In this section we collect and discuss the statements, needed for the acceptance of the thesis formulated in the Introduction:

Adaptive time-frequency approximations of signals unify most of the univariate computational approaches to EEG analysis, and offer compatibility with its traditional (visual) analysis, used in clinical applications.

### 2.1 *Lemma*: Matching pursuit sub-optimal solution to the problem of adaptive approximation is suitable for EEG analysis

Sub-optimal solution of an intractable problem (section 1.7) must have its price. Mathematical examples of failures in pattern recognition, due to the greedy strategy (7) applied by the matching pursuit, were presented e.g. in [20, 21]. These cases did not relate to structures encountered in biomedical signals, but the lack of a relevant counterexample does not deny it's existence. Figure 2 gives an example of a case referring directly to the transient oscillatory activity of the kind which may be actually present in the EEG [22].

Signal (*R*^{0}) in Figure 2 is composed from two Gabor functions, both of them actually present in the dictionary *D* used for the decomposition (Section 1.9). In spite of that, we observe (in the right column) that the first function(*g*_{0}) fitted to the signal is completely different from either of the two functions, from which the signal was composed! According to (7), MP algorithm has chosen function *g*_{
i
}giving the largest product in a *single* step. Of course, taking into account the next steps, this decision was definitely not optimal. Choosing the two Gabors, which were exactly represented in *D*, would explain 100% of signal's energy in only 2 iterations, contrary to the residues produced in the left column of Figure 2 as a consequence of the first choice.

However, such an effect occurs only if both Gabors present in the signal have not only the same frequencies, but also *exactly* the same phase. Such a "coincidence" would be possible in a biological signal *only* if both the structures were produced by the same generator. And still, MP represents them jointly only if their time centers are close enough. Their larger displacement would result in separate representation even in such a synchronized case. Therefore, we may argue that this effect, mathematically classified as a failure of the sub-optimal procedure, is actually a welcome feature in the analysis of physiological signals. It was explored in [23] for detecting the synchronized spiking activity, as opposed to series of unrelated EEG spikes.

Even if we were able to calculate the optimal adaptive decomposition minimizing error (6), it would have one more disadvantage: namely, the set of *M* functions chosen to optimally represent signal *f*, may significantly differ from the optimal set of *M* + 1 functions chosen for the same signal *f* (from the same dictionary *D*). With the iterative MP solution (7), {}*i* = 1..*M* will be the same in both decompositions. Considering that the features related to the sub-optimality of the MP solution turn to be advantages rather than drawbacks in the analysis of biomedical signals, we conclude that matching pursuit algorithm is the correct implementation for an adaptive time-frequency approximation of EEG.

### 2.2 *Lemma*: Matching pursuit decomposition is asymptotically well defined and does not depend on arbitrary settings

Calculating representation (5) of a signal *f* via matching pursuit in given dictionary *D* is a deterministic procedure, given by (7). Therefore, for a given signal, representation (5) depends only on the dictionary *D*. For the Gabor dictionaries discussed in section 1.9, the only free parameter is their size. It influences MP decomposition in a predictable way: larger dictionaries improve the quality of the decomposition, that is less waveforms are needed to achieve the same error *ε* in (6). However, starting from a certain density of dictionary's waveforms in the space of their parameters {*u*, *w*, *s*}, this improvement reaches asymptotically a saturation. At this point, which we shall tentatively term a "sufficient density of a Gabor dictionary", decompositions in larger dictionaries, or in different stochastic realizations of the same dictionary, are indistinguishable (of course this relates only to the structures coherent with the dictionary, c.f. [18]).

This qualitative reasoning – although still not backed up by a quantitative proof – is well supported by numerical experiments and years of real-world applications. Problems with mathematical derivations stem from the non-linearity of the MP procedure (7) and variable content of the analyzed signals. Therefore, the notion of "sufficient density" of a Gabor dictionary is up to now defined only empirically. Nevertheless, for the following we shall use it as a requirement for (5). For reasonable dimensions of signals, such densities are achieved with the dictionary sizes easily implementable on today's computers.

### 2.3 *Conjecture*: Relevant content of the EEG signal is coherent with Gabor time-frequency dictionaries

This statement depends on the meaning of "relevant EEG content", which, unfortunately, does not have a strict definition.

EEG originates from postsynaptic potentials of firing neurons; only if a large ensemble is working synchronously, the signal can reach a level measurable from the scalp electrodes. A single electrode records a potential from an arbitrary number of such ensembles, spatially filtered by different conductivities of cortex, dura mater, skull and scalp. Recorded EEG is not only affected by possibly non-linear interactions between these ensembles, but the signal may also contain extra-cerebral – biological or not – potentials (artifacts).

Rhythmic activity has been recognized as a prominent feature of the signal since the beginnings of EEG recordings. It appears as transient – rather than permanent – oscillations. These oscillations are produced by large masses of neurons (typically 10^{4} – 10^{7} [24]). Synchronization and desynchronization of these neural masses cannot be instantaneous, so the oscillations exhibit some waxing and waning. Gabor functions provide a concise general model for such signals.

"Relevant content of EEG" can be also translated to "relevant structures used to date in EEG analysis". That would refer mainly to the visual analysis, discussed in lemma 2.6 and 2.7.

Finally, non-permanence of this statement – underlined by using "conjecture" rather than "lemma" – relates also to the fact that, if any other structures would appear to be important in this context, they can be added to the dictionaries without a negative impact on the convergence of the MP procedure (7).

### 2.4 *Conjecture*: All the relevant EEG structures, which can be found via linear expansions in different bases, are included in the MP parameterization

As discussed in section 1.5, exploratory value of linear expansions relates to *g*_{
i
}with the largest weights in expansion (1) – only those are usually treated as reflecting the significant signal's structures. Accepting Conjecture 2.3 we assume, that all relevant EEG structures are well represented in the Gabor dictionary. Also, waveforms used in linear expansions (1) can be more or less exactly assimilated by smooth functions (10). E.g. some wavelets contain discontinuities, but in the context of spatially-filtered EEG (c.f. Lemma 2.3) smooth Gabor functions should be more adequate for its description. Therefore, we may expect that if a signal contains a mixture of oscillatory and transients activities, and each of them would be highlighted in a different expansion (1), then all of them will be efficiently represented in the MP approximation (5).

As an example, we may quote a study of pharmaco-EEG [25], where traditional spectral estimators of spindling activity between 12 and 14 Hz were replaced by the energy of relevant structures estimated from MP expansion. It provided full compatibility with the traditional approach and better accordance with physiological expectations, owing to the increased sensitivity of MP-based estimator.

However, the correspondence to which we refer in this Conjecture, relates only to EEG *structures*, which should be parameterized by the MP decomposition as efficiently as in any of the linear expansions (1). But certain *statistical* properties of the signal can be derived only from its full representation e.g. in wavelet or Fourier bases. On the other hand, properties of MP expansion can be used to construct completely new measures, like e.g. the Gabor atom density (GAD), proposed for prediction of epileptic seizures in [26].

### 2.5 *Lemma*: Matching pursuit provides the most general solution to the problem of non-unique estimates of time-frequency energy density, applicable for EEG analysis

All the time-frequency distributions from section 1.6 are estimates of the same magnitude, that is energy density of the signal. However, estimates derived from different approaches (e.g. Wigner, RID, CWT or spectrogram) differ significantly, mainly due to the variable contribution of cross-terms. Also, estimates obtained from the same approach may look quite different depending on their parameters, like e.g. the length of the time window in spectrogram, different wavelets in CWT or smoothing kernels *φ* in RID.

Estimate (13), derived from the MP decomposition, contains the auto-terms, corresponding to structures present in expansion (5), and no cross-terms (section 1.10). Therefore, for a signal coherent with the dictionary used in the MP decomposition, estimate (13) approximates the intersection of its different quadratic estimates of energy density. This common part corresponds to the generalized auto-terms, contrary to the cross-terms variable between estimates. Therefore, if we accept Conjecture 2.3, this lemma is also proved. Even if we doubt the coherence of EEG signal with the Gabor dictionary, this approach is currently the best candidate for a unifying approach, owing to the uniqueness discussed in Lemma 2.2, since all the other quadratic estimates depend on an arbitrary choice of parameters.

Recent studies of brain electrical activity, recorded from the scalp [14, 27] and from the cortex [28], confirmed also that (13) provides resolution superior to the previously applied quadratic methods.

### 2.6 *Lemma*: Among the contemporary methods of signal analysis, adaptive approximations with time-frequency dictionaries provide the best correspondence to visual EEG analysis

The knowledge, verified through decades of clinical applications, consists mainly of associations between appearances of rhythmic activities and certain waveforms in EEG with physiologically relevant states or pathologies, like e.g. level of alertness, sleep depth or epileptic seizure. These rhythms and waveforms constitute the dictionary of clinical EEG analysis. Due to a limited repeatability of their visual detection, the field is often considered an art rather than science [24]. Mapping this dictionary into mathematical terms will allow for application of repeatable scientific methods.

**EEG rhythms** (*δ*, *θ*, *α*, *β*, *γ*) are approximately assigned to frequency bands. Therefore, if we assume that those frequency bands are fixed, we can detect their presence from the relative power of the corresponding frequency band in the Fourier spectrum. However, power spectrum is a property of the whole analyzed epoch, and does not provide the information on the time extent of the given oscillatory phenomenon.

Appearance and disappearance of a rhythm can be clearly detected from a time-frequency map of signal's energy density, obtained from one of the quadratic methods discussed in section 1.6. However, stating that oscillations with frequency *f* last from the time *t*_{1} to *t*_{2} requires an analysis (post-processing) of a redundant map. Results will depend on the combination of the applied post-processing method and the chosen estimate of the energy density.

On the contrary, unequivocally defined (Lemma 2.2) MP expansion (5) gives *explicitly* the time span and frequency of detected oscillations.

**Transient EEG structures** – sleep spindles, K-complexes, epileptic spikes and many others – are obviously undetectable from the spectral estimates. They may exhibit certain, more or less typical, time-frequency signatures; we may imagine using the complex transforms (3) or (4) for their detection, but as previously mentioned we are confronted with continuum of possible transforms and post-processing algorithms.

Adaptive approximations (5) provide today the only method of an *explicit* description of a significant variety of waveforms. As presented in [23], this description is efficient also for non-oscillating structures like epileptic EEG spikes.

### 2.7 *Conjecture*: The correspondence from Lemma 2.6 can bridge the art of visual EEG analysis and reproducible methods of signal processing

Adaptive time-frequency approximation is currently the only signal processing method, offering description of structures, present in a signal, explicitly in terms of their physical parameters like amplitude, frequency or time width. Similarly to the way in which an expert evaluates EEG, the algorithm concentrates on most prominent, local structures, rather than general properties of the whole analyzed epoch.

Visual analysis of EEG relies primarily on detecting occurrences of rhythms and other waveforms, called graphoelements. In some cases, their classical definitions are given already in time-frequency terms. For example, sleep spindles are defined (c.f. [11]) as structures of least 0.5 sec duration, frequency between 12 and 14 Hz and amplitude above 15–25 *μV*. Such a definition can be directly applied to filter the decomposition (5) in the space of the parameters of functions , fitted to the analyzed signal. Waveforms , conforming to such ranges of parameters, represent sleep spindles – as verified by comparison with their visual detection in [29].

However, not all the graphoelements have such strict, mathematical definitions. The very name "graphoelement" (rather than waveforms) suggests, that their visual detection may rely on different aspects of its shape. Nevertheless, most of these shapes can be effectively approximated by Gabor functions. In such cases we may a posteriori adapt the ranges of the parameters of

, to maximize the concordance with visual detection. This resembles the classical paradigm of expert systems, however, using the MP parameterization (5) has the major advantage of operating in the space of physically meaningful and universal parameters, like time width, amplitude or frequency. An example of a successful implementation of such an approach and its broader discussion can be found in [23].