the whole genome. [0097] Overall, the patterning of barcode differences supports the emerging view that selective sweeps prune mitochondrial diversity within species and mitochondrial and nuclear co- evolution are tightly linked. [0098] Fractal dimension has also been used to quantify the structures of a wide range of objects in biology and medicine and the signal prediction engine may be applied to signals relating to these objects. Fractal analysis has found widespread application in the field of neuroscience and is being used in many other areas. [0099] Network traffic patterns can also be predicted by the signal prediction engine.

Ethernet local area network (LAN) traffic has been found to exhibit statistical self-similarity. Commonly used traffic models are not able to capture this fractal behavior, although that behavior has serious implications for the design, control, and analysis of high-speed, cell-based networks. A critical characteristic of this self-similar traffic is that there is no natural length of a "burst": at every time scale ranging from a few milliseconds to minutes and hours, similar- looking traffic bursts are evident; we find that aggregating streams of such traffic typically intensifies the self-similarity ("burstiness") instead of smoothing it. The signal prediction engine may further be operable to predict traffic patterns in other traffic applications, including, for example, traffic in transportation systems or in any other system having traffic flows in a network of traffic channels. [00100] Additionally, the signal prediction engine is operable to generate a signal

corresponding to many of the Earth's processes that satisfy fractal statistics, where examples range from the frequency-size statistics of earthquakes to the time series of the Earth's magnetic field. The scaling property of fractal signal applies to descriptions of many geological features. Based on well-log measurements, Earth's physical properties have been found to exhibit fractal behaviour. For phenomena like earthquakes that show a fractal distribution of frequency of occurrence this relationship can be used to infer the recurrence interval of a given size earthquake in an area. It may also give insight into the dynamics of earthquakes. Estimation of earthquake recurrence interval may be beneficial to seismic hazard assessment, in engineering, and hazard mitigation. Fractals can further generate estimations of surface area or roughness. For example, they have been applied to studies of the effect of surface roughness on inter-grain permeability. Examples of phenomena with a fractal distribution or geometry in geology include frequency of earthquakes of various sizes, fault parameters, size of clasts in fragmented material, crater size, ore body size, styolite surfaces, dendritic branching and various geomorphic surfaces. Additionally, the signal prediction engine may be applied to the many archaeological patterns that are fractal. [00101] The signal prediction engine may further be applied to vibrations in domains with boundaries or interfaces of irregular geometry. It is known that objects of irregular shape or geometry are bad resonators. Thus, optimal object geometries can be developed using the output of the signal prediction engine. For example, noise reducing surfaces or chambers may be developed by shares that absorb incident acoustic energy in the audio spectrum. An exemplary application is the reduction or normalization of traffic noise. The absorption and acoustic properties of such structures are linked to the spectral properties of the Laplacian operator in complex domains, and wave localization. [00102] Although the above has been described with reference to certain specific

embodiments, various modifications thereof will be apparent to those skilled in the art without departing from the scope of the claims appended hereto. The entire disclosures of all references recited above are incorporated herein by reference.

Claims

1. A method for predicting a substantially self-similar signal comprising:

(a) obtaining an observed substantially self-similar signal;

(b) discretizing the observed substantially self-similar signal into a plurality of

intervals;

(c) performing a mathematical transformation to generate a statistical value for each of the plurality of intervals at a plurality of time scales;

(d) generating, by one or more processors, at least one additional statistical value from an across-scale relationship of at least a subset of the statistical values; and

(e) predicting the substantially self-similar signal from the at least one additional statistical value.

2. The method of claim 1, wherein the substantially self-similar signal is predicted at the same time scale as the observed substantially self-similar signal.

3. The method of claim 1, wherein the at least one additional statistic value is of a coarser time scale than the observed substantially similar signal.

4. The method of claim 1, wherein the predicting applies an additional mathematical

transformation derived from the mathematical transformation.

5. The method of claim 1, wherein the mathematical transformation comprises one of the following: a discrete wavelet transform, a Fourier transform, statistical moment estimates and autocorrelation.

6. The method of claim 1, wherein the statistical values are generated by applying a model comprising one of following: hidden Markov tree model, linear regression and multi Markov-switching Multifractal model.

7. The method of claim 6, wherein parameters of the model for the values are estimated by statistical estimation methods.

8. The method of claim 7, wherein the statistical estimation methods comprise maximum likelihood estimation.

9. The method of claim 7, wherein the statistical estimation methods comprise Bayesian estimation.

10. The method of claim 7, wherein the estimation is generated from Monte Carlo

simulation.

11. The method of claim 1, wherein the substantially self-similar signal is predicted at a time scale using multi-step ahead prediction.

12. The method of claim 1, wherein the additional statistical values are generated at a coarser scale than the statistical values.

13. The method of claim 1, wherein the observed self-similar signal is a physiological signal.

14. The method of claim 1, wherein the observed self-similar signal is a financial signal.

15. The method of claim 1, wherein the observed self-similar signal is a network or

transportation traffic signal.

16. The method of claim 1, wherein the statistical value is generated based on a discrete wavelet transform.

17. The method of claim 1, wherein the self-similar signal is predicted using a vector

autoregressive model.

18. The method of claim 17, wherein the vector autoregressive model further comprises a down sampling method with moving implementation.

19. The method of claim 17, wherein the vector autoregressive model further comprises a redundant wavelet transform.

20. The method of claim 1, wherein the generating the at least one additional statistical value from an across-scale correlation further comprises using a covariance function.

21. A system for predicting a substantially self-similar signal comprising a signal prediction engine operable to:

(a) obtain an observed substantially self-similar signal;

(b) discretize the observed substantially self-similar signal into a plurality of intervals; (c) perform a mathematical transformation to generate a statistical value for each of the plurality of intervals at a plurality of time scales;

(d) generate at least one additional statistical value from an across-scale relationship of at least a subset of the statistical values; and

(e) predict the substantially self-similar signal from the at least one additional

statistical value.

22. The system of claim 21, wherein the substantially self-similar signal is predicted at the same time scale as the observed substantially self-similar signal.

23. The system of claim 21, wherein the at least one additional statistic value is of a coarser time scale than the observed substantially similar signal.

24. The system of claim 21, wherein the predicting applies an additional mathematical

transformation derived from the mathematical transformation.

25. The system of claim 21, wherein the mathematical transformation comprises one of the following: a discrete wavelet transform, a Fourier transform, statistical moment estimates and autocorrelation.

26. The system of claim 21, wherein the statistical values are generated by applying a model comprising one of following: hidden Markov tree model, linear regression and multi Markov-switching Multifractal model.

27. The system of claim 26, wherein parameters of the model for the values are estimated by statistical estimation methods.

28. The system of claim 27, wherein the statistical estimation methods comprise maximum likelihood estimation.

29. The system of claim 27, wherein the statistical estimation methods comprise Bayesian estimation.

30. The system of claim 27, wherein the estimation is generated from Monte Carlo

simulation.

31. The system of claim 21, wherein the substantially self-similar signal is predicted at a time scale using multi-step ahead prediction.

32. The system of claim 21, wherein the additional statistical values are generated at a coarser scale than the statistical values.

33. The system of claim 21, wherein the observed self-similar signal is a physiological

signal.

34. The system of claim 21, wherein the observed self-similar signal is a financial signal.

35. The system of claim 21, wherein the observed self-similar signal is a network or

transportation traffic signal.

36. The system of claim 21, wherein the statistical value is generated based on a discrete wavelet transform.

37. The system of claim 21, wherein the self-similar signal is predicted using a vector

autoregressive model.

38. The system of claim 37, wherein the vector autoregressive model further comprises a down sampling method with moving implementation.

39. The system of claim 37, wherein the vector autoregressive model further comprises a redundant wavelet transform.

40. The system of claim 21, wherein the generating the at least one additional statistical value from an across-scale correlation further comprises using a covariance function.