Factor matrix. The main considerations are the procedures of factor analysis. Criteria for determining the number of factors. Prodovzhennya

Variance analysis of factors

Factor matrix

Zminna Factor A Factor B

As can be seen from the matrix, factor influence (or vagi) A and B for different survivors can significantly vary. Factor attraction A is possible T 1 indicates the tightness of the connection, which is characterized by a correlation coefficient equal to 0.83, then. good (thick) stowage. Factorne navantazhenya B for the same benefit gives r k= 0.3, which indicates a weak bond. As has been reported, factor B correlates very well with the survivable effects T2, T4 and T6.

Physicians that factor influences of both A and B are applied to survivable means that do not lie in their groups, with a density of connection no more than 0.4 (which is weak), you can take into account that the intercorrelation matrix is ​​presented above there are two independent factors, which in their own right mean six living benefits (case T 7).

Zminna T 7 could be seen as an independent factor, since the fragments with each other have no significant correlational influence (more than 0.4). But, in our opinion, there is no trace of work, since the factor “doors are not guilty of rust” does not have any immediate impact on the survivors. designs doors.

In this manner, when confirmed technical department When designing the design of car doors, certain factors will be included as auxiliary benefits, for which it is necessary to find constructive solutions in terms of engineering characteristics.

One principle that is important is the power of the correlation coefficient between the variables: the square shows which part of the dispersion (dispersion) of the signs of the two variables is significant, how much the overlap is significant huddle. So, for example, two variables T 1 and T 3 with a correlation of 0.8 overlap at a step of 0.64 (0.8 2), which means that 64% of the variance of this variable flaxen, tobto. run away. You can also say that mellowness These changes are the same as 64%.

It is clear that factor importance in the factor matrix is ​​also a correlation coefficient, and between factors and changeable ones (surviving factors).

Zminna Factor A Factor B

Therefore, the square of the factor intensity (dispersion) characterizes the level of strength (or overlap) of a given variable and of a given factor. What is significant is the degree of overlap (variance D) of both factors with the change (sustainability) of T1. And it is necessary to calculate the sum of squares of Terezov officials from the first change, then. 0.83 x 0.83 + 0.3 x 0.3 = 0.70. Thus, the potency of variable T1 with both factors becomes 70%. Tse dosit means not overkill.

At the same time, low sleepiness can be seen about those that change is dying out or appears clearly different from other changes included in the analysis. It is important to note that this change does not agree with factors for one of the reasons: either it is dying out differently (as, for example, change T 7), or there may be a great peace of extinction, or there are signs that create dispersion.

It should be noted that the significance of the skin factor is also determined by the amount of dispersion between the variables and factor influences. In order to calculate the actual importance of the factor, it is necessary to know in the skin column of the factor matrix the sum of the squares of the factor importance for the skin variable. Thus, for example, the dispersion of factor A (DA) is 2.42 (0.83 x 0.83 + 0.3 x 0.3 + 0.83 x 0.83 + 0.4 x 0.4 + 0, 8 x 0.8 + 0.35 x 0.35). The analysis of the significance of official B showed that D B = 2.64, then. the importance of official B vischa, lower official A.

Since it is possible to divide the value of the factor into the number of variables (in our application there are two), then the value is removed to show what part of the dispersion (or information) in the output correlation matrix is ​​the storage factor. For factor A? ~ 0.34 (34%), and for factor B -? = 0.38 (38%). Having summed up the results, we subtract 72%. Thus, the two factors combined account for more than 72% of the variance in the output matrix indicators. This means that as a result of factorization, part of the information in the output matrix was sacrificed to the random two-factor model. As a result, 28% of the information that could have been updated had the six-factor model been adopted was wasted.

Where has the peace been granted, the doctors, that all the changes have been looked at, what can be done about the design of the doors, the doctors? Most likely, the values ​​of the correlation coefficients of the changes that one official expects are much underestimated. Based on the results of the analysis, it was possible to turn to form other values ​​of correlation coefficients in the intercorrelation matrix (div. Table 2.2).

In practice, such a situation often arises when the number of independent factors is large, so that they all focus on the most important problems, either from a technical or economic point of view. There are a number of ways to reduce the number of factors. The most familiar of them is Pareto analysis. In this case, these factors are selected (in the world of varying significance), which contribute up to 80-85% of their total significance.

Factor analysis can be used in the implementation of the structured function method (QFD), which is widely used behind the cordon when forming technical specifications for a new virus.

As can be seen from the correlation matrix, the changes that lead to satisfaction with the work are more correlated with each other, and the changes that lead to satisfaction with the booth also correlate with each other. The correlations between the two types of variables (variables associated with satisfaction at work and those associated with satisfaction at work) are relatively small. It seems plausible that there are two apparently independent factors (two types of factors) that appear in the correlation matrix: one related to satisfaction at work, and the other to satisfaction with home life.

Factory navantazhenya

Another stage of factor analysis is the co-production of the officials and the way of the main components, and the way of the main officials. The result for our butt is a solution based on two factors. Let's take a look at the correlations between the two changeable factors (or “new” changeable ones). These correlations are called factor correlations.

Table 3. 16

Table of factor factors (principal component method)

As is obvious from Table 3.16, the first official correlates with changes, less than the other (shards of the value of the skin changes of the first official are greater, less than the other). This is obvious, since, as it was said above, the factors are seen sequentially and there is less random dispersion (div. section The importance and number of factors that can be seen, Stor. 61).

Methods of wrapping factors

The third stage of factor analysis is the wrapping of factor factors, which is the result of the first stage. Typical methods of wrapping and strategies varimax, quartimax, і equimax. The method of these methods is to remove a reasonable (interpreted) vantage matrix, so that factors are clearly indicated by high vantage values ​​(for example, more than 0.7) for some changes and low ones for others. Qiu's new model is sometimes called simple structure.

Idea of ​​wrapping method varimax bula described more (div. divided Principal component method, Stor. 60). This method can also be applied to the butt. As before, our job is to find a wrapper that maximizes the dispersion along the new axes; or, in other words, to remove the matrix of attention to the skin factor in such a way that the smells are eliminated as much as possible, and their simple interpretation is possible. Below is a table showing the rotation factor.

Table 3. 17

Table of factors navantazhenya (obertannya – varimax)

As can be seen from Table 3.17, the first factor in the values ​​of high demand for work is associated with satisfaction at work, and the other factor is associated with satisfaction with the booth. From which you can develop a concept in which satisfaction, acquired from the help of a nurse, is composed of two parts: satisfaction with the booth and the work. In such a manner, crushed classification the following changes. Based on the classification, the first factor can be called the factor of satisfaction with the work (or the factor of social values) and, obviously, the other – the factor of satisfaction with the booth (or the factor of special values).

Interpretation of the results of factor analysis

The final stage of factor analysis is the alternative interpretation of the factors contained in the result of the wrapping. Here, the researcher needs good theoretical preparation and knowledge of the experimental results already accumulated in this research area.

In fact, the interpretation of the officials lies in the seen significant factor factors (referent variables) in the skin of the officials. There are no exact criteria that allow differentiation of significant factor factors (influence). For example, in large samples (hundreds of people or more) significant values ​​of 0.3 or more are important. When the selection is changed to several dozen individuals, as significant, values ​​of the order of 0.4–0.5 are obtained.

The interpretation of officials always proceeds smoothly; In some cases, it is only very clear (for example, in different data that correspond to different types of scales), and in some cases, the authors are completely convinced of it, since the factor includes tests, which are important Let's go to bed.

Ideally (the subdivision of the variables does not differ from the normal one), the interpretation of the results of the factor analysis can be printed from the analysis of the correlation matrix, then proceed to the factor values ​​(seeing the referent values) none). The coming period is a compilation of the results of the correlation matrix and the identification of factors to determine the meaning of your values. And, it turns out, the last stage is the analysis of the negative properties of the place and the nature of those variables (sign) that are observed, which may have the highest correlation with this factor. The naming of factors is based on the determination of those referent variables that have the maximum value and may have the highest correlation with the factor. For example, if tests that evaluate the efficiency of obtaining loose material place high emphasis on this factor, then the remaining factor can be called the “mechanical memory” factor.

The offensive butt is grounded on the fortune-telling data, which is related to the development of satisfaction with life. Let’s assume that the nutrition is directed 100 times a day, so we can grow up. The tool contains 10 points designed to measure satisfaction at work, satisfaction with one’s hobbies, satisfaction with home life and overall satisfaction in other areas of life. The responses to the power supply were entered into the computer and scaled so that the average for all points was approximately 100.

The results are located in the Factor.sta data file. You can open this file using the additional option File - Open; Most likely, this data file is located in the /Examples/Datasets directory. Below is a breakdown of the changes to this file (to view the list, select All change specifications from the Data menu).

Meta analysis . The method of analysis is to determine the relationship between satisfaction in various areas of activity. However, it is important to understand the number of factors that go into different areas of activity and their significance.

Select analysis. Select Factor Analysis from the Analysis menu - Rich Exploratory Analysis to display the start panel of the Factor Analysis module. Click the Changes button on the start panel (div. below) and select 10 changes for your file.

Other options . To perform standard factor analysis, this dialog box contains everything you need. For taking off take a quick look Other commands available from the start panel, you can select the correlation matrix (in the Data File field) as an input file. In the PD field, you can reverse line-by-line or pairwise or middle substitution for missing data.

Indicate the method for viewing factors. Press the OK button to go to the next dialog box with the name Set the method for viewing factors. In this dialogue window, you can view descriptive statistics, select multiple regression analysis, select a method for viewing factors, select the maximum number of factors, minimum weight values, and other There are differences in the specifics of the methods for seeing factors. Now let's move on to the Descriptions tab.

Pereglyad descriptive statistics. Now press the button Revisit corr./middle/st. Whose window is in order to open the window Review of descriptive statistics.

Now you can look at the descriptive statistics graphically or at the additional table of results.

Calculation of the correlation matrix. Click the Correlations button next to the Additional tab to display a table of results with correlations.

All correlations in this table of results are positive, and all correlations are significant. For example, the variables Hobby_1 and Miscel_1 correlate at 0.90. Any correlations (for example, correlations between satisfaction at work and satisfaction at home) appear to be fairly small. It looks like this, because the matrix has a clear structure.

Vision method. Now press the Click button in the dialog window Review descriptive statistics to return to the dialog window Set the method for viewing factors. You can choose from a number of viewing methods in the Additional tab (Additional tab of the dialog box Set the method of viewing factors for the description of the skin method, as well as the Introductory review from the description of the method of the Main components and the method of the Main factors). In this application, the method of Head Components is used for washing, the Max field. the number of factors can be set to 10 (the maximum number of factors for your application) and the Min field. priv. value is 0 (minimum value of this command).

To continue the analysis, press the OK button.

Review of results. You can view the results of factor analysis in the dialogue window Factory analysis results. Now select the Dispersion explained tab.

Images of power . The assigned meaning of power is their color for the painter with a praised decision about how many traces deprive the factors (interpret) of the description from the initial examination. Now click on the Power Values ​​button to display the table with power values, total variance, accumulated moisture values, and accumulated values.

As is obvious from the table, the actual value for the first official is 6.118369; tobto. part of the dispersion, explained by the first official, is approximately 61.2%. It is important to note that these values ​​were found to be easily equalized here, since 10 variables are subject to analysis, and therefore the sum of all important values ​​is equal to 10. Another factor includes approximately 18% of the variance. Other officials seek revenge for a little more than 5%halal dispersions.Select the number of factors. The section Introductory review briefly describes the way in which weight values ​​can be extracted to improve nutrition, as many factors as possible are included in the model. Consistent with the Kaiser criterion (Kaiser, 1960), you must deprive the factors of their values ​​of great 1. From the above table it appears that the criterion leads to the selection of two factors.

Criterion for rocky wasp . Now click on the Stone Osipus Graph button to display the graph of the water values ​​based on the Quettel Osipus criterion (Cattell, 1966). The schedule, the feeds are lower, will be supplemented in sections to combine the current values ​​in order to create a criterion for the beginning. Cattell hardens, primed using the Monte Carlo method, to the point where the constant decline of its values ​​increases and after which the level of other power values ​​eliminates the random “noise”. On the graph below, this point can correspond to factor 2 or 3 (as shown by the arrows). So try resentment and wonder, which gives a more adequate picture.

Now let's look at the factors of vantage.

Factory navantazhenya . As described in the Introductory review section, factor influence can be interpreted as a correlation between factors and variables. So the stench is the worst important information, on which the interpretation of factors is based. One can immediately admire the (incorrect) factorial attraction of all ten officials. In the Navigation tab of the Factor Analysis Results dialog box, in the Factor Wrapping field, set the value without wrapping and click the Factor Vantage button to display the Vantage table.

It is clear that the number of factors was observed in such a manner that the current factors included less and less dispersion (div. section Introductory review). It is not surprising that the first factor is the most important one. It is significant that the signs of factor attractions may be more significant in order to show that changes with protracted attractions on that same factor interact with this factor in protracted order. However, you can multiply all the vantage points in the column by -1 and increase the signs. Otherwise, the results appear unchanged.

Wraparound of the factor solution. As described in the section Introductory review, the effective orientation of the factors in the factor space is sufficient, and even the wrapping of the factors creates correlations as well as other wrappings. Then, it seems natural to rotate the factors in such a way as to select the simplest factor structure for interpretation. In fact, the term is a simple structure of inventions and meanings by Thurstone (1947) as the head rank for describing minds, if the factors are designated as high in importance for the activities of change and low - for others, as well as when there are a number of great overriding interests tazhen, tobto. There are a lot of changes with the essentials and more than one factor. The most standard computational method for wrapping a simple structure is the varimax wrapping method, Kaiser's polymorphism (Kaiser, 1958). Other methods proposed by Harman (Harman, 1967) are the quartimax, biquartimax and equimax methods (div. Harman, 1967).

Vibir wrap . Let's first look at the number of factors that you want to use for interpretation. Previously, it was believed that the most plausible and attractive number of factors was two, but based on the criterion of osp it was likely that solutions with three factors were also valid. Click the Scroll button to open the dialog box Set the method for viewing factors, and change the field Maximum number of factors in the Swidky deposit from 10 to 3, then click the OK button to continue the analysis.

Now we can wrap it using the varimax method. In the Navigation tab of the dialog box Results of factor analysis, in the Wrapping of factors field, set Weekend Varimax.

Click the Factory Vanishing button to display the results of the Factory Vanishings in the table.

Reflection of the decision when three factors are combined. The table shows a general emphasis on the first factor for all changes, including those that lie before the day. Factor 2 may increase the importance of importance for all changes, including those related to satisfaction at work. Factor 3 has only one significance for the Home_1 variable. The fact that the third factor is highly important makes one wonder why we can’t get such a good result without the third factor?

A look at the decision when two factors are combined . Click the Select button again in the Factor Analysis Results dialog box to return to the Specify the method for viewing factors dialog box. Change the Maximum number of factors field in the Shvidky tab from 3 to 2 and click OK to go to the Factor analysis results dialog box. In the Navigation tab, in the Factor rotation field, set the Varimax output value and click the Vantage Factor button.

Factor 1, as can be seen from the table, is of greatest importance to those who are satisfied with their work. The least important thing is for those who are happy with the bed. Other considerations acquire intermediate values. Factor 2 may be of greatest importance for changes associated with satisfaction in the workplace, lower importance for satisfaction at work, and average importance for other changes.

Interpretation of the solution for two-factor wrapping . How can this model be interpreted? Everything looks like this, as two factors are most clearly identified as the factor of satisfaction with work (factor 1) and the factor of satisfaction with home life (factor 2). Satisfaction with one’s hobbies and various other aspects of life appear to be due to both factors. This model conveys to the general sense that satisfaction with work and home life, along with this choice, can be independent of one another, but resentment can contribute to satisfaction with other aspects of life.

Diagram of a decision based on the combination of two factors . To draw diagrams for the dispersion of two factors, click the button 2M graph of navigation in the Navigation tab of the dialog box Results of factor analysis. The diagram shown below simply shows two options for skin changes. Please note that the distribution diagram well illustrates two independent factors and 4 variables (Hobby_1, Hobby_2, Miscel_1, Miscel_2) with overlapping values.

Now let's see how well the quariation matrix can be constructed, which guards against two-factor solutions.

The redundant correlation matrix has been created. Click on the button Created excess correlations in the Variance explained tab to view the two tables with the created correlation matrix and the matrix of excess correlations (minus the created correlations, I caution to be).

Entries in the table of Zalishkov correlations can be interpreted as a “sum” of correlations, for which two factors cannot be distinguished. Naturally, the diagonal elements of the matrix are subject to standard modification, since the corresponding factors cannot correspond to the square root of one minus the corresponding strengths for two factors (It is clear that the variability of the variable is dispersion, which can be explained by the selected number of factors). If you carefully look at this matrix, you can see that there are actually no excess correlations here, large 0.1 or less than -0.1 (in fact, only a small number of them are close to this value). It is worth noting that the first two factors include approximately 79% of the total variance (div. accumulation % of the total values ​​in the results table).

"The secret" of the distant butt . The butt, which you have thoroughly trained, actually gives the highest level of two-factor design, close to ideal. This represents a larger part of the variance, which has a reasonable interpretation and produces a correlation matrix with moderate correlations (excessive correlations). In fact, real data rarely allows such a simple solution to be removed, and in fact, this anonymous data was removed using an additional random number generator with a normal division, available in the system. In a special manner, two orthogonal (independent) factors were “introduced,” which generated correlations between the variables. This method of factor analysis consists of two factors, such as the factor of satisfaction with work and the factor of satisfaction with home life. In this way, if the object (and not piecemeal, like the butt, data) combined two factors, then you, having seen them, could learn more about the hidden or latent structure of the object.

Other results . First of all, let us make some short comments until you get other results.

Partnerships . To remove the complexity of the decision, click the Strength button in the tab. The dispersion of the dialog box is explained. Results of factor analysis. It is clear that the consistency of a variable is the portion of the dispersion that can be created for a given number of factors. The wrapping of the factor space does not affect the amount of strength. Very low strengths for one or two of the variables (of which there are many in the analysis) may indicate that the price of the changes is not a good explanation of the model.

Coefficients value. Factor coefficients can be used to calculate the value of skin warning factors. The coefficients themselves are of little interest, depending on the factorial significance of the coefficient during further analysis. To display coefficients, click the Factor Value Coefficients button in the Values ​​tab of the Factor Analysis Results dialog box.

the significance of the factors. The factor values ​​can be viewed as precise values ​​for the skin sample of the respondent (that is, for the skin test of the output data table). The Factor Values ​​button in the Values ​​dialog box The results of factor analysis allow us to calculate factor values. These values ​​can be saved for later pressing the Save Values ​​button.

The last comment. Factor analysis is a complicated procedure. Kozhen, who is constantly vikorist factor analysis With a lot (for example, 50 or more) of changes, there may be many applications of “pathological behavior”, such as: negative influences and uninterpreted decisions, special matrices, etc. If you rely on factor analysis to identify significant factors with a large number of variables, you should carefully consider report book(For example, Harman's book (Harman, 1968)). Thus, there are many critical decisions in factor analysis due to their subjective nature (the number of factors, the method of wrapping, interpretation of significance), be prepared before you need clear evidence, first of all You feel like you are singing to someone. The module Factor analysis of business divisions is specially designed to make it easy for the user to interact interactively between in different numbers factors, wrappers, etc., to try and compare different solutions.

This butt is taken from the advanced PPP system STATISTICA by StatSoft

Factor analysis is the core of mathematical statistics. Yogo Tsili, yak I meted mathematical statistics, Polegs at the Rosrobtz models, to understand the method, to allow the analizuvati masivi of the sprinklers of the Dannya, the Fizynikovo Forma Forma.

One of the most typical forms for the presentation of experimental data is a matrix, the columns of which represent various parameters, authorities, tests, etc., and the rows represent specific objects, phenomena, modes, which are described by a set of specific parameter values. In fact, the dimensions of the matrix appear to be quite large: for example, the number of rows of this matrix can range from several tens to several hundred thousand (for example, with sociological restrictions), and the number of columns can range from one - two hundred. A straightforward, “visual” analysis of a matrix of this size is impossible, which is why in mathematical statistics there are a lot of approaches and methods used to “compress” the output information contained in the matrix until it is accessible for viewing according to size, draw from the output information the greatest possible ", throwing up "in a different row", "in a row".

When analyzing the data presented in the matrix form, two types of tasks emerge. The tasks of the first type may involve the creation of a “short description” of a subset of objects, while the tasks of the other type may reveal the interaction between the parameters.

It is important to note that the main incentive for the appearance of meanings in the task lies not only and not so much in briefly encoding a large array of numbers, but in a much more principled setting, which is of a methodological nature: if it is possible to briefly describe a large array of numbers, then in fact, what is revealed is the melodious objective regularity that makes a short description possible; Even the search for objective patterns is the main method for which, as a rule, data are collected.

The known approaches and methods for processing the data matrix vary depending on what type of data processing is expected to be carried out, and thus, until the matrix of what size is stagnant.

As for the problem of a short description of the connections between the parameters with an average number of these parameters, then in this case a similar correlation matrix contains several tens or hundreds of numbers and itself cannot serve as a short what is the description of the essential connections between the parameters, and is guilty of this by succumbing to further processing.

Factor analysis is defined as a set of models and methods used to “compress” the information that fits into the correlation matrix. The basis of various factor analysis models is the following hypothesis: guarded or simulated parameters and rather than indirect characteristics of the investigated object or phenomenon, there are actually internal (directly taken) parameters or power, the number of which There are few signs that indicate the values ​​of the guarded parameters. These internal parameters are usually called factors. The task of factor analysis is to present parameters as linear combinations of officials and, possibly, some additional, “not real” quantities - “recode”. What is surprising is the fact that, although the factors themselves are not known, such a distribution can be rejected and, moreover, such officials can be taken into account. For a skin object, the value of the skin factor can be indicated.

Factor analysis, regardless of the methods, begins with the processing of the intercorrelation table, based on independent tests, resulting in a correlation matrix, and ends with the derivation of the factor matrix, then. table that shows the importance of skin test factors. Table 1 is a hypothetical factor matrix that includes only two factors.

Factors are considered in the top row of the table from the most significant to the least significant, and their skin tests for 10 tests are given in similar items.

Table 1

Hypothetical factor matrix

Coordinate axes. It is customary to represent factors geometrically as coordinate axes, so that some skins can be used to test images as points. Rice. 1 explains this procedure. On this graph of the skin from 10 tests carried out in Table 1, you can see the points of two factors that correspond to axes I and II. Thus, test 1 represents a point with coordinates 0.74 axis I and 0.54 axis II. The points that represent the results of 9 tests were generated in a similar way, with the same values ​​as in the table. 1.

Please note that the positions of the coordinate axes are not fixed by the data. The output correlation table shows the position of the tests (the point in Fig. 1) shodo one of one. The same points can be placed on a plane depending on the position of the coordinate axes. For these reasons, when conducting a factor analysis, consider wrapping the axes until you find the most pleasant and interpretable image.

Rice. 1. Hypothetical factor analysis, which shows the differences between two group skin factors in 10 tests.

In Fig. 1 take off after wrapping axis I" and II" shown with dotted lines. This wrapping is consistent with the criteria established by Thurstone. positive variety and simple structure. The first transfers the wrapping of the axes to the stage, in which all significant negative aspects are turned off. Most psychologists believe that negative factor importance is logically inconsistent with aptitude tests, which means that the more an individual scores on a specific factor, the lower the result for the specific factor. him test. The criterion of simple structure, in essence, means that the skin test is due to the influence of the fewest factors.

Officials provide a summary of both criteria that can be easily and unambiguously interpreted. Since a test has a high impact on one factor and does not have a significant impact on other factors, we can find out about the nature of that factor that replaced this test. However, since the test has average or low scores on six factors, it tells us little about the nature of any of them.

In Fig. 1 it is clearly visible that after the wrapping of the coordinate axes, all verbal tests (1-5) grow even close to the “I” axis, and numerical tests (6-10) are closely grouped around the II axis. nuth axes, shown in Table 2. The importance factors in Table 2 do not have negative values, although they are of unimportantly small values, which is clearly visible before the selection. . Числові тести, навпаки, мають високі навантаження за фактором ІІ "і зневажливо низькі - за фактором І". Таким чином, обертання координатних осей суттєво спростило ідентифікацію та називання обох факторів, а також опис факторного складу кожного тесту. На практиці число факторів часто виявляється There are more than two, which, obviously, complicates its geometrical manifestations and statistical analysis, but does not change the essence of the procedure in question.

Table 2

Factor matrix after wrapping

The activities of the descendants are based on a theoretical model as the principle of axes wrapping. In addition, we must take into account the immutability and confirmation of the same factors in independent or updated studies.

Interpretation of officials. Having taken care of the procedure for wrapping the factor solution (or, to put it simply, a factor matrix), we can proceed to the interpretation and naming of factors. This stage of work will increasingly require psychological intuition, rather than statistical training. In order to understand the nature of a particular factor, we are missing nothing but to identify the tests that have a high importance for this factor, and try to reveal the psychological processes that are hidden for them. The more tests with high significance for this factor are revealed, the easier it is to reveal its nature. 3 table 2, for example, it is immediately clear that factor I is “verbal”, and factor II is numerical. Hover over the table. 2 factorial implications reflect the correlation of the skin test with the factor.

FACTOR ANALYSIS

The idea of ​​factor analysis

When investigating complex objects, phenomena, systems of factors that signify the power of these objects, it is often impossible to control them completely, and sometimes it is unknown to indicate their number and place. However, other quantities may be available for vibrating, either as they or otherwise lie within the factors that are important to us. Moreover, if the influx of an unknown factor that affects us manifests itself in several extinct signs and powers of the object, these signs can reveal a close connection with each other and a hidden number of factors can be beneficial then fewer, fewer extinct species.

To identify the factors that are indicated by the dimmed signs of objects, factor analysis methods are used.

As an example of factor analysis, one can indicate the importance of the specificity of the basis of psychological tests. Powerful particularities are not subject to direct influence. They can be judged solely by the behavior of the person and the nature of their diet. To explain the results of the research, they are subjected to factor analysis, which allows us to identify those specific powers that influence the behavior of the individual.
Basically different methods Factor analysis leads to the following hypothesis: rather than mediating the characteristics of the object under investigation, internal (attached, latent, unguarded) parameters and power emerge , the number of which is small and which indicate the values ​​of the guarded parameters. These internal parameters are usually called factors.

Meta-factor analysis - concentrate the output information, expressing the large number of analyzed signs through a smaller number of large internal characteristics of the phenomenon, which, prote, is not susceptible to the middle world

It has been established that monitoring and monitoring the level of occult factors makes it possible to identify the advanced stages of the object at the early stages of the development of the defect. Factor analysis makes it possible to determine the stability of correlations between other parameters. The very correlations between parameters, as well as between parameters and underlying factors, provide basic diagnostic information about processes. The use of the tools of the Statistica package when performing factor analysis includes the need for additional computational tools and to perform the analysis in a clear and understandable way for the researcher.

The results of factor analysis will be successful if it is possible to interpret the findings of factors based on the meaning of the indicators that characterize those factors. This stage of work is quite convincing; It requires a clear indication of the place of the indicators obtained before the analysis and the basis of what the officials saw. Therefore, during the preliminary selection of indicators for factor analysis, the traces are replaced by them, and not carried out until the largest number of them is included in the analysis.

The essence of factor analysis

Let us introduce some of the main principles of factor analysis. Say hello to the matrix X The basis of the variable parameters of the object is the covariate (correlation) matrix C, de R- Number of parameters, n- Be careful about the number. The path of linear transformation X=QY+U you can change the size of the output factor space X to the level Y, when R"<<R. This indicates the transformation of the point, which characterizes the state of the object in j-peaceful space, into a new space of worlds with a smaller size R Obviously, the geometric proximity of the two or impersonal points in the new factorial space means the stability of the object.

Matrix Y identify factors that are not taken into account, which in essence are hyperparameters that characterize the greatest hidden power of the analyzed object. Foreign officials most often choose statistically independent ones, which facilitates their physical interpretation. Vector sign to beware X It is possible to sense the consequences of changing these hyperparameters.

Matrix U consists of excess factors, which include mainly the elimination of the sign x(i). Rectangular matrix Q place factorial emphasis, which indicates a linear connection between signs and hyperparameters.
The factors of importance are the significance of the coefficients for the correlation of skin signs and skin symptoms. The closer the connections between the given signs and the analyzed factor, the greater the significance of the factor importance. A positive sign of factor avantagement indicates a direct (and a negative sign – a reverse) link between the given sign and the factor.

Thus, the data on factor involvement allow us to formulate principles about the set of output signs that reflect another factor, and about the visibility of the surrounding signs in the structure of the skin factor.

The factor analysis model is similar to the multivariate regression and variance analysis model. The fundamental importance of the factor analysis model is that the vector Y is a factor that is not taken into account, and in regression analysis it is a parameter registration. On the right side of the line (8.1) there are unknown unknowns: the matrix of factor factors Q and the matrix of the value of the factor factors Y.

To find the matrix of factor factors, the vicoristic level is QQ t = S–V, where Q t is the transposed matrix Q, V is the matrix of approaches of excess factors U, then. . The equation is determined by iteration when a certain zero proximity of the covariance matrix V(0) is specified. After finding the matrix of factor influences Q, the underlying factors (hyperparameters) are calculated for the equals
Y=(Q t V -1)Q -1 Q t V -1 X

The statistical analysis package Statistica allows you to interactively calculate the matrix of factor influences, as well as the values ​​of several further specified main factors, most often the first two main components of the output parameter matrix.

Factor analysis in the Statistica system

Let's take a look at the history of factor analysis from the analysis of the results of a questionnaire survey of business practitioners. It is necessary to identify the main officials who signify the joy of working life.

At the first stage, it is necessary to select variables for factor analysis. Vicoristic and correlation analysis, the investigator is able to identify the interconnections of the following signs, which, in its turn, makes it possible to see the new and excessive accumulation of signs by combining strongly corrosive signs k.

If factor analysis is carried out on all variables, the results may not be entirely objective, since some variables are determined by other data and cannot be regulated by the organization’s experts.

To understand which trace indicators to turn on, we will look at the correlation coefficient matrix in Statistica: Statistics/ Basic Statistics/ Correlation Matrices/ Ok. At the start window of the Product-Moment and Partial Correlations procedure (Fig. 4.3), the One variable list button is displayed to expand the square matrix. Select all changes (Select all), Ok, Summary. Let's remove the correlation matrix.

If the correlation coefficient changes no more than 0.7 to 1, this means a strong correlation of indicators. In this case, you can turn off one change due to the strong correlation. And finally, since the correlation coefficient is small, you can turn off the change through those so that nothing will be added to the final sum. In our sample, there is no strong correlation between any variables, and factor analysis will be carried out for a full set of variables.

To run factor analysis, you need to click on the Statistics / Multivariate Exploratory Techniques / Factor Analysis module. The Factor Analysis window will appear on the screen.

For analysis, we select all the different electronic tables; Variables (changes): select all, Ok. In the Input file row (input data file type) indicate Raw Data. The module has two types of output data – Raw Data and Correlation Matrix.

The MD deletion section specifies the method for processing missing values:
* Casewise - method of excluding missing values ​​(depending on the rules);
* Pairwise - a guy's way of excluding missing values;
* Mean substitution – substitution of the middle value to replace missing values.
The Casewise method means that in a spreadsheet that contains data, all rows that have one missing value are ignored. There is a price for all the important ones. The Pairwise method ignores missing values ​​not for all exchanged bets, but only for selected pairs.

Choose a method for processing missing values ​​Casewise.

Statistica collects missing values ​​in this way, such as imputation, calculates the correlation matrix and applies it to a selection of factor analysis methods.

The upper part of the window contains information. Here it is reported that the missing values ​​are handled using the Casewise method. 17 precautions were taken and 17 precautions were accepted for further enumeration. The correlation matrix was calculated for 7 variables. The lower part of the window contains three tabs: Quick, Advanced, Descriptives.

The Descriptives tab has two buttons:
1- look at correlations, average and standard variations;
2- run a multiple regression.

By pressing the first button, you can see average and standard displays, correlations, approaches, and various graphs and histograms.

In the Advanced tab, on the left side, select the Extraction method for factor analysis: Principal components. On the right side, select the maximum number of factors (2). Either the maximum number of factors is specified (Max no of factors), or the minimum value: 1 (eigenvalue).

We press Ok, and Statistica will quickly increase the calculations. The Factor Analysis Results window appears on the screen. As stated earlier, the results of factor analysis are expressed by a set of factor values. This can be accessed from the Loadings tab.

Upper part of the window – information:
Number of variables (number of variables to be analyzed): 7;
Method (method of seeing factors): Principal components;
Log (10) determinant of correlation matrix (tenth logarithm of the determinant of the correlation matrix): -1.6248;
Number of factors extracted: 2;
Eigenvalues ​​(vsnі values): 3.39786 and 1.19130.
At the bottom of the window there are functional buttons that allow you to easily view the analysis results, both numerically and graphically.
Factor rotation is a function wrapper in which you can select different axle rotations from the drop-down window. By additionally rotating the coordinate system, it is possible to obtain an impersonal solution, for which it is necessary to select a solution to be interpreted.

There are different methods for wrapping coordinates into space. The Statistica package supports all such methods presented in the factor analysis module. So, for example, the varimax method suggests a permutation of coordinates: a wrapper that maximizes the dispersion. The varimax method provides a simplified description of the factor matrix, reducing the values ​​to 1 and 0. This shows the dispersion of the squares of the official. The factor matrix, based on the varimax wrapping method, is largely invariant with respect to the choice of different multipliers.

Wrapping by the quartimax method is used in a similar way to simplifying only in relation to the rows of the factor matrix. Does Equimax occupy an intermediate position? When wrapping factors around this method, you will immediately try to simplify both the columns and the rows. The considered wrapping methods are based on orthogonal wrappings, then. As a result, uncorrelated factors emerge. Methods of direct obliminu and promax wrapping are carried out to oblique wrappings, which result in correlative factors. The term?normalized? The names of the methods indicate those in which the factor coefficients are normalized so that they are divided by the square root of the corresponding variance.

With all the methods used, the result of the analysis without wrapping the coordinate system – Unrotated – is immediately obvious. If the result of rejection appears to be interpretable and beyond our control, then we can rely on it. However, you can wrap the axes and see other solutions.

Click on the "Factor Loading" button and see the numerical factor loading.

It is clear that factor importance is the significance of the coefficients of correlation between skin and skin-related factors.

The value of factor significance greater than 0.7 shows that the sign of change is closely related to the analyzed factor. The closer the connections between the given signs and the analyzed factor, the greater the significance of the factor importance. A positive sign of factor avantagement indicates a direct (and a negative sign? A reversal) link between the given sign and the factor.
Then, from the table of factor factors, two factors were identified. The first signifies OSB – a sign of social well-being. Other changes made by another official.

In row Expl. Var (Fig. 8.5) the dispersion is induced, which falls on another factor. In the row Prp. Totl is induced part of the dispersion that falls on the first and other factors. Also, the first official accounts for 48.5% of the total variance, while the other official accounts for 17.0% of the entire variance, which accounts for other uninformed officials. The result showed two factors explaining 65.5% of the total variance.

Here we also have two groups of factors - OSB and the decision of the changes that appear to be ZHR - it is important to change the job. Perhaps it is possible to understand this situation better based on the collection of additional data.

Select and clarify a number of factors

Once you have obtained information about how many dispersions you see in the skin factor, you can turn to nutrition, how many factors you can remove from the trace. Due to its nature, the solution is more complete. Here are some basic recommendations, and following them in practice gives the best results.

The number of hidden factors (hyperparameters) is determined by the way of calculating the power numbers (Fig. 8.7) of the X matrix in the factor analysis module. For this, in the Explained variance tab (Fig. 8.4), you need to click the Scree plot button.

The maximum number of solar factors can be compared to the number of power numbers of the parameter matrix. With an increase in the number of officials, the problems of their physical interpretation significantly increase.

You can now select more factors with significant values ​​greater than 1. In fact, this means that if a factor does not see a dispersion that is equivalent to the dispersion of the same variable, then it is omitted. This criterion is widely disputed. In this case, based on this criterion, only 2 factors (two main components) must be saved.

You can find such a place on the graph, where the decline in its value of the evil to the right is maximized. It is transferred that the right hand at this point is found to be less “factorial osip”. Apparently, up to this criterion, you can remove 2 or 3 factors from the application.
3 fig. It is clear that the third official contains a larger portion of the igneous dispersion.

Factor analysis of parameters makes it possible to identify at an early stage a breakdown in the work process (caused by a defect) in various objects, which is often impossible to mark with careless attention to parameters. We note that the disruption of correlation connections between parameters occurs significantly earlier than a change in one parameter. Such a combination of correlation links makes it possible to quickly identify factor analysis of parameters. For this purpose, the array and registered parameters are sufficient.

It is possible to give further recommendations for the use of factor analysis regardless of the subject area.
* The skin factor may include at least two different parameters.
* The number of parameters changes may be greater for the number of changes.
* A number of factors may be discussed based on the physical interpretation of the process.
* First of all, make sure that the number of factors is much less than the number of changes.

The Kaiser criterion also saves even a lot of officials, while the criterion of stony wasp saves even a few officials. However, the criteria are generally good for normal minds, if the number of factors is very small and the number of variables is very small. In fact, nutrition is more important if the decision taken can be interpreted. Therefore, it is necessary to consider a number of decisions with more or less factors, and then select the one that makes the most sense.

The range of output signs may be represented in the same scales of vibration, which allows calculating vicoristic correlation matrices. In another case, there arises the problem of “vaga” of various parameters, which leads to the need for stagnation when calculating covariance matrices. You may be faced with the additional problem of repeatability of the results of factor analysis for changing the number of signs. To indicate that the problem is identified, the Statistica package simply has to switch to a standardized form for presenting parameters. In this case, all parameters become equal at the stage of their connection with the processes in the object of investigation.

If there are too many variables in the set of output data and they have not been removed by correlation analysis, then it is not possible to calculate the return matrix (8.3). For example, if the change is the sum of two other variables selected for this analysis, the correlation matrix for such a set of variables cannot be generalized, and factor analysis in principle cannot be calculated. In reality, this is true if one tries to freeze the factor analysis to the exclusion of very stale variables that are sometimes lost, for example, in the sample of feeders. Then you can individually reduce all correlations in the matrix by adding a small constant to the diagonal elements of the matrix, and then standardize them. This procedure should lead to a matrix that can be standardized, and then factor analysis can be established before it. Moreover, this procedure does not reflect the set of factors, and the estimates are less accurate.

A system with changeover mills (SMS) is a system that is based only on the input inflow and the output of a specified constant parameter, the initial mill. Regulations or attenuator? This is the case of the simplest UPS, in which the transmission coefficient can be discretely and smoothly changed according to some kind of law. The investigation of the UPS should be carried out for linearized models in which the transient process, due to a change in the parameter, is considered to be completed.

Attenuators based on G-, T-, and P-like connections of series- and parallel-connected diodes have achieved the greatest expansion. The support of diodes under the infusion of ceramic jet can be changed over wide ranges, which allows changing the frequency response and damping in the path. The independence of the phase phase when regulating the extinction of such attenuators is achieved with the help of reactive lancets included in the basic structure. Obviously, with different connections between the supports of parallel and subsequent diodes, the same level of weakening that is introduced can be removed. Otherwise, the change in the phase phase will be different.

It is possible to simplify the automated design of attenuators, which includes the advanced optimization of the coring lances and the parameters of the ceramic elements. As a result of monitoring the UPS vikoristovatemo electrically quenched attenuator, the equivalent circuit is shown in Fig. 8.8. The minimum level of extinguishing is ensured between the small support element Rs and the large support element Rp. In the case of an increased support of the element Rs and a changed support of the element Rp, the weakening that is introduced increases.

The extent of changes in phase change in frequency and attenuation for circuits without correction and with correction is shown in Fig. 8.9 and 8.10 daily. With a corrected attenuator in the attenuation range of 1.3-7.7 dB and a mixture of frequencies 0.01-4.0 GHz, a change in the phase shift of no more than 0.2 ° is achieved. In the attenuator without correction of the phase change in the same mix of frequencies and range, the attenuation reaches 3°. Thus, the phase destruction is changed for a maximum of 15 times correction.

What is important is the correction and control of independent variable factors that influence the extinguishment and change of the phase phase. This makes it possible, using the additional Statistica system, to carry out a factorial and regression analysis of the UPS by establishing physical regularities between the parameters of the Lantzug and other characteristics, as well as simplifying the search for optimal parameters of the circuit.

The output data was formed in this way. For the correction parameters and control supports, which differ from the optimal ones on the larger and smaller sides on the frequency network 0.01-4 GHz, there was a calculation of the attenuation that was introduced, and a change in the phase shift.

Methods of statistical modeling, including factorial and regression analysis, which have not previously been used for the design of discrete devices with changeable mills, make it possible to identify the physical laws of the operation of system elements. This corresponds to the structure of the device, resulting from the specified optimality criterion. Zokrem, in whose section the phase-invariant attenuator is considered as a typical butt of the system with changeable mills. The revealed and interpretation of factor influences that flow into various characteristics allows us to change the traditional methodology and completely simplify the search for correction parameters and regulation parameters.

It has been established that the use of a statistical approach to the design of such devices is justified both in assessing the physics of their work and in priming the principle circuits. Statistical modeling allows for the essential speed of experimental research.

Results

• Caution for underlying factors and related factor influences is necessary to identify the internal laws of processes.
• By identifying the critical values ​​of the control areas between factor values, we can trace and consolidate the results of factor analysis for similar processes.
• The definition of factor analysis is not limited by the physical features of the processes. Factor analysis is an important method for monitoring processes and is essential before designing systems for various purposes.