03d-sensitivity-analysis.qmd

title	Nonlinear Modeling: Sensitivity Analysis
subtitle	Introduction to Statistical Modelling
author	Prof. Joris Vankerschaver

```{r, include=FALSE} library(tidyverse) library(gridExtra) library(latex2exp) theme_set(theme_bw() + theme(text = element_text(size = 14))) ``` # Sensitivity Analysis ## Why sensitivity analysis? - Verify what _sources of uncertainty_ contribute most to variance (uncertainty) of model output. - Sources of uncertainty in model can be - Model parameters, initial conditions, inputs - Model structure - Better understand changes in model predictions due to the above ## Why sensitivity analysis? - Detect what _model parameters_ contribute most to model output uncertainty - Want to reduce model uncertainty, so best to focus on most influential parameters - Gives idea of correlation between parameters - Helps in choice of what parameters to estimate (in parameter estimation) ## Why sensitivity analysis? - Gives information about interesting location, time, ... to collect experimental data - Basis for experimental design - Gives information on insensitive model parameters - Useful in model reduction of overparametrized models ## Local vs global 1. Local sensitivity analysis - Determine sensitivity at **one certain point** in parameter space - Not very computationally intensive 2. Global sensitivity analysis - Determine sensitivity in **delimited area** of parameter space - Usually gives a mean sensitivity - Can become extremely computationally intensive - Each technique has advantages and disadvantages - Each technique gives different type of information ## Examples of sensitivity analysis: water quality model :::: {.columns} ::: {.column width="50%"} - Hundreds of parameters - Each model simulation takes days to run - Identifying highly sensitive parameters is critical {fig-align="center" width=75% align="center"} ::: ::: {.column width="50%"} {width=75%} {width=75%} ::: :::: \scriptsize Source: *Developing a cloud-based toolbox for sensitivity analysis of a water quality model* (S. Kim et al, Environmental Modeling and Software, 2021) ## Examples of sensitivity analysis: cell signaling Toll-like signaling pathway: - Cellular response to external stimuli (e.g. infection) - Central role for NF-$\kappa$B transcription factor - Shuttles back and forth between cytoplasm and nucleus {fig-align="center" width=60%} \scriptsize Source: Images from _Fundamentals of Systems Biology_, M. Covert, CRC Press, 2014. ## Examples of sensitivity analysis: cell signaling Hoffmann-Levchenko (2005): Computational model for NF-$\kappa$B - 25 ODEs, 36 parameters - Models protein production, degradation, transport - Important role for **parameter estimation** and **sensitivity analysis** {fig-align="center" height="50%"} \scriptsize Source: Images from _Fundamentals of Systems Biology_, M. Covert, CRC Press, 2014. ## Examples of sensitivity analysis: cell signaling Sensitivity analysis: which parameters affect the model the most? - Transcription rate: affects output a lot (**sensitive**) - Degradation rate: relatively **insensitive** Gives rough idea, needs to be corroborated with full model. :::: {.columns} ::: {.column width="50%"} {width=90%} ::: ::: {.column width="50%"}  ::: :::: \scriptsize Source: Images from _Fundamentals of Systems Biology_, M. Covert, CRC Press, 2014. # Local sensitivity analysis ## Local sensitivity analysis How sensitive is model output ($y$) to changes of model parameter ($\theta$) *at one single point* in parameter space? - **(Absolute) local sensitivity**: partial derivative of variable with respect to parameter at single point in parameter space $$ S(\theta, x) = \frac{\partial y}{\partial \theta}(\theta, x) $$ - If $k$ parameters, then also $k$ sensitivity functions: $$ S_i(\theta, x) = \frac{\partial y}{\partial \theta_i}(\theta, x), \quad i = 1, \ldots, k. $$ ## Local sensitivity analysis: absolute sensitivity **Problem**: often very hard to compute partial derivative analytically. **Solution**: compute derivative **numerically** through finite difference method: - Forward difference: $$ \left.\frac{\Delta y}{\Delta \theta_j}\right|_+ = \frac{y(x,\theta_j+\Delta\theta_j)-y(x,\theta_j)}{\Delta\theta_j} $$ - Backward difference: $$ \left.\frac{\Delta y}{\Delta \theta_j}\right|_- = \frac{y(x,\theta_j)-y(x,\theta_j-\Delta\theta_j)}{\Delta\theta_j} $$ ## Local sensitivity analysis: absolute sensitivity - How to choose perturbation $\Delta\theta_j$? - Too large: approximation is not good - Too small: numerical instabilities. - In practice, choose $\Delta \theta_j$ **small** and **fixed**, e.g. $$ \Delta \theta_j = 10^{-6}. $$ ::: {.callout-tip} ## Convergence Both the forward and the backward difference agree with the derivative up to **first order** in $\Delta \theta_j$: $$ \frac{\partial y(x)}{\partial \theta_j} = \left.\frac{\Delta y(x)}{\Delta \theta_j}\right|_+ + \mathcal{O}(\Delta \theta_j), \quad \frac{\partial y(x)}{\partial \theta_j} = \left.\frac{\Delta y(x)}{\Delta \theta_j}\right|_- + \mathcal{O}(\Delta \theta_j). $$ ::: ## Local sensitivity analysis: absolute sensitivity - Third option: central difference $$ \frac{\Delta y(x)}{\Delta \theta_j} = \frac{y(x,\theta_j+\Delta\theta_j) - y(x,\theta_j-\Delta\theta_j)}{2\Delta\theta_j} $$ ::: {.callout-tip} ## Convergence The central difference agrees with the derivative up to **second order** in $\Delta \theta_j$: $$ \frac{\partial y(x)}{\partial \theta_j} = \frac{\Delta y(x)}{\Delta \theta_j} + \mathcal{O}((\Delta \theta_j)^2). $$ ::: ## Local sensitivity analysis: relative sensitivity Absolute sensitivity is influenced by magnitude of variable and parameter. - Problematic if we want to compare sensitivities of different combinations of outputs and parameters - Use **relative sensitivity**. ## Local sensitivity analysis: relative sensitivity Different definitions, depending on what's important: 1. Relative sensitivity w.r.t. parameter: $$ \frac{\partial y(t)}{\partial \theta_j} \cdot \theta_j $$ Compare sensitivity of same variable w.r.t. *different parameters* 2. Relative sensitivity w.r.t. variable $$ \frac{\partial y_i(t)}{\partial \theta_j} \cdot \dfrac{1}{y_i} $$ Compare sensitivity of *different variables* w.r.t. same parameter ## Local sensitivity analysis: relative sensitivity 3. Total relative sensitivity $$ \frac{\partial y_i(t)}{\partial \theta_j} \cdot \dfrac{\theta_j}{y_i} $$ Compare all sensitivities (of *different variables* w.r.t. *different parameters*) ## Local sensitivity analysis - Relative sensitivities allow to **rank sensitivities**. Important for: - Choice parameters for parameter estimation - Choice parameters for model reduction - Choice for additional measurement or experimental determination of parameter (reduce sources of uncertainty) - Ranking **depends on value of parameter**, can be different at different position in parameter space - How to compare continuous sensitivity functions? - Interest in specific values of independent variable - Where measurements are available - Where measurements will be collected ## Local sensitivity analysis - Create generic model with - Time $t$ as independent variable - Outputs $y_i$, $i=1,\ldots,v$ - Parameters $\theta_j$, $j=1,\ldots,p$ - Moments of measurements $t_k$, $k=1,\ldots,N$ - Total relative sensitivity of variable $y_i$ w.r.t. parameter $\theta_j$ at moment $t_k$ $$ S_{i,j,k} = \frac{\partial y_i(t_k)}{\partial \theta_j} \cdot \frac{\theta_j}{y_i} $$ ## Local sensitivity analysis Importance parameter is determined by its impact on _all_ variables \newline $\rightarrow$ sum and average over all variables \newline $\rightarrow$ take sign into account (square and root)\newline __root mean square sensitivity for parameter__ $\theta_j$ $$ \delta_{j,k}^{rmsq}=\sqrt{\dfrac{\sum_{i=1}^vS_{i,j,k}^2}{v}} $$ \ \newline $\delta_{j,k}^{rmsq}$ can be very variable from moment to moment \newline $\rightarrow$ sum and average over all time points\newline __time mean root mean square sensitivity for parameter__ $\theta_j$ $$ \delta_j^{rmsq} = \dfrac{1}{N}\sum_{k=1}^N \delta_{j,k}^{rmsq} $$ ## Local sensitivity analysis - Gives one single measure for sensitivity of parameter - Use this measure to determine importance of parameter - Obtained value depends on - nominal parameter value: nonlinear models give different values at different location in parameter space (see also global sensitivity analysis) - choice of time points is arbitrary: this can lead to different set of parameters that are best estimated using dataset (see also identifiability) - Modifications can be defined based on application/goal # Side track: Monte Carlo simulation ## These slides have been moved to the slide deck `03f-monte-carlo.pdf` # Global sensitivity analysis ## Global sensitivity analysis (GSA) - Measure for sensitivity in delimited area in parameter space - PDFs for parameters need to be chosen/found (same as for uncertainty analysis) 3 techniques will be discussed: - Standardized regression coefficients - Screening techniques - Variance decomposition ## GSA: Standardized regression coefficients - Linear regression of Monte Carlo simulations - Each line is simulation of variable $y$ for different parameter set $\Theta$, i.e., other point in parameter space ```{r, include=FALSE} source("./scripts/03a-parameter-estimation/SP-GSA.R") ``` ```{r, echo=FALSE} #| out-width: 3in #| out-height: 2in #| fig-width: 4.5 #| fig-height: 3 #| fig-align: center plot_mc_do ``` - Figure: 100 simulations of dissolved oxygen, with $k_1, k_2$ sampled uniformly between $0.1$ and $0.8$. ## GSA: Standardized regression coefficients - Consider outcomes at fixed time $T$ - Quantify effect of parameters $\theta_1, \ldots \theta_p$ through linear model $$ y_{t = T} = b_1 \theta_1 + \cdots + b_p \theta_p + \epsilon $$ - Regression coefficient $b_i$ gives contribution of parameter $\theta_i$ in explaining variance of $y_{t = T}$ ```{r, echo=FALSE} #| out-width: 4.5in #| out-height: 2in #| fig-width: 7 #| fig-height: 3 #| fig-align: center p1 <- ggplot(curves_t15, aes(x = k1, y = DO)) + geom_point() p2 <- ggplot(curves_t15, aes(x = k2, y = DO)) + geom_point() grid.arrange(p1, p2, ncol = 2) ``` ## GSA: Standardized regression coefficients - Correct for spread on both parameter and output - Recalculate coefficients $b_i$ to $SRC$s $$ SRC_{\theta_i} = b_i \cdot \dfrac{\sigma_{\theta_i}}{\sigma_y} $$ - Sample standard deviations from - vector $y_{t = T}$ for output - parameter samples for parameter $\theta_i$. ## GSA: Standardized regression coefficients - For linear model in parameters that were examined, the total variance is explained by $SRC$s $$\sum_i SRC_{\theta_i}^2=1$$ - For nonlinear models (in the parameters) _not_ all variance will be explained. The part that is explained is given by determination coefficient $$R^2=\dfrac{\sum_{i=1}^n(\hat{y}_i-\overline{y})^2}{\sum_{i=1}^n(y_i-\overline{y})^2}$$ - $\hat{y}_i$: prediction by regression model - Technique only valid if $R^2 > 0.7$ ## GSA: Screening techniques - Goal: obtain idea of importance of model parameters using only a limited number of simulations - Example of technique: Morris screening - Calculation of *elementary effect* for $\theta_i$ $$ EE_{\theta_i} = \dfrac{y(\theta_i+\Delta)-y(\theta)}{\Delta} $$ - $\Delta$ is a predetermined step size in parameter - Remark analogy with local sensitivity, however, step size much larger ## GSA: Morris screening - Assume 2 parameters $\theta_1$ and $\theta_2$ - Choose regions in parameter space to compute elementary effects - Summarize EE using mean and variance {fig-align="center"} ## GSA: Morris screening - Vector of $EE$ (in this case 3) - Statistical analysis of this vector - $\mu_{EE_{\theta_i}}$: indication on average effect of this parameter over entire parameter space; large value means important parameter and vice versa - $\sigma_{EE_{\theta_i}}$: information about linear behavior of parameter; large value means nonlinear parameter or parameter involved in interactions with other parameters - Do same for $\theta_2$ ## GSA: Morris screening - Normally 2 simulations needed per $EE$ - 1991: Morris introduced more efficient way - Number of simulations needed for $p$ parameters: - Naive: $(2 \times \text{\#EE})^p$ - Morris: $(p + 1) \times \text{\#EE}$ {height="70%"} ## GSA: Variance decomposition - Goal: find share of each model parameter in variance of model output - Used for models that are strongly nonlinear or nonmonotonous - Example of model with 3 parameters: $$ \sigma_y^2=\sigma_1^2+\sigma_2^2+\sigma_3^2+\sigma_{12}^2+\sigma_{13}^2+\sigma_{23}^2+\sigma_{123}^2 $$ - Normalisation (i.e., divide by $\sigma_y^2$) gives _sensitivity indices_ $$ 1=S_1+S_2+S_3+S_{12}+S_{13}+S_{23}+S_{123} $$ - Indicate which fraction of total variance is determined by certain parameter or parameter combination ## GSA: Variance decomposition - _Total sensitivity indices_ \begin{eqnarray*} S_{T1} & = & S_1 + S_{12} + S_{13} + S_{123} \\ S_{T2} & = & S_2 + S_{12} + S_{23} + S_{123} \\ S_{T3} & = & S_3 + S_{13} + S_{23} + S_{123} \end{eqnarray*} - Give total contribution of a certain parameter, including interaction effects - Watch out: some contributions are counted multiple times, hence sum of all total sensitivity indices is no longer 1 ## GSA: Variance decomposition - Two techniques: - FAST (Fourier Amplitude Sensitivity Test): uses Fourier decomposition of model output; can determine first order effects (total effects $\rightarrow$ extendedFAST); computationally intensive ((ten) thousands of simulations) - Sobol indices: uses multiple integrals, both first order and higher order effects; computationally expensive; less efficient than FAST