Identifying indicators of dynamics. Application Program.

Costescu, Mihai-radu

Time series analysis has some specific features. Characteristic indicators to the series, considering the fact that time is that variable that defines these indicators, are also known as indicators of the dynamics. These indicators can be divided into three groups: absolute, relative and average.

The first group represents the absolute growth or reduction, which is determined as the difference between the absolute successive levels of one of the variables of the series and a given level considered as a basic level of comparison.

There are two variants of this indicator:

a) Fixed base. It is calculated as the difference between absolute levels of the series and a certain level remained constant. The original level of the series, symbolized by x0, is typically used as a referential level. Relationship calculation is:

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

that is:

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Fixed base growth or reduction shows the increase (+) or decrease (-) to the absolute levels as a basis for comparison.

b) Base chain. It is calculated as the difference between each successive level and its predecessor:

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

that is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Base chain growth or reduction shows the increase (+) or decrease (-) of the levels of the series from one period to another or from one point to another.

Comparing the relations of the two variants of calculation, we can see that:

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From the group of relative indicators we consider the index of dynamics and the rate of dynamics.

Dynamic index is calculated as the ratio of absolute levels, taken in succession, and a certain level considered as a basis for comparison.

In this case we can also see two variants:

a) Fixed base. It is calculated as the ratio of absolute levels and a constant level using the relation:

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that is:

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b) Base chain. It is calculated as the ratio between each successive level and its predecessor, using the relation:

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that is:

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Comparing the two relations, we can see that:

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Dynamic indices can also be expressed as a percentage.

Growth rate is in turn two versions:

a) Fixed base, computed with:

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

or, more detailed:

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b) Base chain, which mainly has the same calculation method, except that, in its calculation, are used base chain indicators:

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

or:

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Growth rate is expressed only as a percentage and shows the increase (+) or decrease (-) levels of a phenomena.

Finally, the third group, the average indicators, is divided into two groups:

a) Average absolute indicators, including:

a1) Average absolute levels

If the time series is expressed in periods of time, the average is calculated as an arithmetic mean.

If the time series is expressed in moments of times, the average is calculated as a chronological average; if the intervals between moments are equal it is used the simple chronological average; if the intervals between moments are not equal it is used a weighted chronological average.

Chronological averaging involves:

> calculating mobile averages, as average of two consecutive terms, one of which is repeating:

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> calculating chronological averages:

[check] as a simple arithmetic average of moving averages, if the intervals between moments are equal:

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[check] as a weighted arithmetic average, where the weights are the intervals between moments of times if these intervals are different:

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a. 2) The average growth as a simple arithmetic average of the growths with base chain:

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b) Average relative indicators, including:

b. 1) Average index dynamics:

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b.2) The average rate of growth:

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

The average is only expressed in percent.

To determine these indicators, a procedure easy to be attached to a Turbo Pascal program was created.

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('I',i+1,'/1');
      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('I',i+1,'/',i,'');
      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, Editura Universitaria, Craiova, 2001

2. Costescu, M.R., Ionascu, C., Prelucrarea electronica a informatiei, Editura Universitaria, Craiova, 2001

Mihai-Radu COSTESCU,

University of Craiova,

Faculty of Social Sciences, Political Sciences Specialization

Email: cmihairadu@yahoo.com