1. Experimental research methods
1.1. Planning an experiment in research
Under the «experiment» is to be understood a series of operations performed on the object of study to obtain information about its properties. The experiment, in which the researcher is able to change the terms of the research, called «active experiment». If the researcher is unable to change its terms, but only their records, this is «passive experiment».
The most important task of processing methods obtained during the experiment data is the task of constructing a mathematical model of the phenomenon, a process or an object. It can be used in the analysis of processes and the design of objects. You can get a good approximate mathematical model, when is used specifically active experiment. Another objective of treatment received during the experiment data optimization problem is, that this combination of the influence of independent variables in which the selected indicator optimality takes extreme value.
Experience – a single experimental part.
Plan of the experiment – a set of data to determine the number, terms and procedure of the experiment.
Planning experiment – the choice experiment plan that meets the specified requirements, a set of actions aimed at developing strategies experimentation (from receipt of prior information to obtain workable mathematical model and determine the optimal conditions). This deliberate experiment control, implemented in part–knowledge of the mechanism of the phenomenon.
During the measurements, the following data and formalizing results as a mathematical model, there are errors and loss of information contained in the original data. Application of experimental design to determine the error of the mathematical model and judge its adequacy. If the accuracy of the model is insufficient, the use of methods of experimental design lets you upgrade mathematical model of conducting additional experiments without loss of prior information and cost.
The purpose of planning the experiment – of these terms and conditions of the experiments in which is possible to obtain reliable and accurate information about the object with the least expenditure of labor, and present this information in a compact and convenient form of quantitative assessment of accuracy.
Let studied property of an object depends on Y n independent variables (Х1, Х2, ..., Хn) and we want to clarify the nature of this relationship – Y =
F (Х1Х2, ..., Хn), which we have only a general idea. The value of Y – called "review", and the dependence Y = F (Х1, Х2, ..., Хn) – "response function".
Feedback should be quantified. However, there may be signs of quality and Y. In this case, the possible use of rank approach. Example rank approach – score on the exam when one number is estimated complex combination obtained information about the student's knowledge.
Independent variables Х1, Х2, ..., Хn – other factors also need to have a quantitative assessment. If there are qualitative factors, then each of their level must be assigned a certain number. It is important to choose only factors as independent variables, ie only those that can be changed without affecting other factors. Factors to be mixed. Moves to a mathematical model to hold the preliminary analysis of the significance of factors (degree of influence on the function), their ranking and exclude minor factors.
The range of changes in factors set domain of Y. If we assume that each factor corresponds coordinate axis, then the resulting space is called the factor space. When n = 2 domain of Y is a rectangle, for n = 3 – cube, for n> 3 – hypercube. When choosing a range of factors to take into account changes compatible, ie control to any of these bands – that combination of factors would be realized in the experiments would not lead to the absurd. For each factor indicate threshold:
Ximin ≤ Xi ≤ Ximax, i = 1, ... n
Regression analysis of response function is designed to produce mathematical model in the form of the regression equation
Y = F (X1, X2, ... Xn; B1, B2, ... Bm) + e
where, В1, ..., Вm – coefficients; e – error.
The main planning methods applied at different stages of the study, use:
• planning of the experiment, the main value of which is separation from the totality of the factors essential factors to be further detailed study;
• design of experiments for analysis of variance, that is drawing up plans for objects with qualitative factors;
• planning regression experimentthat allows taking regression model (polynomial and others);
• planning extreme experiment in which the main task – experimental optimization research object;
• planning the study of dynamic processes, etc.
Currently, methods of planning of the experiment incorporated in specialized packages are widely available on the market of software products, such as: StatGrapfics, Statistica, SPSS, SYSTAT and others.
Presenting experiments
When using experimental design techniques need to find answers to 4 questions:
• What combination of factors and how these connections should be taken to determine the response function?
• How to find coefficients В0, В1, ..., Bm?
How to evaluate the accuracy of representation response function?
• How to use the presentation to find the optimal values of Y?
Geometric representation response function in the factor space X1, X2, ..., n is called surface response (Fig.1.1.1).
Figure 1.1.1: Surface review
If Influence on Y only one factor X1, the response function of the – pretty simple Y and graph Y = F (X) (Figure 1.1.2). With it you can customize the look of the mathematical expression response function. If we do not believe that the experiments are reproduced well, it is usually repeated several times experiments and dependence are given spread empirical data:
Figure 1.1.2: Construction of the response function of one variable on experience in this
If two factors, it is necessary to conduct experiments at different ratios of these factors. The resulting response function in 3–dimensional space (Fig. 1.1.1) can be analyzed by conducting a series of cross sections at fixed values of other factors (Fig.1.1.3). Graphic isolation sections can be approximated by a set of mathematical expressions.
Fig.1.1.3: The intersection of the surface response at fixed rating (s) and variables (b, c)
When three or more factors task becomes practically unresolved. If solutions are found, then use a set of expressions is difficult, and often not realistic.