非常推薦一篇文章：CCSI - Survey and evaluate Uncertainty Quantification Methodologies
本文是基于上文的學(xué)習(xí)整理筆記。侵刪。

0 不確定性分析

研究一個(gè)課題時(shí)，我們通常會(huì)有一個(gè)重要的假設(shè)前提，即不存在不確定因素，方案評(píng)價(jià)時(shí)能得到完全信息。但是，未來實(shí)際發(fā)生的情況與事先的估算、預(yù)測(cè)很可能有相當(dāng)大的出入。
為了提高經(jīng)濟(jì)評(píng)價(jià)的準(zhǔn)確度和可信度，盡量避免和減少投資決策的失誤，有必要對(duì)投資方案做不確定性分析，為投資決策提供客觀、科學(xué)的依據(jù)。
【此處有例子 -《技術(shù)經(jīng)濟(jì)學(xué)”教案》】

UQ方法論

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

關(guān)于不確定性（Uncertainty）

一個(gè)物理過程/模型。

分析物理過程中不確定性

三種不確定：輸入，參數(shù)，模型本身。

For example,
模型本身的局限性（無法完全準(zhǔn)確地描述過程）
模型簡化（同上）
參數(shù)不確定性（參數(shù)測(cè)量？）
數(shù)值計(jì)算（數(shù)值方法固有誤差-兩種-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 一類是指事物內(nèi)在的不確定性（Aleatory Uncertainty），另一類是指對(duì)事物的認(rèn)知不完整所導(dǎo)致的不確定性( 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的六個(gè)任務(wù)【計(jì)劃】

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.

主要會(huì)用到的方法論包括：

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
[確定性模型- 維基百科，自由的百科全書]
(https://zh.wikipedia.org/zh-hans/確定性模型)
確定性模型（Deterministic model）是指不包含任何隨機(jī)成份的模型。 對(duì)于確定性模型，只要設(shè)定了輸入和各個(gè)輸入之間的關(guān)系，其輸出也是確定的，而與實(shí)驗(yàn)次數(shù)無關(guān)。
3.1.2.1 Polynomial Chaos Methods（多項(xiàng)式逼近方法）
多項(xiàng)式逼近方法是近年來非常流行的計(jì)算方法,其基本思想就是將精確解在隨機(jī)參數(shù)空間進(jìn)行多項(xiàng)式展開。
例如,使用Laguerre多項(xiàng)式處理Gamma隨機(jī)參數(shù)輸入,使用Jacobi多項(xiàng)式處理Beta分布的隨機(jī)參數(shù)輸入等.
不確定性量化的高精度數(shù)值方法和理論
3.1.2.2 Quasi Monte Carlo Methods

UQ中的降維

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：輸入有很多隨機(jī)變量，但是主要影響輸出的 只有其中幾個(gè)。

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 - 有時(shí)可以自動(dòng)識(shí)別這些變量，從而固定它們的值，使之不會(huì)作為隨機(jī)變量影響模型
2. selecting variable subsets：identify and eliminate the variables with the least contribution to the output variability. （Kullback-Leibler divergence）

【剛剛發(fā)現(xiàn)自己拖到最后一秒寫完的Abstract是狗屎，希望教授不要認(rèn)真對(duì)待它。。。后悔莫及，以后再也不要不懂裝懂寫一些蠢東西了??】

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.