Full factorial experiment

The full factorial experiment (PFE) is a combination of several measurements that satisfy the following conditions:

The number of measurements is 2 ⁿ , where n is the number of factors;
Each factor takes only two values - upper and lower;
In the process of measurement, the upper and lower values of the factors are combined in all possible combinations.

The advantages of the full factorial experiment are

simplicity of solving the system of parameter estimation equations ;
statistical redundancy of the number of measurements, which reduces the influence of the errors of individual measurements on the evaluation of parameters.

Preliminaries

Approximation of a non-linear function of two variables by a plane

System Parameter Assessment

In practice, it is often necessary to evaluate the parameters of a certain system, that is, to construct its mathematical model and find the numerical values of the parameters of this model. The initial data for building a model are the results of an experiment , which is a combination of several measurements performed according to a certain plan. In the simplest case, the plan is a description of the measurement conditions, that is, the values of the input parameters (factors) during the measurement.

As an example of systems, the evaluation of parameters of which is relevant from a practical point of view, can serve various technological processes. To illustrate, consider the photolithography process.

Photolithography is a drawing of a picture on a surface by a photographic method. It consists of the following stages: surface preparation, applying a photosensitive emulsion ( photoresist ), drying, installing a stencil or plate with a negative pattern, exposure (exposure to light) by ultraviolet rays, etching (manifestation). Since the technological details of photolithography are not important in this context, we will consider the thickness of the photosensitive emulsion d (in microns) and the exposure time t (in seconds) as the main factors affecting the lithography process. The output parameter (response) of the process is its resolution R , that is, the maximum number of distinguishable lines that can be drawn on one millimeter of the surface. This value is determined by applying a special test image to the surface.

So, the technological process of photolithography is described by some function of the form

\ R=f(d,t).

{\ displaystyle \ R = f (d, t).}

Building a model of the technological process allows us to identify the behavior of the response of the system depending on changes in the factors and thereby find ways to optimize the technology. For this particular case - choose the emulsion thickness and exposure time that will provide the best image quality.

In the general case, the response of the system is described by some function. $n$ ${\ displaystyle n}$ variables

\ y=f(x_{1},x_{2},...,x_{n}).

{\ displaystyle \ y = f (x_ {1}, x_ {2}, ..., x_ {n}).}

The mathematical model of the system is obtained as a result of the approximation of this function by some other function, for example, linear

\ y=a_{0}+a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}

{\ displaystyle \ y = a_ {0} + a_ {1} x_ {1} + a_ {2} x_ {2} + ... + a_ {n} x_ {n}}

,

Where $a_{0},a_{1},a_{2}...a_{n}$ ${\ displaystyle a_ {0}, a_ {1}, a_ {2} ... a_ {n}}$ - the required parameters of the model.

The figure graphically shows the process of building a linear model of the photolithography process, where $x_{1}$ ${\ displaystyle x_ {1}}$ - emulsion film thickness, $x_{2}$ ${\ displaystyle x_ {2}}$ - exposure time, $y$ ${\ displaystyle y}$ - permission obtained in these conditions. Function $y=f(x_{1},x_{2})$ ${\ displaystyle y = f (x_ {1}, x_ {2})}$ nonlinear, but sufficiently close to the point $A_{0}$ ${\ displaystyle A_ {0}}$ it can be replaced by a tangent plane $y=a_{0}+a_{1}x_{1}+a_{2}x_{2}$ ${\ displaystyle y = a_ {0} + a_ {1} x_ {1} + a_ {2} x_ {2}}$ . In the area shown in the figure, the maximum model error is $\Delta y$ ${\ displaystyle \ Delta y}$ .

Knowing the coefficients of the model $a_{0},a_{1},a_{2}$ ${\ displaystyle a_ {0}, a_ {1}, a_ {2}}$ , it is possible with a certain accuracy to predict the value of the function (and hence the behavior of the system) in the vicinity of the point $A_{0}$ ${\ displaystyle A_ {0}}$ . In determining the values of the coefficients $a_{0},a_{1},a_{2}$ ${\ displaystyle a_ {0}, a_ {1}, a_ {2}}$ and is the goal of the experiment.

Experiment Matrix

The location of the experimental points in the two-dimensional factor space

Suppose the initial parameters of the technological process are: film thickness 55 μm, exposure time - 30 s, that is

\ x_{1C}=55;\quad x_{2C}=30.

{\ displaystyle \ x_ {1C} = 55; \ quad x_ {2C} = 30.}

Take the upper and lower values of both factors so that they are located symmetrically relative to the current value, for example

\ x_{1B}=60;\quad x_{2B}=35;

{\ displaystyle \ x_ {1B} = 60; \ quad x_ {2B} = 35;}

\ x_{1H}=50;\quad x_{2H}=25;

{\ displaystyle \ x_ {1H} = 50; \ quad x_ {2H} = 25;}

We will create a table in which the values of both factors are in all possible combinations and we will take measurements at these points (the response values are given conditionally):

{\begin{array}{|c|c||c|}\hline x_{1}&x_{2}&y\\\hline 50&25&140\\50&35&210\\60&25&170\\60&35&220\\\hline \end{array}}

{\ displaystyle {\ begin {array} {| c | c || c |} \ hline x_ {1} & x_ {2} & y \\\ hline 50 & 25 & 140 \\ 50 & 35 & 210 \\ 60 & 25 & 170 \\ 60 & 35 & 220 \\\ hline \ end {array}}}

Assuming that the linear process model is

\ y=a_{0}+a_{1}x_{1}+a_{2}x_{2}

{\ displaystyle \ y = a_ {0} + a_ {1} x_ {1} + a_ {2} x_ {2}}

,

Based on the results obtained, a system of four equations with two variables can be constructed. This system is shown below, as well as its abbreviated notation as a matrix. The matrix of this type is called the experiment matrix .

{\begin{cases}a_{0}+50a_{1}+25a_{2}=140\\a_{0}+50a_{1}+35a_{2}=210\\a_{0}+60a_{1}+25a_{2}=170\\a_{0}+60a_{1}+35a_{2}=220\end{cases}};\left({\begin{array}{c c c|c}x_{0}&x_{1}&x_{2}&y\\\hline 1&50&25&140\\1&50&35&210\\1&60&25&170\\1&60&35&220\end{array}}\right).

{\ displaystyle {\ begin {cases} a_ {0} + 50a_ {1} + 25a_ {2} = 140 \\ a_ {0} + 50a_ {1} + 35a_ {2} = 210 \\ a_ {0} + 60a_ {1} + 25a_ {2} = 170 \\ a_ {0} + 60a_ {1} + 35a_ {2} = 220 \ end {cases}}; \ left ({\ begin {array} {ccc | c} x_ {0} & x_ {1} & x_ {2} & y \\\ hline 1 & 50 & 25 & 140 \\ 1 & 50 & 35 & 210 \\ 1 & 60 & 25 & 170 \\ 1 & 60 & 35 & 220 \ end {array}} \ right).}

In the experiment matrix, the second and third columns are the values of the factors, the fourth column is the response values of the system, and the first column contains the units corresponding to the unit coefficients of the free member of the model $a_{0}$ ${\ displaystyle a_ {0}}$ . We will consider this column as some virtual factor. $x_{0}$ ${\ displaystyle x_ {0}}$ which always takes single values.

System Solution

Transition to normalized coordinates

To facilitate the solution of the system, we will carry out the normalization of factors. We assign the normalized value +1 to the upper values of the factors, the normalized value −1 to the lower values, the normalized value 0 to the mean value. In the general form, the normalization of the factor is expressed by the formula

{\tilde {x}}_{i}={\frac {2(x_{i}-x_{iC})}{x_{iB}-x_{iH}}}={\frac {2x_{i}-x_{iB}-x_{iH}}{x_{iB}-x_{iH}}}.

{\ displaystyle {\ tilde {x}} _ {i} = {\ frac {2 (x_ {i} -x_ {iC})} {x_ {iB} -x_ {iH}}} = {\ frac {2x_ {i} -x_ {iB} -x_ {iH}} {x_ {iB} -x_ {iH}}}.}

Given the normalization of factors, the system of equations and the matrix of the experiment will take the following form:

{\begin{cases}{\tilde {a}}_{0}-{\tilde {a}}_{1}-{\tilde {a}}_{2}=140\\{\tilde {a}}_{0}-{\tilde {a}}_{1}+{\tilde {a}}_{2}=210\\{\tilde {a}}_{0}+{\tilde {a}}_{1}-{\tilde {a}}_{2}=170\\{\tilde {a}}_{0}+{\tilde {a}}_{1}+{\tilde {a}}_{2}=220\end{cases}};\left({\begin{array}{c c c|c}{\tilde {x}}_{0}&{\tilde {x}}_{1}&{\tilde {x}}_{2}&y\\\hline +1&-1&-1&140\\+1&-1&+1&210\\+1&+1&-1&170\\+1&+1&+1&220\end{array}}\right).

{\ displaystyle {\ begin {cases} {\ tilde {a}} _ {0} - {\ tilde {a}} _ {1} - {\ tilde {a}} _ {2} = 140 \\ {\ tilde {a}} _ {0} - {\ tilde {a}} _ {1} + {\ tilde {a}} _ {2} = 210 \\ {\ tilde {a}} _ {0} + { \ tilde {a}} _ {1} - {\ tilde {a}} _ {2} = 170 \\ {\ tilde {a}} _ {0} + {\ tilde {a}} _ {1} + {\ tilde {a}} _ {2} = 220 \ end {cases}}; \ left ({\ begin {array} {ccc | c} {\ tilde {x}} _ {0} & {\ tilde { x}} _ {1} & {\ tilde {x}} _ {2} & y \\\ hline + 1 & -1 & -1 & 140 \\ + 1 & -1 & + 1 & 210 \\ + 1 & + 1 & -1 & 170 \\ + 1 & + 1 & + 1 & 220 \ end {array}} \ right).}

Since the sum of the terms in the second and third columns of the matrix is zero, the free term of the model can be found by adding all four equations:

4\ {\tilde {a}}_{0}=140+210+170+220=740;

{\ displaystyle 4 \ {\ tilde {a}} _ {0} = 140 + 210 + 170 + 220 = 740;}

{\tilde {a}}_{0}=185.

{\ displaystyle {\ tilde {a}} _ {0} = 185.}

To find any other coefficient of the model, you need to change the signs in the equations so that in the corresponding column were one unit, then add up all four equations:

4\ {\tilde {a}}_{1}=-140-210+170+220=40;

{\ displaystyle 4 \ {\ tilde {a}} _ {1} = - 140-210 + 170 + 220 = 40;}

{\tilde {a}}_{1}=10.

{\ displaystyle {\ tilde {a}} _ {1} = 10.}

4\ {\tilde {a}}_{2}=-140+210-170+220=120;

{\ displaystyle 4 \ {\ tilde {a}} _ {2} = - 140 + 210-170 + 220 = 120;}

{\tilde {a}}_{2}=30.

{\ displaystyle {\ tilde {a}} _ {2} = 30.}

Thus, the linear model of the technological process in the vicinity of the point (55, 30) has the form

\ y=185+10{\tilde {x}}_{1}+30{\tilde {x}}_{2}.

{\ displaystyle \ y = 185 + 10 {\ tilde {x}} _ {1} +30 {\ tilde {x}} _ {2}.}

In general, the system solution will look like

{\tilde {a}}_{k}={\frac {1}{2^{n}}}\sum _{i=1}^{2^{n}}y_{i}{\tilde {x}}_{ki}

{\ displaystyle {\ tilde {a}} _ {k} = {\ frac {1} {2 ^ {n}}} \ sum _ {i = 1} ^ {2 ^ {n}} y_ {i} { \ tilde {x}} _ {ki}}

Return to abnormal factors

The transition from normalized to non-normalized factors is carried out by inverse transformation

{x}_{i}={\tilde {x}}_{i}{\frac {x_{iB}-x_{iH}}{2}}+{x}_{iC}={\tilde {x}}_{i}{\frac {x_{iB}-x_{iH}}{2}}+{\frac {x_{iB}+x_{iH}}{2}}.

{\ displaystyle {x} _ {i} = {\ tilde {x}} _ {i} {\ frac {x_ {iB} -x_ {iH}} {2}} + {x} _ {iC} = { \ tilde {x}} _ {i} {\ frac {x_ {iB} -x_ {iH}} {2}} + {\ frac {x_ {iB} + x_ {iH}} {2}}.}

To find the model parameters for unrated coordinates, we substitute the expressions for the normalized coordinates into the model equation:

\ y={\tilde {a}}_{0}+{\tilde {a}}_{1}{\tilde {x}}_{1}+{\tilde {a}}_{2}{\tilde {x}}_{2}=

{\ displaystyle \ y = {\ tilde {a}} _ {0} + {\ tilde {a}} _ {1} {\ tilde {x}} _ {1} + {\ tilde {a}} _ { 2} {\ tilde {x}} _ {2} =}

\ ={\tilde {a}}_{0}+{\tilde {a}}_{1}{\frac {2(x_{1}-x_{1C})}{x_{1B}-x_{1H}}}+{\tilde {a}}_{2}{\frac {2(x_{2}-x_{2C})}{x_{2B}-x_{2H}}}=

{\ displaystyle \ = {\ tilde {a}} _ {0} + {\ tilde {a}} _ {1} {\ frac {2 (x_ {1} -x_ {1C})} {x_ {1B} -x_ {1H}}} + {\ tilde {a}} _ {2} {\ frac {2 (x_ {2} -x_ {2C})} {x_ {2B} -x_ {2H}}} =}

\ ={\tilde {a}}_{0}-{\tilde {a}}_{1}{\frac {2x_{1C}}{x_{1B}-x_{1H}}}-{\tilde {a}}_{2}{\frac {2x_{2C}}{x_{2B}-x_{2H}}}+{\frac {2{\tilde {a}}_{1}}{x_{1B}-x_{1H}}}x_{1}+{\frac {2{\tilde {a}}_{2}}{x_{2B}-x_{2H}}}x_{2}=

{\ displaystyle \ = {\ tilde {a}} _ {0} - {\ tilde {a}} _ {1} {\ frac {2x_ {1C}} {x_ {1B} -x_ {1H}}} - {\ tilde {a}} _ {2} {\ frac {2x_ {2C}} {x_ {2B} -x_ {2H}} + {\ frac {2 {\ tilde {a}} _ {1}} {x_ {1B} -x_ {1H}}} x_ {1} + {\ frac {2 {\ tilde {a}} _ {2}} {x_ {2B} -x_ {2H}}} x_ {2} =}

\ ={\tilde {a}}_{0}-{\tilde {a}}_{1}{\frac {x_{1B}+x_{1H}}{x_{1B}-x_{1H}}}-{\tilde {a}}_{2}{\frac {x_{2B}+x_{2H}}{x_{2B}-x_{2H}}}+{\frac {2{\tilde {a}}_{1}}{x_{1B}-x_{1H}}}x_{1}+{\frac {2{\tilde {a}}_{2}}{x_{2B}-x_{2H}}}x_{2}.

{\ displaystyle \ = {\ tilde {a}} _ {0} - {\ tilde {a}} _ {1} {\ frac {x_ {1B} + x_ {1H}} {x_ {1B} -x_ { 1H}}} - {\ tilde {a}} _ {2} {\ frac {x_ {2B} + x_ {2H}} {x_ {2B} -x_ {2H}} + {\ frac {2 {\ tilde {a}} _ {1}} {x_ {1B} -x_ {1H}}} x_ {1} + {\ frac {2 {\ tilde {a}} _ {2}} {x_ {2B} - x_ {2H}}} x_ {2}.}

Comparing the last expression with the expression for the linear model in non-normalized coordinates

\ y=a_{0}+a_{1}x_{1}+a_{2}x_{2}

{\ displaystyle \ y = a_ {0} + a_ {1} x_ {1} + a_ {2} x_ {2}}

,

get the expressions for the model parameters:

\ a_{0}={\tilde {a}}_{0}-{\tilde {a}}_{1}{\frac {x_{1B}+x_{1H}}{x_{1B}-x_{1H}}}-{\tilde {a}}_{2}{\frac {x_{2B}+x_{2H}}{x_{2B}-x_{2H}}};

{\ displaystyle \ a_ {0} = {\ tilde {a}} _ {0} - {\ tilde {a}} _ {1} {\ frac {x_ {1B} + x_ {1H}} {x_ {1B } -x_ {1H}}} - {\ tilde {a}} _ {2} {\ frac {x_ {2B} + x_ {2H}} {x_ {2B} -x_ {2H}}};}

\ a_{1}={\frac {2{\tilde {a}}_{1}}{x_{1B}-x_{1H}}};

{\ displaystyle \ a_ {1} = {\ frac {2 {\ tilde {a}} _ {1}} {x_ {1B} -x_ {1H}}};}

\ a_{2}={\frac {2{\tilde {a}}_{2}}{x_{2B}-x_{2H}}};

{\ displaystyle \ a_ {2} = {\ frac {2 {\ tilde {a}} _ {2}} {x_ {2B} -x_ {2H}}};}

In general

a_{0}={\tilde {a}}_{0}-\sum _{i=1}^{n}{\tilde {a}}_{i}{\frac {x_{iB}+x_{iH}}{x_{iB}-x_{iH}}};

{\ displaystyle a_ {0} = {\ tilde {a}} _ {0} - \ sum _ {i = 1} ^ {n} {\ tilde {a}} _ {i} {\ frac {x_ {iB } + x_ {iH}} {x_ {iB} -x_ {iH}}};}

$a_{k}={\frac {2{\tilde {a}}_{k}}{x_{kB}-x_{kH}}}.$ ${\ displaystyle a_ {k} = {\ frac {2 {\ tilde {a}} _ {k}} {x_ {kB} -x_ {kH}}}.}$

For the example above

\ a_{0}=185-10\cdot {\frac {60+50}{60-50}}-30\cdot {\frac {35+25}{35-25}}=-105;

{\ displaystyle \ a_ {0} = 185-10 \ cdot {\ frac {60 + 50} {60-50}} - 30 \ cdot {\ frac {35 + 25} {35-25}} = - 105; }

\ a_{1}={\frac {2\cdot 10}{60-50}}=2;

{\ displaystyle \ a_ {1} = {\ frac {2 \ cdot 10} {60-50}} = 2;}

\ a_{2}={\frac {2\cdot 30}{35-25}}=6.

{\ displaystyle \ a_ {2} = {\ frac {2 \ cdot 30} {35-25}} = 6.}

Finally, we get the model in natural coordinates:

\ y=-105+2x_{1}+6x_{2}

{\ displaystyle \ y = -105 + 2x_ {1} + 6x_ {2}}

.

Complete factor experiment

PFE matrix in general view

In general, the matrix of a full factorial experiment with n factors is

\left({\begin{array}{c c c c | c}{\tilde {x}}_{0}&{\tilde {x}}_{1}&...&{\tilde {x}}_{n}&y\\\hline +1&-1&...&-1&y_{1}\\+1&-1&...&+1&y_{2}\\...&...&...&...&...\\+1&+1&...&+1&y_{2^{n}}\end{array}}\right).

{\ displaystyle \ left ({\ begin {array} {cccc | c} {\ tilde {x}} _ {0} & {\ tilde {x}} _ {1} & ... & {\ tilde {x }} _ {n} & y \\\ hline + 1 & -1 & ... & - 1 & y_ {1} \\ + 1 & -1 & ... & + 1 & y_ {2} \\ ... & ... &. .. & ... & ... \\ + 1 & + 1 & ... & + 1 & y_ {2 ^ {n}} \ end {array}} \ right).}

Properties of the PFE matrix

The matrix PFE has the following properties:

The number of rows in the matrix is 2 ⁿ ;
The zero column of the matrix consists of units:

{\tilde {x}}_{0i}=1;

{\ displaystyle {\ tilde {x}} _ {0i} = 1;}

In columns 1 ... n there are all possible 2 ⁿ combinations of the values −1 and +1;
The last column contains the measurement results obtained for the values of the factors recorded in the corresponding rows in columns 1 ... n .
The sum of the elements of the zero column is always 2 ⁿ :

\sum _{i=1}^{2^{n}}{\tilde {x}}_{0i}=2^{n};

{\ displaystyle \ sum _ {i = 1} ^ {2 ^ {n}} {\ tilde {x}} _ {0i} = 2 ^ {n};}

The sum of the elements of any column, except the zero and the last one, is zero:

\sum _{i=1}^{2^{n}}{\tilde {x}}_{ki}=0\qquad (k=1...n);

{\ displaystyle \ sum _ {i = 1} ^ {2 ^ {n}} {\ tilde {x}} _ {ki} = 0 \ qquad (k = 1 ... n)}

The last two expressions can be combined into a single relation:

\sum _{i=1}^{2^{n}}{\tilde {x}}_{ki}=2^{n}\delta _{k0}\qquad (k=0...n),

{\ displaystyle \ sum _ {i = 1} ^ {2 ^ {n}} {\ tilde {x}} _ {ki} = 2 ^ {n} \ delta _ {k0} \ qquad (k = 0 .. .n),}

Where $\delta _{ij}$ ${\ displaystyle \ delta _ {ij}}$ - unit matrix $(i,j=0...n)$ ${\ displaystyle (i, j = 0 ... n)}$ ;

The sum of the squares of the elements of any (except the last) column is always 2 ⁿ :

\sum _{i=1}^{2^{n}}{\tilde {x}}_{ki}^{2}=2^{n}\qquad (k=0...n);

{\ displaystyle \ sum _ {i = 1} ^ {2 ^ {n}} {\ tilde {x}} _ {ki} ^ {2} = 2 ^ {n} \ qquad (k = 0 ... n );}

The sum of the products of the corresponding elements of any two columns (except the last one) is zero:

\sum _{i=1}^{2^{n}}{\tilde {x}}_{ki}{\tilde {x}}_{mi}=0\qquad (k,m=0...n;k\neq m);

{\ displaystyle \ sum _ {i = 1} ^ {2 ^ {n}} {\ tilde {x}} _ {ki} {\ tilde {x}} _ {mi} = 0 \ qquad (k, m = 0 ... n; k \ neq m);}

The last two expressions can be written as the orthogonality of the columns of the matrix:

\sum _{i=1}^{2^{n}}{\tilde {x}}_{ki}{\tilde {x}}_{mi}=2^{n}\delta _{km}\qquad (k,m=0...n);

{\ displaystyle \ sum _ {i = 1} ^ {2 ^ {n}} {\ tilde {x}} _ {ki} {\ tilde {x}} _ {mi} = 2 ^ {n} \ delta _ {km} \ qquad (k, m = 0 ... n);}

Calculation of linear model coefficients

The coefficients of the linear model in normalized coordinates are calculated by the formulas:

{\tilde {a}}_{k}={\frac {1}{2^{n}}}\sum _{i=1}^{2^{n}}y_{i}{\tilde {x}}_{ki}\qquad (k=0...n);

{\ displaystyle {\ tilde {a}} _ {k} = {\ frac {1} {2 ^ {n}}} \ sum _ {i = 1} ^ {2 ^ {n}} y_ {i} { \ tilde {x}} _ {ki} \ qquad (k = 0 ... n);}

The coefficients of the linear model in natural (non-normalized) coordinates are calculated by the formulas:

a_{0}={\tilde {a}}_{0}-\sum _{i=1}^{n}{\tilde {a}}_{i}{\frac {x_{iB}+x_{iH}}{x_{iB}-x_{iH}}};

{\ displaystyle a_ {0} = {\ tilde {a}} _ {0} - \ sum _ {i = 1} ^ {n} {\ tilde {a}} _ {i} {\ frac {x_ {iB } + x_ {iH}} {x_ {iB} -x_ {iH}}};}

a_{k}={\frac {2{\tilde {a}}_{k}}{x_{kB}-x_{kH}}}\qquad (k=1...n).

{\ displaystyle a_ {k} = {\ frac {2 {\ tilde {a}} _ {k}} {x_ {kB} -x_ {kH}}} \ qquad (k = 1 ... n).}

Transformation of natural factors into normalized and backward

{\tilde {x}}_{i}={\frac {2x_{i}-x_{iB}-x_{iH}}{x_{iB}-x_{iH}}};

{\ displaystyle {\ tilde {x}} _ {i} = {\ frac {2x_ {i} -x_ {iB} -x_ {iH}} {x_ {iB} -x_ {iH}}}}

{x}_{i}={\tilde {x}}_{i}{\frac {x_{iB}-x_{iH}}{2}}+{\frac {x_{iB}+x_{iH}}{2}}.

{\ displaystyle {x} _ {i} = {\ tilde {x}} _ {i} {\ frac {x_ {iB} -x_ {iH}} {2}} + {\ frac {x_ {iB} + x_ {iH}} {2}}.}

Sources

Model building and boundary testing of electronic devices: Method. instructions / Comp. A.N. Zhirabok, V.N. Lyakhov. - Vladivostok: Publishing house FESTU, 2006. - 32 p.

Adler Yu. P., Markova E. V., Granovskiy Yu. V. Planning an experiment when searching for optimal conditions. - M .: Science, 1976. - 279 p., Il.

Montgomery DK. Experiment planning and data analysis: Trans. from English - L .: Shipbuilding, 1980. - 384 p., Il.