# Types of plans for experiments. - Modeling of systems

## Types of experiment plans.

An experiment in which all possible combinations of factor levels are realized is called the full factorial experiment (PFE). If the chosen planning model includes only the linear terms of the polynomial and their products, then the experiment plan is used to estimate the coefficients of the model, with all factors k varying on two levels, that is, 4 = 2. Such plans are called type 2 plans, where N = 2 * is the number of all possible tests.

The initial stage of the experiment planning to obtain the coefficients of the linear model is based on the variation of the factors at two levels: the lower and the upper x 1sh - symmetrically located relative to the main level /= 1, k. Geometric interpretation is shown in Fig. 6.2, a. Since each factor takes only two values ​​ X1 П = Хю-Ах1 and х 1ш = Хм + Ахь then to standardize and simplify the recording of the conditions of each test and to process the sample experimental data, the scales along the factor axes are chosen so that the lower level corresponds to -1, the upper one corresponds to 4-1, and the main one to zero. This is easily achieved using a transformation of the form where x { is the encoded value of the/th factor; x, - the natural value of the factor; x 10 - zero level; A, = (; c 1c -x & lt; and )/2 is the factor variation interval.

Example 6.2. Suppose that the variable j, such as the temperature T y ° C, acts as the j-th factor, that is, X | -T, and the main level x * 100 ° C and the variation interval Ax, & gt; = 20 ° C are chosen. Then the coded values ​​of x, according to the levels, will accordingly be (80-100)/20 = - 1 for the lower, (120-100)/20 = +1 for the upper, (100-100)/20 = 0 for the main.

The location of the points for the Type 2 FPE is shown in Fig. 6.1, as well as in Fig. 6.2, 6. Writing out combinations of factor levels for each experimental point of the square, we obtain the plan O of the full factorial experiment of type 2:  Fig. 6.2. Geometric interpretation of the full factorial experiment of type 2: a - without scaling; 6 - when scaling on the axes Fig. 6.3. Geometric interpretation of the full factorial experiment of type 2

In this case, plans can be written in abbreviated form with the help of conditional alphabetic row designations. To do this, the ordinal number of the factor is put in correspondence with the lowercase letter of the Latin alphabet: x 1 - + a, x 2 -> gt and t. .

Then for each line of the plan, Latin letters are written only for the factors that are on the upper levels; The test with all the factors at the lower levels is denoted as (1). The plan entry in the alphabetical notation is shown in the last line.

Example 63. Geometric interpretation of the PFE 2 is shown in Fig. 6.3, a his plan is below: A full factorial experiment makes it possible to determine not only the regression coefficients corresponding to linear effects, but also the regression coefficients corresponding to all the interaction effects. The effect of interaction of two (or more) factors appears with the simultaneous variation of these factors, when the effect of each of them on the output depends on the level at which other factors are located.

To estimate the free term b 0 and to determine the interaction effects b 12 > 6 13 , ..., b 12b , ... the experiment B plan is expanded to the X scheduling matrix by adding the corresponding dummy variable & quot ;: of the single column x 0 and the columns of the products x x x 2 , x x x < sub> h ..., x x x g x g , ..., as shown, for example, for PPE type 2 in Table. 6.1.

Table 6.1 As it is seen from the considered plans of experiments of types 2 and 2, the number of tests in the PFE considerably exceeds the number of determined coefficients of the linear model of the experimental design, that is, the PFE has a large redundancy and therefore there arises the problem of reducing their number.

Consider the construction of plans for the so-called fractional factorial experiment. Let there be a simple complete factorial experiment of type 2. Using the planning matrix shown in Table 1, 6.2, we can calculate the coefficients and present the results in the form of an equation Table 6.2 If the process can be described by a linear model in the selected intervals of level variation, it is sufficient to determine three coefficients: 6 0 , b 1 and B 2 . Thus, there remains one degree of freedom that can be used to minimize the number of trials. With the linear approximation 6 12 -> 0 and the column vector x x x 2 (Table 6.2) can be used for the new factor x 3 . We put in Table. 6.2 this factor in the brackets over the interaction In this case, the separate estimates that took place in the PPE of type 2 will no longer be and the estimates are shifted as follows: When postulating a linear model, all pair interactions are taken into account. Thus, instead of eight tests in a full factorial experiment of type 2, only four must be carried out. The rule for the fractional factorial experiment is formulated as follows: to reduce the number of tests, the new factor is assigned the value of the column vector of the matrix belonging to the interaction, which can be neglected.

In an experiment of four trials, half the PPE type 2, the so-called "half-replica", is used to evaluate the effect of the three factors. If you equate x and - x 1 x 2 , semi-replica & quot ;. To denote fractional replicas in which the linear effects are equated to the effects of interaction, use the conventional designation 2 ~. For example, semi-replica from 2 is written as 2, and "quarter-replica" - 2.

Example 6.4. When constructing semi-replicas 2 x 3 can be equated to x x x 2 or -L ^ 2 - Two semi-replicas 2 "are shown in Table. 6.3. For the product of three columns of the left matrix, the relation +1 a of the right matrix

t. ie all the signs of the columns of products are the same and in the first case they are +1, and in the second case, 1.

In addition to symmetrical two-level plans of type 2 *, multilevel plans are also used in the planning of experiments, in which factors vary by 3, 4, ..., t levels and are designated as 3.4 , t are the planes. Multilevel asymmetric plans, in which factors vary at different levels, are constructed in various ways: by combining complete and fractional factor plans of type 2 *, by converting symmetrical plans into nonsymmetric plans, etc. The plans considered are called regression analysis plans for a multifactor experiment [10, 22].

Table 6.3 When the planning model is analyzed by methods of variance analysis, dispersion analysis plans are applied. If all possible sets of conditions are realized in the statement of the experiment, then complete classification of the variance analysis is indicated. If there is a reduction in the search of options, this is an incomplete classification of the variance analysis. Reduction of the search can be done randomly (without restriction on randomization) or in accordance with some rules (with restrictions on randomization). Most often as such plans use block plans and plans such as the Latin square [10, 18, 21].

Now we turn to the issues related directly to the planning of experiments with the machine model M and of the particular system 5.

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