计算机试验设计与数据分析

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"计算机试验设计与分析，涵盖了计算机科学与数据分析系列的主题，旨在融合计算机科学与统计学，探讨如何设计有效的计算机实验以及如何对实验结果进行深入的数据分析。该系列书籍由知名大学的教授担任编辑，旨在促进计算机科学与统计、数值及概率方法的交叉融合，通过出版参考著作、教材和手册推动这一领域的进步。目前已出版的书籍涉及贝叶斯人工智能、数据挖掘模式识别算法以及使用MATLAB进行探索性数据分析等主题。" 在计算机试验中，设计阶段是至关重要的，因为它决定了实验的有效性和可重复性。设计计算机试验时，通常需要考虑以下几个关键因素： 1. **实验目的**：明确试验的目标，是验证某个假设、评估算法性能还是优化系统配置。 2. **实验变量**：确定自变量（控制变量）和因变量（观测结果），并确保实验过程中只改变一个或有限个自变量以保持实验的可控性。 3. **实验设计**：选择合适的实验设计方法，如完全随机设计、随机化区组设计或拉丁方设计，以减小潜在的偏倚和误差。 4. **数据收集**：设置数据收集策略，包括采样频率、数据记录方式和数据质量控制。 5. **重复性**：通过重复实验以获取足够的样本量，以提高结果的统计显著性和可信度。数据分析阶段则包括以下步骤： 1. **数据清洗**：处理缺失值、异常值和不一致数据，确保后续分析的准确性。 2. **描述性统计**：计算平均值、中位数、标准差等统计量，了解数据的基本特征。 3. **可视化**：通过图表展示数据分布、趋势和关联，帮助发现潜在模式。 4. **假设检验**：运用t检验、ANOVA或卡方检验等统计方法，检验实验结果的显著性。 5. **模型构建**：可能涉及回归分析、机器学习算法等，建立预测模型或解释关系模型。 6. **效果评估**：使用性能指标如准确率、精确率、召回率等评估模型或算法的表现。 7. **解释与推断**：基于分析结果，得出结论，解释观察到的现象，并进行科学推断。 8. **结果验证**：通过额外实验或已有文献验证分析结果，确保其可靠性和通用性。这个系列的书籍将为读者提供一个全面理解计算机试验设计与数据分析的平台，无论是在软件开发、人工智能研究还是数据挖掘领域，都能从中受益。通过阅读这些书籍，读者可以学习到如何有效地设计和分析计算机实验，从而更好地利用统计和计算工具解决问题。

6 Design and Modeling for Computer Experiments

Factorial design: A factorial design is a set of level-combinations with

main purpose of estimating main eﬀects and some interactions of the factors.

A factorial design is called symmetric if all factors have the same number of

levels; otherwise, it is called asymmetric.

Full factorial design: A design where all the level-combinations of the

factors appear equally often is called a full factorial design or a full design.

Clearly, the number of runs in a full factorial design should be n = k



j=1

where q

is the number of levels of the factor j and k is the number replications

for all the level-combinations. When all the factors have the same number of

levels, q, n = kq

. The number of runs of a full factorial design increases ex-

ponentially with the number of factors. Therefore, we consider implementing

a subset of all the level-combinations that have a good representation of the

complete combinations.

Fractional factorial design: A fraction of a full factorial design (FFD) is

a subset of all level-combinations of the factors (Dey and Mukerjee (1999)).

A FFD can be expressed as an n × s matrix, where n is the number of runs

and s the number of factors, and the jth column of the matrix has q

levels

(entries). For example, the design in Example 1 below can be expressed as

D =

⎡

⎢

⎣

80 90 5 a

80 120 7 b

80 150 9 c

90 90 7 c

90 120 9 a

90 150 5 b

100 90 9 b

100 120 5 c

100 150 7 a

⎤

⎥

⎦

(1.1)

where there are four factors, each having three levels. Totally, there are

81 = 3

possible level-combinations, but the design D chooses only nine of

them. How to choose a good subset is the most important issue in FFD. A

carefully selected combination known as the orthogonal array is recommended

in the literature and has been widely used in practice.

Orthogonal array: An orthogonal array (OA) of strength t with n runs

and s factors, denoted by OA(n, s, q, r), is a FFD where any subdesign of n

runs and m (m ≤ r) factors is a full design. Strength two orthogonal arrays

are extensively used for planning experiments in various ﬁelds and are often

expressed as orthogonal design tables. The reader can refer to Hedayat, Sloane

and Stufken (1999) for the details.

Orthogonal design tables: An orthogonal design table, denoted by L

×···×q

), is an n ×s matrix with entries 1, 2, ··· ,q

at the j

column such

that:

1) each entry in each column appears equally often;

2) each entry-combination in any two columns appears equally often.

Introduction 7

When all the factors have the same number of levels, it is denoted by L

which is an orthogonal array OA(n, s, q, 2). The ﬁrst ﬁve columns of Table 1.1

give an orthogonal table L

), where the design serves for an experiment of

9 runs and 4 factors, each having three levels: ‘1,’ ‘2,’ and ‘3.’ In fact, we can

use other three symbols, such as ‘α,’ ‘β,’ and ‘γ’ to replace ‘1,’ ‘2,’ and ‘3,’ in

Example 1 Suppose that there are four factors in a chemical engineering ex-

periment and each chooses three levels as follows:

A (the temperature of the reaction): 80

C, 90

C, 100

B (the time length of the reaction): 90 min., 120 min., 150 min.;

C (the alkali percentage): 5%, 7%, 9%;

D (operator): a, b, c.

There are 3

= 81 level-combinations. If we use L

) to arrange a fractional

factorial design, the columns of L

) can be ﬁlled for factors A, B, C,and

D by any choices of 4! = 24 ones. Suppose that the four factors are put in

columns in a natural order. Then, replace the ‘abstract’ three levels of the

four columns by the factor’s levels and obtain a fractional factorial design on

the right-hand side of Table 1.1. There are nine level-combinations chosen.

The ﬁrst one is (80

C, 90 min., 5 %, a), and the last one is (100

C, 150 min.,

7%,a). Note that the factor D (operator) is not of interest, but we have to

consider its eﬀect as the operator may inﬂuence the output of the experiment.

Such a factor is called a noise factors.

TABLE 1.1

Orthogonal Design L

) and Related Design

No 1234 ABCD

1 1111 80

C 90 min. 5% a

1222 80

C 120 min. 7% b

1333 80

C 150 min. 9% c

2123 90

C 90 min. 7% c

2231 90

C 120 min. 9% a

2312 90

C 150 min. 5% b

3132100

C90min.9%b

3213100

C 120 min. 5% c

3321100

C 150 min. 7% a

Isomorphism: Exchanging columns and rows of an orthogonal design table

still gives an orthogonal design table. Two orthogonal designs are called

isomorphic if one can be obtained from the another by exchanging rows and

columns, and permuting levels of one or more columns of the design matrix. In

8 Design and Modeling for Computer Experiments

the traditional factorial design two isomorphic factorial designs are considered

to be equivalent.

ANOVA models: A factorial plan is based on a statistical model. For one

factor experiments with levels x

, ··· ,x

, the model can be expressed as

= μ + α

+ ε

= μ

+ ε

,j=1, ··· ,q,i=1, ··· ,n

, (1.2)

where μ is the overall mean of y, μ

is the true value of the response y at

,andε

is the random error in the ith replication at the jth level x

.All

the ε

’s are assumed to be independently identically distributed according to

N(0,σ

). Then the mean μ

can be decomposed into μ

= μ + α

, where α

is called the main eﬀect of y at x

and they satisfy α

+ ···+ α

=0. The

number of runs in this experiment is n = n

+ ···+ n

. The model (1.2) is

called an ANOVA model.

An ANOVA model for a two factor experiment, factor A and factor B,can

be expressed as

ijk

= μ + α

+ β

+(αβ)

+ ε

ijk

i =1, ··· ,p; j =1, ··· ,q; k =1, ··· ,K, (1.3)

where

μ = overall mean;

= main eﬀect of factor A at level α

;

= main eﬀect of factor B at level β

;

ijk

= random error in the kth trial at level-combination α

;

(αβ)

= interaction between A and B at level-combination α

under the

restrictions:



i=1

=0;



j=1

=0;



i=1

(αβ)



j=1

(αβ)

=0.

There are p−1 independent parameters α

s, q−1 β

s, and (p−1)(q−1) (αβ)

The total number of these parameters is pq − 1. For an s factor factorial

experiment with q

, ··· ,q

levels, we may consider interactions among three

factors, four factors, etc., and the total number of main eﬀects and interactions

becomes (



j=1

−1), which exponentially increases as s increases. Therefore,

the sparsity principle and hierarchical ordering principle are proposed.

Sparsity principle: The number of relatively important eﬀects and inter-

actions in a factorial design is small.

Hierarchical ordering principle: Lower order eﬀects are more likely to

be important than higher order eﬀects; main eﬀects are more likely to be

important than interactions; and eﬀects of the same order are equally likely

to be important.

10 Design and Modeling for Computer Experiments

Clearly, we wish Cov(

β) to be as small as possible in a certain sense. That

suggests maximizing M(D

) with respect to D

. However, M is an m × m

matrix, so we have to ﬁnd a scale criterion of maximization of M, denoted by

φ(M(D

)), and we can then ﬁnd a φ-optimal design that maximizes φ(M)

over the design space. Many authors have proposed many criteria, such as

• D-optimality: maximize the determinant of M;

• A-optimality: minimize the trace of M

−1

;

• E-optimality: minimize the largest eigenvalue of M

−1

The concepts of ‘determinant’, ‘trace’, and ‘eigenvalue’ of a matrix are re-

viewedinSectionA.1.Inmultivariatestatisticalanalysisthedeterminant

of the covariance matrix is called the generalized variance. D-optimality is

equivalent to minimizing the generalized variance and also is equivalent to

minimizing the volume of the conﬁdence ellipsoid of (β −

β)



M(β −

β) ≤ a

for any a

> 0. The A-optimality is equivalent to minimizing the sum of the

variances of

, ··· ,

. The reader can ﬁnd more optimalities and related

theory in Atkinson and Donev (1992), Pukelsheim (1993), etc.

When the true model is known, optimal designs are the best under the

given optimality. Optimal designs have many attractive properties, but they

lack robustness to model misspeciﬁcation. When the underlying model is

unknown, optimal designs may have a poor performance. If one can combine

eﬃciency of the optimal design and robustness of the uniform design, the new

design may have a good performance. Yue and Hickernell (1999) provide some

discussion on this direction.

1.3 Computer Experiments

1.3.1 Motivations

Much scientiﬁc research involves modeling complicated physical phenomena

using a mathematical model

y = f(x

, ··· ,x

)=f(x), x =(x

, ··· ,x

)



∈ T, (1.8)

where x consists of input variables, y is an output variable, the function f

may or may not have an analytic formula, and T is the input variable space.

Model (1.8) may be regarded as a solution of a set of equations, including

linear, nonlinear, ordinary, and/or partial diﬀerential equations, and it is of-

ten impossible to obtain an analytic solution for the equations. Engineers or

scientists use models to make decisions by implementing the model to pre-

dict behavior of systems under diﬀerent input variable settings. Hence, the

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