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[悬赏]响应面方法,分析与解释 (已翻译29%)

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英文原文:Response Surface Methodology, Analysis & Interpretation
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admin 发布于 2017-06-26 11:14:36 (共 7 段, 本文赏金: 25元)
参与翻译(2人): greenflute 廿九_ 默认 | 原文

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1. 简介

响应面模型(Response Surface Models,简称RSM),是简单线性回归的一个变种,由于使用了非线性关系的二阶效应,使之成为了在确定对某个现象(最大值、最小值、拐点)的特定响应时确定最佳变量组合的比较流行的技术之一。

RSM 在理解多重预测变量与一个或多个预测结果之间的关系时特别有用。

RSM 在那些过程和统计优化占重要角色的工业领域尤其受欢迎,例如化工,酶,以及制造业,生物实验室,纺织工业等等。传统的优化方法例如 COST (每次只改变单个东西)或者 OFAT (一次一个因素)由于没有考虑各个因素之间的交互,经常对优化计算形成误导。另外,实际情况下,对多个因素的改变会造成特定的响应,这就需要用到RSM了。

下面的例子中,采用了OFAT方式进行了15次实验(三个因素各5次)。从下面收集到的试验结果来看,是能够推导线性或者平方关系的,但实际上,在这种情形下是无法确定因素之间的互想影响关系的。

Figure 1: Source: ASQ Reliability Division (Slideshare)

RSM 使用了一系列的实验来在响应中寻找最优解,它的关注点在于寻找新的因数以便能较好地捕获响应现象中的因素互动。

greenflute
翻译于 2017-07-01 15:34:17
 

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2. Data

The below pilot was conducted with KPIs associated with the weaving process of a textile mill. The target of the below exercise is to get optimum settings of yarn quality and loom settings to maximize number of meters weaved (/loom yield).

The following are prerequisites for the RSM methodology:

  • Independent variables selected must be controllable in nature. Variables that cannot be controlled, though significant, must be kept constant before the experimentation process. This can be done based on a or prior-experience
  • Pre-processing such as Outlier treatment, missing value treatment and necessary transformations must be duly performed
  • Correlation analysis/step-wise VIF must be subsequently performed to drop the variables that exhibit high correlation with each other (correlation coeff  > 0.60 can be dropped)


3. 方法

假设满足了先决条件,则使用以下方法生成最佳响应面。

图2:方法一瞥

廿九_
翻译于 2018-02-02 10:05:16
 

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4. Results and Interpretation

4.1 Selection of screening design

The foremost step in the process of Optimum surface determination is the creation of a good screening design based on the data available. In case, historical data is available, a custom design can be created based on the historical data collected. The below screening design is a 2-level full factorial design (with all possible combinations of factors to estimate the main and interaction effects) to define the experimental space.

Number of factors – n

Number of levels - 2

Number of experimental runs required – 2<sup>n</sup>

The 2-levels of factors (based on High and Low values) are coded based on realistic range of the factors (Table 1). Corresponding response (meters weaved) must be recorded for each factor combination settings.

Table 1: Table with coded values of 2-level of factors

 

 A% 

 P 

 RPM 

 Wp/ph 

 WP Eff  Loss 

 WT   tstp 

 RunT 

KPICKS

Lea Strength

Imperfections

Hairiness

 Twist   Factor 

Elongation

 Low   (-1)

55

70

250

0

4.5

2

110

26

7

22.67

3

4

3.2

 High   (+1)

70

100

350

9.7

5.3

4

128.5

44.2

10

52.8

6

8

8

 

Figure 3: Factorial Design Matrix (Selection of Screening Design)

Note:

  1. The below design has no replication (running design more than once is easier for data analysis). Replication is essential to ensure the homogeneity of variance across the experimental space by excluding the factor combinations that result in inconsistency in response variables from the factor space.
  2. Inclusion of center points will improve the final factorial design matrix

The design must be checked for orthogonality by calculating the pair-wise estimate of correlation of factors (Figure 4). Additionally, many algorithms have been developed to calculate the minimum number of runs required to ensure orthogonality to minimize the cost of multiple experimental runs.

Figure 4: Correlation Plot of screening design



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4.2 Predictor Screening

The next step involves identifying the screening of predictors for their ability to predict an outcome. This has been performed using the Predictor Screening in JMP. The screening algorithm uses the bootstrap random forest partitioning technique to evaluate the contribution of predictors on the predictand. Additionally, the predictor screening can identify predictors that might be weak predictors when used alone but strong when used in combination with other predictors. This is especially useful in this methodology, as the interaction element is also taken into account.

An aggressive approach of predictor screening was used in this sample run. In practical situations, however, multiple iterations need to be performed to drop variables. Model performance measures must be checked with each individual iteration.

Below, only the top 4 contributing predictors of the first iteration was selected for the steps ahead. The results of the predictor screening can be verified by fitting a linear model based on the same data. A good predictor will have a p-value < 0.05 (based on the confidence level considered) to be statistically significant.

Figure 5: Result of Predictor Screening

Figure 6: Result of Linear model



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4.3 Fitting model for RSM

The next step requires us to fit the selected predictors to an RSM model.

  1. RSM will estimate the main effect, the interaction effect (cross-linear effect – XY and quadratic effect – X<sup>2</sup>)
  2. The significant effects contributing to the response must be selected based on the results of the ANOVA table
  3. We may notice aliasing or confounding of the main effects, that can be identified in the alias matrix and the design diagnostics.  

X -> Y + XYZ

If multiple-way interactions are noticed to be active and strong, the design can be concluded to be poor and tweaking will be required accordingly.

Prediction profiler is checked to maximize the desirability function by setting maximum meters dyed (loom yield). The procedure helps identify optimum settings for all the contributing variables.

  • The controls given in the prediction profiler lets us see how the prediction model changes as the individual settings of predictors are changed
  • Model sensitivity can be gauged in the profiler

The maximum yield (21.947 m) with corresponding settings of A%, RPM, Lea Strength & T.F can be observed in Figure 7.

Figure 7: Output of Prediction Profiler



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4.4 Visualization of Response Surfaces

Surface Plot and Contour plots help us visualize the response surface in a 2-D and 3-D plot.

The gridded Surface Plot is a representation of the surface plot in a 3-D Space. The plot determining optimum operating conditions reaching maximum from the best-fitted model with a map of contour lines, follows a direction of movement along the path of maximum response from a reference point.                                                                             

Figure 8: Surface Plot

The gridded Contour Plot generated the contours of a response variable in a rectangular coordinate system as shown in Figure 9. A contour plot shows a 3D surface in two dimensions with contours delineating changes in the 3-D space. The contour lines represent lines of equal response and can be visualized as response contours two factors at a time. The remaining predictors (contributing least to response) are fixed at a given point. In the above example, lea strength and TF have been fixed at 1 based on the results of the prediction profiler. 

Figure 9: Contour Profiler

Finally, the optimum coded value have to be converted into the uncoded form based on the same high-low transformation. It is recommended to test the model performance based on available validation data or by replicating the results in an actual set-up, if cost is not a restraint. 


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