03e-case-study-streeter-phelps.qmd

title	Nonlinear modeling: Case study: river discharge
subtitle	Introduction to Statistical Modelling
author	Prof. Joris Vankerschaver
format	beamer header-includes \usepackage{siunitx} \usepackage{numprint} \newcommand{\BOD}{\mathsf{BOD}} \newcommand{\DO}{\mathsf{DO}}

```{r, include=FALSE} library(tidyverse) library(gridExtra) library(latex2exp) theme_set(theme_bw() + theme(text = element_text(size = 14))) ``` # Model overview ## Setting ::: {.callout-tip} ## Streeter-Phelps model Use water pollution as water quality monitoring tool. Describes how dissolved oxygen decreases in a river along a certain distance by degradation of biological oxygen demand. ::: - Aerobic bacteria gradually remove organic pollution downstream of pollution source - Reactions - Aerobic removal of biochemical oxygen demand - Oxygen transfer between atmosphere and water - Assumption: plug-flow stream - Simple dynamical model (nonlinear) ## Setting Typical values for rivers - Biochemical oxygen demand (BOD) - Not polluted: BOD < 1mg/l - Mildly polluted: 2mg/l < BOD < 8mg/l - Dissolved oxygen (DO) - Maximal saturation: DO = 12.9mg/l - Typical value in freshwater stream: DO $\approx$ 9mg/l - Threat to aquatic life: DO < 5mg/l ## Model Constant flow rate: location doesn't matter, dynamic model **in time** \begin{align*} \frac{d \BOD}{dt} & = \BOD_{in} - k_1 \BOD \\ \frac{d \DO}{dt} & = k_2(\DO_{sat} - \DO) - k_1 \BOD \end{align*} where - $\BOD_{in}$: $\BOD$ flux of waste discharge (mg $\cdot$ l$^{-1}$ $\cdot$ min$^{-1}$) - $\DO_{sat}$: dissolved oxygen concentration at saturation - $k_1$: deoxygenation rate (min$^{-1}$) - $k_2$: reaeration rate, rate at which oxygen can be absorbed from the atmosphere (min$^{-1}$) ## Model inputs - Initial conditions: - $\BOD_{t=0} = 7.33$mg/l - $\DO_{t=0} = 8.5$mg/l - Initial model inputs: - $\BOD_{in} = 1$mg$\cdot$l$^{-1}$ $\cdot$min$^{-1}$ - $\DO_{sat} = 8.5$mg$\cdot$l$^{-1}$ - $k_1 = 0.3$min$^{-1}$ (**unknown**) - $k_2 = 0.4$min$^{-1}$ (**unknown**) ## Model trajectories ```{r, include=FALSE} source("scripts/03a-parameter-estimation/SP-simulate.R", local = knitr::knit_global()) ``` ```{r} #| echo: false #| out-height: 3in #| fig-width: 6 #| fig-height: 4.5 #| fig-align: center plot.exact.trajectories() ``` # Parameter estimation ## Parameter estimation - simplex method ```{r, include=FALSE} source("scripts/03a-parameter-estimation/SP-estimate.R", local = knitr::knit_global()) ``` ```{r} #| echo: false #| message: false #| out-height: 3in #| fig-width: 6 #| fig-height: 4.5 #| fig-align: center plot.trajectory("Nelder-Mead") ``` ## Parameter estimation - BFGS ```{r} #| echo: false #| message: false #| out-height: 3in #| fig-width: 6 #| fig-height: 4.5 #| fig-align: center plot.trajectory("L-BFGS-B") ``` ## Optimal parameter result ```{r} #| echo: false #| message: false #| out-height: 3in #| fig-width: 6 #| fig-height: 4.5 #| fig-align: center plot.fitted.trajectory() ``` # Sensitivity analysis ## Absolute sensitivity functions Absolute sensitivity of $\DO$ with respect to $k_1$ and $k_2$. ```{r} #| echo: false #| message: false #| out-height: 2in #| fig-width: 5 #| fig-height: 3 #| fig-align: center source("scripts/03a-parameter-estimation/SP-sensitivity-absolute.R", local = knitr::knit_global()) ``` ## Difference exact approximate ```{r} #| echo: false #| message: false #| out-height: 3in #| fig-width: 6 #| fig-height: 4.5 #| fig-align: center source("scripts/03a-parameter-estimation/SP-sensitivity.R", local = knitr::knit_global()) ``` ## Relative sensitivity functions ```{r} #| echo: false #| message: false #| out-height: 3in #| fig-width: 6 #| fig-height: 4.5 #| fig-align: center source("scripts/03a-parameter-estimation/SP-sensitivity-relative.R", local = knitr::knit_global()) ``` ## Conclusions - Note difference in values with absolute sensitivities - $\DO$ seems more sensitive to $k_2$ than to $k_1$ - Extrema at slightly different time points: information concerning correlation between both parameters - Maximal sensitivity at same time point: parameters strongly correlated (impact of change in parameters similar) - Studying sensitivity very valuable: sensitivity used in many techniques for model analysis ## Aside: quality of estimation - DO measurements at different times: measurement error (obtained manually) is 0.05 mg$\cdot$l$^{-1}$ - Estimate simultaneously $k_1$ and $k_2$ (assume all other parameters and initial conditions constant) - Gives $k_1 = 0.353$min$^{-1}$, $k_2 = 0.389$ ## Quality of estimation: FIM ```{r, echo=FALSE, message=FALSE} source("scripts/03a-parameter-estimation/SP-fisher.R", local = knitr::knit_global()) ``` Fisher information matrix: $$ \mathsf{FIM} = \sum_{i=1}^N \left(\frac{\partial y}{\partial \theta}(t_i)\right)^T Q_i \left(\frac{\partial y}{\partial \theta}(t_i)\right) $$ - Measurement noise $\sigma_{DO} = 0.05 \si{mg.l^{-1}}$ - Weight "matrix" in the objective function $Q = \sigma_{DO}^{-2}$. Gives: $$ \mathsf{FIM} = \left[\begin{matrix} \phantom{-} 3.91 \cdot 10^{4} & -2.96 \cdot 10^{4} \\ -2.96 \cdot 10^{4} & \phantom{-} 4.56 \cdot 10^{5} \end{matrix}\right] $$ ## Quality of estimation: confidence intervals Error covariance matrix: $$ \mathsf{C} = \mathsf{FIM}^{-1} = \left[ \begin{matrix} 2.69 \cdot 10^{-5} & 1.75 \cdot 10^{-6} \\ 1.75 \cdot 10^{-6} & 2.31 \cdot 10^{-6} \end{matrix} \right] $$ 95% confidence intervals: \begin{align*} k_1 & : 0.353 \pm 0.011\\ k_2 & : 0.389 \pm 0.003 \end{align*} Covariance: $$ \mathsf{cor}(k_1, k_2) = 0.22 $$ ## Quantitative analysis - Calculate $\delta_{rmsq}$ for $\DO$ and different parameters - Illustration: different measuring schemes for $\DO$: - Scheme 1: $t_k = 0 : 0.1 : 2$ - Scheme 2: $t_k = 0 : 2 : 20$ - Scheme 3: $t_k = 0 : 2 : 10$ - Scheme 4: $t_k = 10 : 2 : 20$ ```{r, echo=FALSE, message=FALSE} source("scripts/03a-parameter-estimation/SP-delta-rmsq.R", local = knitr::knit_global()) f <- function(v) { round(v, 2) } ``` \begin{center} \scriptsize \begin{tabular}{c|c|c|c|c} & \textsf{Scheme 1} & \textsf{Scheme 2} & \textsf{Scheme 3} & \textsf{Scheme 4} \\ \hline $k_1$ & `r round(d1_k1, 2)` & `r f(d2_k1)` & `r f(d3_k1)` & `r f(d4_k1)` \\ $k_2$ & `r round(d1_k2, 2)` & `r f(d2_k2)` & `r f(d3_k2)` & `r f(d4_k2)` \\ \scriptsize $\DO_{sat}$ & `r round(d1_sat, 2)` & `r f(d2_sat)` & `r f(d3_sat)` & `r f(d4_sat)` \\ \hline & $k_2 < k_1 < \DO_{sat}$ & $k_1 < k_2 < \DO_{sat}$ & $k_1 < k_2 < \DO_{sat}$ & $k_1 < k_2 < \DO_{sat}$ \\ \end{tabular} \normalsize \end{center} ## GSA: Morris screening: parameter ranges - Global sensitivity for 6 parameters and initial conditions in designated ranges: \begin{gather*} k_1: [0.1; 0.6] \quad k_2: [0.1; 0.6] \quad \DO_{sat} : [10; 12] \\ \BOD_{in} : [0.1; 2] \quad \DO_{t=0}: [6; 10] \quad \BOD_{t=0}: [6; 10] \end{gather*} ## GSA: Morris screening ```{r} #| echo: false #| message: false #| out-height: 2in #| fig-width: 5 #| fig-height: 3 #| fig-align: center source("scripts/03a-parameter-estimation/SP-morris.R", local = knitr::knit_global()) ``` Among the selected parameters: - At time 0, $\DO$ is only sensitive to $\DO_0$ - At time 25, $\DO$ is sensitive to $k_2$ and $\DO_{sat}$. ## GSA: Monte Carlo - Monte Carlo: 100 simulations with varying $k_1$ and $k_2$ - Parameter ranges: $k_1, k_2$ uniformly sampled from $[0.1, 0.8]$ - Interested in parameter effect at $t = 2$ and $t = 15$ ```{r, include=FALSE} source("./scripts/03a-parameter-estimation/SP-GSA.R") ``` ```{r} #| echo: false #| message: false #| out-height: 2in #| fig-width: 5 #| fig-height: 3 #| fig-align: center grid.arrange( plot_mc_bod, plot_mc_do, ncol = 1) ``` ## GSA: Monte Carlo ```{r} #| echo: false #| message: false #| out-height: 3.5in #| fig-width: 8 #| fig-height: 5 #| fig-align: center grid.arrange( p_k1_bod, p_k2_bod, p_k1_do, p_k2_do, ncol = 2) ``` ## GSA: Standardized regression coefficients Previous plots show: - $BOD$ is sensitive to $k_1$ at $t = 2, 15$ - $DO$ is sensitive to $k_2$ at $t = 15$ (and somewhat at $t = 2$) Regression coefficients for $DO$ at $t = 15$: $$ \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} -0.342 \\ 6.374 \end{bmatrix}. $$ Standardized regression coefficients (using $\sigma_{k_1} = 0.211$ and $\sigma_{DO} = 1.887$): $$ SRC_{k_1} = -0.038, \quad \text{and} \quad SRC_{k_2} = 0.713. $$