We analysed the change as a function of time, by using annual groups of line intensities for each instrumental setup as independent data points d

i
. The baseline is taken as time t
0 = 2012.45, which is the mean time point of our original data, 16.31 years after the discovery of the event (Nakano et al. 1996). The total process follows an exponential decline (see Equation (7)). As the total timescale τ
rec is so much longer than the epoch of our investigation, a free parameter, giving the curvature of the exponential, cannot be derived unambiguously and numerically stably. Thus we use the Taylor series linear approximation here. We derive independent model regression points m

i
 of the type
2



mi=c(ti−t0)+n,where∑idi−mi2σi2→Min,

with c being the average annual change. As mentioned above the errors σ

i
 vary strongly for some lines (see also Table 2). Thus Equation (2) does not resemble the χ
2 definition where 



σi∝mi

 anymore. Thus standard regression algorithms used widely do not apply (York 1966; Giordano & Iavernaro 2021; Lecuna et al. 2020). The regression analysis follows Tellinghuisen (2020). The derived value of n normalizes each of the data sets with respect to the line strengths at t
0. This is slightly different from using just a weighted mean for the time t
0, but is numerically more stable against the strong year-on-year variations of the errors, which are seen especially for the helium lines. For the regression the mpfit library (Markwardt 2009) was used, with the variable errors handled according to York (1966) in the implementation of Tellinghuisen (2020). The errors given by the mpfit library for the c parameter represent the statistical error with two parameters for the calculation of the degrees of freedom. However, we are primarily interested in the significance of any slope, rather than the parameter c (respectively, its normalized counterpart C

k
 from Equation (3)) and its potential contribution to the statistical error budget. To derive the significance of the slope, a C program was written to perform a Monte Carlo (MC) simulation. The value n from Equation (2) defines the normalized values 



d¯i≔di/n

 and 



σ¯i≔σi/n

 given in Table 2. Each data point 



d¯i

 was varied independently 10 million times in agreement with its individual Gaussian error distribution, and a new regression for the normalized change C

k
 ∀ k ∈ [1, 107] was calculated with model points M

i
 types as
3



Mi=Ck(ti−t0)+Ni,where∑id¯i−Mi2σ¯i2→Min.

Moreover, a similar set of parameters from the same MC simulated data points, assuming that there was no change in time-defining model points M, was derived
4



Mi=Ni,where∑id¯i−Mi2σ¯i2→Min.

The fractional area of overlap A between the “sloping” and “non-sloping” distribution functions yields the statistical significance (1 − A) of the slope as a single parameter. This significance is lower than what would be derived from the standard deviation of the inclination (relative change per year) given by the fit with two free parameters. Figure 7 shows an example of such a pair of histograms. A priori this solution of the MC simulation, caused by the wide spread of errors between the individual data points, does not have to be distributed as a Gaussian. Tests indeed showed that it deviates from solutions with large slopes. As the result for our cases only shows very small slopes, there is only a marginal deviation from a Gaussian. As the assumption of a regular standard deviation holds we are able to use the error function erf from the integral of the Gaussian according to Equation (5) to derive the size of the intersect
5



A=1−erf−x2,

and from this the level x × σ of significance.
Citation: Markwardt 2009
Citation function: Uses; Uses