Determination of dynamic indicators. Applied program.

Costescu, Mihai-Radu

The analysis of chronological series presents some specific elements. The characteristic indicators of the chronological series, taking into account that the variable considered for keeping a track of these indicators is the time, are known as well as indicators of dynamics. These indicators can be divided into three groups: absolute, relative and medium.

In the first group, we present the absolute benefit (deficit). It is determined as the difference between absolute levels of one variable of the series successively taken and an arbitrary level considered as the basis of the comparison.

Two variants of this indicator are to be distinguished:

a) With fixed basis. Is calculated as the difference between the absolute levels of the series and an arbitrary level which is maintained constant. As referential level is usually used the initial level of the series, symbolized with x0. The calculation formula is:

[[DELTA].sub.n/0] = [X.sub.n] - [X.sub.0],

that is:

[[DELTA].sub.1/0] = [X.sub.1] - [X.sub.0], [[DELTA].sub.2/0] = [X.sub.2] - [X.sub.0], [[DELTA].sub.3/0] = [X.sub.3] - [X.sub.0], ...

The absolute benefit (deficit) with fixed basis highlights the increase (+) or the decrease (-) of the absolute levels compared to the level used as the basis for comparison.

b) With chain basis. Is calculated as the difference between each level, successively taken, and its previous, using the formula:

[[DELTA].sub.n/n-1] = [X.sub.n] - [X.sub.n-1],

that is:

[[DELTA].sub.1/0] = [X.sub.1] - [X.sub.0], [[DELTA].sub.2/1] = [X.sub.2] - [X.sub.1], [[DELTA].sub.3/2] = [X.sub.3] - [X.sub.2], ...

The absolute benefit (deficit) with chain basis highlights the increase (+) or the decrease (-) of the levels of the series from one period from another or from one moment to another.

By comparing the calculation formulas of the two variants, it appears that:

[[DELTA].sub.n/0] = [summation] [[DELTA].sub.k/k-1].

In the group of the relative indicators we present the rate and the dynamics rhythm.

The dynamics rate is calculated as the ratio between the absolute levels, successively taken and an arbitrary level used as the basis for comparison.

In the case of this indicator, two variants are to be distinguished, as well:

a) With fixed basis. Is calculated as the ratio between the absolute levels and an arbitrary level maintained constant, using the formula:

[I.sub.n/0] = [x.sub.n]/[x.sub.0]

that is:

[I.sub.1/0] = [x.sub.1]/[x.sub.0], [I.sub.2/0] = [x.sub.2]/[x.sub.0], [I.sub.3/0] = [x.sub.3]/[x.sub.0]...

b) With chain basis. Is calculated as the ratio between each level, successively taken, and its previous, using the formula:

[I.sub.n/n-1] = [x.sub.n]/[x.sub.n-1]

that is:

[I.sub.1/0] = [x.sub.1]/[x.sub.0], [I.sub.2/1] = [x.sub.2]/[x.sub.1], [I.sub.3/2] = [x.sub.3]/[x.sub.2] ...

By comparing the two calculation formulas, it appears that:

[I.sub.n/0] = [PI][I.sub.k/k-1].

The dynamics rates can be expressed in percentages, as well.

The dynamics rhythm has, in its turn, two variants:

a) With fixed basis, calculated with the formula:

[R.sub.n/0] = ([I.sub.n/0] - 1). 100

or, by developing:

[R.sub.n/0] = ([I.sub.n/0] -1) x 100 = [x.sub.n] - [x.sub.0]/[x.sub.0] x 100 = [[DELTA].sub.n/0]/[x.sub.0] x 100.

b) With chain basis, which has, basically, the same calculation methodology, except that, in its calculation, the indicators with chain basis are used:

[R.sub.n/n-1] = ([I.sub.n/n-1] - 1) x 100,

or:

[R.sub.n/-1] = [[DELTA].sub.n/n-1]/[x.sub.n-1] x 100

The dynamics rhythm is expressed exclusively in percentages and highlights the increase (+) or the decrease (-) of or the levels of a phenomenon, in percentages.

Finally, the third group, medium indicators, is divided into two subgroups:

a) Absolute medium indicators, including: a1) The average of the absolute levels.

If the chronological series is of intervals, the average is calculated as an arithmetic mean.

If the chronological series is of moments of time, the medium level is calculated as a chronological average; if the intervals between the moments are equal, then the simple chronological average is used; if the intervals between the moments are not equal, then the weighted chronological mean is used.

The calculation of the chronological average presupposes:

I. The calculation of mobile averages, as arithmetic means of two consecutive terms, from which one is repeated:

[[bar.x].sub.1] = [x.sub.1] + [x.sub.2]/2; [bar.[x.sub.2]] = [x.sub.2] + [x.sub.3]/2; [bar.[x.sub.3]] = [x.sub.3] + [x.sub.4]/2;... [[bar.x].sub.n-1] = [x.sub.n-1] + [x.sub.n]/2;

II. Calculation of the chronological average:

1. as simple arithmetic mean of mobile averages, if the intervals between the moments are equal:

[[bar.x].sub.c] = [[bar.x].sub.1] + [[bar.x].sub.2] + [[bar.x].sub.3] + ... + [bar.x].sub.n-1]/n-1

2. as weighted arithmetic mean of mobile averages, the weights being the intervals between the moments, if these intervals are different:

[[bar.x].sub.c] = [[bar.x].sub.1]] x [t.sub.1] + [[bar.x].sub.2]] x [t.sub.2] + [[bar.x].sub.3]] x [t.sub.3] + ... + [[bar.x].sub.n-1]] x [t.sub.n-1]/[t.sub.1] + [t.sub.2] + [t.sub.3] + ... + [t.sub.n-1]

a. 2) The medium benefit determined as a simple arithmetic mean of the benefits with chain basis:

[bar.[DELTA]] = [summation][[DELTA].sub.k/k-1]/n sau [bar.[DELTA]] = [[DELTA].sub.n/0]/n.

b) Relative medium indicators, including:

b. 1) The dynamics medium rate:

[bar.I] = [nth root of [PI][I.sub.k/k-1]] = [nth root of [I.sub.n/0]].

b.2) The dynamics minimum rhythm:

[bar.R] = ([bar.I] - 1) x 100.

The medium rhythm is expressed exclusively in percentages.

In order to determinate these indicators, a Turbo Pascal procedure has been realized, easy to attach to a program, presented below.

procedure indicatorii_dinamicii;
 type vect=array[1..20] of real;
 var dbf,ibf,rbf,dbl,ibl,rbl,val,t,medm:vect;
 dmed,imed,rmed,med:real;
 raspuns:char;
 i,n,eps:integer;

procedure citire(var eps,n:integer;var val,t:vect);
 var k:integer;
 begin {procedure citire}
 eps:=0;
 write('Introduceti numarul valorilor din sir: ');
 readln(n);
 for k:=1 to n do
 begin
 write('v(',k,') = ');
 readln(val[k]);
 end;
write('Seria este de momente de timp (y/n)?');
readln(raspuns);
if raspuns = 'y'
 then
 begin
 write('Intervalele dintre momente sunt egale (y/n)?');
 readln(raspuns);
 if raspuns = 'n'
 then
 begin
 for k:=1 to n-1 do
 begin
 write('intervalul ',k,' = ');
 readln(t[k]);
 end;
 eps:=1;
 end;
 end;
end;{procedure citire}

procedure scriere(n:integer;dbf,ibf,rbf,dbl,ibl,rbl:vect);
var i:integer;
begin {procedure scriere}
writeln;
writeln;
writeln('Indicatori cu baza fixa');
writeln('Sporuri/deficite');
for i:=1 to n-1 do
 begin
 write(' D',i+1,'/1 ');
 end;
writeln;
for i:=1 to n-1 do
 write(' ',dbf[i]:7:2,' ');
writeln;
writeln('Indici');
for i:=1 to n-1 do
 begin
 write(' ');
 end;
writeln;
for i:=1 to n-1 do
write(' ',ibf[i]:7:2,' ');
writeln;
writeln('Ritmuri (%)');
for i:=1 to n-1 do
 begin
 write(' R',i+1,'/1 ');
 end;
writeln;
for i:=1 to n-1 do
 write(' ',rbf[i]:7:2,' ');
writeln;
writeln;
writeln('Indicatori cu baza mobila');
writeln('Sporuri/deficite');
for i:=1 to n-1 do
 begin
 write(' D',i+1,'/',i,' ');
 end;
writeln;
for i:=1 to n-1 do
 write(' ',dbl[i]:7:2,' ');
writeln;
writeln('Indici');
for i:=1 to n-1 do
 begin
 write(' ');
 end;
writeln;
for i:=1 to n-1 do
 write(' ',ibl[i]:7:2,' ');
writeln;
writeln('Ritmuri (%)');
for i:=1 to n-1 do
 begin
 write(' R',i+1,'/',i,' ');
 end;
writeln;
for i:=1 to n-1 do
 write(' ',rbl[i]:7:2,' ');
writeln;
writeln;
end; {procedure scriere}

function media_ponderata(n:integer;val,t:vect):real;
var k:integer;
 sum,ttot:real;
 medmob:vect;
begin {function media_ponderata}
 sum:=0;
 ttot:=0;
 for k:=1 to n-1 do
 begin
 medmob [k]:=(val [k] +val [k+1])/2;
 sum:sum+medmob[k]*t[k];
 ttot:=ttot+t[k];
 end;
 media_ponderata:=sum/ttot;
end;{function media_ponderata}

function media(n:integer;val:vect):real;
 var k:integer;
 sum:real;
 medmob:vect;
 begin {function media}
 sum:=0;
 for k:=1 to n-1 do
 begin
 medmob [k]:=(val [k] +val [k+1])/2;
 sum:sum+medmob[k]*t[k];
 end;
media:=sum/(n-1);
end;{function media}

begin {procedure indicatorii_dinamicii}
citire(eps,n,val,t);
for i:=2 to n do
 begin
 dbf[i-1]:=val[i]-val[1];
 ibf[i-1]:=val[i]/val[1];
 rbf[i-1]:=(ibf[i-1]-1)*100;
 end;
for i:=2 to n do
 begin
 dbl[i-1]:=val[i]-val[i-1];
 ibl[i-1]:=val[i]/val[i-1];
 rbl[i-1]:=(ibl[i-1]-1)*100;
 end;

scriere(n,dbf,ibf,rbf,dbl,ibl,rbl);
dmed:=dbf[n-1]/(n-1);
imed:=exp(ln(ibf[n-1])/(n-1));
rmed:=(imed-1)*100;
if eps = 1 then med:=media_ponderata(n,val,t)
 else med:=media(n,val);
writeln;

writeln(' Media = ',med:7:2);
writeln(' Sporul/deficitul mediu al dinamicii = ',dmed:7:2);
writeln(' Indicele mediu al dinamicii = ',imed:7:2);
writeln(' Ritmul mediu al dinamicii = ',rmed:7:2);
end;{procedure indicatorii_dinamicii}

Bibliography:

1. Costescu, M.R., Vasilescu, N., Ionascu, C., Statistica si elemente de teoria sondajului. Editia a II a revizuita si imbunatatita (Statistics and survey theory elements. Edition II revised and improved), Universitaria Publishing House, Craiova, 2001

2. Costescu, M.R., Ionascu, C., Prelucrarea electronica a informatiei (Information electronic processing), Universitaria Publishing House, Craiova, 2001.

Mihai-Radu COSTESCU, University of Craiova, Faculty of Social Sciences, Political Sciences Specialization

Email: cmihairadu@yahoo.com