Let us consider the graphs of real functions of time, X(t) and Y(t), recorded at the same observation instants belonging to the discrete time set {t₁,…t_n}⊂ given in Fig. 1. We suppose that the studied phenomenon involves in general the censoring of the observed system (individual, cell, chromosome…) by loss, death, destruction… Hence, we only get from the experiments the empirical distribution of the data recorded on a sample of individuals different at each time (like in the LD50 experiments). We call monotony signature of X (resp. Y) the sequence of the values “+1” and “−1” corresponding to their successive monotony intervals: sgnX_i = +1 (resp. −1) corresponds to the increase or constancy (resp. decrease) of the function X on its ith monotony interval, and the monotony signature of X (resp. Y) for nine successive intervals of monotony in Fig. 1 equals:{sgnX_i}_i = 1,9 = (+1,−1,+1,+1,−1,−1,+1,+1,−1) (resp. {sgnY_i}_i = 1,9 = (−1,−1,+1,+1,+1,−1,+1,+1,−1).

We can decide that the monotony signature of the observed X profile is significantly different from the Y reference profile (Fig. 1), after testing the similarity of these signatures due to a common causality between X and Y (hypothesis H1) against a random choice of the values of the successive sgnX_i's (hypothesis H0), by using, when sgn_iY = −1, the probability P__i decreasing from a value taken on the support in (denoted above S(f_i)) of the distribution f_i of X_i to a value of X_i + 1 on S(f_i + 1) (cf. Fig. 2 for i = 1):

The sequence of the probabilities {P__i}_{i = 1,…,n–1} of decay of X on its ith monotony interval is called the weighted monotony signature of X.

Proposition 1 In the Gaussian case, where the distribution of X_i is N(x_i,σ_i), we have:

Proof 1 If the density distribution of errors is Gaussian, the formula (3) becomes (by neglecting the index i):

P_ = ʃ_IR f(ξ) F(aξ + b) dξ,

where f (resp. F) is the distribution (resp. cumulative distribution) function of the standard Gaussian law N(0,1), a = σ₁/σ₂ and b = (μ₁ – μ₂)/σ₂. By changing the integration variable ξ in z = F(ax + b), we have:

dz = af(aξ + b) dξ and ξ = (g(z) – b)/a,

where g = F⁻¹. Then we can write P_ under the following formula:

Because the cumulative distribution functions of the uniform law on [–2σ, 2σ] and of the Gaussian law N(0,σ) are very close (cf. Fig. 3, right), the results concerning the calculations of P_ are similar. Then, in any case of errors, we have chosen uniformly 100 couples of values (x_i,x_i + 1)_{i = 1,…,100} in [0,10]². Then, we simulated 100 samples of 100 couples of values (ξ_ik,ξ_(i + 1)k)_{k = 1,…,100} by using 100 couples of Gaussian distributions {N(x_i,1), N(x_i + 1,1)}_{i = 1,…,100}, and we calculated the difference Δ_i between the probability P__i calculated from the integral formula (1) and the empirical frequency P__i* obtained from the observation of the events {ξ_ik > ξ _(i + 1)k}_{k = 1,…,100}. The result is given in Fig. 3 (left), showing as expected that the empirical distribution f_Δ* of the random variable Δ_i = P__i – P__i* is asymptotically (in sample size) Gaussian N(0,1).

Proposition 2 In the uniform case, let denote by [0,δ] (resp. [D₁,D₂]) and x₁ (resp. x₂) the interval and the mean of the uniform law of X₁ (resp. X₂). Then, there are six different configurations (cf. Fig. 4):

1) D₁ < 0 ≤ δ ≤ D₂, then P_ = [(δ – D₁)² – D₁²]/2δD = δ/2D – D₁/D I

2) D₁ < 0 ≤ D₂ < δ, then P_ = 1 – D₂²/2δD II

4) 0 ≤ D₁ ≤ D₂ < δ, then P_ = 1-(D₂²-D₁²)/2δD = (2δ-(D₂ + D₁))/2δ IV

5) D₁ > δ, then P_ = 0 V

6) D₂ < 0, then P_ = 1 VI

Proof 2 In the uniform case, the formula (3) becomes:

Hence, we have:

inf(δ,D2)

inf(δ,D2)

Then, by considering all the possibilities of values of the extrema inf(δ,D₂) and sup(0,D₁), we get the six different formulas (5) (cf. Fig. 4) ■

Fig. 5 below gives the localization X (resp. Y) of the physiologic crossing-overs along chromosome 3 for human females, with the blue curve in Fig. 5 (resp. males, with red bars) [4]. By comparing the two monotony signatures as given, we reject the hypothesis that female and male crossing-overs have similar localization by chance with independency between X and Y (P < 10⁻⁷), which is in favor of the existence of the same frailty domains along the male and female chromosome, explaining the frequency of crossing-over co-occurrences.

During successive sessions of choice, participants belonging to a defined group are choosing one picture from an “absurd questionnaire” consisting of 50 pairs of pictures, in each of which one picture has to be chosen [5]. The results given in Fig. 6 are used to search for evidence in favor of the influence of group dynamics on individual choices of the pictures proposed in the questionnaire. The swaps between the two pictures of a pair, measured for each pair of pictures between the initial choice (called A choice) across sessions could be interpreted as a manifestation of group dynamics: for instance, the “honeymoon” (dependence on the leader) and the successive “fight-flight” (reaction against the dependence on the leader) attitudes could be represented by a greater number of group dynamics swaps (further away from pure randomness). It seems intuitive that an increase of simultaneous swaps could be linked to an increased group activity or group dynamics event. Statistics carried out on the totality of swaps A→B or B→A evidenced a significant increase in the swap numbers between S01/S02 and S02/S03 session transitions, and between S03/S04 and S04/S05 session transitions, incompatible with a random fluctuation. In other words, the number of changes in choices increased significantly between the 1st and the 3rd sessions, and then remained essentially constant until the end of the training, except between the 3rd and the 5th sessions. During session S04, the number X of A→B swaps and Y of B→A swaps showed similar fluctuations close to significance against similarity obtained by chance with independency between X and Y (P = 0.6), while there is no significant change in the evolution of the sum X + Y. This is consistent with the existence of a collective unconscious behavior.

After 25 days of observation of a person at home [6–8], we get the results given in Fig. 7 corresponding to his/her staying in the different rooms of a smart flat in which different sensors recorded his/her activity at home. They allow us to calculate different temporal profiles assigning the observed person in different clusters corresponding to a normal or a pathologic behavior. Alarms can be triggered when passing from a normal type of nychthemeral activity to a pathologic one. For example, in a degenerative neural pathology like Alzheimer's disease, we can observe an abnormal activity in the same room, called perseveration, which is a pathologic repetition of actions in general already successful (“errare humanum est, perseverare diabolicum”) [9]. We model this phenomenon of persistence in a pathologic activity by setting atypical extended occupancy periods in a room or by performing repetitively a specific daily routine, in comparison to more standard scenarios encountered in everyday life.

A way to compare different temporal evolutions of room occupancies is to follow the evolution in time of the number of events of entrance in room, for different rooms and 25 successive days. The fluidity of the activity can be related to the number of entrances in connected rooms like the bedroom and the toilets, because a decorrelation between these numbers could sign a stereotyped activity by repeating for example pathological entrances in the toilets or non-entrances from the bedroom (e.g., in case of anuria, nocturia, or pollakiuria). The data in Fig. 7 show that only three discrepancies within a period of 25 days allow us to reject a similarity between entrance activities No. 3 and No. 4 by chance, with independency of these activities (P = 2 × 10⁻⁷), the same monotony signature for connected rooms being then the sign of a normal behavior.

In [9,10], the authors recall that weightlessness is known to affect cellular functions by as yet undetermined processes, but with a role of the cytoskeleton and microtubules, which behave as a complex system that self-organizes by a combination of reaction, diffusion, collective transport and self-organization of any present colloidal particles [9]. This self-organization does not occur when samples are exposed to a brief early period of weightlessness [10]. During both space-flight and ground-based (clino-rotation) experiments on the effect of weightlessness on the transport and segregation of colloidal particles, a significant difference between the velocity of colloidal particle beads has been observed (Fig. 8), as well as the rejection of the similarity by chance with independency between gravity and non-gravity monotony signatures during the evolution of the microtubule organization in embryonic cells (P = 3 × 10⁻⁵). This suggests, depending on factors such as cell and embryo shapes, that major biological functions associated with microtubule-driven particle transport and self-organization might be strongly perturbed in velocity amplitude, but not in sequencing, by weightlessness.