Indications of stress can be measured using self-rated questionnaires, temporarily coarse-grained invasive physiological means, such as blood-sampled biomarkers, or temporarily fine-grained or continuous physiological or physiological and behavioural means [12] . These approaches are not mutually exclusive and reveal that stress is a multidimensional construct. We focus on continuous physiological and behavioural measures as they are both objective compared to the questionnaire approach, that is subjective, and capture the short-term temporal dynamics of the stress response as opposed to coarse-grained measures such as blood sampling. Another direct advantage is that the presented approach is not invasive.

However, there are at least two fundamental problems with measuring stress. First, stress is not measurable in the sense of physics as it has no physical units. Second, indirect stress indicators such as self-reporting, bodily fluid analyses [13] , or multi-dimensional complexity analysis [10] , require micro-to-macroscopic translations that are not a priori invertible. We use the notion of invertibility to indicate, for example, that even if we could easily detect microscopic structures by the presence of certain molecules, a process linked to macroscopic observable stress states might not induce such states under certain circumstances.

Let us consider an example. The corticotropic stress response is one of the pronunciations of stress, among many others [1] , and part of the regulatory system response. Nevertheless, it is neither the stress itself nor does it have the same unit. This holds true for any of the measurables that we therefore call surrogate data. Furthermore, the attempt to quantify stress faces the difficulty that the various surrogate data are situated at vastly different scales: from microscopic chemical reactions all the way up to physiological changes such as heart rate or skin conductance and further to macsroscopic topological measurements such as torso motions or positional changes measurable when the stress response implies flight-or-fight, let alone the subjective psychological phenomena that accompany stress.

We accept that it is not necessary to express stress in physical units for its pronunciations can be measured by its effects on physiology and behaviour. These effects expressed via surrogate data are measurable physical quantities.

The neurochemical stress-related changes are not directly amenable to continuous and non-invasive measuring. Thus, we would rather focus on macroscopic stress equivalents, i.e., physiological and behavioural measures, that can be measured non-invasively over time. We have provided evidence that some surrogate physiological and behavioural data, including heart rate, skin blood perfusion, blood oxygenation, skin temperature, movement and steps frequency obtained using body-attached sensors exhibit approximately normally distributed and self-similar time series [10] . This allows for complexity analysis of such data and for simultaneous multimodal projections in place of observational statistics. We introduce an application of spectral theory to implement a compound analysis of multiple macroscopic characterisations of physiological and behavioural surrogate data.

The presented concepts of the stress coefficients and the compound stress indices are as follows. We introduce adaptability, c.f., subsection Adaptability, that is a product of complexity total variation and frequency reflecting both the qualitative and quantitative changes of the time-dependent acquired data.

Further, we introduce a notion of stress coefficient based on adaptability in subsection Stress Coefficient. It is a constant of which we propose that it may reflect the relation between stressors and response based upon adaptability and the environmental demands [14] .

Adaptability is measured through various stress coefficients that can be compared and their congruency determined, c.f., subsection Congruency. Congruency is constructed to indicate that reactivity and demands, in other words adaptibility, are at equilibrium. An example of a stress coefficient can be produced by comparing adaptability of, e.g., heart rate with steps frequency.

We present the structure of the theory at Figure 1.

The fundamental principle we adopt is that stress, not directly measurable in the sense of physics, can be constitutively modelled by multidimensional Euclidian spaces derived from complexity analysis and its consequent indexing of macroscopic surrogate data.

There is a consequence of the outlined principle. We use physical quantities on an intermediate level formed by sensory data acquisitions. These time-series have associated SI units. Their projection on the complexity indices removes the units disparity.

Our approach is designed to provide a deeper understanding of physiological and behavioural patterns, an understanding that cannot be obtained using standard observational statistics such as, e.g., mean, variance and correlation of the raw data. An illustration of the presented approach is shown at Figure 2.

We define adaptability as the product of the total variation of uniform segments of complexity measures of self-similar normally distributed physiological and behavioural data multiplied by the frequency of this input. The frequency is given by the reciprocal of the count of uniform segments of time considered. The total variation, given in one dimension by arc-length, is denoted TV. The complexity measure is given by the Hurst index [16] . The total variation of the considered segmented physiological and behavioural data is given by

TV ( H ( Z ) ) = max { i = 1 , ⋯ , m } ∈ P [ 0 , T ] ∑ i = 1 m | 〚 H ( Z ( t i ) ) 〛 | (1)

In the above expression { Z ( t i ) , i = 1, ⋯ , m } forms subsequent time partitioning of the surrogate data, assuming that ( 0 , T ) = ∪ i = 0 m − 1 ( t i , t i + 1 ) , t 0 = 0 , t m = T , P [ 0 , T ] is the set of all possible such partitioning, and 〚   〛 denotes a jump of the enclosed quantity. H ( Z ( t i ) ) is the Hurst index of the sensory surrogate data corresponding to the time interval ( t i , t i + 1 ) .

Adaptability is defined by

v ( H ( Z ) ) T = def TV ( H ( Z ) ) f ( H ( Z ) ) (2)

where the frequency of the jumps in the complexity of the data, f ( H ( Z ) ) T , is defined by

f ( H ( Z ) ) T = def 1 T card { 〚 H ( Z ( t ) ) 〛 | t ∈ ( 0 , T ) } (3)

The frequency is simply a number of data samples we consider over a certain period of time. We consider one sample encapsulating two hours of surrogate data acquisition in the presented results. The reason we consider larger time periods than an hour is that we need a sufficient amount of data to estimate the complexity indices.

We use frequency considering that different processes have different time-scale characteristics. An example are heart rate and skin temperature time-series, the changes being much faster for heart rate compared to skin temperature.

We measure the strength of jumps of the Hurst exponent computed from the segments of the acquired time-series using total variation. Such a measure has no units. Considering the strength of the jumps projected on one of the y-axes we consider the range of a given measure applied to a given time-series. We express the units of total variation of complexity measures, given by (1), in terms of the complexity measure range.

The total variation disregards the sign of jumps. Nevertheless it does indicate the complexity travels on the y-axis. Figure 3 is for illustration purposes of congruency only. The simplified version encoded in (3.2) accounts for the strength of the jumps of the two different complexities that should be about the same if demand is met through the corresponding response. This is the meaning of the construct. This approach is weaker to that shown at Figure 3 where the signs of jumps are highlighted.

Adaptability is designed to measure, using certain properties of surrogate time-series, the capabilities of a given system to adjust the complexity of their

physiological and behavioural patterns. We can always assume that the total acquisition time, T, is finite, and that the number of sampling equidistant subintervals is non-zero and would be bounded in real data collection. Consequently, it is the total variation that plays the fundamental role.

Assume, given acquired one dimensional sensory data Z ,

TV ( H ( Z ) ) = 0 , for all partitions given by P [ 0 , T ] , T > 0 . (4)

Given (4) holds, the complexity of the data has to be constant regardless of the number of samples we analyse, i.e., regardless of the time resolution we apply. This implies, almost surely, that H ( Z i ) is constant for any i = 1 , ⋯ , m . This would imply only two possibilities. Either the system producing Z does not register any stressors because of the lack of them or it cannot react to demands, assuming the surrogate data chosen are adequate. The former circumstances are unlikely, the latter circumstance implies that the Hausdorff-Besicovitch dimension of Z is equal to one and that is incompatible with the existence of a living organism.

Assume on the other hand that we increase the frequency but the adaptability stays bounded. That would imply that the total variation has to decrease proportionally to the frequency. Consequently, the variations in the complexity measure with respect to the increasingly higher resolution cannot change much. This is a pronunciation of self-similarity of the sensory data Z . These considerations lead to the multiplicative structure of the adaptability definition (2).

Summarising the two above arguments it seems plausible to assume that the higher the reactivity expressed by a higher adaptability index the higher the chances of a leaving organism to deal with demands and to provide the corresponding responses.

A characterisation of demand/response requires at least two players as soon as the system has to respond to a variety of qualitatively different stressors. An example: a requirement of higher steps frequency must be accompanied by higher breathing rate as well as higher heart rate. The stress coefficient we introduce below is a dimensionless renormalisation constant relating adaptivity measures of two, possibly similar/correlated, surrogate variables.

We choose to use all the available data regardless of their implicit characterisation.

Consider two acquired approximately self-similar time-series, X and Y . The stress coefficient, σ ( X , Y ) , for the sensory data X , Y , is given by,

v ( H ( X ) ) = σ ^ ( X , Y ) v ( H ( Y ) ) (5)

where v ( H ( ⋅ ) ) is given by (2) and H denotes the Hurst exponent. Notice that stress renormalization constant is not symmetric, i.e., σ ^ ( X , Y ) ≠ σ ^ ( Y , X ) . Since

σ ^ ( X , Y ) = σ ^ ( Y , X ) − 1

then, σ ^ ( ⋅ , ⋅ ) > 0 ,

| log ( σ ^ ( X , Y ) ) | = | log ( σ ^ ( Y , X ) ) | .

Thus we implement

σ ( X , Y ) = def | log ( σ ^ ( X , Y ) ) | (6)

as the stress coefficient for the reason of symmetry.

Two examples of various stress coefficients given by (5) are presented by Table A4 and Table A5 in Appendix.

Suppose a stress coefficient σ is equal to one. This can occur if and only if the underlying surrogate time-series Y and X have the same distributions. The reason is that for the stress coefficient to be equal to one both sums of the jump magnitudes of the Hurst exponents must equate. Assume further that both processes have the same data acquisition frequency. These processes posses ideal congruency. Surely, congruency is unattainable by human physiological and behavioural patterns. However, we use this construct to obtain a fixed point against which we can measure stress corresponding to various physiological and behavioural patterns.

It follows from (5) that σ ( X , Y ) > 1 if the adaptability of the process X is higher than that of Y and smaller than one in the opposite case.

What precedes deserves some further comments and interpretation. We mentioned that the lack of temporal congruency may indicate that the adaptability of the controlling response may be inadequate given the imposed demands. However, this may not be entirely true. Let us consider some more extreme acute stress reactions, i.e., fight-or-flight scenario, to be adequate reactions. In these stress situations, the individual is likely to show congruency of complexity patterns. However, this is true only as long as we consider the adequate parameters, meaning orthosympathetic arousal and high physiological and behavioural patterns. This is different in another classical stress situation physiological and behaviourally characterized by freezing with nearly no macroscopic behaviour at all, but with most likely very high orthosympathetic arousal. In an observational sequence containing a freezing episode, this would appear as a major incongruence between complexity of some physiological parameters such as, e.g., heart rate and that of physiological and behavioural parameters such as step frequency. Overall, we may argue that high congruency over time between different physiological and behavioural parameters is a feature of harmony and low physical and psychological stress levels. As a corollary, lower levels of congruency over time or major congruency leaps may be indicative of low harmony or disorder such as chronic stress.

The congruency concept is illustrated by Figure 3. The left plot indicates congruent structure while the other one does not. Congruency indicates that when complexity of a demand increases the complexity of some related responses should increase as well. Time-local lack of congruency indicates complexity of the controlling response is inadequate relative to the imposed demands. This is shown at the left graph. The graph shown on the right indicates temporal dis-congruencies. At around the fifth hour of the sensory data acquisition the complexity of steps frequency is very low while the complexity of the heart rate is very high. Five hours later the relation of the complexities is reversed. In other words, the congruency accounts for the fact that a positive jump in the complexity of demand is or is not accompanied by the positive jump in the complexity of the corresponding response and vice versa. The circled complexity characteristics exhibit just the opposite. Considering just heart rate/steps frequency is likely an oversimplification, though. An individual actively escaping a danger could increase both heart rate and steps frequency simultaneously in a congruent manner. It is plausible to assume that over a longer period of time such a singular event can be filtered out to obtain a more stable longitudinal congruency image that might be characteristic for a given individual.

The Compound Spectral Stress Index (CSSI) is constructed to achieve two objectives: 1) to compress stress coefficients tables information, e.g., Table A4 and Table A5, into a single number, and 2) to provide a non-biased demand-response distance from the ideal congruency. These objectives are addressed using the spectral norm of the covariance matrix difference between ideal congruency and a given stress coefficient matrix.

The ideal congruency matrix, B , is given by B = { b i , j } i , j = 1 m , for some m ∈ ℕ , where B is the identity matrix corresponding to assumed absence of stress. The real stress coefficients matrices, A , are constructed to be the upper triangular matrix images of tables such as those shown at Table A4 and Table A5.

We define the Compound Spectral Stress Index, τ ( A , B ) , by

τ ( A , B ) = def max { ρ ( ( A − B ) T ( A − B ) ) } (7)

where ρ denotes the spectrum of the enclosed matrix. This means that CSSI is given by the spectral distance of A from the reference congruency given by the matrix B . The spectral distance, or equivalently the spectral norm, is a reliable metric yielding a distance measure given by the largest singular value of the matrix A − B , i.e., by the largest eigenvalue of its covariance form used in (7).

It is important to keep the ordering of the sensory sequencing constant to obtain consistent CSSIs.

An example of CSSIs pertaining to eight healthy subjects is given by Figure 4.

The posteriori correlation with questionnaire based subjective self-evaluations suggests that the increase of the CSSI corresponds to the decreasing adaptive capacity over a given time frame. The smaller the value of CSSI the higher is the subjet’s capability to deal with various forms of stressors.

We show the high variability among normal individuals that is captured by our analyses in real-life situations based on thousands of data points. That is different compared to other studies that use a higher number of individuals, but only just a few data points. The data yielding the CSSIs of the eight individuals are based on the tables shown in Appendix.