On fitting data for parameter estimates: residual weighting and data representation

Abstract

We describe a fundamental problem in fitting rheological data to infer parameters and thus structural information about a material. The results may depend greatly on subjective choices in the fitting method making such inferences non-unique and therefore highly uncertain or wrong. We study experimental data and demonstrate that the most commonly used fitting scheme for oscillatory linear viscoelastic data is deceptively wrong for the purpose of inference, i.e., viscoelastic moduli (G′,G″) weighted by the experimental data. Our results establish best practices for fitting rheological data, linear viscoelastic or otherwise, with strong implications for inverse problems of structural inference.

APPENDIX

Least squares estimate under data transformation

As per the notation of Eq. (1), for S₁, S₂ : G′, G″; w = S, the objective function is given as

$$ {\mathrm{RSS}}_1=\sum \limits_{i=1}^N\left\{\kern0.48em {\left(1-\frac{{\hat{G\prime}}_i\left({\omega}_i,\overrightarrow{p}\right)}{G{\prime}_i\left({\omega}_i\right)}\right)}^2+{\left(1-\frac{{\hat{G{\prime\prime}}}_i\left({\omega}_i,\overrightarrow{p}\right)}{G{{\prime\prime}}_i\left({\omega}_i\right)}\right)}^2\right\} $$

(19)

Similarly for S₁, S₂ : G^∗, tan δ; w = S, the objective function is given as

$$ {\mathrm{RSS}}_2=\sum \limits_{i=1}^N\left\{\kern0.48em {\left(1-\frac{{{\hat{G}}_i}^{\ast}\left({\omega}_i,\overrightarrow{p}\right)}{{G_i}^{\ast}\left({\omega}_i\right)}\right)}^2+{\left(1-\frac{\tan\;{\hat{\delta}}_i\left({\omega}_i,\overrightarrow{p}\right)}{\tan\;{\delta}_i\left({\omega}_i\right)}\right)}^2\right\} $$

(20)

RSS₂ can be written in terms of {G′, G″} using the nonlinear transformation shown in Table 2 as

$$ {\mathrm{RSS}}_2=\sum \limits_{i=1}^N\left\{\kern0.48em {\left(1-\frac{\sqrt{G\prime {\hat{\mkern6mu}}_i{\left({\omega}_i,\overrightarrow{p}\right)}^2+G\prime \prime {\hat{\mkern6mu}}_i{\left({\omega}_i,\overrightarrow{p}\right)}^2}}{\sqrt{G{\prime}_i{\left({\omega}_i\right)}^2+G{{\prime\prime}}_i{\left({\omega}_i\right)}^2}}\right)}^2+{\left(1-\frac{{\hat{G{\prime\prime}}}_i\left({\omega}_i,\overrightarrow{p}\right)\times G{\prime}_i\left({\omega}_i\right)}{{\hat{G\prime}}_i\left({\omega}_i,\overrightarrow{p}\right)\times G{{\prime\prime}}_i\left({\omega}_i\right)}\right)}^2\right\} $$

(21)

Clearly, RSS₁ ≠ RSS₂ for the same weighting under a nonlinear data transformation. Now, we consider the S₁, S₂ : η′, η″; w = S case for which

$$ {\mathrm{RSS}}_3=\sum \limits_{i=1}^N\left\{{\left(1-\frac{{\hat{\eta \prime}}_i\left({\omega}_i,\overrightarrow{p}\right)}{\eta {\prime}_i\left({\omega}_i\right)}\right)}^2+{\left(1-\frac{{\hat{\eta {\prime\prime}}}_i\left({\omega}_i,\overrightarrow{p}\right)}{\eta {{\prime\prime}}_i\left({\omega}_i\right)}\right)}^2\right\} $$

(22)

RSS₃ can be written in terms of {G′, G″} using the linear data transformation shown in Table 2 as

$$ {\displaystyle \begin{array}{l}{\mathrm{RSS}}_3=\sum \limits_{i=1}^N\left\{{\left(1-\frac{{\hat{G{\prime\prime}}}_i\left({\omega}_i,\overrightarrow{p}\right)}{G{{\prime\prime}}_i\left({\omega}_i\right)}\right)}^2+{\left(1-\frac{{\hat{G\prime}}_i\left({\omega}_i,\overrightarrow{p}\right)}{G{\prime}_i\left({\omega}_i\right)}\right)}^2\right\}\\ {}\kern0.75em \end{array}} $$

(23)

Clearly, RSS₁ = RSS₃ under a linear data transformation where the weighting transforms according to the law of error propagation, i.e., Eq. (7).

Derivation of phase angle uncertainty

Here, we derive the phase angle uncertainty u_δ given by Eq. (11). In particular, we establish an upper bound on the phase angle uncertainty u_δ by looking at the signal zero-crossing locations locally, and then quantifying the uncertainty in these locations due to the finite resolution of the rheometer. In Fig. 9, we show point A as the apparent zero-crossing location for a qualitative torque signal when the signal is perfectly sinusoidal. Due to the resolution uncertainty, the actual value of torque at point A can lie anywhere on the vertical line BC where the length of BC is $2\;{u}_{T_0}$. Assuming that the signal still follows the sinusoidal shape with same frequency, the real zero-crossing location lies along the line ED. So, the real zero-crossing can occur earlier (along the line EA) or later (along the line AD) with respect to apparent zero-crossing location A. The maximum uncertainty in the zero-crossing location occurs at points E and D and its magnitude is given by ${u}_{t\;\left({T}_0\right)}$.

Fig. 9

Actual magnitude of torque at apparent zero-crossing location A of a perfectly sinusoidal signal (solid line) can lie anywhere along the line BC where the length AB represents the torque resolution uncertainty ${u}_{T_0}$ of the instrument. As a result, the actual zero-crossing can be delayed and lie along AD (signal shown as dashed line), or it can be expedited and lie along AE (signal shown as dash-dot line). The maximum uncertainty in the zero-crossing location corresponds to either point D or E and has the same magnitude given by Eq. (25)

The sinusoidal waveform EC in Fig. 9 has an angular frequency ω and an amplitude T₀. Labeling the time as t = 0 at E, the amplitude at C (=${u}_{T_0}$) is given by

$$ {u}_{T_0}={T}_0\sin \left(\omega\;{u}_{t\;\left({T}_0\right)}\right), $$

(24)

which provides an expression for the zero-crossing location uncertainty in torque signal as

$$ {u}_{t\;\left({T}_0\right)}=\frac{1}{\omega }{\sin}^{-1}\left(\frac{u_{T_0}}{T_0}\right)\kern2.999999em \mathrm{for}\kern0.24em {u}_{T_0}\le {T}_0. $$

(25)

Equation (24) is valid when ${u}_{T_0}$ ≤ T₀. For a signal smaller than the resolution limit, the waveform cannot be resolved by the rheometer. In that case, the zero-cross location uncertainty reaches its maximum at

$$ {u}_{t\;\left({T}_0\right)}=\frac{\pi }{2\omega}\kern5.399997em \mathrm{for}\kern0.24em {u}_{T_0}>{T}_0. $$

(26)

Similar to Eqs. (25)–(26), the uncertainty in the angular displacement zero-crossing location ${u}_{t\;\left({\theta}_0\right)}$ is given by

$$ {u}_{t\;\left({\theta}_0\right)}=\frac{1}{\omega }{\sin}^{-1}\left(\frac{u_{\theta_0}}{\theta_0}\right)\kern2.999999em \mathrm{for}\kern0.24em {u}_{\theta_0}\le {\theta}_0 $$

(27)

$$ {u}_{t\;\left({\theta}_0\right)}=\frac{\pi }{2\omega}\kern5.399997em \mathrm{for}\kern0.24em {u}_{\theta_0}>{\theta}_0. $$

(28)

The zero-crossing location uncertainties in torque and displacement add up if the real zero-crossing locations are on opposite sides of their respective apparent zero-crossing locations. On the other hand, they subtract if the real zero-crossing locations are on the same side of their apparent zero-crossing locations. The upper bound on phase angle uncertainty is given for the first case as

$$ {u}_{\delta }=\omega \left({u}_{t\;\left({T}_0\right)}+{u}_{t\;\left({\theta}_0\right)}\right), $$

(29)

which on simplification using Eqs. (25)–(28) gives

$$ {u}_{\delta }=\left\{\begin{array}{l}{\sin}^{-1}\left(\frac{u_{T_0}}{T_0}\right)+{\sin}^{-1}\left(\frac{u_{\theta_0}}{\theta_0}\right)\kern6.839995em \mathrm{for}\kern0.24em {u}_{T_0}\le {T}_0\&{u}_{\theta_0}\le {\theta}_0\\ {}\frac{\pi }{2}+{\sin}^{-1}\left(\frac{u_{\theta_0}}{\theta_0}\right)\kern8.999993em \mathrm{for}\kern0.24em {u}_{T_0}>{T}_0\&{u}_{\theta_0}\le {\theta}_0\\ {}{\sin}^{-1}\left(\frac{u_{T_0}}{T_0}\right)+\frac{\pi }{2}\kern9.119993em \mathrm{for}\kern0.24em {u}_{T_0}\le {T}_0\&{u}_{\theta_0}>{\theta}_0\\ {}\pi \kern12.47999em \mathrm{for}\kern0.24em {u}_{T_0}>{T}_0\&{u}_{\theta_0}>{\theta}_0\end{array}\right\} $$

(30)

Uncertainty weighted fits ignoring data covariances

If covariances are ignored in Eq. (4), we get results which are presented in Figs. 10 and 11 (analogous to Figs. 7 and 8). Even if covariances are ignored, conclusions do not change and still a significant reduction of subjectivity effects in fitting is observed while using data uncertainty weighting. When accounting for covariances, the fit parameters ${G}_{\mathrm{N}}^0$ and ${\tau}_{\mathrm{d}}^{\infty }$ vary by up to a factor of 1.02 and 1.01 respectively for w = u_S and w = ${\hat{u}}_S$ (Fig. 8); this factor while ignoring covariances becomes 1.02 for both cases as shown in Fig. 11.

Fig. 10

Fit parameter estimates when neglecting data covariances for all weightings

Fig. 11

Zoomed-in data of Fig. 10 only showing the parameter estimates from fits weighted by (a) experimental data uncertainty w = u_S and (b) estimated data uncertainty $w={\hat{u}}_S$. These have been calculated by ignoring covariances in Eq. (4)

Model parameterization using viscosity and characteristic timescale

Parameter uncertainty may depend on which model parameterization is chosen. For the coupled reptation and contour-length fluctuations model used here, rather than ${G}_{\mathrm{N}}^0$ and ${\tau}_{\mathrm{d}}^{\infty }$ as the model parameters, we can use zero-shear viscosity η₀ and ${\tau}_{\mathrm{d}}^{\infty }$ as the model parameters. The zero-shear viscosity η₀ is related to ${G}_{\mathrm{N}}^0$ and ${\tau}_{\mathrm{d}}^{\infty }$ for this case as (Doi 1981; Roovers 1986)

$$ {\eta}_0=\frac{1}{3}{G}_{\mathrm{N}}^0{\tau}_{\mathrm{d}}^{\infty}\left[{\left(1-\frac{\nu }{\sqrt{Z}}\right)}^3+\frac{\nu^3}{5Z\sqrt{Z}}\right] $$

(31)

For the terminal regime dataset considered here, we find the η₀ parameterization has smaller uncertainty than the ${G}_{\mathrm{N}}^0$ parameterization, i.e., η₀ shows reduced effects of subjective choices of data representation and residual weighting compared to ${G}_{\mathrm{N}}^0$. Figure 12 shows that the fit estimates of η₀ for all representations and weightings vary by a factor of only 1.04 as compared to 1.64 for ${G}_{\mathrm{N}}^0$. Uncertainty weighted fits (accounting for covariances) further reduce the subjectivity effect for η₀. As shown in Fig. 13, η₀ varies by up to a factor of 1.01 for w = u_S and 1.0009 for w = ${\hat{u}}_S$.

Fig. 12

Results of Fig. 7 recast in terms of τ^∞_d and η₀ (rather than G⁰_N), using various data representations and residual weightings. For fits weighted by experimental data uncertainty w = u_S and estimated data uncertainty$w={\hat{u}}_S$, data covariances were accounted for using Eq. (4)

Fig. 13

Zoomed-in data of Fig. 12 only showing the parameter estimates from fits weighted by (a) experimental data uncertainty w = u_S and (b) estimated data uncertainty $w={\hat{u}}_S$. Data covariances were accounted for using Eq. (4). Note that η₀ varies by only up to a factor of 1.0009 in (b)

This fortuitous reduction in subjectivity for η₀ is due to the available experimental data being in the terminal regime of a viscoelastic liquid where viscous effects dominate. A different result may be expected for a viscoelastic liquid with high-frequency experimental data and dominating elastic effects. The ultimate choice of parameterization depends on how the information is used, e.g., inference of different molecular or microstructural aspects related to either ${G}_{\mathrm{N}}^0$ or η₀.