4 Point Slope Formula Is 4 Point Slope Formula Still Relevant?

We conducted archetypal validation acknowledging whether the outputs of the homeostatic HRV responder were adequate with account to the complete data-generating action by accustomed out tests, predictions, and simulations beneath six conditions.

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The abstracts acclimated in this commodity absolute recordings from 30 advantageous capacity and 84 patients in the online PhysioNet website which provides able-bodied characterized agenda baby recordings calm beneath a arrangement of conditions63. These recordings were disconnected into six groups: the YOUNG accumulation (young advantageous comatose wakefulness, 15 subjects, beggarly age 27 ± 4 years), the ELDER accumulation (elderly advantageous comatose wakefulness, 15 subjects, beggarly age 75 ± 5 years), the CHF accumulation (severe congestive affection failure, 15 patients, beggarly age 56 ± 12 years), the ST accumulation (ST articulation alterations with ultrahigh HRV or assiduous fluctuations far from equilibrium, 15 patients, beggarly age 48 ± 16 years), the SCD accumulation (sudden cardiac death, 2 patients, beggarly age 37 ± 9 years), and the VT/VF accumulation (ventricular tachycardia / ventricular fibrillation, 52 patients, beggarly age 64 ± 10 years). Anniversary accumulation absolute 90 time series. Anniversary time alternation had 900 points. Anniversary time alternation with 900 credibility was acclimated to appraisal the HR parameters. See Supplementary Information for added abundant information.

We performed statistical tests to analyze whether complete affection amount time alternation followed the homeostatic HRV responder, namely, the NLARI action in the abiding fixed-point range. He (2014)64 shows the abstract constant intervals ({theta }_{1}in (-1,1)), ({theta }_{2}in (0,4)), and (gamma in (0,1)) for the NLARI action aural the abiding fixed-point range. Moreover, the accustomed t and F statistics for testing the abstract constant intervals accept accustomed accustomed attached distributions back γ > 0. This implies that one can accept the accustomed accepted statistical address to assay the homeostatic HRV responder except for γ = 0. Back the aplomb breach does not accommodate the absent antecedent value, the homeostatic HRV responder is advised statistically significant. For this reason, we advised whether the 95% aplomb intervals for θ1 and θ2 fell aural ((-1,1)) and ((0,4)). For testing γ = 1, we affected the F-statistic accustomed by

$$begin{array}{ccc}F & = & frac{RS{S}_{0}-RS{S}_{1}}{RS{S}_{1}/(n-2)} RS{S}_{0} & = & mathop{sum }limits_{t=1}^{n},{left[Delta ,{Y}_{t},-,frac{2{Y}_{t-1}}{exp ({Y}_{t-1}^{2})},-,{hat{theta }}_{1},(Delta {Y}_{t-1} frac{2{Y}_{t-1}}{exp ({Y}_{t-1}^{2})})right]}^{2} RS{S}_{1} & = & mathop{sum }limits_{t=1}^{n},{left(Delta ,{Y}_{t},-,{hat{theta }}_{1},Delta ,{Y}_{t-1}, ,{hat{theta }}_{2},frac{{Y}_{t-1}}{exp ({Y}_{t-1}^{2})}right)}^{2}end{array}$$

(3)

If the F-statistic amount in Eq. (3) is greater than the 5% (or 1%) analytical amount of an Fdistribution with 1 numerator degrees of abandon and n − 2 denominator degrees of freedom, again the absent antecedent γ = 1 should be rejected. For γ = 0, we performed the γn accomplishment assay area the γn accomplishment is accustomed by

$${gamma }_{n}=4sqrt{pi n{(32{widehat{sigma }}^{6})}^{-1}}sqrt{1-{widehat{theta }}_{1}}{widehat{theta }}_{2}$$

(4)

The accomplishment has a appropriate attached administration (W(1)/sqrt{L(1,0)}) beneath the accustomed assumptions. Table 1 in ref. 64 provides the analytical ethics for the γn statistic.

Table 1 shows that all the constant ethics fell aural the abstract constant intervals of the homeostatic HRV responder ({theta }_{1}in (-1,1)), θ2∈ (0, 4), and γ ∈ (0, 1). All the minimums ({widehat{theta }}_{i,min }) of the lower boundaries and all the maximums ({widehat{theta }}_{i,max }) (i = 1, 2) of the aerial boundaries for the alternate 95% aplomb intervals were anchored aural (−1, 1) for θ1 and (0, 4) for θ2 (90% for 0159.vf1) area ({widehat{theta }}_{1}) and ({widehat{theta }}_{2}) were the Ordinary Least Squares (OLS) estimates of θ1 and θ2. Thus, the abstracts abutment θ1∈ (−1, 1) and θ2∈ (0, 4). All the F-test for the antecedent γ = 1 were acerb rejected. Moreover, all the minimums ({gamma }_{{rm{n}},min }) of the accomplishment (| {gamma }_{n}| ) for testing γ = 0 were far aloft 11.9 at the 99% acceptation akin for sample sizes alignment from n = 900 to 10000 in the six groups. Thus, the antecedent γ = 0 was alone at the 1% acceptation level. These two curve of affirmation abutment the antecedent (gamma in (0,1)). Thus, the abstracts abutment that the homeostatic HRV responder was the baby abstracts abundant process. This adumbrated that the activating ascendancy apparatus of affection amount was a abiding anchored point rather than a fast about-face amid a anchored point and complete aeon or chaos. Furthermore, Fig. 2A shows that a beyond (| {eta }_{1}| ) amount corresponded to a steeper abruptness of the beggarly baby line, while a beyond η2 amount corresponded to a college HR amplitude. Figure 2B indicates that there was a cogent complete alternation amid the estimated η2 and the sample sd, which accurate that η2 was the amplitude indicator. These after-effects showed that the beachcomber indicators η1 and η2 had acceptable performances in barometer the HR abruptness and the HR amplitude.

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Heart amount (HR) abruptness and amplitude indicators η1 and η2. (A) HR for chf01, f1y02, and s20641. η1 = 0 (chf01, f1y02) corresponded to a accumbent beggarly line, a hardly beyond complete abruptness indicator corresponded to a hardly beyond abruptness of the beggarly baby band (s20641) (an upward-trending beggarly band for η1 = 0.0013: 131–300; a downward-trending beggarly band for η1 = −0.0005: 301–700; and an upward-trending beggarly band for η1 = 0.0012: 701–900). A hardly beyond amplitude indicator corresponded to a beyond amplitude (η2 = 0.0451 for chf01, η2 = 0.0630 for f1y02; η2 = 0.1722 for s20641:1–900 points). (B) Scatter plots assuming a able complete alternation amid the sample accustomed aberration from the beggarly sd and estimated η2 for the YOUNG and ELDER groups (no cardiac diseases), CHF (congestive affection failure), ST (ST articulation alteration), SCD (sudden cardiac death), and VT/VF (ventricular tachycardia/ventricular fibrillation) groups.

A acceptable out-of-sample anticipation achievement provides able affirmation for the abstracts abundant process. To actuate whether the homeostatic HRV responder has a acceptable out-of-sample forecasting performance, the HRV responder was acclimated to adumbrate HRV dynamics beneath six groups. We apparent for the aboriginal time that the noise-driven homeostatic HRV responder predicted best of the HRV time alternation in the YOUNG, ELDER, and CHF groups and some HRV time alternation in the VT/VF group. For example, the homeostatic HRV responder (Fig. 3B) based on in-sample abstracts apparent agnate patterns as the aboriginal ones fatigued by out-sample abstracts (Fig. 3A) area a lower η2 amount resulted in a lower HRV amplitude, which provides affirmation that η2 is an amplitude indicator. However, the noise-driven homeostatic HRV responder could not adumbrate HRV trajectories apprenticed by capricious dissection stress.

Prediction achievement of the noise-driven HRV responder. The HRV responder aural the abiding fixed-point ambit is alleged the homeostatic HRV responder. The predicted traces by the noise-driven homeostatic HRV responder (B) mimicked complete HRV traces for 0209.vt1 (ventricular tachycardia), f1y07 and f1y06 (young healthy), and chf06 and chf01 (congestive affection failure). (A) Moreover, the amplitude of HRV decreased as the amplitude indicator η2 decreased.

Furthermore, Fig. 4 shows that the noise-driven homeostatic HRV responder auspiciously predicted not alone HRV dynamics (Fig. 4A) but additionally a continued anamnesis for a apology adjournment of 10 (Fig. 4B) (a action has all-embracing assurance if its autocorrelation action (ACF) decays added boring than an exponential adulteration as the lag increases). This aftereffect accustomed the award that a time adjournment of an alike cardinal in apology could advance to abiding ample fluctuations36.

Predictions of HR dynamics and continued memory. (A) The out-of-sample predicted HRV trace apparent agnate trace to the complete HRV from an aged accountable (f1o04: 1801–3600 points). (B) The out-of-sample predicted ACF trace bigger mimicked the complete ACF trace based on the baby abstracts from the aforementioned accountable (f1o04: 1801–3600 points) with a apology adjournment of 10. The predicted ACF ethics were affected application the abstracts generated by the noise-driven homeostatic HRV responder as a adeptness of the accidental variable. The predicted and complete HRV responders in (A,B) had the aforementioned constant ethics estimated application the baby abstracts from the aforementioned accountable (f1o04: 1–1800 points). A action has continued anamnesis if its autocorrelation action (ACF) decays added boring than an exponential adulteration (e.g., at a abstract amount or an oscillation) as the lag increases. These after-effects appearance the adeptness of the noise-driven homeostatic HRV responder to adumbrate the HRV dynamics and oscillatory continued anamnesis for a continued alike apology delay.

We advised the simulation achievement of the homeostatic HRV responder beneath six altitude (groups). We apparent that the noise-driven homeostatic HRV responder bootless to simulate HRV anchored at the bend of adherence (a abreast aught adherence coefficient) in a subset of ST, SCD, and CHF groups. Instead, the stimulus-driven homeostatic HRV responder could accurately simulate best of the HRV time alternation in the VT/VF accumulation as apparent in Fig. 5.

Fitting achievement of the stimulus-driven homeostatic HRV responder. (A) The complete HRV trace plots of 0213.vf1 (ventricular fibrillation), 0067.vt1 (ventricular tachycardia), 0261.vt1 (ventricular tachycardia), 0209.vf1 (ventricular fibrillation), 0040.vt2 (ventricular tachycardia), and 0217.vf1 (ventricular fibrillation). (B) The adapted traces produced by the stimulus-driven homeostatic HRV responder. These adapted traces actor complete HRV, implying that the stimulus-driven homeostatic HRV responder can call dissection arrhythmias back cardiovascular perturbations are unpredictably dissection stimuli or acute vibration. These arrhythmias reflect the dissection accompaniment of cardiovascular and autonomic afraid systems.

As apparent above, the homeostatic HRV responder could accept a acceptable predictive and simulation achievement for abounding HRV time series. Notably, the estimated ambit may be inconsistent if the adherence accessory is abutting to 1. In this case, the constant amount can be approximated to accomplish a acceptable fit by accustomed out simulation as apparent in 0217.vf1 in Fig. 5 (see Supplementary Information).

The adherence of the NLARI-HR archetypal after-effects from the appulse of uncertainties on the achievement of HR constant estimates. Sources of ambiguity in HR constant estimates are due primarily to the randomness of baby samples and changes in their inputs. In Section 3.2, we authenticate that about all baby time alternation abstracts lie aural the abiding fixed-point ambit at a 95% aplomb breach assuming that the aftereffect of ambiguity acquired by randomness on HR ambit ((alpha ,beta ,gamma )) was small. Thus, we focus actuality on constant acuteness assay to quantify ambiguity and the advancement of ambiguity acquired by changes in the inputs back encountering abrupt relationships amid inputs and outputs. Acuteness is usually abstinent by the arrangement of the allotment change in an achievement (or a abased variable) to the allotment change in an ascribe (or an absolute variable). Actuality we accede the acuteness of HR ambit in Eq. (1) to a arrangement of abrupt baby accidental perturbations ({epsilon }_{it},{rm{{prime} }}) with beggarly ωi and about-face ({sigma }_{i}^{2}), area ({varepsilon }_{it}{rm{{prime} }}={epsilon }_{it}{rm{{prime} }}-{omega }_{i}) is a white babble absolute of the white babble εt for i = 1, ⋯ , n. The abashed NLARI action is accustomed by

$${X}_{t}=(omega {omega }_{i}) (2-alpha ){X}_{t-1}-{(1-alpha )}_{t-2} beta frac{-({X}_{t-2}-{mu }_{t-2})}{exp {({X}_{t-2}-{mu }_{t-2})}^{2}} ({varepsilon }_{t} {varepsilon }_{it}{prime} )$$

(5)

area ({mu }_{t}=[(omega {omega }_{i})/alpha ]t). The acuteness of HR ambit to changes in the inputs can be affected application the afterward formula:

Definition 2.  The acuteness of HR constant ϕ = α, β, γ, σ, η2 to changes in inputs Δφi = Δωi or ϕ = ω, α, β, γ, η1 to changes in inputs Δφi = Δσi is accustomed by

$$S({varphi }_{i},phi )={{rm{l}}{rm{i}}{rm{m}}}_{Delta {varphi }_{i}to 0}left|frac{Delta {phi }_{i}/phi }{Delta {varphi }_{i}/varphi }right|approx left|frac{Delta {phi }_{i}/phi }{Delta {varphi }_{i}/varphi }right|$$

(6)

for i = 1, ⋯ , n. We say that the HR constant ϕ about to the perturbations Δφi is altogether aloof if (S({varphi }_{i},phi )=0), aloof if (S({varphi }_{i},phi ) < 1), assemblage acute if (S({varphi }_{i},phi )=1), and acute if (S({varphi }_{i},phi ) > 1).

Due to (Var({varepsilon }_{t} {varepsilon }_{it}{prime} )={sigma }^{2} {sigma }_{i}^{2}), (Delta {sigma }_{i}=sqrt{{sigma }^{2} {sigma }_{i}^{2}}-sigma ). Agenda that (Delta {omega }_{i}=(omega {omega }_{i})-omega ={omega }_{i}). Then, we accept (S({omega }_{i},phi )=| ({phi }_{i}-phi )omega /({omega }_{i}phi )| ) and (S({sigma }_{i},phi )=| ({phi }_{i}-phi )sigma /[(sqrt{{sigma }^{2} {sigma }_{i}^{2}}-sigma )phi ]| ). We conducted simulations to account the acuteness (S({omega }_{i},phi )) and (S({sigma }_{i},phi )) based on the HR constant estimates of f1y02 from the YOUNG accumulation and s20641 from the ST group, which represented a anchored (the adherence accessory γ = 0.1787 for f1y02, 900 samples) and a nonstationary (γ = 0.0171 for s20641, 900 samples) affection amount time series, respectively. The acuteness of the boilerplate admiration of HR ambit over 3000 again simulations is presented in Fig. 6A,B for ϕ = α, β, γ, σ, η2 to changes in the perturbations Δωi and in Fig. 6C,D for ϕ = ω, α, β, γ, η1 to changes in the perturbations Δσi for f1y02 and s20641, area ωi = 0.00004i and σi = 0.007i for i = 1, ⋯ , 20. Therefore, the constant acuteness assay indicates that the HR ambit were aloof or altogether aloof for the NLARI action aural the abiding fixed-point range, admitting the apology accessory and the adherence accessory about to changes in the accustomed aberration of perturbations were acute for the NLARI action on the abuttals of the abiding fixed-point range.

Sensitivity of HR parameters. (A,B) The acuteness (S(omega ,phi )) of the HR constant ϕ = α, β, γ, σ, η2 to changes in the beggarly of abrupt perturbation Δωi (i = 1, ⋯ , 20) based on the HR constant estimates of affection amount time alternation f1y02 and s20641. (C,D) The acuteness (S(sigma ,phi )) of ω, α, β, γ, η1 to changes in the accustomed aberration of abrupt perturbation Δσi for f1y02 and s20641. Affection amount time alternation f1y02 (900 samples) followed the NLARI action aural the abiding fixed-point range, while affection amount time alternation s20641 (900 samples) followed the NLARI action on the abuttals of the abiding fixed-point range.

The baby time scales or ascertainment scales in the abstract ambit from a few abnormal (describing oscillations in the respiratory rate) to account (describing oscillations affiliated to vasomotion and modifications to borderline resistances). Thus, it is all-important to accomplish time calibration assay to analyze the acceptable time calibration covered by the NLARI-HR model. That is, we charge to appraise the furnishings of altered time scales on HR constant estimates. Enlarging the time calibration of a time alternation can be accomplished by accumulation the time alternation application ({Y}_{t}^{(m)}=frac{1}{m}{sum }_{i=1}^{m}{Y}_{(t-1)m i}) for (t=1,cdots ,[T/m]) (([T/m]) denotes the accumulation allotment of T∕m)36. Accede the ascertainment scales alignment from 1 to 10 abnormal and from 1 to 10 account and complete f1y02, f1o04, chf01, s20641, and 0030.vf1 (which were admired as the adumbrative affection amount time alternation of the YOUNG, ELDER, CHF, ST, and VT/VF groups, respectively) on the aboriginal calibration m = 1. We conducted simulations to appraisal the basal HR ambit ((alpha ,beta ,omega ,sigma )) and acquired the boilerplate HR constant ethics over 3000 again simulations. From Fig. 7, we see that as the time calibration increases, the myocardial electrical attrition accessory (α) and PNS-SNS action accessory (β) converged to the aforementioned amount of 1 on the minute scale, in which the beyond the PNS-SNS action accessory was, the faster the acceleration of aggregation (e.g., s20641, with the aboriginal amount β = 0.0171, had the slowest aggregation speed, while 0030.vf1, with the better amount β = 0.9274, had the fastest aggregation speed). The accustomed aberration of metabolic perturbations (σ) converged to a baby value, admitting the beggarly of metabolic disturbances (ω), except for 0030.vf1, which had a baby value, was continuously continued at the minute scale. These after-effects announce that HR ambit on the minute calibration cannot absolutely abstract physiological advice from affection amount abstracts because of ultra-enlarged scales.

Time scales covered by the HR model. The polylines on the left- and right-hand ancillary reflect changes in HR constant estimates on boilerplate at the additional scales m = 1, ⋯ , 10 (s) and the minute scales m = 1, ⋯ , 10 (min), respectively, area α is the myocardial electrical attrition coefficient, β is the PNS-SNS action coefficient, ω is the the beggarly of metabolic perturbations, and σ is the accustomed aberration of metabolic perturbations. The amount beneath presents HR constant ethics of m = 1 for f1y02, f1o04, chf01, s20641, and 0030.vf1 (900 samples).

It is important to analyze the achievement of the present adjustment with the classical parametric ability ashen assay in assessing the action of the autonomic afraid system. The ability ashen body assay of HR time alternation in the absolute low-frequency (VLF: 0 to 0.05 Hz), low-frequency (LF: 0.05 to 0.15 Hz), and high-frequency (HF: 0.15 to 1.0 Hz) bands provides a quantitative noninvasive apparatus to appraise the ANS modulation. It is broadly accustomed that a aerial LF agency an access in the action of the SNS, while a aerial HF shows an access in the action of the PNS. Thus, the arrangement of LF to HF ability (LF/HF ratio) is activated to appraisal the arrangement amid SNS and PNS activity3,65. The apology accessory was admired as the PNS-SNS action metric. Actuality we compared the apology accessory (β) with the parametric ability ashen body (PSD) for altered abundance bands, acquired by agency of the autoregressive archetypal of adjustment p = 16. The apology accessory estimates and their PSDs were almost constant for some affection amount abstracts (such as f1y02, chf01, and f1o04 in Fig. 8), but decidedly inconsistent with its PSD for some affection amount abstracts (such as 0030.vf1 in Fig. 8). For example, in the four affection amount time series, s20641 had the everyman LF/HF ratio, which adumbrated that its PNS action was far greater than its SNS activity; on the added hand, it apparent ample fluctuations abroad from the average, with a absolute low-frequency assuming that its PNS action was far beneath than its SNS activity. This animosity did not action for the apology accessory to admeasurement PNS-SNS activity. For example, s20641, which had a baby apology coefficient, was constant with ample fluctuations abroad from the boilerplate amount assuming a anemic PNS to SNS activity.

Comparison of the apology accessory with the PSD in complete data. (A) The ability ashen body (PSD) of 0030.vf1, f1y02, chf01, f1o04, and s20641 from the VT/VF, YOUNG, CHF, ELDER, and ST groups. (B) HR trajectories and the apology accessory (β) for 0030.vf1, f1y02, chf01, f1o04, and s20641. The apology accessory estimates were almost constant with their PSDs for f1y02, chf01, and f1o04, but decidedly inconsistent with the PSD for s20641. Affection amount time alternation s20641 apparent absolute low-frequency fluctuations, which was inconsistent with its absolute low LF/HF arrangement but constant with its baby apology coefficient.

To allegorize the abnormality empiric in Fig. 8, we performed simulation abstraction to allegorize the abnormality empiric in Fig. 8 and begin that PSD admiration was absolute nonstationary for a nonstationary, a nonlinear stationary, or a nonlinear nonstationary time series. The simulation aftereffect accustomed that the classical parametric ability ashen assay was not acceptable for nonstationary and nonlinear academic processes. We agenda that an adaptive ascertainment calibration could abate the nonstationarity. Then, simulations were agitated out at the ascertainment calibration of m = 3 to appraisal the PSD beneath altered apology coefficients. Figure 9 presents four realizations of the PSD for anniversary case application the time alternation generated by the NLARI action in Eq. (2). It is apparent that as the apology accessory increases, the LF/HF arrangement tends to (i) abatement for 0 < β ≤ 0.1 (which was constant with the ascertainment that s20641 had a absolute low LF/HF arrangement in β = 0.0171 in Fig. 8), (ii) abatement or access for 0.2 ≤ β ≤ 0.4 (which was constant with the ascertainment that the apology coefficients of f1o04, chf01, and f1y02 almost corresponded to their LF/HF ratios in Fig. 8), and (iii) access for 0.5 ≤ β ≤ 1.6 (which was constant with the actuality that 0030.vf1 had a ample LF/HF arrangement for a ample β = 0.9274 in Fig. 8). The PSD realizations appearance the achievability that in a almost abiding fixed-point range, a low LF/HF arrangement reflected a aerial PNS. However, adverse to antecedent expectations, a aerial LF/HF arrangement ability be due to overactive PNS rather than overactive SNS.

Comparison of the apology accessory with the PSD in simulation data. The realizations of the PSD were acquired by the autoregressive archetypal of adjustment p = 16 at the time calibration of m = 3 for the apology accessory β from 0.005 to 1.6. The aboriginal time alternation (m = 1) were generated by the NLARI action in Eq. (2), area the time adjournment κ2 = 1, the attrition accessory α = 0.7786, the accustomed aberration σ = 0.0275, and the sample admeasurement T = 1500. It is apparent that a baby apology accessory (beta in [0.01,0.1]) could accord to a absolute low LF/HF arrangement (s20641 with β = 0.0171); a almost ample apology accessory (beta in (0.1,0.5)) could accept a low LF/HF arrangement (f1o04 with β = 0.4365), a almost low LF/HF arrangement (f1o04 with β = 0.1097), a almost aerial LF/HF arrangement (chf01 with β = 0.1297); and a ample apology accessory (beta in [0.6,1.6]) could trend to a aerial LF/HF arrangement (0030.vf1 with β = 0.9274).

To analyze whether there is the affiliation amid the frequencies and the apology coefficient, it is all-important to abolish the impacts of randomness and nonstationarity of HR time alternation on PSD estimation. We performed simulations to appraisal the boilerplate abundance over 3000 again times. The abundance of a time alternation that exhibits aberrant variations changes for altered time scales (Fig. 10A). For this reason, we advised whether there was a accord amid the frequencies and the apology accessory at ascertainment scales alignment from m = 1 to 50. Figure 10B shows a complete alternation amid the abundance and the apology coefficient, but the complete alternation did not abide in low-frequency bands, which was constant with the ascertainment that aerial LF/HF arrangement did not absolutely represent a aerial SNS action in Fig. 9. Additionally, we agenda that the abundance became aloof to an added apology accessory on an over aerial time scale.

Relationship amid the frequencies and the apology coefficient. (A) The trajectories of the time alternation generated by the NLARI action on altered time scales alignment from m = 1 to 50. As the time calibration increases, the trajectories of the time alternation became added smooth. (B) A artifice of abundance of the time alternation adjoin the apology accessory β. The time alternation were generated by the NLARI action in Eq. (2) application the attrition accessory α = 0.9489, β = 0.02i for i = 1, ⋯ , 50, the accustomed aberration σ = 0.022, the sample admeasurement 3000, and again times 3000. It is apparent that the abundance of the time alternation was absolutely accompanying to the apology coefficient, but the abundance became aloof to an added apology accessory for an over continued time scale.

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