## Abstract

In a recent publication, we carried out an extensive analysis of an unsupervised method of determining optimum shift factors in time-temperature-superposition of accelerated-aging data that involves minimizing the vertical arclength to obtain the master curve. For synthetic Arrhenius data with a variety of noise distributions, the work showed that, in conjunction with bootstrap-resampling, the method can produce reliable estimates of the mean activation energy along with uncertainty quantification. The present work applies the above method to six different datasets taken from the published literature and demonstrates accurate prediction of mean activation energy from the data as-is without the need for any pre-processing or fitting. It also compares uncertainty margins computed by second-order bootstrap with that by linear regression theory and shows that the former appears to provide consistent margins in the presence of common noise types in real data, including intra-isotherm measurement-errors, sample-to-sample variations, and intrinsic deviation from perfect Arrhenius behavior.

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## Funding

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

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## Appendix. Linear Regression to compute uncertainty margins in Arrhenius activation energy

### Appendix. Linear Regression to compute uncertainty margins in Arrhenius activation energy

The regression method involves least-square-fitting of the ln(*a*_{T}) vs. *X*_{T} = {(*RT*_{ref})^{−1} − (*RT*)^{−1}} plot, whose slope (\( \hat{\beta} \)) directly provides a mean estimate of the activation energy *E*_{a}. From linear regression theory (Johnson and Wichern 2007), under the assumption of homoscedastic Gaussian noise, the regression slope \( \hat{\beta} \) follows the student *t*-distribution with *n* − 2 degrees of freedom (when the intercept term is included), i.e.,

where *β* is the (unknown) true slope, and the standard error of the sampling distribution of \( \hat{\beta} \) is given by \( SE\left(\hat{\beta}\right)=\sqrt{SSE/\left\{\left(n-2\right) SSX\right\}} \), where \( SSE={\sum}_{i=1}^n{e}_i^2 \) is the sum of squared residuals and \( SSX={\sum}_{i=1}^n{\left({X}_i-\overline{X}\right)}^2 \) the sum of squared deviation of the *X* values from its sample mean \( \overline{X} \). [In our case, the *X* variables are *X*_{i} = (*RT*_{ref})^{−1} − (*RT*_{i})^{−1}, *T*_{i} is the absolute temperature of the *i*^{th} isotherm, and *n* the number of isotherms.] Thus, the margin of error in the regression estimate of activation energy (∆*E*_{reg}) at 100(1 − *α*)% level of confidence can be obtained as:

where *t*_{α/2, n − 2} is the critical value notation indicating the 100(1 − *α*/2) percentile of the *t* distribution with (*n* − 2) degrees of freedom.

To apply the method of regression within the present framework, we perform the following steps: (i) determine the bootstrap-mean activation energy \( \left({\overline{E}}_{minarc}\right) \)using the minimum-arclength algorithm; (ii) shift the isotherms according to Eq. (4) (using \( {E}_a={\overline{E}}_{minarc} \)) and least-square-fit the resulting master curve with an appropriate function (in our case, chosen appropriately from Eq. (5a–d)), resulting in a fit-function, let us call \( {y}_{fit}\left(t;{\overline{E}}_{minarc}\right) \); (iii) determine the deviation from perfect Arrhenius by optimizing parameters {*e*_{T}} of Eq. (6) that yields the least-square-deviation of the sample data from \( {y}_{fit}\left(t;{\overline{E}}_{minarc}\right) \); (iv) perform linear regression analysis on the resulting ln(*a*_{T}) vs. *X*_{T} = {(*RT*_{ref})^{−1} − (*RT*)^{−1}} (where the shifts {*a*_{T}} are given by Eq. (6)).

As a numerical consistency check, one expects the shift factor at *T* = *T*_{ref} to be ≈1, i.e., by design the curve ln(*a*_{T}) vs. *X*_{T} = {(*RT*_{ref})^{−1} − (*RT*)^{−1}} should have negligible intercept. To this end, we would like to report that in all cases considered in this work, whether synthetic or real data, the intercept term was statistically insignificant, i.e., large *p* value, as expected.

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Maiti, A. Time-temperature-superposition analysis of diverse datasets by the minimum-arclength method: long-term prediction with uncertainty margins.
*Rheol Acta* **60, **155–162 (2021). https://doi.org/10.1007/s00397-021-01262-8

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### Keywords

- Time-temperature superposition
- Bootstrap
- Uncertainty quantification
- Long-term prediction
- Lifetime estimation