Factor analysis in STATISTICA. Correlated matrix Correlated matrix for factor analysis

Dispersion analysis of factors

factor matrix

Minliva Factor A Factor B

Yak can be seen from the matrix, factorial navantazhennya (abo wagi) A and B for older people living in can be meaningfully derived. The factor of navantazhennya A for vimoga T 1 shows a tricky sound, as it is characterized by a coefficient of correlation, which is expensive 0.83, so it is good (low) fallowness. Factorna navantazhennya B for the same vimogi yesє r k= 0.3, which shows a weak tone. As soon as it was transferred, factor B is even better correlated with the living vimogs T 2, T 4 and T 6.

I will look at how factorials are added to A, so that B is injected on to not overlapping until the last group of living vimogs with a tight connection no more than 0.4 (which is weak), it can be important, if the matrix of intercorrelation is represented in the body. independent factors, such as the number of surviving vimogs (behind a vignette T 7).

The change of T 7 can be seen in an independent factor, since none of the survivors is not significantly correlated (ponad 0.4). Ale, in our opinion, it’s not going to work, so as the factor "the door is not guilty of rzhaviti" is not a perfect constructions doors.

In such a rank, with the solidified technical design for the design of the car door design itself, name the removed factors will be inscribed as living conditions, for which it is necessary to know the constructive solution in the view of the engineering characteristics.

Significantly on one principle important is the power of the correlation between the minions: the squares, the show, which is a part of the variance (rozkidu) signs є that is spiritual for two minions, which are very different. So, for example, if two changes T 1 and T 3 with a correlation of 0.8 overlap with a step of 0.64 (0.8 2), then this means that 64% of the variances of this type and the difference between them are out-of-the-box, so that they get out of hand. You can also say that spirituality cich winter roads 64%.

Nagadamo, which factorial navantazhennya in the factor matrix є also the coefficients of the correlation, also the factors and the changing (living vimogs).

Minliva Factor A Factor B

To that, the factorial navantage (variance) characterizing the degree of spilability (or overlap) of a given change and a given official has been squared. Apparently, the steps of the cross-section (variance D) of both factors from the change (which can be lived) T 1. For this, it is necessary to calculate the sum of the squares of the factors from the first change, to 0.83 x 0.83 + 0.3 x 0.3 = 0.70 ... In such a rank, the percentage of change T 1 from the volume of factors to become 70%. Tse to finish the wagome perekrittya.

At the same hour, low spirituality can be indicative of those who are changing in vimiru or imagining, obviously different from those in the analysis. It’s worth it, but it’s given a change not to be taken with the factors for one of the reasons: for there’s a vimiru іnshe witness (like, for example, a change of T 7), for there’s a great pardon for being aware of, or I’m learning to disperse.

It also means that the significance of the skin factor is also determined by the magnitude of the variance between the changes and factories (vagoyu). In order to calculate the value of the factor, it is necessary to know in the skin factor matrix the sum of the squares of the factorial preparation for the skin change. In such a rank, for example, the variance of factor A (D A) 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 value of factor B shows that D B = 2.64, so that the significance of factor B is higher, but not factor A.

If the value of the distribution factor is not influenced by the number of changes (for our applications), then the value will be shown as a part of the variance (or information) γ in the outgoing correlation matrix of the warehouse and the factor. For factor A γ ~ 0.34 (34%), and for factor B - γ = 0.38 (38%). Pidsumuvavshis results, otrimaєmo 72%. In such a rank, two factors, being combined, make up only 72% of the variance of indicators in the input matrix. This means that, as a result of factorization, a part of information at the current matrix of the bull was sacrificed to induce a two-factor model. As a result, 28% of information was missed, as it could have been updated, if the six-factor model was adopted.

Is there a pardon allowed, vrahoyuchi, what are all the changes, how can they be made according to the design of the door, vrahovani? Naybilsh ymovirno, the value of the coefficients of the correlations of the winners, should be referred to one factor, which is often underestimated. Based on the analysis performed, it is possible to turn to the form of the largest values ​​of the correlation coefficients in the intercorrelation matrix (div. Table 2.2).

In practice, it is often necessary to deal with such a situation, when the number of independent factors is great, but all of them are in solution problems and from a technical or economic point of view. There is a number of ways from the number of factors. Naybilsh views from them - Pareto analysis. At the same time, there are officials (in the world of change of significance), who use 80-85% between them of total significance.

Factor analysis can be carried out when implementing the method of structured function of quality (QFD);

FACTOR ANALYSIS

Idea of ​​factor analysis

With pre-existing folding objects, appearances, factor systems, which start the power of these objects, it is even more difficult to change it without a median, and sometimes it is impossible to find the number and sense. If you want to be able to see it, you can be of an affordable value, so why should you lie down as a factor in us. Moreover, if the infusion of an unattractive factor to us is manifested in decile vimiruvanny signs, or the power of the object, there can be signs of a strong connection between themselves and in the background the number of factors in can be much less, there is no more number in the winter.

For identifying factors, how to begin to identify signs of objects, methods of factor analysis are used.

As a rule, the factor analysis can be used to determine the power of the individual on the basis of psychological tests. The power of specialness does not lend itself to direct recognition. You can judge about them by the behavior of people, or by the nature of the views on food. To clarify the results in the past, we give factor analysis, which allows the appearance of the special power, which is infused into the behavior of the individual.
Basically new methods factor analysis to lie on the offensive hypothesis: it is more likely that the parameters are more likely to be influenced by the indirect characteristics of the pre-sensed object; The internal parameters are taken into account by factors.

Meta-factorial analysis - concentrate the on-going information, there is a large number of visible signs through a fewer number of more internal characteristics of the phenomenon, which, however, cannot be seen without

It was established that the vision and further caution behind the local factors give the possibility of detecting a failure to be detected at even early stages of development of a defect. Factorial analysis allows the stability of the correlative links among the same parameters. The very same correlations between parameters, as well as parameters and specific factors, are the main diagnostic information about the processes. Stagnation of tools of the Statistica package when determining the factor analysis of a test is necessary to register additional numerical data and to rob the analysis directly and sensibly for a corystuvach.

The results of the factorial analysis will be successful, as well as given the interpretation of the emerging officials, and the changes in performance that characterize the officials. The given stage of the robot is even more advanced; won a clear statement about the zesty sense of the indicators, which were obtained for analysis and on the basis of those seen by officials. That is, with the preliminary review of indicators for the factor analysis of the keruvatis by the wickedness, and not by the pragmatism before the inclusion of a larger number in the analysis.

The essence of factor analysis

Guided by the main position of the factor analysis. Come on for the matrix NS in the dynamic parameters of the object, the covariance (correlation) matrix C, de R- number of parameters, n- the number with the caution. by way linear re-implementation X=QY+U it is possible to change the size of the specific factorial space NS to the level Y, at the same time R"<<R... The process of converting the point, which characterizes the mill of the object in j in a new space, in a new space in a small space R". Obviously, the geometrical closeness of the two, or otherwise, pointless points in the new factorial space means the stability of the object's camp.

matrix Y To revenge unsupervised factors, which are on the basis of hyperparameters, which characterize the greatest power of the analyzed object. Officials in the country are most often chosen statistically independent, but will fall back on physical interpretation. Vector with warning signs NS a low sense of succession of changes in hyperparameters.

matrix U stock up on surplus factors, which include x(i). rectangular matrix Q to take revenge on the factorial navantazhennya, which visually start the line of links between signs and hyperparameters.
Factor navantazhennya - the value of the coefficients of the correlation of the skin with the visual signs of the skin with the evolving factors. Chim tisnishe links of a given signifies a given factor, that is, a meaningful factor navantazhennya. The positive sign of the factorial navantazhennya vkaz on the direct (and the negative sign on the ring) of the link of the given sign of the factor.

In such a rank, the data about the factorial navantazhennyh allow formulating visnovki about the set of vyhіdnyh signs, which represent the same factor, and about the vase of the surrounding signs in the structure of the skin factor.

The model of factorial analysis is similar to the model of variable regression and dispersion analysis. The principle principle of the model of factor analysis is that the vector Y is at the expense of unwarranted officials, and in regression analysis - at the same time the parameters. In the right part of the line (8.1), the matrix of the factorial navnyennya Q and the matrix the value of the zagalnye factors Y.

For a given matrix of factorial factors, the value of the equation is QQ t = S-V, de Q t - matrix Q is transposed, V is the matrix of covariance of surplus factors U, tobto. Rivnyannya virishuєuє Given the matrix of factorial factors, the level of Q is calculated by the leading factors (hyperparameters) for the values
Y = (Q t V -1) Q -1 Q t V -1 X

The package of statistical analysis Statistica allows in the dialogue mode to enumerate the matrix of factorial parameters, as well as the values ​​of decliners for the assigned head officials, most often two - behind the first two head components in the input matrix of parameters.

Factor analysis in Statistica systems

The post-presentation of the factor analysis on the basis of the processing of the results of the questionnaire survey of the executives of the enterprise is discernible. It is necessary to visit the main officials, as they begin to work life.

At the first stage, you need to make a change to carry out the factor analysis. Vikoristovuchi is a correlative analysis, a glimpse of a wiggle is connected to a prelude to a sentiment, well, in his own devil, yes, there is a chance to see a new one, and a hackless typing is a sign of a way to know a lot.

If you carry out a factorial analysis for all the changes, then the results may not come out as objective ones, so as the ones of the changes are based on the data, and they cannot be regulated by the spirits of the given organization.

For the sake of intelligence, such as indicators of success, I will follow the explicit data of the matrix of correlation coefficients in Statistica: Statistics / Basic Statistics / Correlation Matrices / Ok. At the start screen of the Product-Moment and Partial Correlations procedure (Fig. 4.3), the One variable list button is selected for opening a square matrix. Vibiraєmo all changes (select all), Ok, Summary. We will accept a correlation matrix.

As the coefficient of correlation varies between 0.7 and 1, this means a strong correlation of indicators. In a whole range, you can include one change with a strong correlation. I navpaki, as a function of the correlation of mali, can be changed through those who can’t reach the zalous sumi. In our view of strong correlation, we will not be sparing, and factor analysis will be carried out for a new set of winners.

To run factor analysis, it is necessary to test the Statistics / Multivariate Exploratory Techniques module / Factor Analysis (factor analysis). The Factor Analysis module appears on the screen.

For the analysis of vibramo all changing electronic tables; Variables (Змінні): select all, Ok. The Input file row (type for the input data file) appears as Raw Data. The module can have two types of output data - Raw Data and Correlation Matrix - a correlation matrix.

For the MD deletion section, the method for processing missing values ​​is set:
* Casewise - how to make a missing value (for a change);
* Pairwise - a guy with the ability to turn on missing values;
* Mean substitution - substitution of the mean to replace the missing value.
The way Casewise is cleared in the fact that in the electronic table, for revenge the data, ignore all the rows, in which one would like to be missing a value. It should be borne by everyone. The Pairwise way to ignore missing values ​​is not for all minions, but rather for a vibrant bet.

Vibrating way of handling Casewise missing values.

Statistica to process the missing values ​​in this way, which is said to be a correlation matrix and proponed to a variety of methods of factor analysis.

The upper part of the window is informational. Here you will see how missing values ​​are processed by the Casewise method. Estimated 17 points and 17 points accepted for the next counting. The correlation matrix is ​​calculated for 7 changes. The lower part of the window reveals 3 tabs: Quick, Advanced, Descriptives.

The Descriptives contribution has two buttons:
1 look at the correlations, middle and standard views;
2 will cause many regressions.

By pressing the first button, you can marvel at the middle and standard views, correlations, covariances, as well as the development of graphs and histograms.

In the Advanced contribution, in the left part, the Extraction method of the factor analysis: Principal components (the method of the head components). The right part has the maximum number of factors (2). Specify either the maximum number of factors (Max no of factors), or the least value: 1 (eigenvalue).

Onslaught Ok, and Statistica will quickly calculate. Factor Analysis Results will appear on the screen. As said earlier, the results of the factorial analysis rotate with a set of factorial navantages. For that, let's go to the Loadings tab.

The upper part of the window - information:
Number of variables: 7;
Method (method of seeing factors): Principal components (head components);
Log (10) determinant of correlation matrix: -1.6248;
Number of factors extracted: 2;
Eigenvalues: 3.39786 і 1.19130.
At the lower part of the window there are functional buttons, which allow you to study the results of the analysis, numerically and graphically.
Factor rotation - the wrapping of the factors, in this particular window you can vibrate the rotation of the axes. For the additional rotation of the coordinate system, it is possible to correct a non-linguistic solution, which requires a vibration to be interpreted.

Explore different methods of wrapping coordinates to space. The Statistica package promotes all such methods presented in the factor analysis module. So, for example, the varimax method is used to transform coordinates: wrapping, maximizing variance. In the varimax method, it is not easy to describe 100% of the factor matrix, all values ​​are up to 1 or 0. At the same time, the variance of the squares of the factor can be seen. The factor matrix, which can be disregarded for the additional method of wrapping varimax, is invariably invariant in a larger world to the variety of different types of changes.

Wrapping by the method of quartimax is used to set the analogue for the same reasoning only in terms of the ratio to the rows of the factor matrix. Equimax interim loan? in case of wrapping the factors behind the cym method, one hour to try to get rid of the hundredth, and the rows. The methods of wrapping have been developed to be applied to orthogonal wrapping, so that as a result, unrelated officials appear. Methods of direct region and PROMAX wrapping are carried out to oblique wraps, as a result of which there are correlated between factories. Termin? Normalized? in the names of the methods, we order those that are factorialized to be normalized, so that they can be divided by the square root with a variety of variances.

With all the proponated methods, the result can be analyzed without wrapping the coordinate system - Unrotated. As soon as the result is lost, the result will be interpreted, and if we are vlashtovuvati, then we can be impressed. I’m dumb, you can wrap the axis and be amazed at the decision.

Klatsaєmo on the button "Factor Loading" and we are amazed at the factorial nagging numerically.

Nagadaєmo, scho factor navantazhennya - the value of the coefficients of the correlation of the skin with the changes of the skin with the evolving factors.

The value of the factorial navantazhennya, more than 0.7 is shown, which is given as a sign of a change in tying with this factor. Chim tisnishe links of a given signifies a given factor, that is, a meaningful factor navantazhennya. The positive sign of the factorial navantazhennya vkaz on the direct (and the negative sign? On the vorotnu) of the link of the given sign of the factor.
Also, from the tables of factorial navantages, two factors are shown. The first sign of RSD is the perception of social well-being. Reshta is changed by another factor.

Row Expl. Var (Fig. 8.5) shows the variance, like a fit, on the same factor. Row Prp. Totl shows a fraction of the variance, like the attack on the first and other factor. Also, the first factor is 48.5% of the total variance, and the other factor is 17.0% of the total variance, all the solutions fall on the first non-traumatic factor. In the bag, two emerging factors explain 65.5% of all variance.

Here, there are also two groups of officials - OSB and those without wives, who are seen by the ZSR - bazhannya change to the robot. Mabut, maє sense doslijuvati tse bazhannya gruntovnіshe on the basis of the collection of additional donations.

Vibration and refinement of the number of factors

Since only the information about those, because of the variance of the visible skin factor, can be turned to nutrition, about those, about those who have been oversubscribed. For its nature, the price is quite high. Alle є deyakі zalnovzhivanі recommendations, and in practice, attention to him yields the most beautiful results.

The number of foreign factors (hyperparameters) starts with the way of calculating the absolute numbers (Fig. 8.7) of the matrix X in the module of factorial analysis. For the whole depository Explained variance (Fig. 8.4), you need to click the Scree plot button.

The maximum number of foreign factors can be added to the number of external numbers of the matrix of parameters. Even with the increase in the number of factors, difficult physical interpretations are growing.

A selection can be made only of factors, with power values ​​greater than 1. According to the essence, it means that if the factor does not see the variance, which is equivalent, take, the variance of the same change, then it should be omitted. The whole criterion of victoriousness is widespread. In the guided sight, on the basis of the criterion, only 2 factors (two head components) should be saved.

You can also know the place on the graph, de-shedding the power of evil to the right, to get along as much as possible. To transfer, the right-hander from the center of the point is located only "factorial osip". Depending on the criterion, it is possible to add 2 or 3 factors in the butt.
3 fig. it can be seen that the third factor slightly increased the part of the backward dispersion.

Factorial analysis of parameters allowing for damage at the early stage of the breakdown of the working process (detection of a defect) in new cases, as it is often unwise to go through the middle without being careful about the parameters. It should be explained that the breakdown of the correlative links between the parameters of the winners is much earlier than the change of one parameter. Similarly, the creation of correlative links allows you to quickly develop factor analysis of parameters. For a whole lot of mothers there are registered parameters.

It is possible to give general recommendations on the test of factor analysis independently from the subject area.
* The skin factor can be affected by no less than two parameters.
* The number of parameters is due to be greater than the number of parameters.
* A number of officials are guilty of obruntovuvatisya, involved in the physical interpretation of the process.
* Be sure to wait until the number of officials is less than the number of officials.

The Kaiser's criterion is one of several factors, but at that time there are few factors. However, offense to the criterion is generally good with normal minds, since there are altogether few factors and a lot of changes. In practice, we are more important є nutrition about those, if the decision can be interpreted. Let that be encouraged to see the solution with a larger and smaller number of factors, and then choose one of the best comprehensible.

The space of the outward signs is guilty of being represented in one-sided scales of vimir, since K. Tse is allowed when calculating vikoristovuvati correlation matrices. In general, there is a problem of "vag" of different parameters, which is necessary to make it necessary when calculating covariate matrices. The problem of repetition of the results in factorial analysis in case of a change in the number of signs may appear. Slide to mean that the problem is simply to be seen in the Statistica package by way of the transition to the standardized form of the presented parameters. With all the parameters, the link with the processes in the object is equal to each other.

Even though the set of financial data excessive changes and a correlation analysis has not been carried out, it is impossible to calculate the inverted matrix (8.3). For example, if there is a change in the sum of two of them, they are being viewed for the whole analysis, then the correlation matrix for such a set of minions cannot be brutal, and the factorial analysis cannot, in principle, be vicinities. In practice, it’s possible to find out, if you start to get a factorial analysis to the depleted winter weather, you’ll be able to trample, for example, in the sample of questionnaires. It is possible to individually reduce all the correlations in the matrix by adding a small constant to the diagonal elements of the matrix, and also to standardize it. The procedure should be adjusted to the matrix, as it can be turned, and to that it is possible to fix the factorial analysis. Moreover, the procedure does not add to the set of factors, and the estimates are less accurate.

The system for changing mills (SPS) is a system that does not only look at the input flow, but as a fixed time parameter, which is the starting point. Regulations pidsiluvach or attenuator? the butt of the simplest ATP, in which the transmission function can be discretely or smoothly changes according to some kind of law. The pre-development of the ATP will be carried out for linearized models, in some transitional process, tying with a change in parameter, I will start, we will be completed.

Athenyuatori, vikonanі on the basis of G-, T- and P-similar data after and in parallel with the inclusion of diodes nabuli widened. Opіr dіodіv under the flowing curvy struma can change in wide fringes, which allows changing the frequency response and extinguishing in the tract. The independence of the phase damping during the regulation of the extinguishing in such an attenuator can be reached behind the aid of the reactive lancers included in the base structure. Obviously, in case of a different pairing of parallel supports of parallel and last days, there can be one or the same level of the introduced weakening. Ale the change of phase zsuvu will grow.

Before the end of the day, the possibility of using automated design of attenuators, which enables the sub-optimization of the coring lance and the parameters of the coring elements. Yak doslidzhuvanoy SPS will be victoristovuvati electrically keroving the attenuator, the diagram of the replacement is shown in Fig. 8.8. The minimum level of extinguishing will be ensured in the case of a small support of the Rs element and a great support of the Rp element. In the world of reducing the support of the element Rs and the change in the support of the element Rp, the weakening is not reduced.

Deposits of the phase change in frequency and extinction for the circuit without correction and with correction in Fig. 8.9 and 8.10 as appropriate. Coriguvati has an attenuator in the range of 1.3-7.7 dB and a smooth frequency of 0.01? 4.0 GHz, the phase shift is reached, not more than 0.2 °. An attenuator without a correction of the phase shift in the same smoothing of frequencies and ranges has a weakening of 3 °. In such a rank, the phase change of changes for the rate of correction of the correction is 15 times.

We will take into account the parameters of the correction and management of independent servants, so that we will add a phase change to the extinguishing and control phase. At the same time, with the help of the Statistica system, it is possible to carry out a factorial and regression analysis of the SPS by establishing physical regularities between the parameters of the lancet and the extreme characteristics, as well as by asking for a joke of the optimal parameters of the circuit.

Vyhіdnі danі was formed in such a way. For the parameters of the correction and control pillars, what are the optimal ones for a large and smaller side at a frequency of 0.01? 4 GHz, the bullets are calculated without the weakening and change of the phase shift.

The methods of statistical modeling, growth, factorial and regression analysis, which have not been previously used for the design of discrete attachments with changing mills, allow the development of physical laws of the robotic elements of the system. The center of the stem structure is attached to the outgoing line from the given criterion of optimality. Zokrema, in the given distribution, the phase-invariant attenuator was seen as a typical butt of the system with winter camps. The development and interpretation of factorial navantages, which inject into the development of preliminarily characteristics, allows for changing the traditional methodology and quite simply to simplify the adjustment of the parameters in the correction and control parameters.

It has been established that the statistical approach to the design of additional annexes is valid for assessing the physics of their robots, as well as for priming the principles of their schemes. Statistical modeling allows for the quick rate of testing of experimental data.

results

• Keeping an eye on the out-of-the-box factors and related factorial options - there is a need to find out the internal laws of the processes.
• Noting the value of the critical values ​​of the control over the appearance and the factorial navantages for the accumulation and use of the results of the factor analysis for the same type of processes.
• The stagnation of factorial analysis is not bounded by the physical features of the processes. Factorial analysis є as a forceful method of monitoring processes, so it is fixed to the design of systems of the most reasonable value.

As if to carry out factorial analysis in order to be suitable, and not to be satisfied with the attitudes behind the suggestions ("little Giff", as they called the standard gentleman's set of methodology) The axis here is where we can be misunderstood: a procedure like about the pardon: the correlation matrix is ​​not positive definite. What does it mean, why should you be trying to deal with a problem?
On the right, in the process of factorization, the procedure of inverting the manifestation of the so-called vocal matrix in relation to the ratio to the correlation. Here, there is a simple analogy with specific working numbers: having multiplied the number by the same number, we are guilty of rejecting one (for example, 4 і 0.25). However, for the deyaky numbers of the vertebrates before them, it is not іsnuє - it is not a matter of multiplying zero by nos, so that the result is one. With matrices and history. The matrix, multiplied by the one-to-one matrix, is a single matrix (one in diagonal, and all values ​​are zero). However, for the deyaky matrices, they are not very healthy ones, but it means that it becomes untenable to carry out a factorial analysis for such types of matrices. Z'yasuvati daniy the fact can, for the additional help of a special number, be called a viznachnik (determinant). As far as the matrix is ​​pragmatic to zero or negative, we are stuck with a problem.
What are the reasons for this situation? Most often, there are winners in most cases when there is a line of fallowness and winter. It sounds wonderful, the sprinkles of the same amount of time and sound, vikorist and rich methods. However, at times, since such deposits are no longer effective, they are hard to determine, algorithms of complex analysis are given. The offensive butt is visible. Come with us є such a set of danikh:
data list free / V1 to V3. begin data. 1 2 3 2 1 2 3 5 4 4 4 5 5 3 1 end data. compute V4 = V1 + V2 + V3.
Remaining zmіnnna is the exact sum of the first three. If the question is similar to the situation in real life? If it is included in the set of winter sisters according to subtests and the whole test; if the number of winter races is greater than the number of preliminaries (especially when the number of changes is strong, for there may be some encirclement of values). In general, the exact line of fallowness can be vinicati vipadkovo. Often, an artifact of the procedure for vyryvannya - for example, if there are hundreds of times in the middle of the warning (say, the number of vocal types); Yak bachimo, the whole situation is widened.
If, when carrying out a factor analysis in the SPSS of the determined array of determinants and vocal correlative matrices, then the package will tell you about the problem.
How can you find a group of winners, how do you create multicollinearity? Appear, the good old method of head components, unaffected on the line of fallowness, prodovzhu pratsyuvati і shhos vidu on the mountain. Just beat it up, but the spirits, from the minuscule ones, are coming up to 0.90-0.99, and the absolute number of deyakie factors is getting even small (or negative), not a good sign. Substitute for the same wrapping varimax and wonder how a group of mincers indulged at once in a malignant call of the commodity. Name it іntage її on the tse factor є is invisibly great (0.99, for example). Even though the set of vintages is small, wistfully different, the possibility of artifact lineage is turned off and the vibration is high, then the appearance of such a sound can be considered not less than the result of value. It is possible for such a group to show in regressive analysis: that change, as it showed the most modernization, growth of fallow lands, and all of them tried to be used as predictors. R, so that the function of multiple correlations is guilty of the whole fall, but we are equal. Having covered the background of the diagnosis of multicollinearity beforehand, it is possible to make a non-merciless set, which will establish the exact line of fallowness.
Well, nareshty, there are several other reasons that the correlation matrix is ​​not positively valued. Tse, in the first place, the presence of a great number of inconsistencies. In some way, you will be able to get the most out of the obvious information, and then you will change the processing of the passes in a pairwise way. As a result, there may be an "illogical" matrix, but the model of factor analysis will be too tough for the model of factor analysis. In another way, if you have violated the factorization of the correlative matrix, pointed in the literature, you can get confused with the negative influx of rounded numbers.

Basic provisions

Factorial analysis is one of the new portions of the complex statistical analysis. A collection of methods has been developed to clarify the correlation between the various parameters. The result of the correlation analysis is a matrix of correlation coefficients. With a small number of signs (minus), it is possible to carry out a visual analysis of the whole matrix. With an increase in the number of signs (10 or more), visual analysis does not give positive results. It turns out that all the differences in the basic connections can be explained by the different public officials, such as the functions of the illusory parameters, while the factors themselves can be made unconscious through the aliases. The founder of factorial analysis is the American doctrine L. Thurstone.

Current statistics and factorial analysis of reasonability of methods, based on a really sensible connection between signs, allowing for the emergence of latent (prikhovani) common characteristics of the organizational structure of the process of development and mechanization

Butt: it is assumed that n cars are estimated by 2 signs:

x 1 - car part,

x 2 - triviality of the working motor resource.

Due to the correlation of x 1 і x 2 in the coordinate system, it appears that it is not straightforward to reach the number of points purchased, and it is formally displayed with the new axes і (Fig. 5).

fig. 6

characteristic particularity F 1 i F 2 polyagaє in the fact that the stench passes through the purchase of points and into its own line x 1 x 2.Maximum

the number of new axes will be equal to the number of elementary marks. The subdivisions of factorial analysis have shown that the method can be successfully stuck in the problems of grouping and classification of objects.

Submission of information in factorial analysis.

To carry out the factorial analysis, the information is guilty, but it is presented in the matrix view with the size m x n:

The rows of the matrix show the signs with the warning (i =), and the hundred with the signs (j =).

Signs that characterize the object may vary in size. In order to bring it to the same level of confusion and to prevent it, the matrix of the given data should be normalized by introducing a single scale. The most widely used method of standardization є standardization. From the wintry to the wintry

Average value j signs,

Medium-square vidhilennya.

This re-creation is called standardization.

The basic model of factor analysis

The basic model of factorial analysis of ma viglyad:

z j - j-th sign (value of vipadkov);

F 1 , F 2 , ..., F p- zagalny factors (vypadkovі values, normally rospodіlenі);

u j- characteristic factor;

 j1 ,  j2 , …,  jp – navantazhennya factor, scho characterize the rate of inflow of the skin factor (parameters of the model, scho to value);

Officials from the country may not be very important for the analysis of all the signs. Characteristic officials show that it is necessary to be introduced only to the given, I know, the specifics of the sign, which cannot be turned through the factories. factorial navantazhena  j1 ,  j2 , …,  jp characterize the value of the inflow of that chi іnshogo factor in the variation of this sign. Basically, the factorial analysis is based on the factor navantage. variance S j 2 skin signs, can be divided into 2 warehouses:

Persha chastina pumovlyu dіyu zagalnyh factors - spіlnіst h j 2;

another part of the collection of the characteristic factor -th - d j 2.

All changes are presented in a standardized view, so the variance - gopr_snak S j 2 = 1.

As far as those characteristic officials do not correlate with themselves, then the variance of the j-ї signs can be represented in the viglyad:

de - a part of the variance of the sign, k-th factor.

Additional input of any factor into the total variance of the road:

Introduce these factors into the total variance:

The results of the factorial analysis are presented manually in the view of the tables.

A- matrix of factorial navantages. It can be trimmed in different ways, in Danish hour the broadest version of the method of the head components or the head bureaucrats.

The procedure is calculated using the method of head factors.

The decision of the head components to be built up to a step-by-step re-creation of the matrix of external tributes X :

NS- the matrix of vicarious tributes;

Z- matrix of standardized sign values,

R- matrix of paired correlations:

Diagonal matrix of power (characteristic) numbers,

 j know the decisions of a characteristic ryvnyannya

E- single matrix,

 j is an indicator of the dispersion of the cutaneous head component,

for wash the standardization of the latest tributes, todi = m

U- a matrix of power vectors, which are known from the equation:

Really tse means solution m line systems for skin

Tobto the skin moisture number is derived from the ryvnyan system.

to know V- a matrix of normalized vectors.

The matrix of the factorial view A is calculated by the formula:

Because of the known values ​​of the head components for one of the equivalent formulas:

The number of industrial enterprises was estimated for three characteristic signs:

middling viroblenya per one teacher x 1;

level of profitability x 2;

Equity funds x 3.

Result of representations in standardized matrices Z:

by matrix Z the matrix of paired correlations is trimmed R:

It is known that the matrix of paired correlation matrices (for example, using the Faddeev method):

I will be more characteristic of the ryvnyannya:

Virishuchi tse rivnyannya is known:

In such a rank, the list of elementary signs x 1, x 2, x 3 can be used for the most important three head components, moreover:

F 1 I will explain approximately all options,

F 2 -, and F 3 -

All three main components explain the options to increase by 100%.

Virishuchi qiu system is known:

The systems for  2 and  3. For  2 solution of the system will be similarly:

Power vector matrix U nabuvaє viglyadu:

Skin element of the matrix is ​​suitable for the sum of squares of elements of the j-th

stovpchik, we can make a standard matrix V.

Apparently, it’s guilty that we’re equal = E.

The matrix of the factorial representation is rendered from the matrix representation

=

Behind the zm_stom skin element of the matrix A presenting the private functionality of the matrix in relation to the global acquaintance x j i head components F r. All the elements to that.

Umova's vivacity r- number of components.

Additional insertions of the skin factor into the total dispersion signify the following:

Factor analysis model for nabude viglyad:

We know the values ​​of the head components (matrix F) behind the formula

The center of the distribution of the value of the head components is located at points (0,0,0).

The further analysis of the results of the design is followed by the decision about the number of significant signs and the leading components of the meaning of the names of the leading components. The preset design of the head components, the designation for them is called sub'actively based on the performance of the imaging matrix A.

The nutritional formula for the names of the head components is easy to understand.

meaningfully w 1 - without insignificant car performance, until elements close to zero are turned on,

w 2 - no significant car performance,

w 3 - the number of significant performance factors, which do not take part in the formulated name of the head component.

w 2 - w 3 - a multiplicity of car functions, which takes the same fate as the name.

Calculated efficiency of information for the skin head official

The set of zyasovnyh signs vazhaєmo zadovіlnym, if the value of the functional information lies in the boundaries of 0.75-0.95.

a 11 =0,776 a 12 =-0,130 a 13 =0,308

a 12 =0,904 a 22 =-0,210 a 23 =-0,420

a 31 =0,616 a 32 =0,902 a 33 =0,236

For j = 1 w 1 = ,w 2 ={a 11 ,a 21 ,a 31 },

.

For j = 2 w 1 ={a 12 ,a 22 }, w 2 ={ a 32 },

For j = 3 w 1 ={a 33 }, w 2 ={a 13 ,a 33 },

significant signs x 1 , x 2 , x 3 The warehouse of the head component starts at 100%. with the most significant additions signs x 2, sense of profitability. correct for name signs F 1 bude efficiency of production.

F 2 start as a component x 3 (fond_ddacha), called її the effectiveness of the main viceroy.

F 3 start with components x 1 ,x 2-in the analysis, you may not look at that, I will explain all 10% of the foreign options.

Literature.

Popov A.A.

Excel: Practical handbook, DESS kom.-M.-2000.

Dyakonov V.P., Abramenkova I.V. Mathcad7 in mathematics, physics and the Internet. View "Nomidzh", M.-1998 distribution 2.13. Victory regression.

L.A. Soshnikova, V.N. Tomashevich і ін. Bagatom Statistical Analysis in Economics ed. V.N. Tomashevich. - M. -Nauka, 1980.

Coleman V.A., O.V. Staroviriv, V.B. Turundayevskiy Theory of Imovings and Mathematical Statistics. -M. - Vishcha school-1991.

Before Iberla. Factorial analysis.-M. Statistics.-1980.

It is the essence of statistical procedures, which are directly related to the vision from a given set of changing subdivisions, which are closely related (correlated) between themselves. The change, which is included in one subdivision and correlates with oneself, but in the meaningful world of independent from the other submultiples, establish a factor. Meta-factorial analysis - the identification of officials is clearly not spared by officials for the help of helpless spared minions. In an additional way, the number of visual factors is calculated є the calculation of the correlation matrix, which is close to the visual factor, is correct. Qia matrix be called published correlation matrix. For this purpose, you can see how the matrix is ​​viewed from the original correlation matrix (from which analysis is done), you can calculate the difference between them. The Zalishkova matrix can be used for "bad weather", that is, those who are looking at the correlation cannot be rejected with sufficient accuracy on the basis of obvious factors. The methods of head components and factorial analysis do not have such a significant criterion, which allows one to judge the correctness of the solution. Another problem of the field is that when it comes to seeing the factors of the winery, there are no options for wrapping it up, but it is based on the quiet ones themselves, which give different solutions (factor structures are often used in the first way). Residual vibration between possible alternatives to all the middle endless numbers of mathematically equal solutions to lie in the wisdom of understanding the results of the interpretation. And the examples of the objective criterion for the assessment of young decisions are not important, the propagation of the selection of solutions can be unfounded and unconvincing.

It is essential that there are no clear statistical criteria for factorization. Protest, low values, for example, less than 0.7, to indicate the value of a speedy number of factors.

Met Kofіtsієnt vzamozvyazyu between deyakimyu and zagalny factor, which turns the world into the inflow of a factor into a sign, to be called a factorial navantazhennyam given sign from this zagalny factor.

The matrix, which can be stored in factorial numbers, is the number of hundred percent, as the number of large numbers of foreign factors, and the number of rows, as the number of external signs, is called a factor matrix.

The basis for the calculation of the factor matrix is ​​the matrix of paired coefficients of correlation of external signs.

The correlation matrix of the physical steps is interconnected between the skin pair of signs. Similarly, the factor matrix of the phase of the lineal connection of the skin signs with the skin zagalny factor.

The magnitude of the factorial navantazhennya does not change the modulus of one, but the sign її to speak about a positive and negative link is a sign of a factor.

What is more is the absolute value of the factorial navantazhennya signs according to the deyakogo factor, the bigger the world is the factor viznazhennya indicator.

The value of the factorial navantazhennya on a deyakoy factor, close to zero, to speak about those, so the factor is practically not injected into giving signsє.

The factor model gives the ability to calculate the contribution of factors to the initial variance of all signs. Pidsumovuchi squares of factorial nadkhodzhen for the skin factor for all the signs, recognizable of the addition to the backgone dispersion of the system and sign: what is the part of the cich nadhojen, which is more significant, important is the Danish factor.

At the same time, it is possible to discover and the optimal number of foreign factors, to complete the good description of the system of external signs.

The value (world of manifestation) of the factor in the surrounding object is called the factorial vagoyu of the given factor. Factornі vagi allow to rank, orderly ob'kti according to the skin factor.

The more factorial vagu of the deyakogo ob'єktu, the more that side of the manifestation is manifested more in the new one, as it is tolerated by these factors.

Factorial vagi can be either positive or negative.

Due to the fact that the factor є with standardized values ​​from the mean values, equal to zero, the factorial vagi, close to zero, seem to be about the middle level of the factor, positive - about those, about the steps of the middle, negative - about those. then won nizcha for the middle.

Practically, if the number of already known head components (or factors) is not greater, not m/ 2, as they explain the variance of not less than 70%, and the onset component of the addition to the total variance is not more than 5%, the factor model is used to achieve good results.

If you want to know the value of the factors and to protect them in the viglyadi of the pre-wintry ones, hit the vimikach Scores ... (Value) factor value, as a rule, lie between -3 to +3.

Factor analysis - more tugging and folding apparatus, lower method of head

component, that is why it is stuck in the same way that the results are

component analysis is not completely vlashtovuyut. Ale oskilki ci two methods

check the same data, it is necessary to adjust the results of the component and

factor analysis, that is, matrices of options, as well as an equal regression on

head components and local officials, comment on the similarity and insight

results.

The maximum number of officials m for a given number, the sign R start to fail

(P + m)<(р-m)2,

At the end of all the procedure of factor analysis, behind the addition of mathematical re-implementation, the factor fj is perceived through visual signs, so that the parameters of the linear diagnostic model can be recognized in an explicit view.

The method of leading components and factorial analysis є the supremacy of statistical procedures, which are directly related to the view from a given set of changing submultiples, which are tightly knitted (correlated) between themselves. The change, whether to enter into the same subdivision and correlate with oneself, but in the meaningful world of independent from the winners from the other subdivisions, are approved by the bureaucrats 1 ... Meta-factorial analysis - the identification of officials is clearly not spared by officials for the help of helpless spared minions.

Zagalna viraz for j The th factor can be written as follows:

de Fj (j change from 1 to k) - the main factories, Ui- characteristic, Aij- Constant, how to be victorious in the line combination k chinnik_v. Characteristics of officials can not be correlated with one with one or with other factors.

The procedures of the factorial-analytical processing, which are stagnant before they are taken out of the data, development, or the structure (algorithm) of the analysis, are based on one and the same basic stages: 1. Preparation of the data matrix. 2. Evaluating the matrix of interconnection of signs. 3. Factorization(At the same time, it is necessary to specify a number of factors that can be seen in the course of the factorial solution, and the method of calculation). On the whole stage (like on the offensive one), you can also assess how well the factorial solution to the close of the given data is taken away. 4. Overtannya - the reincarnation of factors, which lays down on the interpretation. 5. Number of factorial values by cutaneous factor for cutaneous caution. 6. interpretation of the tribute.

The winery of factorial analysis is tied to the need for one-hour analysis of the great number of coefficients of correlation between different scales. One of the problems associated with the methods of the head components and factor analysis of the field is in the fact that the criteria, which allowed us to reconsider the correctness of the known decision, is not considered. For example, with regressive analysis, it is possible to set indicators for fallow winter, set as an empirical way, with indicators calculated theoretically on the basis of a proponated model, and victorious to determine the correlation between them as criteria for the correctness of the calculation. In the discriminant analysis, the correctness of the decision is based on the fact that the belongingness of the probes is transferred to the quiet class (as well as real goodness, which is a great thing in life). Unfortunately, in the methods of the head components and factor analysis, there is no such critical criterion, but it is possible to judge the correctness of the decision. (the factorial structures start out cheaply in the same way). Residual vibration between possible alternatives to all the middle endless numbers of mathematically equal solutions to lie in the wisdom of understanding the results of the interpretation. And the examples of the objective criterion for the assessment of young decisions are not important, the propagation of the selection of solutions can be unfounded and unconvincing.

The third problem of the field is that the factorial analysis is often used to stop using the meta vryatuvati is badly thought out until it becomes clear that a statistical procedure is not necessary for the result. The combination of methods in the head components and factor analysis allows for the chaotic information to vibrate in the order of the concept (which is the most common reputation).

Another group of terms is referred to matrices, which are expected and interpreted as part of the solution. turn factors - the process of making a decision is the easiest to interpret for a given number of factors. There are two main classes of turns: orthogonalі skis... For the first time, all officials will vibrate orthogonal (not one for one) and will factorial navantage matrix It is a matrix of interconnections and is promoted by changing factors. The quantity navantazhen vіdobrazhaє stupіn zv'yazku kozhnoї sposterіgaєtsya zmіnnoї i cutaneous factor i іnterpretuєtsya yak koefіtsієnt korelyatsії mіzh sposterіgaєtsya i zmіnnoї factor (latentnoї zmіnnoї), and in that the furrows zmіnyuєtsya od -1 to 1. Rіshennya, otrimane pіslya orthogonal rotation, on іnterpretuєtsya osnovі analіzu The matrix of factorial navantages is due to the fact that the factors in the maximum level are tied to the fact that there is a change in the factor. In such a rank, the skin factor appears to be given by a group of primordial wines, which may be based on the newest factorial options.

If the obliqueness of the wrapping is displayed (i.e., it is a priori allowed for the possibility of correlating factors in oneself), then there will be more than a few additional matrices. Factor correlation matrix revenge the correlation between factors. Factorial navantage matrix, Zgadana vishche, split into two: structural matrix of interconnections mіzh factors і changing і factorial display matrix, Yaka small liniyyny interconnection between the skin is made possible to change the skin factor (without urahuvannya injected imposition of some factors on the other, so that the correlative factors rotate between oneself). The creation of oblique wrapping of the interpretation of factors is based on the formation of the primary wines (as it was before, as described above), and even more on the basis of the matrix of the factorial image.

Nareshty, for both turns to be calculated matrix of factor values, Yaka vikorystovuєtsya in special regression type for calculating factor values ​​(factor scores, indicators for factors) for skin care on the basis of the value for them of the primary snakes.

Determine the methods of head components and factorial analysis, meaningfully the same. In the course of the analysis of the head component method, there will be a model for the best explanation (maximum revision) of the variance of the experimental data that are taken for all the time. As a result, "components" are seen. In factorial analysis, it is transferred that the skin change is explained (determined) by the number of hypothetical zagalny factors (injected into all the changes) and characteristic factors (for the skin change of its own). The first numerical procedures are determined by such a rank, which is caused by the factors of variance, which are taken into account as a result of the reduction in the definition, as well as by the variance, which can be explained by specific factors, and the analysis of such variance, which can be explained more easily. As a result, there are enterprises called officials. However, since it’s already lost, from the zesty-psychological point of view, there is no difference in mathematical models of the statistical value, so for the time being, special explanations are not given, about how it’s too easy to go, we’ll find the factor of the term , so and according to the ratio to factors.

Checking vibrations and missing data. Chim is more vibrant, tim is more vigor of indicators in the interconnection. It is even more important for the mother to have a large enough vibe. The necessary size of the vibration can also be found in each step in the interconnection of indicators in the population in the whole and in the number of factors: in case of strong and reliable interconnection, there is a small amount of even less

So, vibirka, size like 50 viprobovanikh, estimate yak duzhe nasty, 100 - nasty, 200 - average, 300 - good, 500 - even better and 1000 - weird ( Comrey, Lee, 1992). Vyhodyachi from the mirkuvan tsikh, as the zagalny principle, you can recommend until you get the vibrations of at least 300 viprobes. For a version, rely on a sufficient number of marker winters with high factorial nominations (> 0.80) to reach vibrations close to 150 samples ( Guadagnoli, Velicer, 1988). normalcy for skin changes asymmetry(Some of the curve of a pre-licked rose is broken to the right, or even in some proportion to the theoretically normal curve) і ekssesu(The step of knotty uphill or bend down the "ringing" of the obvious connection, visually represented in the frequency diagrams, in the context of the "ringing" of the graph of proficiency, characteristic of a normal connection). As long as the change is in asymmetry and eksses, then it can be reconfigured by introducing a new change (as the function is unambiguous as seen) in such a rank, so that the new change was raised normally (the report: Tabachnik, Fidell, 1996, ch. 4).

Vlasnі vectors and vіdpovіdnі power numbers
for the staring butt

Since the correlation matrix is ​​diagonalizable, then for the rejection of the results of factor analysis it is possible to use the matrix algebra of power vectors and power values ​​(div. Dodatok 1). If the matrix is ​​diagonalizable, then the whole sutta of information about the factorial structure takes place in the її diagonal form. In the factorial analysis of the power, the numbers show the variance, which can be explained by factors. The factor with the greatest power value will explain the greatest variance and t. D., Do not reach the factors with small or negative power values, since you don’t want to go through the analysis. Matrix of factorial navantages є matrix of interconnections (interpreted as coefficients of correlation) and factors and changes. The first hundred percent - the goal of the correlation between the first factor and the skin wrinkle in terms of: part of the ticket (-.400), comfort of the complex (.251), Turning temperature (.932), Water temperature(.956). Another hundred percent is the correlation between other factors and skin changes: part of the ticket (.900), comfort of the complex(-.947), water temperature (.348), Water temperature(.286). The factor is interpreted on the basis of those who are strongly tied to him (i.e., Mayut according to a new high-ranking) of the winters. So, the first factor with the leading rank is "climatic" ( temperature of water and water), In that hour, yak another "economical" ( The parity of the ticket and the comfort of the complex).

Interpreting factors, then I will respect those who change, which may be very important for the first factor ( Turning temperatureі Water temperature( part of the ticketі comfort of the complex), Vzaєmopov'yazanі negatively (from a cheap resort you can ochіkuvati great comfort). The first factor is called unipolar (all changes are grouped at one pole), and the other is bipolar(Winter fell on two prototypes behind the zmist group - two poles). The sign of the plus sign, the positive pole, and the negative pole with the minus sign. At the same time, name the poles "positive" and "negative" when interpreting the factor, there is no doubt about the judgmental sense of "trash" and "good". The vibration of the sign is displayed for an hour and it is calculated in the form of a badge. orthogonal wrap

Wrap up to get stagnant to see the factors for maximizing the high proportions and minimizing the low ones. Know the numerical methods of wrapping, but more often than not the victorious turn varimax This is a procedure for maximizing variances. The whole turn maximizes the variance of factorial naphtha, the higher the increase in the number, and the lower one for the skin factor. Qia meta reach for help transformation matrix Λ:

transformation matrix- the whole matrix of sine and cosine kuta Ψ, for which turn. (Zvidsi and the name of the re-creation - turn That is, from the geometric point of view, the rotation of the axes about the cob of coordinates of the factorial space is observed. spirit of change- the whole variance, insured with additional factorial navantazhen. The price is a quadratic multiplier correlation of the miniscule, transferred to the model of the official. The income is calculated as a sum of squares of factorial navantages (SKN) for change for all officials. Table. 4 qualifications for vouchers for vouchers road (-.086) 2 + (. 981) 2 = .970, i.e. 97% variance vouchers for vouchers explained by factors 1 and 2.

The fraction of the variance of the factor for all the variables - the price of the SKN by the factor, is subdivided into the number of changes (in the case of orthogonal wrapping) 7 ... For the first official, the part of the dispersion of the road:

[(-.086)2+(-.071)2+(.994)2+(.997)2]/4 = 1.994/4 = .50,

that is, the first factor will explain 50% of the variance of the changes. Another factor will explain 48% of the variance of the winter and (due to the orthogonality of the wrapping) two factors in the sum explain 98% of the variance of the winter.

Linkage mіzh factorial navantazhennym, spіnotami, SKN,
variance and covariates of orthogonal factors for rotation

Part of the dispersion of the solution, which is explained by the factor, is a part covariance- the price of SKN for the factor, divided by the sum of spilot (the amount of SKN for the change). The first factor will explain 51% of the variance of the solution (1.994 / 3.915); other - 49% (1.919 / 3.915); two factors at once explain the whole covariance.

Eigenval - displays the value of the variance of a given number of factors. In terms of quality, it is recommended to vipisati all formulas for rejecting rosary values ​​according to changes. For example, for the first respondent:

1.23 = -.086(1.12) + .981(-1.16)

1.05 = -.072(1.12) - .978(-1.16)

1.08 = .994(1.12) + .027(-1.16)

1.16 = .997(1.12) - .040(-1.16)

Abo in algebraic form:

Z parosti vouchers = a 11F 1 + a 12F 2

Z comfort of the complex = a 2l F 1 + a 22F 2

Z temperature = a 31F 1 + a 32F 2

Z drive temperature = a 41F 1 + a 42F 2

If you want more money, you can use it more with more experience, but there is a change in the factor. Comri i Li ( Comrey, Lee, 1992) allowance, for a new option, for a change of 0.71 (explain 50% variance), - miracles, 0% variance) - even good, 0%) - good, 0%) - good і 0.32 (explain 10% variance) - weak ...

Let’s admit that you spend (until the deyakoy world is "bad") until you see a hundred people growing up in inches and centimeters. With such a rank, you have two wines. If you want to continue, for example, pouring in the food additives on the grower, you will be able to keep it alive offense wink? Ymovirno, nemaє, comrade K. Zrostannya є one characteristic of the people, right from the fact that in some odinitsy people are seen.

Occurrence between minors can be found for help diagrams rosciyuvannya... The line of regression is given a graphical representation of fallowness. As soon as there is a new change on the basis of the line of regression, the image on the diagram, then it will be possible to include in itself the most suttavi rice of both of them. Also, in fact, we have passed the number of winters and replaced two by one. Significantly, there is a new factor (change) in the action and the linear combination of two different types of change.