非常推荐一篇文章：CCSI - Survey and evaluate Uncertainty Quantification Methodologies
本文是基于上文的学习整理笔记。侵删。

0 不确定性分析

研究一个课题时，我们通常会有一个重要的假设前提，即不存在不确定因素，方案评价时能得到完全信息。但是，未来实际发生的情况与事先的估算、预测很可能有相当大的出入。
为了提高经济评价的准确度和可信度，尽量避免和减少投资决策的失误，有必要对投资方案做不确定性分析，为投资决策提供客观、科学的依据。
【此处有例子 -《技术经济学”教案》】

UQ方法论

3.0 Uncertainty Quantification Methodologies

• 3.1 Forward Uncertainty Propagation
  • 3.1.1 Random Sampling Methods
  • 3.1.2 Deterministic sampling Methods
• 3.2 Sensitivity Analysis Methods
  • 3.2.1 Local Sensitivity Analysis Methods
  • 3.2.2 Global Sensitivity Analysis Methods
• 3.3 Response Surface Methods
• 3.4 Dimensional Reduction Methods
  • 3.4.1 Reduce the Number of Stochastic Variables
  • 3.4.2 Transform the Data to Lower Dimensionality

关于不确定性（Uncertainty）

一个物理过程/模型。

分析物理过程中不确定性

三种不确定：输入，参数，模型本身。

For example,
模型本身的局限性（无法完全准确地描述过程）
模型简化（同上）
参数不确定性（参数测量？）
数值计算（数值方法固有误差-两种-truncation error/rounded?）

the computational model may not include all of the correct reactions or physical processes.
Simplifications in the model, such as in the length scale of interactions of sorbent and CO2, may lead to uncertainties.
Individual parameters in the model may not be known precisely.
Numerical error may also become an issue in computational fluid dynamics simulations although it should not be an issue in the simpler equilibrium-based models.

Parameter-related uncertainties: aleatoric and epistemic uncertainties 一类是指事物内在的不确定性（Aleatory Uncertainty），另一类是指对事物的认知不完整所导致的不确定性( Epistemic Uncertainty)。

A preliminary list of uncertainties is provided in Table 1. These uncertainties include a combination of aleatoric and epistemic uncertainties, although uncertainties in model processes are typically considered as epistemic in nature.

2.2.1 Model Uncertainties
A preliminary list of uncertainties is provided in Table 1. These uncertainties include a combination of aleatoric and epistemic uncertainties, although uncertainties in model processes are typically considered as epistemic in nature.

2.2.2 Parameter Uncertainty
Many of the parameters in the process model have a range of plausible values rather than fixed values. A selected subset of model parameters and their input ranges are provided in Table 2. Typically, the parameters listed in Table 2 would be treated as representing aleatoric uncertainty.

分析UQ的六个任务【计划】

6.1.1 Compile CCSI process model characteristics
6.1.2 Compile relevant UQ methodologies and methods
6.1.3 Compile existing UQ tools6.1.4 Document results from each
6.1 subtasks
6.2 Demonstrate UQ methodology on MEA simulations.6.2.1 Define UQ objective and available experimental data for MEA
6.2.2 Identify parameters and probability distributions in MEA6.2.3 Define/implement UQ framework for MEA6.2.4 Perform UQ studies on MEA6.2.5 Release UQ framework (version 0) and complete report
This report documents progress on Task 6.1 through the end of September, 2011.

主要会用到的方法论包括：

3.0 Uncertainty Quantification Methodologies
A typical UQ study begins with defining a UQ process, which is a detailed plan of actions relevant for a given application. An example UQ process for large-scale multi-physics models such as those of the carbon capture simulation models may consist of the following steps:

• 1. Problem definition: what are the major UQ objectives; what model to use; what version; basic assumptions; quantities of interest; etc.
• 2. Model verification and testing: what is the impact of numerical errors?
• 3. Identify uncertain inputs: carry out initial selection of uncertain inputs, along with characterization of their prior uncertainty (typically ranges of uncertainty).
• 4. Identify observational/experimental data and integrate data into the model for refining the uncertain parameter distributions.
• 5. Uncertain parameter screening: identify the main drivers of output uncertainty for more detailed analysis when the parameter dimension is high.
• 6. Response surface analysis: build a surrogate surface to speed up uncertainty and quantitative sensitivity analysis.
• 7. Uncertainty and quantitative sensitivity analysis, risk analysis, full system calibration/validation, predictability assessment.
• 8. Documentation and review.
     A variety of different uncertainty methods are required to address the broad range of uncertainties identified in the CCSI process model. These methods can be classified in the following broad categories:
• Forward uncertainty propagation
• Sensitivity analysis (SA)
• Response surface tools for models with long simulation times
• Dimensional reduction tools for large numbers of uncertain variables The following subsections provide introductory statements concerning a variety of different methods used in uncertainty analysis. All of the techniques have positive and negative features, and no single technique is optimum for all situations. Therefore, these techniques are introduced as candidates for inclusion into an UQ toolkit. The choice of a particular method is deferred until a specific uncertainty analysis is designed.

3.1 Forward Uncertainty Propagation

Forward propagation techniques

Most of the uncertainty forward propagation techniques require assignment of a statistical distribution for each of the model parameters considered to be uncertain. Many techniques already exist for developing the statistical distributions.

Data-based methods include standard statistical techniques such as *maximum likelihood estimation, minimum distance estimation, method of moment estimation, and Bayesian inference. *

the difficulty of attaining the proper data to support assigning representative statistical distributions.
3.1.1 Random Sampling Methods
simplest sample selection technique

Independent parameters - statistical distribution,
New sample points are generated without taking into account the previously generated sample points.

3.1.2 Deterministic sampling Methods
[确定性模型- 维基百科，自由的百科全书]
(https://zh.wikipedia.org/zh-hans/确定性模型)
确定性模型（Deterministic model）是指不包含任何随机成份的模型。 对于确定性模型，只要设定了输入和各个输入之间的关系，其输出也是确定的，而与实验次数无关。
3.1.2.1 Polynomial Chaos Methods（多项式逼近方法）
多项式逼近方法是近年来非常流行的计算方法,其基本思想就是将精确解在随机参数空间进行多项式展开。
例如,使用Laguerre多项式处理Gamma随机参数输入,使用Jacobi多项式处理Beta分布的随机参数输入等.
不确定性量化的高精度数值方法和理论
3.1.2.2 Quasi Monte Carlo Methods

UQ中的降维

a large number of input stochastic variables
BUT only a few of the input variables dominate the variability in the model output.
模型中：输入有很多随机变量，但是主要影响输出的 只有其中几个。

Two basic approaches are to
Reduce the Number of Stochastic Variables
or to use

Feature extraction methods to transform the data in the high-dimensional space to a space of fewer dimensions.

3.4.1 Reduce the Number of Stochastic Variables

1. Simple sensitivity analysis - 有时可以自动识别这些变量，从而固定它们的值，使之不会作为随机变量影响模型
2. selecting variable subsets：identify and eliminate the variables with the least contribution to the output variability. （Kullback-Leibler divergence）

【刚刚发现自己拖到最后一秒写完的Abstract是狗屎，希望教授不要认真对待它。。。后悔莫及，以后再也不要不懂装懂写一些蠢东西了😂】

3.4.2 Transform the Data to Lower Dimensionality

1. linear【main】： as in principal component analysis, but many nonlinear dimensionality reduction techniques also exist. See (Fodor, 2002) for a review of such methods.
   There is information loss involved in the transformation, but in some cases the reduced model still explains most of the model variability.