On discrete Walsh functions in the - Semantic Scholar

PREPRINT: European Conference on Computer Aided System Theory Las Palmas, Gran Canaria, 2009

Walsh matrices in the Design of Industrial Experiments* Claudio Moraga (1, 2), Héctor Allende (3, 4) (1)

(2)

European Centre for Soft Computing, 33600 Mieres, Spain Dortmund University of Technology, 44221 Dortmund, Germany (3) Technical University Federico Santa María Valparaíso, Chile (4) University Adolfo Ibañez Viña del Mar , Chile

Abstract Discrete Walsh functions are well known in digital signal processing, telcommunications, and logic design. In this paper we show that they also appear “naturally” in matrices representing forms of interaction among different factors involved in the design of industrial experiments.

1.. Introduction J.L. Walsh introduced in 1923 a set of orthogonal functions [17] that carry his name, as the multiplicative closure of the Rademacher functions [13] disclosed a year earlier. The Walsh functions constitute a complete orthogonal system for the Hilbert space L2[0,1). Therefore every square-integrable function f : [0,1) → has a Walsh-Fourier Series representation [11]. Furthermore, Walsh functions have been identified as character functions of the dyadic group D, which is defined by the set of 0-1 sequences {(x1, x2, …) | xi  {0,1}} with the componentwise addition modulo 2 as group operation [7], [15]. Even though there has been substantial work on the continuous Walsh functions (see e.g. [16], [14]), the discrete Walsh functions have become very well known in the digital world for their applications in telecommunications [8 ], signal processing [1], [12] and logic design [6], [9]. Discrete Walsh functions have appeared in the literature under different orderings. Among the most important, besides the one originally given by Walsh [17], which is known as “sequency ordering”, there is the one introduced by Paley [10], which is directly related to the product of Rademacher functions and the Walsh-Hadamard ordering [9], which is characterized by a Kronecker product structure. Readers looking for more information may like to see [4] for an extended literature review. F. Pichler reported [12] that discrete Walsh functions in sequency order, appear “quite naturally” in the layout of telephone lines to minimize cross modulation effects. (In Germany, this was known as the “Kreuzungsplan von Pinkert”, named after a German telephone engineer around the year 1880.) The present paper shows that they also happen to appear in the design of experiments, according to the method introduced in [2]. An experiment can be defined broadly as an act of observation. This work however, uses a more restrictive definition, and states that an experiment is a series of trials or tests which produce quantifiable outcomes (response variable). In experimental design, the first step is to decide (based on the goals of the experiment) to what factors and alternatives the experimental units are to be exposed, and what project parameters are to be set. Then it should be examined whether any of the parameters cannot be kept at a constant value and account for any undesired variation. Finally, it should be chosen which response variables are to be measured and which should be the subject of complementary experiments. Experiments generally recommendable in the industrial area are called “factorial” [2], [3]. The goal of these experiments is to determine the effect of variations, interaction and extreme values in the parameters which *

Work leading to this paper was partially supported by the Foundation for the Advance of Soft Computing, Mieres, Asturias, (Spain), and by a Fondecyt 1070220 Research Grant (Chile)

characterize the industrial process under observation. According to [2], parameters are called “factors” and their types or values are called “levels”. For this reason it is usual to speak of “factorial design of experiments”. A factorial design uses every possible combination of the alternatives of all the factors. This experiment has the advantage of exploring the effects of all the possible combinations. It thus, discovers the effects of each factor and its interactions with the other ones. For instance, if an experiment is to be designed to determine the effect of certain parameters on the hardness of some building blocks, one of the factors could be the kind of cement used in its production and the levels could be “from volcanic ashes” or “from scoria of the iron and steel industry”. Notice that even though factors and levels may be nominal categories, their effect will be measured, giving a numerical value. This allows the development of an abstract model, which highlights the interaction among factors and their effect on the system. On the other side, one of the disadvantages of factorial experiments from a practical point of view, is the fact that the number of treatment combinations increases rapidly as the number of factor and/or levels increases. In this case, a fraction of the full factorial design can be used. Fractional factorial designs save time and money but provide less information than the full designs.

2. Design of industrial experiments [2], [3] The simplest experiments consider factors with (only) two levels, which may represent extreme values or selected categories. These experiments allow a fast (preliminar) diagnosis on the effects of the factors upon the performance of the system. Consider an experiment which involves two factors with two levels each. Let A and B denote the factors and a1, a2, b1, b2 denote the respective levels. If an experiment is conducted with factor A at level a1 and factor B at level b2, this will be denoted as an a1b2 run of the experiment. Let rij denote the response of the system under test to the factors A and B with levels ai and bj respectively. With this notation the following combined effects may be calculated: i) ii) iii) iv)

the global average effect the slope of A the slope of B the interaction of A and B

= = = =

¼ [r11 + r12 + r21 + r22] ½ [( r21 + r22) – (r11 + r12)] ½ [( r12 + r22) – (r11 + r21)] ½ [( r22 – r21) – (r12 – r11)]

Let X denote a factor with levels x2 and x1. It will be agreed that x2 is higher than x1, where in the case of levels with numerical values, this ordering is taken from ; meanwhile if the levels are nominal categories, the ordering will be defined arbitrarily (but used consistently). The slope of a factor X is a numerical value obtained as the difference between the average response of the system when this factor is at the level x2 and the average response of the system when this factor is at the level x1. The higher the value of the slope, the stronger is the effect of the corresponding parameter upon the system. The interaction between two factors is obtained as the difference between the average response when the levels of both factors are of the same type (both high or both low) and the average response when the levels of both factors are of opposite type. Combination of levels of factors a2b2 a1b2 a2b1 a1b1

Effect 1.

A.

B.

AB

   

 –  –

  – –

 – – 

Fig. 1: Reproduction of Table 2.3 from [2] with rows in reverse order

If the following notation is used: “1” for the global average effect; “A” for the slope of A; “B“ for the slope of B; and “AB” for the interaction of A and B, the structure of the combined effects may be summarized in a table (see Fig. 1), where the rows correspond to the pairs of levels being considered at the input factors, and the columns contain the signs to calculate the effects based on the corresponding responses.

Using the traditional approach of varying one factor at a time, four tests may be performed at the levels indicated in fig 1. The sign table of an experimental design may be directly built as follows: Assign the sign (+) to one of the levels of each factor and the sign (-) to the other. It does not matter which level is chosen for each sign. Build a table with one column for factors and other column per

combination of factors. The table rows are obtained as follows. For one factor column, every row corresponds to a given combination of (+) and (-) values for the respective level alternatives. The set of all the rows contain all the combination of the alternatives of all the factors. For the factor-combination columns, the entries correspond to the pointwise multiplication of the signs of the corresponding one-factor columns. For example, each row in column AB will be filled in by multiplying the corresponding signs of columns A and B. If in the table of Fig. 1 the entries with “+” are interpreted as “+1” and those with “–“, as “-1”, it is easy to recognize the matrix W4 representing the Walsh functions on 4 points, in Hadamard ordering. It becomes apparent the Kronecker structure showing that W4 = W2  W2. Furthermore, the labels of the columns are obtained as [1 B]  [1 A]. This is not a “simple coincidence”, since the structure carries on to experiments with 3 factors and two levels. Table 3.2 of [2] –(with reverse ordering of the rows)– corresponds to W8, which is W2  W2  W2. Example: For the production of wood chipboards, wood shavings are glued with special adhesives. Hardness is a relevant feature of a chipboard. It is assumed that the granularity of wood shavings and the type of adhesive being used affect the hardness of the final product. The following experiment is conducted, where the response is the hardness of the chipboard measured as the weight placed at the center of the board, needed to produce a 5 mm span. Factors

A: Type of adhesive B: Granularity shavings

Combined factor levels

Levels

of

wood

a2: a1: b2: b1:

X.45 W.75b Rough Fine

Response [Kg]

a2b2

r22 = 23

a1b2

r12 = 10

a2b1

r21 = 17

a1b1

r11 = 16

Therefore, the global average effect is ¼ [r11 + r12 + r21 + r22] = ¼ [23 + 10 + 17 + 16] = 16.5 The slope of A is given by ½ [( r21 + r22) – (r11 + r12)] = ½ [(17 + 23) – (16 + 10)] = 7. This value indicates that there is an effect of factor A. The fact that this value is positive indicates that the effect increases as the factor changes from level a1 to level a2. Similarly, the slope of B, expressed as ½ [( r12 + r22) – (r11 + r21)] gives ½ [(10 + 23) – (16 + 17)] = 0, meaning that this factor –(alone)– does not affect the hardness of the chipboard; however the interaction between A and B, calculated as ½ [( r22 – r21) – (r12 – r11)] gives ½ [(23 – 17) – (10 – 16)] = 6, meaning that B interacts with A. It is easy to see that if B is kept at b2, a change of A from a1 to a2 produces an increment of 13 Kg in the hardness, meanwhile if B is kept at b1, a change of A from a1 to a2 only improves that hardness by 1 Kg. The fact that the Walsh matrix appears in experiments with two factors and two levels is not a “simple coincidence”, since the structure carries on to experiments with 3 factors and two levels. Table 3.2 of [2] – (with reverse ordering of the rows)– shown in Figure 2, corresponds to W8, which is W2  W2  W2.

a2b2c2 a1b2c2 a2b1c2 a1b1c2 a2b2c1 a1b2c1 a2b1c1 a1b1c1

1

A

B

AB

C

AC

BC

ABC

+ + + + + + + +

+ – + – + – + –

+ + – – + + – –

+ – – + + – – +

+ + + + – – – –

+ – + – – + – +

+ + – – – – + +

+ – – + – + + –

Fig. 2: Table 3.2 of [2] with reverse ordering of the rows.

If a more detailled analysis is required, the number of levels may be increased to 3. For every factor, two types of comparisons are made: (i) only the difference of the responses to the extreme levels are considered and the response to the middle level is assigned a weight of 0, and (ii) the difference of twice the response to the middle level is calculated against the sum of the responses with respect to the extreme levels. For the general structure, let the factors be A and B, with levels a1, a2, a3 and b1, b2, b3, respectively. Order the columns of the Design Table after [1 B1 B2]  [1 A1 A2], where B1 denotes the column for the effect of parameter B following the comparison (i) and B 2 denotes the column for the effect of parameter B following the comparison (ii). Similarly for A1 and A2. Under these constraints, the corresponding Design Table is shown in Fig. 3. It may be seen that the entries of the table, considered as a matrix, do not represent a Chrestenson matrix [5], which may be considered as a ternary extension of the Walsh matrix. The Chrestenson matrix has complex-valued entries and this could not adequately be interpreted for the comparison of effects of factors and levels upon the responses. The matrix, however exibits a Kronecker structure. It is not difficult to recognize that it represents the Kronecker product of the matrix 1 1 1 1 0  2   1  1 1

with itself. Facto r levels

Effects 1

A1

A2

B1

A1B

A2B

1

1

B2

A1B

A2B

2

2

a3b3 a2b3 a1b3 a3b2 a2b2 a1b2 a3b1

1 1 1 1 1 1 1

1 0 -1 1 0 -1 1

1 -2 1 1 -2 1 1

1 1 1 0 0 0 -1

1 0 -1 0 0 0 -1

1 -2 1 0 0 0 -1

1 1 1 -2 -2 -2 1

1 0 -1 -2 0 2 1

1 -2 1 -2 4 -2 1

a2b1 a1b1

1 1

0 -1

-2 1

1 -1

0 1

2 -1

1 1

0 -1

-2 1

Fig. 3: Design Table for experiments with 2 factors and 3 levels each (The heavy lines frame the blocks of the Kronecker product) The Kronecker structure of the matrices corresponding to the Design Tables speaks for the scalability of the method. Increasing the number of factors –(with the same number of levels)– being considered, requires only the corresponding increased “Kronecker power” –(products with itself)– of the basic matrix. On the other hand, it becomes apparent that the Design Table will grow exponentially on the number of levels. This complexity growth may be alleviated by using fractional experiments, where a subset of factors is kept at some preselected levels meanwhile the others vary. This will lead to corresponding fractional Design Tables of tractable size. (See chapter 4 of [2]) The Kronecker structure of the matrices corresponding to the Design Tables also adds flexibility to the design of experiments, to consider factors with different number of levels. For instance, consider an experiment which involves a factor A with levels a1, a2, a3 and a factor B, with levels b1 and b2. Then the Design Table will be obtained based on the Kronecker product 1 1 1 1 1   1  1  1 0  2   1  1 1 

As shown in figure 4.

a3b2 a2b2 a1b2 a3b1 a2b1 a1b1

1 1 1 1 1 1 1

A1 1 0 -1 1 0 -1

A2 1 -2 1 1 -2 1

B 1 1 1 -1 -1 -1

A1B 1 0 -1 -1 0 1

A2B 1 -2 1 -1 2 -1

Fig. 4: Design Table for an experiment with a factor with two levels and a factor with three levels

3. Conclusions The basic idea for the design of industrial experiments following the method of [2] has been presented. In the case of experiments involving factors with two levels, it is shown that Walsh matrices in Hadamard ordering naturally appear as the structure of the Design Table. Other Kronecker structured matrices may be used to generate Design Tables for experiments involving factors with three levels or experiments where the factors have a different number of levels. The focus of design of experiments, has been the analysis of data from a standard design. It is usual to refer to any factorial design in which one o more level combinations are excluded from the study as a fractional factorial design. To recommend a fraction for a particular problem, is usually hard and it is often impossible to identify the “Best fraction”. Kronecker structured design matrices may provide some basic support, since their blocks may be associated to possible fractional factorial experiments.

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10. 11. 12.

13. 14.

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