table 4.

Table 4.

For this row: K=8, L=-8.

8 3.703 3,46

Finding the theoretical values ​​of the characteristic with (n-2) degrees

t 0.95,n 2=2.365,

those. with probability

assert that

there is a tendency in the dispersion (t K t theor) and there is a tendency in the average, since t L t theor. Therefore, we can talk about the presence of a trend in time

Average method

5.3. Methods for mechanical smoothing of time series

Very often the levels of economic time series fluctuate, with

In this case, the trend of development of an economic phenomenon over time is hidden by random deviations of levels in one direction or another. In order to more clearly identify the development trend of the process under study, including for the further application of forecasting methods based on trend

models, produce smoothing (flattening) time series.

Smoothing always involves some method of local averaging of data, in which non-systematic components cancel each other out.

Time series smoothing methods are divided into two main groups:

1) mechanical alignment of individual levels of a time series with

using actual values ​​of neighboring levels.

2) analytical alignment using a curve drawn

between specific levels of a series so that it reflects the tendency inherent in the series, and at the same time frees it from insignificant

hesitation;

The essence of mechanical smoothing methods is as follows.

The first few levels of the time series are taken, forming smoothing interval. For them, a polynomial is selected, the degree of which should be less than the number of levels included in the smoothing interval; using a polynomial, new, aligned level values ​​in the middle are determined

Simple moving average method.

The simplest smoothing method is moving average, in which

days terms, where m is the width of the smoothing interval. Instead of the average, you can use the median of the values ​​that fall within the smoothing interval.

If it is necessary to smooth out small random fluctuations, then the smoothing interval is taken as large as possible. If it is necessary to preserve smaller fluctuations, the smoothing interval is reduced. All other things being equal, it is recommended to take the smoothing interval odd.

To calculate the smoothed levels of the Y t series, the formula is used:

			Where p m 1 (if odd);

As a result of this procedure, (n-m+1) smoothed values ​​of the series levels are obtained; in this case, the first and last levels of the series are lost (not smoothed out). -

For even values t, after the smoothing procedure, the resulting series is usually centered (the average values ​​of two consecutive moving averages are found).

This method is applicable only for series that have a linear

trend. If the process is characterized by nonlinear development, then a simple moving average can lead to significant distortions.

When the trend of the aligned series has bends and it is desirable for the researcher to preserve the waves, then the weighted method is preferable.

moving average. When constructing a weighted moving average on

Each smoothing interval, the value of the central level is replaced by the calculated one, determined by the weighted arithmetic average formula:

			ytw i

where w i are weight coefficients determined by the method of least

squares, while leveling at each smoothing interval is most often carried out using second- or third-order polynomials11. For example, the weighting coefficients for interval 5 will be

the following: 35 1 [ 3, 12, 17, 12, 3] , and for the interval 7: 21 1 [ 2, 3, 6, 7, 6, 3, 2]

Example. A time series of product output volume (in thousand rubles) is specified. The levels of the Y (t) series are given in Table 5.

Let's choose a smoothing interval m=3 and smooth the simple moving average (third row of the table). After smoothing, an increasing trend is clearly visible.

11 Mikhtaryan V.S., Arkhipova M.Yu. and others. Econometrics: textbook / ed. Mikhtaryan V.S. M.: LLC

"Prospect", 2008, p. 293

Table 5

S(t)avg

S(t)in

smoothing interval

we will conduct

smoothing

weighted

moving average based on a second degree polynomial

(fourth

tables) using the given

higher weight

coefficients.

Exponential smoothing method.

When studying economic data, the influence of later observations on the process is sometimes important. This question is solved by the method

exponential smoothing. In this case, the current value of the temporary

series is smoothed taking into account a smoothing constant (weight), usually

designated. The calculation is carried out using the following formula:

S t Y t (1) S t 1 , (5.4),

Considering the recurrent expansion process for the quantities S t 1, S t 2 and

etc. according to formula (5.4), we obtain:

	) j Y t j (1)t Y 0
S t(1

where j is the number of periods of lag from moment t. According to formula (5.5)

the relative weight of each previous level decreases exponentially with distance from the moment for which the smoothed value is calculated.

Hence the name of this method.

When using the method in practice, problems arise in choosing a parameter and determining the initial level Y 0 . The higher the value

parameter, the less is the influence of previous levels. In each specific case, it is necessary to choose the most acceptable

meaning. Most often this is done by checking multiple values.

The problem of choosing the initial value Y 0 is solved as follows: for Y 0

the first value of the time series or the arithmetic mean is accepted

the first few members of the series.

Let's look at the previous example. Let's do an exponential

smoothing the time series (third row of the table)

The first smoothed value is equal to the first level of the series. The next smoothed value is calculated according to formula (5.3), where

Let's move on to the issue of smoothing time series of economic indicators. Very often, the levels of dynamics series fluctuate, while the trend in the development of an economic phenomenon over time is hidden by random deviations of the levels in one direction or another. In order to clearly identify the development trend of the process under study, including for the further application of forecasting methods based on trend models, time series are smoothed (aligned). Thus, smoothing can be considered as the elimination of the random component  t from a time series model.

The simplest method of mechanical smoothing is simple moving average method. First for the time series y 1 , y 2 , y 3 ,…, y n the smoothing interval is determined t (t< п). If it is necessary to smooth out small random fluctuations, then the smoothing interval is taken as large as possible; The smoothing interval is reduced if smaller fluctuations need to be preserved. All other things being equal, it is recommended to take the smoothing interval odd. For the first T levels of the time series, their arithmetic mean is calculated; this will be the smoothed value of the level of the series located in the middle of the smoothing interval. Then the smoothing interval is shifted one level to the right, the calculation of the arithmetic mean is repeated, etc.

To calculate smoothed levels of a series formula applies

for odd m;

for even T the formula becomes more complicated.

The result of this procedure is p - t + 1 smoothed values ​​of series levels; while the first R and the latest R series levels are lost (not smoothed).

Peculiarity exponential methodsmoothing is that in the procedure for finding the smoothing i of the th level, only the values ​​of the previous levels of the series are used ( i-1, i-2,...), taken with a certain weight, and the weight of the observation decreases as it moves away from the point in time for which the smoothed value of the series level is determined.

If for the original time series y 1 , y 2 , y 3 ,…, y n the corresponding smoothed values ​​of the levels are denoted by S t , t = 1,2, …, P, then exponential smoothing is carried out according to the formula

Here S 0 – quantity characterizing the initial conditions.

In practical problems of processing economic time series, it is recommended to choose the value of the smoothing parameter in the range from 0.1 to 0.3.

Example 4.4. Let's return to Example 1, which looks at Lewplan's quarterly sales volumes. We have already found out that an additive model corresponds to these data, i.e. In fact, sales volumes can be expressed as follows:

Y = U + V + E.

In order to eliminate the influence of the seasonal component, we will use the moving average method. Adding the first four values ​​gives the total sales for 1998. Dividing this sum by four gives the average sales score for each quarter of 1998, i.e.

(239 + 201 +182 + 297)/4 = 229,75;
(201+182+297+324)/4, etc.

The resulting value no longer contains a seasonal component, since it represents the average value for the year. We now have an estimate of the trend value for the middle of the year, i.e. for a point lying in the middle between quarters II and III. If you move forward sequentially at intervals of three months, you can calculate the average quarterly values ​​for the period April - March 1998 (251), July - June 1998 (270.25), etc. This procedure allows you to generate four-point moving averages for the original data set. The resulting set of moving averages represents the best estimate of the desired trend.

Now the obtained trend values ​​can be used to find estimates of the seasonal component. We expect:

Y – U = V + E.

Unfortunately, the trend estimates obtained by calculating the four-point averages refer to several different points in time than the actual data. The first estimate, equal to 229.75, represents the point coinciding with the middle of 1998, i.e. lies in the center of the interval of actual sales volumes in the II and III quarters. The second estimate, equal to 251, lies between the actual values ​​in the third and fourth quarters. We require deseasonalized average values ​​corresponding to the same time intervals as the actual values ​​for the quarter. The position of the deseasonalized averages over time is shifted by further calculating the averages for each pair of values. Let's find the average of the first estimates, centering them on July - September 1998, i.e.

(229,75 + 251)/2 = 240,4.

This is the deseasonalized average for July - September 1999. This deseasonalized value, which is called centered moving average, can be directly compared with the July–September 1998 actual value of 182. Note that this means there are no trend estimates for the first two or last two quarters of the time series. The results of these calculations are given in Table 4.5.

For each quarter, we have seasonal component estimates that include an error or residual. Before we can use the seasonal component, we need to go through the following two steps. Let's find the average values ​​of seasonal estimates for each season of the year. This procedure will reduce some error values. Finally, we adjust the average values, increasing or decreasing them by the same number so that their total sum is zero. This is necessary to average the values ​​of the seasonal component for the year as a whole.

Table 4.5. Estimation of the seasonal component

	Volume of sales Y, thousand pieces	in four quarter	sliding average for four quarter	Centered moving average U	seasonal component Y- U= V+ E
January-March 1998
April June
July-September
October December
January-March 1999
April June
July-September
October December
January-March 2000
April June
July-September
October December
January-March 2001

Table 4.6. Calculation of average values ​​of the seasonal component

Calculated Components		Quarter number
Average value
Seasonal assessment Components						Amount = -0.2
Adjusted seasonal component 1

The correction factor is calculated as follows: the sum of the estimates of the seasonal components is divided by 4. In the last column of the table. 4.5 these estimates are recorded under the corresponding quarterly values. The procedure itself is given in table. 4.6.

The value of the seasonal component once again confirms our conclusions made in example 4.1 based on the analysis of the diagram. Sales volumes for the two winter quarters exceed the average trend value by approximately 40 thousand units, and sales volumes for the two summer periods are below the average by 21 and 62 thousand units. respectively.

A similar procedure is applicable when determining seasonal variation for any period of time. If, for example, the season is the days of the week, to eliminate the influence of the daily seasonal component, a moving average is also calculated, but not by four, but by seven points. This moving average represents the mid-week trend value, i.e. on Thursday; thus, the need for a centering procedure is eliminated.

Ministry of Education of the Russian Federation

All-Russian Correspondence Financial and Economic Institute

Yaroslavl branch

Department of Statistics

Course work

by discipline:

"Statistics"

task number 19

Student: Kurashova Anastasia Yurievna

Specialty "Finance and Credit"

3rd year, periphery

Head: Sergeev V.P.

Yaroslavl, 2002

1. Introduction………………………………………………………………………………3 pages.

2. Theoretical part…………………………………………… …4 pages.

2.1 Basic concepts about dynamics series…………………………...4 p.

2.2 Methods for smoothing and leveling time series………………………………………………………………………………….6 p.

2.2.1 Methods of “mechanical smoothing”………………………6 p.

2.2.2 Methods of “analytical” alignment…………………. 8 pages

3. Calculation part…………………………………………… ……11 p.

4. Analytical part……………………………………………. .16 pp.

5. Conclusion………………………………………………………. 25 pp.

6. List of references……………………………………………………………… 26 pages.

7. Applications………………………………………………………. 27 pp.

Introduction

Complete and reliable statistical information is the necessary basis on which the process of economic management is based. All information of national economic significance is ultimately processed and analyzed using statistics.

It is statistical data that makes it possible to determine the volume of gross domestic product and national income, identify the main trends in the development of economic sectors, estimate the level of inflation, analyze the state of financial and commodity markets, study the standard of living of the population and other socio-economic phenomena and processes.

Mastering statistical methodology is one of the conditions for understanding market conditions, studying trends and forecasting, and making optimal decisions at all levels of activity.

The final, analytical stage of the study is complex, time-consuming and responsible. At this stage, average indicators and distribution indicators are calculated, the structure of the population is analyzed, and the dynamics and relationships between the phenomena and processes being studied are studied.

At all stages of research, statistics uses various methods. Statistical methods are special techniques and ways of studying mass social phenomena.

I. Theoretical part.

1.1 Basic concepts about dynamics series.

Time series are statistical data reflecting the development over time of the phenomenon being studied. They are also called dynamic series, time series.

Each row of dynamics has two main elements:

1) time indicator t;

2) the corresponding levels of development of the phenomenon being studied y;

Time indications in dynamics series are either specific dates (moments) or individual periods (years, quarters, months, days).

The levels of the dynamics series reflect a quantitative assessment (measure) of the development over time of the phenomenon being studied. They can be expressed in absolute, relative or average values.

The dynamics series differ according to the following characteristics:

1) By time. Depending on the nature of the phenomenon being studied, the levels of time series can relate either to certain dates (moments) of time, or to individual periods. In accordance with this, the dynamics series are divided into moment and interval.

Moment dynamics series display the state of the phenomena being studied at certain dates (moments) in time. An example of a moment series of dynamics is the following information on the payroll number of store employees in 1991 (Table 1):

Table 1

List of store employees in 1991

A feature of the moment series of dynamics is that its levels may include the same units of the population being studied. Although there are intervals in a moment series - intervals between adjacent dates in the series - the value of one or another specific level does not depend on the duration of the period between two dates. Thus, the bulk of the store personnel, making up the payroll as of January 1, 1991, who continue to work during a given year, are displayed in the levels of subsequent periods. Therefore, when summing the levels of the moment series, repeated counting may occur.

By means of moment series of dynamics in trade, commodity inventories, the state of personnel, the amount of equipment and other indicators are studied that reflect the state of the phenomena being studied at individual dates (points) in time.

Interval dynamics series reflect the results of the development (functioning) of the phenomena being studied over individual periods (intervals) of time.

An example of an interval series is data on retail turnover of a store in 1987–1991. (tab. 2):

table 2

The volume of retail turnover of the store in 1987 - 1991.

Volume of retail turnover, thousand rubles	885.7	932.6	980.1	1028.7	1088.4

Each level of an interval series already represents the sum of levels over shorter periods of time. In this case, a unit of the population that is part of one level is not included in other levels.

The peculiarity of the interval dynamics series is that each of its levels consists of data for shorter intervals (subperiods) of time. For example, summing up the turnover for the first three months of the year, we get its volume for the first quarter, and summing up the turnover for four quarters, we get its value for the year, etc. Other things being equal, the level of the interval series is greater, the longer the interval is, to which this level belongs.

The property of summing levels over successive time intervals makes it possible to obtain dynamics series for more enlarged periods.

Using interval series, dynamics in trade are used to study changes in the time of receipt and sale of goods, the amount of distribution costs and other indicators that reflect the results of the functioning of the phenomenon under study for individual periods.

Structure of the dynamics series:

Any series of dynamics can theoretically be represented in the form of components:

1) trend – the main tendency for the development of a time series (towards an increase or decrease in its levels);

2) cyclical (periodic fluctuations, including seasonal);

random fluctuations.

1. 2. Methods for smoothing and aligning time series.

Elimination of random fluctuations in the values ​​of series levels is carried out by finding “averaged” values. Methods for eliminating random factors are divided into two groups:

1. Methods of “mechanical” smoothing of fluctuations by averaging the values ​​of the series relative to other nearby levels of the series.

2. Methods of “analytical” alignment, i.e., first determining the functional expression of the tendency of the series, and then new, calculated values ​​of the series.

1.2. 1 Methods of “mechanical” smoothing.

These include:

A. The method of averaging over two halves of a series, when the series is divided into two parts. Then, two values ​​of the average levels of the series are calculated, from which the trend of the series is graphically determined. It is obvious that such a trend does not sufficiently fully reflect the basic pattern of development of the phenomenon.

b. A method of enlarging intervals, in which the length of time intervals is increased and new values ​​of series levels are calculated.

V. Moving average method. This method is used to characterize the development trend of the statistical population under study and is based on the calculation of average levels of the series for a certain period. Sequence for determining the moving average:

The smoothing interval or the number of levels included in it is set. If three levels are taken into account when calculating the average, the moving average is called three-term, five levels are called five-term, etc. If small, random fluctuations in levels in the dynamics series are smoothed out, then the interval (the number of the moving average) is increased. If the waves are to be preserved, the number of members is reduced.

The first average level is calculated using simple arithmetic:

y1 = Sy1/m, where

y1 – 1st level of the row;

m – member of the moving average.

The first level is discarded, and the calculation of the average includes the level following the last level involved in the first calculation. The process continues until the last level of the studied series of dynamics y n is included in the calculation of y.

Based on a series of dynamics constructed from average levels, the general trend in the development of the phenomenon is revealed.

The negative side of using the moving average method is the formation of shifts in fluctuations in series levels due to the “sliding” of aggregation intervals. Smoothing using a moving average can lead to the appearance of “reverse” fluctuations, when a convex “wave” is replaced by a concave one.

Recently, an adaptive moving average has begun to be calculated. Its difference is that the average value of the attribute, also calculated as described above, does not refer to the middle of the series, but to the last period of time in the enlargement interval. Moreover, it is assumed that the adaptive average depends on the previous level to a lesser extent than on the current one. That is, the more time intervals between the level of the series and the average value, the less influence the value of this level of the series has on the value of the average.

d. Exponential average method. The exponential average is an adaptive moving average, calculated using weights that depend on the degree of “remoteness” of individual levels of the series from the average value. The value of the weight decreases as the level moves away along the chronological line from the average value in accordance with the exponential function, therefore such an average is called exponential. In practice, multiple exponential smoothing of the dynamics series is used, which is used to predict the development of the phenomenon.

Conclusion: the methods included in the first group, due to the calculation methods used, provide the researcher with a very simplified, inaccurate idea of ​​the trend in the dynamics series. However, the correct application of these methods requires the researcher to have a depth of knowledge about the dynamics of various socio-economic phenomena.

02/16/15 Viktor Gavrilov

38133 0

A time series is a sequence of values ​​that change over time. I will try to talk about some simple but effective approaches to working with such sequences in this article. There are many examples of such data - currency quotes, sales volumes, customer requests, data in various applied sciences (sociology, meteorology, geology, observations in physics) and much more.

Series are a common and important form of describing data, as they allow us to observe the entire history of changes in the value of interest to us. This gives us the opportunity to judge the “typical” behavior of a quantity and deviations from such behavior.

I was faced with the task of choosing a data set on which it would be possible to clearly demonstrate the features of time series. I decided to use international airline passenger traffic statistics because this data set is very clear and has become somewhat of a standard (http://robjhyndman.com/tsdldata/data/airpass.dat, source Time Series Data Library, R. J. Hyndman). The series describes the number of international airline passengers per month (in thousands) for the period 1949 to 1960.

Since I always have at hand, which has an interesting tool “” for working with rows, I will use it. Before importing data into a file, you need to add a column with a date so that the values ​​are tied to time, and a column with the name of the series for each observation. Below you can see what my source file looks like, which I imported into Prognoz Platform using the Import Wizard directly from the time series analysis tool.

The first thing we usually do with a time series is plot it on a graph. Prognoz Platform allows you to build a chart by simply dragging a series into the workbook.

Time series on a chart

The symbol ‘M’ at the end of the series name means that the series has monthly dynamics (the interval between observations is one month).

Already from the graph we see that the series demonstrates two features:

• trend– on our chart this is a long-term increase in the observed values. It can be seen that the trend is almost linear.
• seasonality– on the graph these are periodic fluctuations in value. In the next article on the topic of time series, we will learn how we can calculate the period.

Our series is quite “neat”, however, there are often series that, in addition to the two characteristics described above, demonstrate another one - the presence of “noise”, i.e. random variations in one form or another. An example of such a series can be seen in the chart below. This is a sine wave mixed with a random variable.

When analyzing series, we are interested in identifying their structure and assessing all the main components - trend, seasonality, noise and other features, as well as the ability to make forecasts of changes in value in future periods.

When working with series, the presence of noise often makes it difficult to analyze the structure of the series. To eliminate its influence and better see the structure of the series, you can use series smoothing methods.

The simplest method of smoothing series is a moving average. The idea is that for any odd number of points in the series sequence, replace the central point with the arithmetic mean of the remaining points:

Where x i– initial row, s i– smoothed series.

Below you can see the result of applying this algorithm to our two series. By default, Prognoz Platform suggests using anti-aliasing with a window size of 5 points ( k in our formula above it will be equal to 2). Please note that the smoothed signal is no longer so affected by noise, but along with the noise, naturally, some useful information about the dynamics of the series also disappears. It is also clear that the smoothed series lacks the first (and also the last) k points. This is due to the fact that smoothing is performed on the central point of the window (in our case, the third point), after which the window is shifted by one point, and the calculations are repeated. For the second, random series, I used smoothing with a window of 30 to better identify the structure of the series, since the series is “high-frequency” with a lot of points.

The moving average method has certain disadvantages:

• A moving average is inefficient to calculate. For each point, the average must be recalculated anew. We cannot reuse the result calculated for a previous point.
• The moving average cannot be extended to the first and last points of the series. This can cause a problem if these are the points we are interested in.
• The moving average is not defined outside the series, and as a result, cannot be used for forecasting.

Exponential smoothing

A more advanced smoothing method that can also be used for forecasting is exponential smoothing, also sometimes called the Holt-Winters method after its creators.

There are several variations of this method:

• single smoothing for series that have no trend or seasonality;
• double smoothing for series that have a trend, but no seasonality;
• triple smoothing for series that have both a trend and seasonality.

The exponential smoothing method calculates the values ​​of a smoothed series by updating the values ​​calculated in the previous step using information from the current step. Information from the previous and current steps is taken with different weights that can be controlled.

In the simplest version of single smoothing, the ratio is:

Parameter α defines the relationship between the unsmoothed value at the current step and the smoothed value from the previous step. At α =1 we will take only the points of the original series, i.e. there will be no smoothing. At α =0 row we will take only smoothed values ​​from previous steps, i.e. the series will turn into a constant.

To understand why smoothing is called exponential, we need to expand the relationship recursively:

It is clear from the relationship that all previous values ​​of the series contribute to the current smoothed value, but their contribution fades exponentially due to an increase in the degree of the parameter α .

However, if there is a trend in the data, simple smoothing will “lag” behind it (or you will have to take the values α close to 1, but then the smoothing will be insufficient). You need to use double exponential smoothing.

Double smoothing already uses two equations - one equation evaluates the trend as the difference between the current and previous smoothed values, then smoothes the trend with simple smoothing. The second equation performs smoothing as in the simple case, but the second term uses the sum of the previous smoothed value and the trend.

Triple smoothing includes one more component - seasonality, and uses another equation. In this case, there are two variants of the seasonal component – ​​additive and multiplicative. In the first case, the amplitude of the seasonal component is constant and does not depend over time on the base amplitude of the series. In the second case, the amplitude changes along with the change in the base amplitude of the series. This is exactly our case, as can be seen from the graph. As the series grows, the amplitude of seasonal fluctuations increases.

Since our first row has both a trend and seasonality, I decided to select triple smoothing parameters for it. In Prognoz Platform, this is quite easy to do, because when the parameter value is updated, the platform immediately redraws the graph of the smoothed series, and visually you can immediately see how well it describes our original series. I settled on the following values:

We will look at how I calculated the period in the next article on time series.

Typically, values ​​between 0.2 and 0.4 can be considered as first approximations. Prognoz Platform also uses a model with an additional parameter ɸ , which dampens the trend so that it approaches a constant in the future. For ɸ I took the value 1, which corresponds to the normal model.

I also made a forecast of the series values ​​using this method for the last 2 years. In the figure below, I marked the starting point of the forecast by drawing a line through it. As you can see, the original series and the smoothed one coincide quite well, including during the forecasting period - not bad for such a simple method!

Prognoz Platform also allows you to automatically select optimal parameter values ​​using a systematic search in the space of parameter values ​​and minimizing the sum of squared deviations of the smoothed series from the original one.

The methods described are very simple, easy to apply, and provide a good starting point for analyzing the structure and forecasting of time series.

Read more about time series in the next article.

Very often, the levels of the dynamics series fluctuate, while the trend of development of the phenomenon over time is hidden by random deviations of the levels in one direction or another. In order to more clearly identify the development trend of the process under study, including for the further application of forecasting methods based on trend models, smoothing(leveling) time series.

Time series smoothing methods are divided into two main groups:

1. analytical alignment using a curve drawn between specific levels of a series so that it reflects the tendency inherent in the series and at the same time frees it from minor fluctuations;

2. mechanical alignment of individual levels of a time series using the actual values ​​of adjacent levels.

The essence of mechanical smoothing methods is as follows. Several levels of the time series are taken, forming smoothing interval. For them, a polynomial is selected, the degree of which should be less than the number of levels included in the smoothing interval; using a polynomial, new, leveled level values ​​are determined in the middle of the smoothing interval. Next, the smoothing interval is shifted one row level to the right, the next smoothed value is calculated, and so on.

The simplest method of mechanical smoothing is simple moving average method.

2.4.1.Simple moving average method.

First for the time series: the smoothing interval is determined. If it is necessary to smooth out small random fluctuations, then the smoothing interval is taken as large as possible; The smoothing interval is reduced if smaller fluctuations need to be preserved.

For the first levels of the series, their arithmetic mean is calculated. This will be the smoothed value of the level of the series located in the middle of the smoothing interval. Then the smoothing interval is shifted one level to the right, the calculation of the arithmetic mean is repeated, and so on. To calculate the smoothed levels of a series, the formula is used:

where (if odd); for even numbers the formula becomes more complicated.

As a result of this procedure, smoothed values ​​of the series levels are obtained; in this case, the first and last levels of the series are lost (not smoothed). Another disadvantage of the method is that it is applicable only to series that have a linear trend.

2.4.2.Weighted moving average method.

The weighted moving average method differs from the previous smoothing method in that the levels included in the smoothing interval are summed with different weights. This is due to the fact that the approximation of the series within the smoothing interval is carried out using a polynomial not of the first degree, as in the previous case, but of degree starting from the second.

The arithmetic weighted average formula is used:

,

wherein the weights are determined using the least squares method. These weights are calculated for different degrees of the approximating polynomial and different smoothing intervals.

1. for polynomials of the second and third orders, the numerical sequence of weights at the smoothing interval has the form: , and for has the form: ;

2. for polynomials of the fourth and fifth degrees and with a smoothing interval, the sequence of weights is as follows: .

For the distribution of weights over the smoothing interval, obtained using the least squares method, see Diagram 1.

2.4.3.Exponential smoothing method.

The same group of methods includes the exponential smoothing method.

Its peculiarity is that in the procedure for finding the smoothed level, only the values ​​of the previous levels of the series are used, taken with a certain weight, and the weight of the observation decreases as it moves away from the point in time for which the smoothed value of the series level is determined.

If for the original time series

the corresponding smoothed values ​​are denoted by , then exponential smoothing is carried out according to the formula:

Where smoothing parameter ; the quantity is called discount factor.

Using the given recurrence relation for all levels of the series, starting from the first and ending with the moment of time, we can obtain that the exponential average, that is, the value of the level of the series smoothed by this method, is a weighted average of all previous levels.