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Confidence sub-contour box: an alternative to traditional confidence intervals

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Abstract

Parameter and initial conditions (factors) estimation is a challenging task in non-linear models. Even if researchers successfully estimate those model factors, they still should estimate their confidence intervals, which could require a high computational cost to turn them into informative values. Some methods in the literature attempt to estimate regions within the search space where factors may jointly exist and fit the experimental data, i.e., confidence contours. However, the estimation of such confidence contours comes up with several issues as the number of factors increases. In this work we propose a method to compute a region within the confidence contour of an inverse problem, we denote this region as the confidence sub-contour box (CSB). To estimate a CSB of an inverse problem, we propose two main algorithms alongside their interpretation and validation, testing their performance with two epidemiological models for different diseases and different kinds of data. Moreover, we exposed some desirable properties of the method through numerical experiments, such as user-defined uncertainty level, asymmetry of the resultant intervals regarding the nominal value, sensitivity assessment related to the interval length for each factor, and identification of true-influential factors. We set available all the algorithms we used in this paper at Mathworks in the GSUA-CSB toolbox.

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Acknowledgements

This study was funded by Departamento Administrativo de Ciencia, Tecnología e Innovación - COLCIENCIAS (Grant Number 111572553478) and Universidad EAFIT (Grant Number 954-000002).

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Authors and Affiliations

  1. School of Applied Sciences and Engineering, Universidad EAFIT, Medellín, Colombia

    Daniel Rojas-Diaz, Alexandra Catano-Lopez, Carlos M. Vélez, Santiago Ortiz & Henry Laniado

Authors
  1. Daniel Rojas-Diaz
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  2. Alexandra Catano-Lopez
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  3. Carlos M. Vélez
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  4. Santiago Ortiz
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  5. Henry Laniado
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Correspondence to Daniel Rojas-Diaz.

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Appendices

Appendix

Appendix 1: Mono-objective example: Dengue

We selected a system of ordinary differential equations with seven states proposed by Lizarralde-Bejarano et al. (2020) to estimate parameters and confidence intervals for a dengue outbreak. This model includes the virus transmission between a population of humans and mosquitoes (see Eq. in 11). We kept the assumptions in Lizarralde-Bejarano et al. (2020) of a total mosquito population variable in time M, where \(M = M_s+M_e+M_i\), and total human population constant in time H, where \(H= H_s+H_e+H_i+H_r\). We list the state variables and parameters of the model in Table 5.

$$\begin{aligned} \begin{array}{lll} \dot{M_s} &{} = &{} \Lambda - \beta _m M_s \frac{H_i}{H} - \mu _m M_s\\[3pt] \dot{M_e} &{} = &{} \beta _m M_s \frac{H_i}{H} - (\theta _m + \mu _m) M_e\\ \dot{M_i} &{} = &{} \theta _m M_e - \mu _m M_i\\ \dot{H_s} &{} = &{} - \beta _h H_s \frac{M_i}{M} + (H_e + H_i + H_r) \mu _h\\ \dot{H_e} &{} = &{} \beta _h H_s \frac{M_i}{M} - (\theta _h + \mu _h) H_e\\ \dot{H_i} &{} = &{} \theta _h H_e - (\gamma _h + \mu _h) H_i\\ \dot{H_r} &{} = &{} \gamma _h H_i - \mu _h H_r \end{array} \end{aligned}$$
(11)
Table 5 Parameter and states for model (11), the model search box is defined in Table 3
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Appendix 2: Multi-objective example: COVID-19

Implemented model for COVID-19 infection (Sen and Sen 2021).

See Table 6.

$$\begin{aligned} \begin{array}{cll} \dot{S} &{}=&{} -\alpha \,S - \frac{\beta \,I\,S}{N} - \frac{\sigma \,S\,A}{N} - \eta \,S\\ \dot{A} &{}=&{} -\tau \,A + \epsilon \,E\\ \dot{C} &{}=&{} \alpha \,S - \mu \,C\\ \dot{E} &{}=&{} -\gamma \,E + \frac{\beta \,I\,S}{N} + \mu \,C + \eta \,S + \frac{\sigma \,S\,A}{N} - \epsilon \,E\\ \dot{I} &{}=&{} \tau \,A + \gamma \,E - \delta \,I\\ \dot{R} &{}=&{} \lambda \,Q\\ \dot{Q} &{}=&{} \delta \,I - \lambda \,Q - k\,Q\\ \dot{D} &{}=&{} k\,Q\\ \end{array} \end{aligned}$$
(12)

with \(N = S + C + E + A + I + Q + R + D\)

Table 6 Parameter and states for model (12), the model search box is defined in Table 4
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Appendix 3: Samples of multiple CSB estimations for dengue and COVID-19 models

See Figs. 8 and 9.

Fig. 8
figure 8

Convergence of CSB intervals for the Dengue model without protecting the \(\hat{X}\) value. For each sample size (N) we estimated the CSB 100 times. Blue squares represents the mean value of the upper bounds (UB) while orange squares represents the mean value of the lower bounds (LB) estimated for each factor. The symmetrical error bars at every square are given by the standard deviation of each set of UB or LB. We obtained the left Y-axis through normalizing the data regarding the OAT intervals. The right Y-axis was obtained normalizing the data regarding the search space ranges

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Fig. 9
figure 9

Convergence of CSB intervals for the COVID-19 model without protecting the \(\hat{X}\) value. For each sample size (N) we estimated the CSB 100 times. Blue squares represents the mean value of the upper bounds (UB) while orange squares represents the mean value of the lower bounds (LB) estimated for each factor. The symmetrical error bars at every square are given by the standard deviation of each set of UB or LB. We obtained the left Y-axis through normalizing the data regarding the OAT intervals. The right Y-axis was obtained normalizing the data regarding the search space ranges

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Appendix 4: Confidence intervals for simpler model

Consider the following non-linear pendulum model with input force \(u=1\) and friction

$$\begin{aligned} \begin{array}{l} \dot{x}_1=x_2\\ \dot{x}_2=\frac{1}{l\,m}-\frac{f\,x_2}{m}-\frac{g\,\sin \left( x_1\right) }{l}. \end{array} \end{aligned}$$
(13)

We selected \(\hat{X}\) as an arbitrary combination of factors within some arbitrary search box and then we proceeded to compute the \(\hat{X}\)-protected CSB intervals with sample size \(N=5000\) alongside the likelihood-based CI. Results are summarized in Table 7 and Fig. 10. We used the same parameterization for this experiment as for the Dengue and the COVID-19 models.

Table 7 Factors and intervals for the pendulum model (13)
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Fig. 10
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Uncertainty analysis of likelihood-based CI and CSB intervals for the pendulum model. The number of simulations for each case were 500

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Appendix 5: Confidence intervals: MSE-based

See Fig. 11.

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UA for likelihood intervals in Tables 8, 9

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Table 8 MSE-based and likelihood confidence intervals for the Dengue model
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Table 9 MSE-based and likelihood confidence intervals for the COVID-19 model
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Appendix 6: Algorithms

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Rojas-Diaz, D., Catano-Lopez, A., Vélez, C.M. et al. Confidence sub-contour box: an alternative to traditional confidence intervals. Comput Stat 39, 2821–2858 (2024). https://doi.org/10.1007/s00180-023-01362-4

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