Factor data analysis. Factor analysis, its types and methods. Factor analysis by the method of chain substitutions

Factor analysis of profit allows you to assess the impact of each factor separately on the financial result as a whole. Read how to conduct it, and also download the methodology.

The essence of factor analysis

The essence of the factorial method is to determine the influence of each factor individually on the result as a whole. This is quite difficult to do, since the factors influence each other, and if the factor is not quantitative (for example, a service), then its weight is estimated by an expert, which leaves an imprint of subjectivity on the entire analysis. In addition, when there are too many factors influencing the result, the data cannot be processed and calculated without special mathematical modeling programs.

One of the most important financial indicators of an enterprise is profit. Within the framework of factor analysis, it is better to analyze the profit margin, where fixed costs absent, or profit from sales.

Find out the reasons for the changes using the Excel model

Download the finished model in Excel. It will help you find out how sales volume, price and sales structure influenced the revenue.

Factor analysis by the method of chain substitutions

In factor analysis, economists usually use the method of chain substitutions, however, mathematically, this method is incorrect and produces highly skewed results, which differ significantly depending on which variables are substituted first and which after (for example, in Table 1).

Table 1... Analysis of revenue depending on the price and quantity of products sold

	Base year		This year		Increase in revenue
		Revenue B 0		Revenue B 0	At the expense of prices In p	Due to the quantity In q
Option 1					P 1 Q 0 -P 0 Q 0	P 1 Q 1 -P 1 Q 0	B 1 -B 0
Option 1
Option 2					P 1 Q 1 -P 0 Q 1	P 0 Q 1 -P 0 Q 0	B 1 -B 0
Option 2

In the first variant, the revenue due to the price increased by 500 rubles, and in the second by 600 rubles; revenue due to the quantity in the first increased by 300 rubles, and in the second by only 200 rubles. Thus, the results vary significantly depending on the order of substitution. ...

You can more correctly distribute the factors influencing the final result depending on the margin (Nat) and the number of sales (Qty) (see Figure 1).

Picture 1

The formula for profit growth due to the margin: P nat = ∆ Nat * (Number (tech) + Number (base)) / 2

The formula for profit growth due to quantity: P count = ∆ Qty * (Nat (tech) + Nat (base)) / 2

An example of two-factor analysis

Consider an example in Table 2.

table 2... Example of two-factor analysis of revenue

	Base year		This year		Increase in revenue
		Revenue B 0		Revenue B 0	Due to the margin In p	quantity In q
					∆ P (Q 1 + Q 0) / 2	∆ Q (P 1 + P 0) / 2	B 1 -B 0
Product "A"

We got the averaged values between the variants of chain substitutions (see table 1).

Three-factor model for profit analysis

The three-factor model is much more complicated than the two-factor model (Figure 2).

Picture 2

The formula used to determine the influence of each factor in the 3-factor model (for example, markup, quantity, nomenclature) on the overall result is similar to the formula in the two-factor model, but more complicated.

P nat = ∆Nat * ((Number (tech) * Nom (tech) + Number (base) * Nom (base)) / 2 - ∆Kol * ∆Nom / 6)

P count = ∆Col * ((Nat (tech) * Nom (tech) + Nat (base) * Nom (base)) / 2 - ∆Nat * ∆Nom / 6)

P nom = ∆Nom * ((Nat (tech) * Kol (tech) + Nat (base) * Kol (base)) / 2 - ∆Nat * ∆Col / 6)

Analysis example

In the table, we gave an example of using the three-factor model.

Table 3... An example of calculating revenue using a three-factor model

Last year	This year	Revenue factors
				Nomenclature
		∆ Q ((N 1 P 1 + N 0 P 0) / 2 - - ∆ N ∆ P / 6)	∆ P ((N 1 Q 1 + N 0 Q 0) / 2 - - ∆ N ∆ Q / 6)	∆ N ((Q 1 P 1 + Q 0 P 0) / 2 - - ∆ Q ∆ P / 6)

If we look at the results of the analysis of proceeds by the factor method, then the greatest increase in proceeds occurred due to the increase in prices. Prices increased by (15/10 - 1) * 100% = 50%, the next in importance was the increase in the range from 3 to 4 units - the growth rate (4/3 - 1) * 100% = 33% and in last place " quantity ", which increased by only (120 / 100-1) * 100% = 20%. Thus, factors affect profit in proportion to the growth rate.

Four-factor model

Unfortunately, for a function of the form Pr = Kol av * Nom * (Price - Seb), there are no simple formulas for calculating the influence of each individual factor on the indicator.

Pr - profit;

Col av - the average quantity per item unit;

Nom - the number of stock items;

Price - price;

There is a calculation method based on the Lagrange finite-increment theorem, using differential and integral calculus, but it is so complex and time-consuming that it is practically not applicable in real life.

Therefore, to isolate each individual factor, more general factors are first calculated according to the usual two-factor model, and then their components in the same way.

General profit formula: Pr = Qty * Nat (Nat - markup on a unit of products). Accordingly, we determine the influence of two factors: quantity and margin. In turn, the number of products sold depends on the nomenclature and the number of sales per item on average.

We get Quantity = Kol av * Nom. And the margin depends on the price and cost, i.e. Nat = Price - Seb. In turn, the influence of the cost price on the change in profit depends on the amount of products sold and on the change in the cost price itself.

Thus, we need to separately determine the influence of 4 factors on the change in profit: Quantity, Price, Seb, Nom, using 4 equations:

Pr = Count * Nat
Quantity = Kol av * Nom
Zatr = Qty * Seb.
Exp = Qty * Price

An example of a four-factor model analysis

Let's look at an example. Initial data and calculations in the table

Table 4... An example of profit analysis using a 4-factor model

Last year
		Count (Wed) Q (Wed 0)				Profit P 0
						Q 0 * (P 0 -C 0)




			∑Q 0 P 0 / ∑Q 0	∑Q 0 P 0 / ∑Q 0

This year
		Count (Wed) Q (Wed 1)
						Q 1 * (P 1 -C 1)



Totals and weighted averages
			∑Q 1 P 1 / ∑Q 1	∑Q 1 P 1 / ∑Q 1

The influence of the factor on the change in profit
Nom N ∆	Number Q ∆	Count (Wed) Q (cf) ∆	Price P ∆		Nat H ∆
∆N * (Q (av 0) + Q (av 1)) / 2 * (H 1 + H 0) / 2	∆Q * (H 1 + H 0) / 2	∆Q (cf.) * (N 1 + N 0) / 2 * (H 1 + H 0) / 2	∆P * (Q 1 + Q 0) / 2	∆С * (Q 1 + Q 0) / 2	∆H * (Q 1 + Q 0) / 2



Totals and weighted averages

Note: the numbers in the Excel table may differ by several units from the data in the text description, because they are rounded to tenths in the table.

1. First, using the two-factor model (described at the very beginning), we decompose the change in profit into a quantitative factor and a margin factor. These are factors of the first order.

Pr = Count * Nat

Number ∆ = ∆Q * (H 1 + H 0) / 2 = (220 - 180) * (3.9 + 4.7) / 2 = 172

Nat ∆ = ∆H * (Q 1 + Q 0) / 2 = (4.7 - 3.9) * (220 + 180) / 2 = 168

Check: ∆Пр = Kol ∆ + Nat ∆ = 172 + 168 = 340

2. We calculate the dependence on the cost parameter. To do this, we split the costs into quantity and cost according to the same formula, but with a minus sign, since the cost reduces the profit.

Cost = Qty * Seb

Ceb∆ = - ∆C * (Q1 + Q0) / 2 = - (7.2 - 6.4) * (180 + 220) / 2 = -147

3. We calculate the dependence on the price. To do this, we split the revenue into quantity and price using the same formula.

Exp = Qty * Price

Price∆ = ∆P * (Q1 + Q0) / 2 = (11.9 - 10.3) * (220 + 180) / 2 = 315

Check: Nat∆ = Price∆ - Seb∆ = 315 - 147 = 168

4. We calculate the impact of the item on profit. To do this, we decompose the number of products sold by the number of units in the range and the average amount per unit of the range. So we will determine the ratio of the factor of quantity and the nomenclature in kind. After that, we multiply the data obtained by the average annual markup and convert it into rubles.

Qty = Nom * Qty (Wed)

Nom ∆ = ∆N * (Q (av 0) + Q (av 1)) / 2 * (H 1 + H 0) / 2 = (3 - 2) (73 + 90) / 2 * (4.7 + 3.9) = 352

Number (avg) = ∆Q (avg) * (N 1 + N 0) / 2 * (H 1 + H 0) / 2 = (73 - 90) * (2 + 3) / 2 * (4.7 + 3.9) = -180

Check: Qty ∆ = Nom ∆ + Qty (cf.) = 352-180 = 172

The above four-factor analysis showed that profit increased compared to last year due to:

increase in prices by 315 thousand rubles;
changes in the nomenclature by 352 thousand rubles.

And decreased due to:

cost growth by 147 thousand rubles;
drop in the number of sales by 180 thousand rubles.

It would seem a paradox: the total number of units sold in the current year compared to the previous year increased by 40 units, but the quantity factor shows a negative result. This is because the growth in sales was due to the increase in nomenclature items. If last year there were only 2 of them, then this year one more has been added. At the same time, in terms of quantity, goods “B” were sold in the reporting year for 20 units. less than in the previous one.

This suggests that the product "C" introduced in the new year partially replaced the product "B", but attracted new buyers, which the product "B" did not have. If next year product “B” continues to lose its position, then it can be removed from the assortment.

As for prices, their increase by (11.9 / 10.3 - 1) * 100% = 15.5% did not greatly affect sales in general. Judging by product "A", which was not affected by structural changes in the assortment, then its sales increased by 20%, despite the price increase by 33%. This means that the rise in prices is not critical for the firm.

Everything is clear with the cost price: it has grown and the profit has decreased.

Factor analysis of sales profit

Evgeny Shagin, CFO RusCherMet Management Company

To carry out factor analysis, you must:

choose a base for analysis - sales revenue, profit;
select the factors, the influence of which needs to be assessed. Depending on the chosen base of analysis, they can be: sales volume, cost price, operating expenses, non-operating income, interest on a loan, taxes;
assess the influence of each factor on the final indicator. In the basic calculation for the previous period, substitute the value of the selected factor from the reporting period and adjust the final indicator taking into account these changes;
determine the influence of the factor. Subtract from the obtained intermediate value of the estimated indicator its actual value for the previous period. If the figure is positive, the change in the factor had a positive impact, negative - negative.

Example of factor analysis of sales profit

Let's look at an example. The report on financial results Let us substitute the value of the sales volume for the current period (571,513,512 rubles instead of 488,473,087 rubles) for the previous period, all other indicators will remain the same (see table 5). As a result, the net profit increased by 83,040,425 rubles. (116,049,828 rubles - 33,009,403 rubles). This means that if in the previous period the company had managed to sell products for the same amount as in this one, then its net profit would have increased by just these 83,040,425 rubles.

Table 5... Factor analysis of profit by sales volume

Index	Previous period, rub.
Index		with substitution meaning factor from the current period
Volume of sales

Gross profit
Operating expenses

Operating profit
Loan interest
Profit before tax

Net profit
1 The value of the volume of sales for the current period. 2 The indicator is recalculated taking into account the adjustment of the sales volume.

Using a similar scheme, it is possible to assess the influence of each factor and recalculate the net profit, and bring the final results into one table (see table 6).

Table 6... Influence of factors on profit, rubles


Volume of sales
Cost price products sold, services
Operating expenses
Non-operating income / expenses
Loan interest

Total	32 244 671

As can be seen from table 6, the greatest impact in the analyzed period was made by the growth in sales (83,040,425 rubles). The sum of the influence of all factors coincides with the actual change in profit for the past period. Hence, we can conclude that the analysis results are correct.

Conclusion

In conclusion, I would like to understand: what do you need to compare profit with in factor analysis? With the last year, with the base year, with the competitors, with the plan? How to understand whether an enterprise has worked well this year or not? For example, a company has doubled its profit for the current year, it would seem that this is an excellent result! But at this time, competitors carried out technical re-equipment of the enterprise and from next year they will oust the lucky ones from the market. And if you compare with competitors, then their income is less, because instead of, say, advertising or expanding the range, they invested in modernization. Thus, everything depends on the goals and plans of the enterprise. From which it follows that the actual profit must be compared, first of all, with the planned one.

FACTOR ANALYSIS

The idea of factor analysis

When studying complex objects, phenomena, systems, the factors that determine the properties of these objects are very often impossible to measure directly, and sometimes even their number and meaning are unknown. But other quantities may be available for measurement, depending in one way or another on the factors of interest to us. Moreover, when the influence of an unknown factor of interest to us manifests itself in several measurable features or properties of an object, these features can reveal a close relationship with each other and the total number of factors can be much less than the number of measured variables.

To identify the factors that determine the measured features of objects, methods of factor analysis are used.

As an example of the application of factor analysis, one can point to the study of personality traits based on psychological tests... Personality traits do not lend themselves to direct measurement. They can only be judged by the behavior of a person or the nature of the answers to questions. To explain the results of the experiments, they are subjected to factor analysis, which allows us to identify those personal properties that affect the behavior of the individual.
At the heart of different methods factor analysis is based on the following hypothesis: the observed or measured parameters are only indirect characteristics of the object under study, in reality there are internal (hidden, latent, not directly observed) parameters and properties, the number of which is small and which determine the values of the observed parameters. These internal parameters are usually called factors.

The purpose of factor analysis is to concentrate the initial information, expressing a large number of considered features through a smaller number of more capacious internal characteristics phenomena that, however, are not directly measurable

It has been established that the isolation and subsequent observation of the level of common factors makes it possible to detect pre-failure states of an object at very early stages of defect development. Factor analysis allows you to track the stability of correlations between individual parameters. It is the correlations between the parameters, as well as between the parameters and general factors that contain the main diagnostic information about the processes. The use of the Statistica package tools when performing factor analysis eliminates the need for additional computing tools and makes the analysis clear and understandable for the user.

The results of factor analysis will be successful if it is possible to interpret the identified factors based on the meaning of the indicators characterizing these factors. This stage of work is very responsible; it requires a clear understanding of the meaningful meaning of the indicators that are involved in the analysis and on the basis of which the factors are identified. Therefore, in the preliminary careful selection of indicators for factor analysis, one should be guided by their meaning, and not by the desire to include as many of them as possible in the analysis.

The essence of factor analysis

Here are a few basic provisions of factor analysis. Let for the matrix NS the measured parameters of the object there is a covariance (correlation) matrix C, where R- number of parameters, n- the number of observations. By linear transformation X=QY+U you can reduce the dimension of the original factor space NS to level Y, wherein R"<<R... This corresponds to the transformation of the point characterizing the state of the object into j-dimensional space, into a new space of dimensions with a lower dimension R". Obviously, the geometric proximity of two or a set of points in the new factorial space means the stability of the state of the object.

Matrix Y contains unobservable factors, which are essentially hyperparameters characterizing the most general properties of the analyzed object. Common factors are most often chosen to be statistically independent, which facilitates their physical interpretation. Vector of Observed Signs NS makes sense of the consequence of changing these hyperparameters.

Matrix U consists of residual factors, which mainly include measurement errors x(i). Rectangular matrix Q contains factor loadings determining the linear relationship between features and hyperparameters.
Factor loads are the values of the correlation coefficients of each of the initial characteristics with each of the identified factors. The closer the relationship of this feature with the factor under consideration, the higher the value of the factor load. A positive sign of the factor loading indicates a direct (and a negative sign - an inverse) relationship of this trait with a factor.

Thus, the data on factor loadings make it possible to formulate conclusions about the set of initial characteristics reflecting a particular factor and about the relative weight of an individual characteristic in the structure of each factor.

The factor analysis model is similar to multivariate regression and analysis of variance models. The fundamental difference between the factor analysis model is that the vector Y is unobservable factors, while in regression analysis these are registered parameters. On the right-hand side of equation (8.1), the unknowns are the matrix of factor loads Q and the matrix of values of common factors Y.

To find the matrix of factor loadings, use the equation QQ t = S – V, where Q t is the transposed matrix Q, V is the covariance matrix of the residual factors U, i.e. ... The equation is solved by iterations while specifying some zero approximation of the covariance matrix V (0). After finding the matrix of factorial loads Q, the general factors (hyperparameters) are calculated according to the equation
Y = (Q t V -1) Q -1 Q t V -1 X

The statistical analysis package Statistica allows in the interactive mode to calculate the matrix of factor loadings, as well as the values of several predefined main factors, most often two - according to the first two main components of the original matrix of parameters.

Factor analysis in the Statistica system

Let us consider the sequence of performing factor analysis using the example of processing the results of a questionnaire survey of employees of an enterprise. It is required to identify the main factors that determine the quality of working life.

At the first stage, it is necessary to select variables for factor analysis. Using correlation analysis, the researcher tries to identify the relationship of the studied features, which, in turn, gives him the opportunity to select a complete and non-redundant set of features by combining strongly correlated features.

If factor analysis is carried out for all variables, then the results may not be entirely objective, since some variables are determined by other data, and cannot be regulated by the employees of the organization in question.

In order to understand which indicators should be excluded, we will construct a matrix of correlation coefficients in Statistica based on the available data: Statistics / Basic Statistics / Correlation Matrices / Ok. In the start window of this Product-Moment and Partial Correlations procedure (Fig. 4.3), the One variable list button is used to calculate the square matrix. Select all variables (select all), Ok, Summary. We get the correlation matrix.

If the correlation coefficient changes in the range from 0.7 to 1, then this means a strong correlation of indicators. In this case, one highly correlated variable can be excluded. Conversely, if the correlation coefficient is small, you can exclude the variable because it adds nothing to the total. In our case, there is no strong correlation between any variables, and factor analysis will be carried out for the full set of variables.

To run factor analysis, you need to call the module Statistics / Multivariate Exploratory Techniques (multivariate research methods) / Factor Analysis (factor analysis). The Factor Analysis module window will appear on the screen.

For analysis, select all the variables of the spreadsheet; Variables: select all, Ok. The Input file line indicates Raw Data. There are two types of source data available in the module - Raw Data and Correlation Matrix - a correlation matrix.

The MD deletion section specifies how to handle missing values:
* Casewise - a way to exclude missing values (by default);
* Pairwise - a pairwise way to exclude missing values;
* Mean substitution - substitution of mean instead of missing values.
The Casewise way is that in a spreadsheet that contains data, all rows that have at least one missing value are ignored. This applies to all variables. The Pairwise method ignores missing values not for all variables, but only for the selected pair.

Let's choose a way to handle missing values Casewise.

Statistica will process the missing values in the way that is indicated, calculate the correlation matrix and offer a choice of several methods of factor analysis.

After clicking the Ok button, the Define Method of Factor Extraction window appears.

The upper part of the window is informational. It says that the missing values were handled by the Casewise method. 17 observations were processed and 17 observations were accepted for further calculations. The correlation matrix is calculated for 7 variables. The lower part of the window contains 3 tabs: Quick, Advanced, Descriptives.

The Descriptives tab has two buttons:
1- view correlations, means and standard deviations;
2- build multiple regression.

By clicking on the first button, you can see the mean and standard deviations, correlations, covariance, build various graphs and histograms.

In the Advanced tab, on the left, select the Extraction method of 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 eigenvalue: 1 (eigenvalue).

Click Ok and Statistica will do the calculations quickly. The Factor Analysis Results window appears. As mentioned earlier, the results of factor analysis are expressed by a set of factor loadings. Therefore, further we will work with the Loadings tab.

The upper part of the window is informational:
Number of variables: 7;
Method (method of identifying factors): Principal components;
Log (10) determinant of correlation matrix: –1.6248;
Number of factors extracted: 2;
Eigenvalues: 3.39786 and 1.19130.
At the bottom of the window there are functional buttons that allow you to comprehensively view the analysis results, numerically and graphically.
Factor rotation - rotation of factors, in this drop-down window you can select different rotations of the axes. By rotating the coordinate system, a set of solutions can be obtained, from which it is necessary to choose an interpreted solution.

There are various methods for rotating the coordinates of space. Statistica offers eight such methods, presented in the Factor Analysis module. So, for example, the varimax method corresponds to a coordinate transformation: a rotation that maximizes variance. In the varimax method, a simplified description of the columns of the factor matrix is obtained, reducing all values to 1 or 0. In this case, the variance of the squares of the factor loads is considered. The factor matrix obtained using the varimax rotation method is more invariant with respect to the choice of different sets of variables.

Rotation by the quartimax method aims at a similar simplification only in relation to the rows of the factor matrix. Equimax occupies an intermediate position? rotating factors by this method simultaneously attempts to simplify both columns and rows. The considered rotation methods refer to orthogonal rotations, i.e. the result is uncorrelated factors. Direct oblimin and promax rotation methods refer to oblique rotations, which result in correlated factors. The term? Normalized? in the names of the methods indicates that the factor loads are normalized, that is, they are divided by the square root of the corresponding variance.

Of all the proposed methods, we will first look at the result of the analysis without rotating the coordinate system - Unrotated. If the result obtained turns out to be interpretable and suits us, then we can stop at this. If not, you can rotate the axes and see other solutions.

Click on the "Factor Loading" button and look at the factor loadings numerically.

Recall that factor loadings are the values of the correlation coefficients of each of the variables with each of the identified factors.

The factor load value greater than 0.7 shows that this feature or variable is closely related to the factor under consideration. The closer the relationship of this feature with the factor under consideration, the higher the value of the factor load. A positive sign of the factor loading indicates a direct (and a negative sign - an inverse) relationship between this trait and the factor.
So, from the factorial loadings table, two factors were identified. The first defines RSD - a sense of social well-being. The rest of the variables are due to the second factor.

The line Expl. Var (Fig. 8.5) shows the variance for a particular factor. The line Prp. Totl shows the proportion of variance attributable to the first and second factors. Consequently, the first factor accounts for 48.5% of the total variance, and the second factor - 17.0% of the total variance, everything else falls on other unaccounted factors. As a result, two identified factors explain 65.5% of the total variance.

Here we also see two groups of factors - OSB and the rest of the set of variables, of which the ZSR stands out - the desire to change jobs. Apparently, it makes sense to investigate this desire more thoroughly based on the collection of additional data.

Selection and refinement of the number of factors

Once you know how much variance each factor has identified, you can return to the question of how many factors should be kept. By its very nature, this decision is arbitrary. But there are some common guidelines, and in practice, following them gives the best results.

The number of common factors (hyperparameters) is determined by calculating the eigenvalues (Fig. 8.7) of the matrix X in the factor analysis module. To do this, in the Explained variance tab (Fig. 8.4), click the Scree plot button.

The maximum number of common factors can be equal to the number of eigenvalues of the parameter matrix. But with an increase in the number of factors, the difficulties of their physical interpretation increase significantly.

First, you can select only factors with eigenvalues greater than 1. In essence, this means that if a factor does not select a variance equivalent to at least the variance of one variable, then it is omitted. This is the most widely used criterion. In the example above, based on this criterion, only 2 factors (two principal components) should be retained.

You can find a place on the chart where the decrease in eigenvalues from left to right slows down as much as possible. It is assumed that only the "factorial talus" is located to the right of this point. In accordance with this criterion, 2 or 3 factors can be left in the example.
Fig. it can be seen that the third factor insignificantly increases the share of the total variance.

Factor analysis of parameters makes it possible to detect at an early stage a violation of the working process (the occurrence of a defect) in various objects, which is often impossible to notice by direct observation of the parameters. This is due to the fact that the violation of correlations between the parameters occurs much earlier than the change in one parameter. Such distortion of correlations makes it possible to detect factor analysis of parameters in a timely manner. To do this, it is enough to have arrays of registered parameters.

It is possible to give general recommendations on the use of factor analysis, regardless of the subject area.
* Each factor must have at least two measured parameters.
* The number of parameter measurements must be greater than the number of variables.
* The number of factors should be justified based on the physical interpretation of the process.
* Always strive to ensure that the number of factors is much less than the number of variables.

The Kaiser criterion sometimes retains too many factors, while the scree criterion sometimes retains too few factors. However, both criteria are quite good under normal conditions when there are relatively few factors and many variables. In practice, the more important question is when the resulting solution can be interpreted. Therefore, usually several solutions with more or fewer factors are investigated, and then one of the most meaningful ones is selected.

The space of the original features should be represented in homogeneous measurement scales, since this allows the use of correlation matrices in the calculation. Otherwise, the problem of "weights" of various parameters arises, which leads to the need to use covariance matrices in the calculation. Hence, an additional problem of repeatability of the results of factor analysis may appear when the number of features changes. It should be noted that this problem is simply solved in the Statistica package by switching to a standardized form of parameter representation. In this case, all parameters become equivalent in the degree of their connection with the processes in the research object.

Poorly conditioned matrices

If there are redundant variables in the initial data set and they have not been eliminated by correlation analysis, then the inverse matrix (8.3) cannot be calculated. For example, if a variable is the sum of two other variables selected for this analysis, then the correlation matrix for that set of variables cannot be inverted, and factor analysis fundamentally cannot be performed. In practice, this happens when trying to apply factor analysis to a set of highly dependent variables, which sometimes happens, for example, in the processing of questionnaires. Then you can artificially lower all the correlations in the matrix by adding a small constant to the diagonal elements of the matrix, and then standardize it. This procedure usually results in a matrix that can be inverted and therefore factor analysis can be applied to it. Moreover, this procedure does not affect the set of factors, but the estimates are less accurate.

Factorial and Regression Modeling of State Variable Systems

A system with variable states (SPS) is a system whose response depends not only on the input action, but also on a generalized constant in time parameter that determines the state. Variable amplifier or attenuator? this is an example of the simplest PCA, in which the transmission coefficient can be discretely or smoothly changed according to some law. The SPS study is usually carried out for linearized models, in which the transient process associated with a change in the state parameter is considered complete.

Attenuators made on the basis of L-, T- and U-shaped connections in series and in parallel connected diodes are most widespread. The resistance of the diodes under the influence of the control current can vary over a wide range, which makes it possible to change the frequency response and attenuation in the path. The independence of the phase shift when regulating the damping in such attenuators is achieved using reactive circuits included in the basic structure. Obviously, with different resistance ratios of parallel and series diodes, the same insertion attenuation level can be obtained. But the change in phase shift will be different.

Let us investigate the possibility of simplifying the automated design of attenuators, which excludes double optimization of correcting circuits and parameters of controlled elements. We will use an electrically controlled attenuator as the investigated SPS, the equivalent circuit of which is shown in Fig. 8.8. The minimum level of attenuation is provided in the case of a low element resistance Rs and a large element resistance Rp. As the element resistance Rs increases and the element resistance Rp decreases, the introduced attenuation increases.

The dependences of the change in the phase shift on the frequency and attenuation for the circuit without correction and with correction are shown in Fig. 8.9 and 8.10 respectively. In the corrected attenuator in the attenuation range of 1.3-7.7 dB and the frequency band of 0.01–4.0 GHz, a change in the phase shift of no more than 0.2 ° was achieved. In an attenuator without correction, the change in the phase shift in the same frequency band and attenuation range reaches 3 °. Thus, the phase shift is reduced by almost 15 times due to the correction.

We will consider the parameters of correction and control as independent variables or factors influencing the attenuation and change in the phase shift. This makes it possible, using the Statistica system, to carry out factorial and regression analysis of the SPS in order to establish physical regularities between the circuit parameters and individual characteristics, as well as to simplify the search for the optimal circuit parameters.

The initial data were formed as follows. For the correction parameters and control resistances differing from the optimal ones up and down on a frequency grid of 0.01--4 GHz, the insertion attenuation and change in the phase shift were calculated.

Statistical modeling methods, in particular, factor and regression analysis, which were not previously used for the design of discrete devices with variable states, allow us to identify the physical patterns of the operation of system elements. This contributes to the creation of the structure of the device based on a given criterion of optimality. In particular, this section has considered a phase-invariant attenuator as a typical example of a state-variable system. The identification and interpretation of factor loads affecting various studied characteristics allows changing the traditional methodology and significantly simplifying the search for correction parameters and control parameters.

It has been established that the use of a statistical approach to the design of such devices is justified both for evaluating the physics of their operation and for substantiating the schematic diagrams. Statistical modeling can significantly reduce the amount of experimental research.

results

Observation of common factors and corresponding factor loadings is a necessary identification of the internal patterns of processes.
In order to determine the critical values of the controlled distances between factor loads, it is necessary to accumulate and generalize the results of factor analysis for processes of the same type.
The application of factor analysis is not limited to the physical characteristics of the processes. Factor analysis is both a powerful method for monitoring processes and is applicable to the design of systems for a wide variety of purposes.

Factor analysis is one of the most powerful statistical tools for analyzing data. It is based on the procedure of combining groups of variables correlating with each other ("correlation pleiades" or "correlation nodes") into several factors.

In other words, the goal of factor analysis is to concentrate the initial information, expressing a large number of considered features through a smaller number of more capacious internal characteristics, which, however, cannot be directly measured (and in this sense are latent).

For example, let's hypothetically imagine a regional legislature with 100 deputies. Among the various issues on the agenda, the following are put up for voting: a) a bill proposing to restore the monument to V.I. Lenin on the central square of the city - the administrative center of the region; b) an appeal to the President of the Russian Federation with a demand to return all strategic industries to state ownership. The contingency matrix shows the following distribution of votes of deputies:

	Monument to Lenin (for)	Monument to Lenin (against)
Appeal to the President (for)	49	4
Appeal to the President (against)	6	41

Obviously, the votes are statistically linked: the overwhelming majority of the deputies who support the idea of restoring the monument to Lenin also support the return of strategic enterprises to state ownership. Likewise, most opponents of the restoration of the monument are at the same time also opponents of the return of enterprises to state ownership. At the same time, thematically, the voting is completely unrelated to each other.

It is logical to assume that the revealed statistical relationship is due to the existence of some hidden (latent) factor. Legislators, formulating their point of view on a wide variety of issues, are guided by a limited, small set of political positions. In this case, we can assume the presence of a hidden split in the deputies according to the criterion of support / rejection of conservative-socialist values. A group of "conservatives" stands out (according to our contingency table - 49 deputies) and their opponents (41 deputies). By identifying such splits, we will be able to describe a large number of separate votes through a small number of factors that are latent in the sense that we cannot detect them directly: in our hypothetical parliament, there has never been a vote in which MPs were asked to determine their attitude to conservative socialist values. We detect the presence of this factor based on a meaningful analysis of the quantitative relationships between the variables. Moreover, if in our example the nominal variables are deliberately taken - support of the bill with the categories “for” (1) and “against” (0), then in reality factor analysis effectively processes interval data.

Factor analysis is very actively used both in political science and in "neighboring" sociology and psychology. One of the important reasons for the great demand for this method is the variety of tasks that can be solved with its help. Thus, there are at least three “typical” goals of factor analysis:

· Reduction of dimension (reduction) of data. Factor analysis, highlighting the nodes of interrelated features and reducing them to some generalized factors, reduces the initial basis of description features. The solution to this problem is important in a situation where objects are measured by a large number of variables and the researcher is looking for a way to group them according to semantic features. The transition from many variables to several factors allows you to make the description more compact, get rid of uninformative and duplicate variables;

Revealing the structure of objects or characteristics (classification). This task is close to the one that is solved by the cluster analysis method. But if the cluster analysis takes the values of several variables as the "coordinates" of objects, then factor analysis determines the position of the object relative to factors (related groups of variables). In other words, using factor analysis, it is possible to assess the similarity and difference of objects in the space of their correlations, or in the factor space. The obtained latent variables are the coordinate axes of the factor space, the objects under consideration are projected onto these axes, which allows you to create a visual geometric representation of the studied data, convenient for meaningful interpretation;

Indirect measurement. The factors, being latent (empirically not observable), do not lend themselves to direct measurement. However, factor analysis makes it possible not only to identify latent variables, but also to quantify their value for each object.

Let us consider the algorithm and interpretation of the statistics of factor analysis using the example of data on the results of parliamentary elections in Ryazan region 1999 (federal district). To simplify the example, let us take electoral statistics only for those parties that have overcome the 5% threshold. The data are taken in the context of territorial election commissions (by cities and districts of the region).

The first step is to standardize the data by converting them to standard points (the so-called L-points, calculated using the normal distribution function).

TEAK (territorial election commission)	"Apple"	"Unity"	Block Zhirinovsky	OVR	Communist Party	THX
Ermishinskaya	1,49	35,19	6,12	5,35	31,41	2,80
Zakharovskaya	2,74	18,33	7,41	11,41	31,59	l b 3 "
Kadomskaya	1,09	29,61	8,36	5,53	35,87	1,94
Kasimovskaya	1,30	39,56	5,92	5,28	29,96	2,37
Kasimovskaya city	3,28	39,41	5,65	6,14	24,66	4,61
The same in standardized points (g-points)
Ermishinskaya	-0,83	1,58	-0,25	-0,91	-0,17	-0,74
Zakharovskaya	-0,22	-1,16	0,97	0,44	-0,14	0,43
Kadomskaya	-1,03	0,67	1,88	-0,87	0,59	-1,10
Kasimovskaya	-0,93	2,29	-0,44	-0,92	-0,42	-0,92
Kasimovskaya city	0,04	2,26	-0,70	-0,73	-1,32	0,01
Etc. (32 cases in total)

	"Apple"	"Unity"	BZ	OVR	Communist Party	THX
"Apple"
"Unity"	-0,55
BZ	-0,47	0,27
OVR	0,60	-0,72	-0,47
Communist Party	-0,61	0,01	0,10	-0,48
THX	0,94	-0,45	-0,39	0,52	-0,67

Even a visual analysis of the matrix of pairwise correlations makes it possible to make assumptions about the composition and nature of the correlation pleiades. For example, positive correlations are found for the Union of Right Forces, Yabloko and the Fatherland-All Russia bloc (the Yabloko-OVR, Yabloko-SPS and OVR-SPS pairs). At the same time, these three variables negatively correlate with the KPRF (support for the KPRF), to a lesser extent with "Unity" (support for "Unity") and even less with the BZ variable (support for the "Zhirinovsky Bloc"). Thus, presumably, we have two pronounced correlation constellations:

("Yabloko" + OVR + SPS) - the Communist Party;

("Yabloko" + OVR + SPS) - "Unity".

These are two different constellations, and not one, since there is no connection between Unity and the Communist Party of the Russian Federation (0.01). It is more difficult to make an assumption with respect to the BZ variable, here the correlations are less pronounced.

To test our assumptions, we need to CALCULATE the eigenvalues, factor scores, and factor loadings for each variable. Such calculations are quite complicated, require serious skills in working with matrices, so here we will not consider the computational aspect. Let's just say that these calculations can be done in two ways: the principal components method and the principal factors method. Principal component analysis is more common; statistical programs use it by default.

Let us dwell on the interpretation of eigenvalues, factor values and factor loadings.

The eigenvalues of the factors for our case are as follows:

bgcolor = white> 5

Factor	Eigenvalue	% total variation
1	3,52	58,75
2	1,14	19,08
3	0,76	12,64
4	0,49	S.22
0,05	0.80
6	0,03	0,51
Total	6	100%

The greater the eigenvalue of a factor, the greater its explanatory power (the maximum value is equal to the number of variables, in our case 6). One of the key elements of factor analysis statistics is the "% total variance" indicator. It shows how much of the variation (variability) of the variables is explained by the extracted factor. In our case, the weight of the first factor exceeds the weight of all other factors combined: it explains almost 59% of the total variation. The second factor explains 19% of the variation, the third explains 12.6%, etc. descending.

Having the eigenvalues of the factors, we can start solving the problem of reducing the dimension of the data. The reduction will occur due to the exclusion of factors with the least explanatory power from the model. And here the key question is how many factors to leave in the model and what criteria to be guided by. Thus, factors 5 and 6 are clearly redundant, which together explain just over 1% of the total variation. But the fate of factors 3 and 4 is no longer so obvious.

As a rule, the model contains factors whose eigenvalues exceed one (Kaiser criterion). In our case, these are factors 1 and 2. However, it is useful to check the correctness of removing four factors using other criteria. One of the most widely used methods is scree plot analysis. For our case, it looks like:

The graph got its name from its resemblance to a mountainside. "Scrap" is a geological term for debris rocks accumulating at the bottom of the rocky slope. "Rock" is really influential factors, "talus" is statistical noise. Figuratively speaking, you need to find a place on the graph where the “rock” ends and the “scree” begins (where the decrease in eigenvalues from left to right slows down greatly). In our case, the choice must be made from the first and second inflections corresponding to two and four factors. Leaving four factors, we get very high accuracy of the model (more than 98% of the total variation), but make it quite complex. Leaving two factors, we will have a significant unexplained part of the variation (about 22%), but the model will become concise and easy to analyze (in particular, visual). Thus, in this case, it is better to sacrifice some degree of accuracy in favor of compactness, leaving the first and second factors.

You can check the adequacy of the resulting model using special matrices of reproduced correlations and residual correlations. The matrix of reproduced correlations contains the coefficients that were restored for the two factors left in the model. Of particular importance in it is the main diagonal, on which the communities of variables are located (in the table are italicized), which show how accurately the model reproduces the correlation of a variable with the same variable, which should be one.

The residual coefficient matrix contains the difference between the original and reproduced coefficients. For example, the reproduced correlation between the ATP and Yabloko variables is 0.88, the initial one is 0.94. Remainder = 0.94 - 0.88 = 0.06. The lower the residual values, the higher the quality of the model.

Reproduced correlations
	"Apple"	"Unity"	BZ	OVR	Communist Party	THX
"Apple"	0,89
"Unity"	-0,53	0,80
BZ	-0,47	0,59	0,44
OVR	0,73	-0,72	-0,56	0,76
Communist Party	-0,70	0,01	0,12	-0,34	0,89
THX	0,88 -0,43		-0,40	0,66	-0,77	0,88
Residual coefficients
	"Apple"	"Unity"	BZ	OVR	Communist Party	THX
"Apple"	0,11
"Unity"	-0,02	0,20
BZ	0,00	-0,31	0,56
OVR	-0,13	-0,01	0,09	0,24
Communist Party	0,09	0,00	-0,02	-0,14	0,11
THX	0,06	-0,03	0,01	-0,14	0,10	0,12

As can be seen from the matrices, the two-factor model, while generally adequate, poorly explains individual relationships. So, the commonality of the BZ variable is very low (only 0.56), the value of the residual coefficient of connection between BZ and "Unity" is too high (-0.31).

It is now necessary to decide how important it is for this particular study to adequately represent the BZ variable. If the importance is high (for example, if the study is devoted to the analysis of the electorate of this particular party), it is correct to return to the four-factor model. If not, two factors can be left.
Taking into account the educational nature of our tasks, we will leave the simpler model.

Factor loadings can be represented as the correlation coefficients of each variable with each of the identified factors 1 as, the correlation between the values of the first factor variable and the values of the Yabloko variable is -0.93. All factor loadings are given in the factor display matrix -

The closer the relationship between the variable and the factor under consideration, the higher the value of the factor load. The positive sign of the factor loading indicates the straight line, and the negative sign indicates the inverse relationship of the variable with the factor.

Given the values of factor loadings, we can construct a geometric representation of the results of factor analysis. On the X-axis we will postpone the loads of variables by factor 1, on the Y-axis - loads of variables by factor 2 and we will get a two-dimensional factor space.

Before proceeding with a meaningful analysis of the results obtained, we will carry out one more operation - rotation. The importance of this operation is dictated by the fact that there is not one, but many variants of the factor loadings matrix, equally explaining the relationship of variables (intercorrelation matrix). It is necessary to choose a solution that is easier to interpret meaningfully. Such is the matrix of loads, in which the values of each variable for each factor are maximized or minimized (close to one or to zero).

Let's consider a schematic example. There are four objects located in factor space as follows:

The loads on both factors for all objects are significantly different from zero, and we are forced to use both factors to interpret the position of the objects. But if we “rotate” the entire structure clockwise around the intersection of the coordinate axes, we get the following picture:

In this case, the loads on factor 1 will be close to zero, and the loads on factor 2 will be close to one (the principle of a simple structure). Accordingly, for meaningful interpretation of the position of objects, we will use only one factor - factor 2.

There are quite a few methods for rotating factors. So, the group of methods of orthogonal rotation always preserves the right angle between the coordinate axes. These include vanmax (minimizes the number of variables with high factor loadings), quartimax (minimizes the number of factors needed to explain a variable), equamax (a combination of the two previous methods). Oblique rotation methods do not necessarily preserve right angles between axes (e.g. direct obiimin). The promax method is a combination of orthogonal and oblique rotation methods. In most cases, the vanmax method is used, which gives good results for most policy research tasks. Also, as with many other methods, it is a good idea to experiment with various technicians rotation.

In our example, after rotation by the varimax method, we obtain the following factor loadings matrix:

Accordingly, the geometric representation of the factor space will look like:

Now you can start meaningful interpretation of the results. The key opposition - the electoral split - according to the first factor is formed by the Communist Party of the Russian Federation, on the one hand, and Yabloko and the Union of Right Forces (to a lesser extent, OVR), on the other. Substantially - based on the specifics of the ideological attitudes of the named subjects of the electoral process - we can interpret this delimitation as a "left-right" split, which is "classic" for political science.

Opposition on factor 2 is formed by OVR and Unity. The latter is adjoined by the "Zhirinovsky block", but we cannot reliably judge its position in the factor space due to the peculiarities of the model, which poorly explains the connections of this particular variable. To explain this configuration, it is necessary to recall the political realities of the 1999 election campaign. At that time, the struggle within the political elite led to the formation of two echelons of the "party of power" - the Unity and Fatherland - All Russia blocs. The difference between them was not of an ideological nature: in fact, the population was asked to choose not from two ideological platforms, but from two elite groups, each of which had significant power resources and regional support. Thus, this split can be interpreted as "power-elite" (or, somewhat simplifying, "power-opposition").

In general, we get a geometric representation of a certain electoral space of the Ryazan region for these elections, if we understand the electoral space as a space for electoral choice, the structure of key political alternatives (“splits”). The combination of these two splits was very typical of the 1999 parliamentary elections.

Comparing the results of factor analysis for the same region in different elections, we can judge the presence of continuity in the configuration of the space of the electoral choice of the territory. For example, the factor analysis of the federal parliamentary elections (1995, 1999 and 2003), which took place in Tatarstan, showed a stable configuration of the electoral space. For the 1999 elections, only one factor was left in the model with an explanatory power of 83% of the variation, which made it impossible to plot a two-dimensional diagram. Factor loadings are given in the corresponding column.

If you look closely at these results, you will notice that in the republic, from election to election, the same basic split is reproduced: “the party of power” - all the others. ”The“ party of power ”in 1995 was the bloc“ Our Home - Russia "(PDR), in 1999 - OVR, in 2003 -" United Russia. "Over time, only the" details "change - the name of the" party in power. " choice.

We conclude this chapter with one practical piece of advice. By and large, the success of mastering statistical methods is possible only with intensive practical work with special programs (already repeatedly mentioned SPSS, Statistica or at least Microsoft Excel). It is no coincidence that the presentation of statistical techniques is conducted by us in the mode of work algorithms: this allows the student to independently go through all the stages of analysis, sitting at the computer. Without attempts at practical analysis of real data, the idea of the possibilities of statistical methods in political analysis will inevitably remain general and abstract. And today the ability to apply statistics to solve both theoretical and applied tasks- a fundamentally important component of the model of a political scientist.

Control questions and tasks

1. What measurement levels correspond to mean values - mode, median, arithmetic mean? What measures of variation are characteristic for each of them?

2. For what reasons is it necessary to take into account the shape of the distribution of variables?

3. What does the statement “There is a statistical relationship between two variables” mean?

4. What useful information about relationships between variables can be obtained from the analysis of contingency tables?

5. What can you learn about the relationship between variables, based on the values of the statistical tests chi-square and lambda?

6. Give a definition of the concept of "error" in statistical research. How can this indicator be used to judge the quality of the constructed statistical model?

7. What is the main purpose of correlation analysis? What characteristics of the statistical relationship does this method reveal?

8. How to interpret the value of the Pearson correlation coefficient?

9. Describe the method of analysis of variance. What other statistical methods use ANOVA statistics, and why?

10. Explain the meaning of the "null hypothesis" concept.

11. What is a regression line, what method is it used to construct?

12. What does the R coefficient show in the final statistics of the regression analysis?

13. Explain the term "multivariate classification method".

14. Explain the main differences between clustering through hierarchical cluster analysis and K-means.

15. How can cluster analysis be used to study the image of political leaders?

16. What is the main problem solved by discriminant analysis? Give the definition of the discriminant function.

17. Name three classes of problems that can be solved using factor analysis. Concretize the concept of "factor".

18. Describe the three main methods of checking the quality of a model in factor analysis (Kaiser criterion, scree criterion, matrix of reproduced correlations).

International migration of financial resources in context of factor analysis

25. J.-B. Say went down in the history of economics as the author of the factor theory of value. What are the main points of this theory?

Feasibility study of a construction project and analysis of collateral for the requested construction loan

Ministry of Agriculture of the Russian Federation

Federal state educational institution

Higher professional education

State University of Land Management

Department economic theory and management

Course work

In the discipline "Analysis and diagnostics of the financial activities of the enterprise"

On the topic: "Factor analysis of the elements of production."

Performed:

student of the 34th group

Maksimova N.S.

Checked:

Chirkova L.L.

Moscow 2009

Introduction …………………………………………………………………………… ..... 3

Chapter 1. Factor analysis of production elements ……………………………………………………………………… ..4

1.1. Factor analysis, its types and tasks ………………………………………………………………………………… ..4

1.2. Deterministic factor analysis. Modeling requirements ……………………………………………………………………… ..8

1.3 Methods and types of deterministic factor analysis ………………… ..10

Chapter 2 . Practical part ……………………………………………………… ..14

2.1. Methods for measuring the influence of factors in the analysis economic activity………………………………………………………………………….14

2.2. Factor analysis financial condition trucking company JSC “Enterprise 1564” …………………………………………….… .20

Conclusion …………………………………………………………………. …… ..24

List of used literature …………………………………………… ......... 25

Appendices ……………………………………………………………………… ..26

Introduction

Factor analysis- a set of methods of multivariate statistical analysis used to study the relationship between the values of variables. With the help of factor analysis, it is possible to identify hidden (latent) variable factors responsible for the presence of linear statistical relationships (correlations) between the observed variables.

The objectives of factor analysis:

reducing the number of variables;
determination of relationships between variables, their classification.

Factor analysis arose at the beginning of the 20th century, initially developed in the tasks of psychology. Charles Spearman and Raymond Cattel made a great contribution to the development of factor analysis.

Factor analysis methods:

principal component analysis
correlation analysis
maximum likelihood method

Factor analysis - determining the influence of factors on the result - is one of the strongest methodological decisions in the analysis of the economic activities of companies for decision-making. For leaders - an additional argument, an additional "angle of view".

However, in practice, it is rarely used for several reasons:

1) the implementation of this method requires some effort and a specific tool ( software product);

2) companies have other “eternal” priorities.

Chapter 1. Factor analysis of production elements

1.1 Factor analysis, its types and tasks.

Factor analysis is understood as a method of complex and systemic study and measurement of the impact of factors on the value of effective indicators.

In general, the following main stages of factor analysis can be distinguished:

1. Statement of the purpose of the analysis.

2. Selection of factors that determine the investigated performance indicators.

3. Classification and systematization of factors in order to provide an integrated and systematic approach to the study of their influence on the results of economic activity.

4. Determination of the form of dependence between the factors and the effective indicator.

5. Modeling the relationship between performance and factor indicators.

6. Calculation of the influence of factors and assessment of the role of each of them in changing the value of the effective indicator.

7. Working with a factor model (its practical use for managing economic processes).

The selection of factors for the analysis of a particular indicator is carried out on the basis of theoretical and practical knowledge in a particular industry. In this case, they usually proceed from the principle: the larger the complex of factors is investigated, the more accurate the results of the analysis will be. At the same time, it should be borne in mind that if this complex of factors is considered as a mechanical sum, without taking into account their interaction, without highlighting the main, determining ones, then the conclusions may be erroneous. In the analysis of economic activity (ACA), an interconnected study of the influence of factors on the value of effective indicators is achieved through their systematization, which is one of the main methodological issues of this science.

Important methodological issue in factor analysis, it is the definition of the form of dependence between factors and performance indicators: whether it is functional or stochastic, direct or inverse, rectilinear or curvilinear. It uses theoretical and practical experience, as well as ways of comparing parallel and dynamic series, analytical groupings of initial information, graphical, etc.

Modeling economic indicators also presents a complex problem in factor analysis, the solution of which requires special knowledge and skills.

Calculation of the influence of factors is the main methodological aspect in AHD. To determine the influence of factors on the final indicators, many methods are used, which will be discussed in more detail below.

The last stage of the factor analysis is the practical use of the factor model for calculating the reserves for the growth of the effective indicator, for planning and predicting its value when the situation changes.

Depending on the type of factor model, there are two main types of factor analysis - deterministic and stochastic.

Deterministic factor analysis is a technique for studying the influence of factors, the connection of which with the effective indicator is of a functional nature, that is, when the effective indicator of the factor model is presented in the form of a product, a quotient, or an algebraic sum of factors.

This view factor analysis is the most common, because, being quite simple to use (compared to stochastic analysis), it allows you to understand the logic of the main factors of enterprise development, to quantify their influence, to understand which factors and in what proportion can and should be changed to improve production efficiency ... We will consider in detail deterministic factor analysis in a separate chapter.

Stochastic analysis is a technique for studying factors, the connection of which with the effective indicator, in contrast to the functional one, is incomplete, probabilistic (correlation). If, with a functional (complete) dependence with a change in the argument, a corresponding change in the function always occurs, then with a correlation connection, a change in the argument can give several values of the increase in the function, depending on a combination of other factors that determine this indicator. For example, labor productivity at the same level of capital-labor ratio may not be the same at different enterprises. It depends on the optimal combination of other factors affecting this indicator.

Stochastic modeling is, to a certain extent, an addition and deepening of deterministic factor analysis. In factor analysis, these models are used for three main reasons:

it is necessary to study the influence of factors for which it is impossible to build a rigidly deterministic factor model (for example, the level of financial leverage);

it is necessary to study the influence of complex factors that cannot be combined in the same rigidly determined model;
it is necessary to study the influence of complex factors that cannot be expressed by one quantitative indicator (for example, the level of scientific and technological progress).

In contrast to the rigidly deterministic stochastic approach, implementation requires a number of prerequisites:

a) the presence of the totality;

b) a sufficient amount of observations;

c) randomness and independence of observations;

d) uniformity;

e) the presence of a distribution of signs close to normal;

f) the presence of a special mathematical apparatus.

The construction of a stochastic model is carried out in several stages:

qualitative analysis (setting the goal of the analysis, determining the population, determining the effective and factor indicators, choosing the period for which the analysis is carried out, choosing the analysis method);
preliminary analysis of the simulated population (checking the homogeneity of the population, excluding anomalous observations, clarifying the required sample size, establishing the distribution laws of the studied indicators);
construction of a stochastic (regression) model (clarification of the list of factors, calculation of estimates of the parameters of the regression equation, enumeration of competing variants of models);
assessment of the adequacy of the model (checking the statistical significance of the equation as a whole and its individual parameters, checking the correspondence of the formal properties of the estimates to the research tasks);
economic interpretation and practical use of the model (determination of the spatio-temporal stability of the constructed dependence, assessment of the practical properties of the model).

In addition to dividing into deterministic and stochastic, the following types of factor analysis are distinguished:

forward and backward;
single-stage and multi-stage;
static and dynamic;
retrospective and prospective (forecast).

In direct factor analysis, research is carried out in a deductive way - from the general to the particular. Reverse factor analysis carries out the study of cause-and-effect relationships by means of logical induction - from particular, individual factors to generalizing ones.

Factor analysis can be single-stage or multi-stage. The first type is used to study the factors of only one level (one level) of subordination without detailing them into their component parts. For example, . In multistage factor analysis, factors a and b are detailed into their constituent elements in order to study their behavior. The detailing of the factors can be continued further. In this case, the influence of factors of various levels of subordination is studied.

It is also necessary to distinguish between static and dynamic factor analysis. The first type is used when studying the influence of factors on performance indicators at the corresponding date. Another type is a technique for studying causal relationships in dynamics.

And finally, factor analysis can be retrospective, which studies the reasons for the increase in performance indicators over the past periods, and promising, which examines the behavior of factors and performance indicators in the long term.

1.2 Deterministic factor analysis. Modeling requirements.

Determinism(from Lat. determino - I define) - the doctrine of the objective, regular and causal conditionality of all phenomena. Determination is based on the provision on the existence of causality, that is, on such a connection of phenomena in which one phenomenon (cause), under well-defined conditions, gives rise to another (consequence). )

It might be useful to read: