INDICATORII SERIILOR CRONOLOGICE

*�,QGLFDWRULL�DEVROX L

Indicatorii de nivel VXQW� FKLDU� WHUPHQLL� XQHL� VHULL� IRUPDWH� GLQ� LQGLFDWRUL� DEVROX L� (y1, ...yt, ..., yn.).

Nivelul totalizat al termenilor

∑

=

n

tty

1

.

0RGLILF ULOH absolute

• FX�ED] �IL[ ��∆t/1):

∆t/1=yt - y1 , unde nt ,2=

• FX�ED] �vQ�ODQ ��∆t/t-1):

∆t/t-1=yt - yt-1 , unde nt ,2=

5HOD LL�XWLOH:

1/m

m

2t1t/t ∆=∆∑

=− , unde nm ≤

∆t/1 - ∆t-1/1 = ∆t/t-1 , unde nt ,3=

Analiza seriilor cronologice

*�Indicatorii relativi

,QGLFH�GH�GLQDPLF � • FX�ED] �IL[ ��It/1):

11/ y

yI t

t = sau 1001

(%)1/ ⋅=y

yI t

t , unde nt ,2=

• FX�ED] �vQ�ODQ �(It/t-1):

11/

−− =

t

ttt y

yI sau 100

1(%)1/ ⋅=

−−

t

ttt y

yI , unde nt ,2=

5HOD LL�XWLOH:

1/2

1/ m

m

ttt II =∏

=− , unde nm ≤

1/1/1

1/−

−

= ttt

t II

I , unde nt ,3=

5LWPXO�GH�GLQDPLF • FX�ED] �IL[ (Rt/1):

1001

11 ⋅

−=

y

yyR

tt sau ( ) ntIR tt ,2,%100%11 =−=

• FX�ED] �vQ�ODQ �(Rt/t-1):

1001

11/ ⋅

−=

−

−−

t

tttt y

yyR sau %100(%)1/1/ −= −− tttt IR , nt ,2=

9DORDUHD�DEVROXW �D�XQXL�SURFHQW�GH�GLQDPLF • FX�ED] �IL[ (At/1):

1/

1/1/

t

tt R

A∆

= sau 100

11/

yAt =

• FX�ED] �vQ�ODQ (At/t-1):

1/

1/1/

−

−−

∆=

tt

tttt R

A sau 100

11/

−− = t

tt

yA

*�Indicatorii medii

Nivelului mediu • SHQWUX�R�VHULH�FURQRORJLF �GH�LQWHUYDOH�GH�WLPS�IRUPDWH�GLQ�LQGLFDWRUL�DEVROX L�

n

yy

n

tt∑

== 1

• pentru o serie de momente cu intervale egale între momente (PHGLD�FURQRORJLF �VLPSO ):

12

......2 132

1

−

+++++++=

−

n

yyyyy

y

y

nni

cr

• pentru o serie de momente cu intervale neegale între momente (mHGLD� FURQRORJLF �SRQGHUDW �:

∑−

=

−−

++

++

++

=1

1

11212

11

2...

2...

22

n

ii

nn

iii

cr

d

dy

ddy

ddy

dy

y

'H�UH LQXW�F ��SHQWUX�VHria de momente cu intervale neegale între datele înregistrate, media FURQRORJLF �SRQGHUDW �HVWH�VLQJXUXO�LQGLFDWRU�PHGLX�FH�FDUDFWHUL]HD] �VHULD��

0RGLILFDUHD�PHGLH�DEVROXW :

11/

−∆

=∆ ∑ −

ntt

sau 1

1

−−

=∆n

yyn

Indicele mediu de�GLQDPLF )(I :

11/

−−∏= n

ttII sau 1

1

−= nn

y

yI

'DF � GLVSXQHP� GH� PDL� PXO L� LQGLFL� PHGLL� GH� GLQDPLF � FH� FDUDFWHUL]HD] � PDL� PXOWH�VXESHULRDGH� VXFFHVLYH� GH� WLPS�� LQGLFHOH�PHGLX� FH� FDUDFWHUL]HD] � vQWUHDJD� SHULRDG � VH� FDOFXOHD] �astfel:

∑⋅⋅⋅⋅= =

k

ii

ki

nn

k

ni

nnIIIII 1 21 ......21 ,

în care:

I - LQGLFHOH�PHGLX�JHQHUDO�GH�GLQDPLF �

iI - LQGLFLL�PHGLL�SDU LDOL�GH�GLQDPLF �

ni - QXP UXO�LQGLFLORU�FX�ED] �vQ�ODQ �FH�LQWU �vQ�FRPSRQHQ D�ILHF UXL�LQGLFH�PHGLX�SDU LDO�

k - QXP UXO�VXESHULRDGHORU��DGLF �DO�LQGLFLORU�PHGLL�SDU LDOL�

5LWPXO�PHGLX�GH�GLQDPLF

( ) %100(%) −= IR

AJUSTAREA SERIILOR CRONOLOGICE (YROX LD�RULF UXL� IHQRPHQ� vQ� WLPS�HVWH� UH]XOWDQWD�XQRU� LQIOXHQ H�GH�QDWXU � VLVWHPDWLF � úL� D�

altora de tip aleator. Componentele sistematice sunt: • trendul �WHQGLQ D�JHQHUDO �� • sezonalitatea�FDUH�VH�PDQLIHVW �VXE�IRUP �GH�DEDWHUL�GH�OD�WHQGLQ D�JHQHUDO �OD�LQWHUYDOH�

UHJXODWH�GH�WLPS�PDL�PLFL�GH�XQ�DQ��VHPHVWUX��WULPHVWUX��OXQ ��GHFDG �� • ciclicitatea� FDUH� VH� SUH]LQW � VXE� IRUP � GH� IOXFWXD LL� vQ� MXUXO� WHQGLQ HL�� vQUHJLVWUDWH� OD�

perioade mai mari de un an. Componentele aleatoare�VH�PDQLIHVW �VXE�IRUPD�XQRU�DEDWHUL�vQWkPSO WRDUH�GH�OD�FHHD�FH�DUH�

VLVWHPDWLF�HYROX LD�YDULDELOHL�DQDOL]DWH� Prin DMXVWDUHD� WHUPHQLORU� XQHL� VHULL� GH� GDWH� VWDWLVWLFH�� VH� vQ HOHJH� RSHUD LD� GH� vQORFXLUH� D�

WHUPHQLORU� UHDOL� FX� WHUPHQL� WHRUHWLFL� FH� H[SULP � OHJLWDWHD� VSHFLILF � GH� GH]YROWDUH� RELHFWLY � D�IHQRPHQHORU�OD�FDUH�VH�UHIHU �GDWHOH�

'LVSHUVLD�WRWDO ( )2

yσ �VH�FDOFXOHD] �GXS �IRUPXOD�

( )n

yyiy

2

2 ∑ −=σ

Dispersia termenilor seriei de la valorile ajustate ( )2/ ryσ � VLQWHWL]HD] � LQIOXHQ D� IDFWRULORU�

reziduali -� IDFWRUL� QHvQUHJLVWUD L� -� �vQ� FD]XO� VHULLORU� FURQRORJLFH� WR L� IDFWRULL� FX� H[FHS LD� IDFWRUXOXL�WLPS��úL�VH�FDOFXOHD] �FX�IRUPXOD�

( )n

Yyiti

ry

2

2/

∑ −=σ ,

în care

itY �UHSUH]LQW �YDORDUHD��WHRUHWLF �D�YDULDELOHL�y vQ�IXQF LH�GH�WLPS�

Dispersia valorilor ajustate de la valoarea medie 2

/ tyσ sLQWHWL]HD] �YDULD LD�SURGXV �QXPDL�

de modificarea factorului timp:

( )n

yYit

ty

2

2/

∑ −=σ

Metode simple de ajustare a seriilor cronologice

Metoda mediilor mobile

6H�IRORVHúWH�SHQWUX�VHULLOH�FDUH�SUH]LQW �RVFLOD LL�VH]RQLHUH�úL�FLFOLFH�� Mediile mobile�VXQW�PHGLL�SDU LDOH��FDOFXODWH�GLQWU-XQ�QXP U�SUHVWDELOLW�GH�WHUPHQL��vQ�FDUH�VH�

vQORFXLHúWH�SH�UkQG�SULPXO�WHUPHQ�FX�WHUPHQXO�FH�XUPHD] �vQ�VHULD�FDUH�WUHEXLH�V �ILH�DMXVWDW �

ÌQ�SUDFWLF �SXWHP�FDOFXOD�PHGLL�PRELOH�GLQWU-XQ�QXP U�LPSDU�GH�WHUPHQL�VDX�GLntr-XQ�QXP U�SDU�GH�WHUPHQL�vQ�IXQF LH�GH�SHULRGLFLWDWHD�LQIOXHQ HL�IDFWRULORU�VH]RQLHUL�

Când ajustarea se face pe baza mediilor mobile calculate dintr-XQ� QXP U� SDU� GH� WHUPHQL��PHGLLOH�PRELOH�VH�RE LQ�vQ�GRX �WUHSWH�

1) medii mobile provizorii ( )tY �FDUH�VH�SODVHD] �vQWUH�WHUPHQLL�VHULHL�

2) medii mobile definitive sau centrate ( )tY ��FDUH�VH�SODVHD] �vQ�GUHSWXO�WHUPHQLORU�VHULHL�úL�UHSUH]LQW �YDORULOH�DMXVWDWH�DOH�WHUPHQLORU�UHVSHFWLYL�GLQ�VHULD�LQL LDO �

0HWRGD�JUDILF

AceVW�SURFHGHX�SUHVXSXQH� UHSUH]HQWDUHD�JUDILF �D� VHULHL�GH�GDWH�HPSLULFH�SULQ�FURQRJUDP �

�KLVWRULRJUDP ���XUPDW �GH�WUDVDUHD�YL]XDO �D�GUHSWHL�VDX�FXUEHL��DVWIHO�vQFkW�V �DLE �DEDWHUL�PLQLPH�ID �GH�SR]L LD�YDORULORU�UHDOH�vQ�JUDILF�

0HWRGD�PRGLILF ULL�PHGLL absolute

$MXVWDUHD� SULQ� DFHVW� SURFHGHX� VH� IRORVHúWH� DWXQFL� FkQG�� SUHOXFUkQG� VHULD� GH� GDWH�� VH� RE LQ�

PRGLILF UL�DEVROXWH�FX�ED] �vQ�ODQ �DSURSLDWH�FD�YDORDUH�XQHOH�GH�DOWHOH�� )XQF LD�GH�DMXVWDUH�

∆−+= )1(1 tyYt , unde nt ,1=

sau

∆+= it tyYi 0 ,

unde: y0�UHSUH]LQW �WHUPHQXO�OXDW�FD�ED] �GH�DMXVWDUH��DFHD�YDORDUH�FDUH�VH�DSURSLH�FHO�PDL�PXOW�GH�

GUHDSWD�VDX�FXUED�WUDVDW �YL]XDO�vQ�JUDILF�� ti UHSUH]LQW �YDULDELOD�GH�WLPS�vQ�UDSRUW�FX�ED]D�GH�DMXVWDUH�IRORVLW ��SR]L LD�SH�FDUH�WHUPHQXO�

UHVSHFWLY�R�DUH�ID �GH�WHUPHQXO�DOHV�FD�ED] ��� 0HWRGD�LQGLFHOXL�PHGLX�GH�GLQDPLF �

$FHVW�SURFHGHX�VH�IRORVHúWH�DWXQFL�FkQG�WHUPHQLL�VHULHL�DX�WHQGLQ D�GH�FUHúWHUH�GH�IRUPD�XQHL�SURJUHVLL�JHRPHWULFH��vQ�FDUH�UD LD�SRDWH�IL�FRQVLGHUDW �FD�HJDO �FX�LQGLFHOH�PHGLX�GH�GLQDPLF � ( )I .

)XQF LD�GH�DMXVWDUH�

11

−⋅= tt IyY

sau

i

i

tt IyY ⋅= 0 ,

unde: y0�UHSUH]LQW �WHUPHQXO�OXDW�FD�ED] �GH�DMXVWDUH� ti UHSUH]LQW �YDULDELOD�GH�WLPS�vQ�UDSRUW�FX�ED]D�GH�DMXVWDUH�IRORVLW ��SR]L LD�SH�FDUH�WHUPHQXO�UHVSHFWLY�R�DUH�ID �GH�WHUPHQXO�DOHV�FD�ED] ���

Metode analitice de ajustare 0HWRGHOH�DQDOLWLFH�DX� OD�ED] �XQ�PRGHO�PDWHPDWLF�� vQ� FDUH� WHQGLQ D�FHQWUDO �D�HYROX LHL� VH�

H[SULP �FD�R�IXQF LH�GH�WLPS� y = f(t)�QXPLW �IXQF LH�GH�DMXVWDUH� în care: t - UHSUH]LQW �YDORULOH�YDULDELOHL�LQGHSHQGHQWH��WLPSXO�� y - UHSUH]LQW � YDORULOH� YDULDELOHL� GHSHQGHQWH� �IHQRPHQHOH�� FDUH� VXQW� SUH]HQWDWH� vQ� VHULD�

FURQRORJLF � $OHJHUHD� WLSXOXL� GH� IXQF LH� FDUH� VH� SRWULYHúWH� FHl mai bine pentru exprimarea trendului se

IDFH�SH�ED]D�XUP WRDUHORU�FULWHULL�DSOLFDELOH�RS LRQDO�

• FULWHULXO� ED]DW� SH� UHSUH]HQWDUHD� JUDILF �� 6H� FRQVWUXLHúWH� FURQRJUDPD� úL� VH� DSUHFLD] �IRUPD�WHQGLQ HL�GH�HYROX LH�

• FULWHULXO�GLIHUHQ HORU��6H�SURFHGHD] �OD�FDOFXOXO�GLIHUHQ HORU�DEVROXWH�FX�ED] �vQ�ODQ �GH�RUGLQXO� XQX�� GRL� HWF�� SkQ � FkQG� RE LQHP� GLIHUHQ HOH� GH� RUGLQ i aproximativ constante DMXVWDUHD�I FkQGX-VH�GXS �SROLQRPXO�GH�JUDGXO i.

'DF �IHQRPHQXO�FHUFHWDW�V-D�GH]YROWDW�vQ�SURJUHVLH�JHRPHWULF ��DGLF �LQGLFLL�FX�ED] �vQ�ODQ �

VXQW� FRQVWDQ L� �It/t-1� � FRQVWDQW��� DGPLWHP� F � VHULD� FURQRORJLF � UHVSHFWLY � SUH]LQW � R� WHQGLQ �H[SRQHQ LDO �

ÌQ� XUPD� DOHJHULL� IXQF LHL� GH� DMXVWDUH� GXS � FULWHULLOH� SUH]HQWDWH� VH� LPSXQH� HVWLPDUHD�SDUDPHWULORU�DFHVWRU�IXQF LL�XWLOL]kQG�PHWRGD�FHORU�PDL�PLFL�S WUDWH��$FHDVW �PHWRG �DUH�FD�IXQF LH�RELHFWLY�PLQLPL]DUHD�VXPHL�S WUDWHORU�DEDWHULORU�YDORULORU�UHDOH�GH�OD�FHOH�DMXVWDWH�GHFL�

( )∑ − 2miniti Yy , unde ti= 1, 2, ... , n

Trend liniar

it btaYi

+= ,

în care:

itY - UHSUH]LQW �YDORULOH�DMXVWDWH�FDOFXODWH�vQ�IXQF LH�GH�YDORULOH�FDUDFWHULVWLFLL�IDFWRULDOH��ti);

a - UHSUH]LQW �SDUDPHWUXO�FDUH�DUH�VHQV�GH�P ULPH�PHGLH�úL�DUDW �FH�QLYHO�DU�IL�DWLQV y�GDF �LQIOXHQ D� WXWXURU� IDFWRULORU�FX�H[FHS LD�FHOXL� vQUHJLVWUDW��DU� IL� IRVW�FRQVWDQW �SH� WRDW �perioada;

b - UHSUH]LQW �SDUDPHWUXO�FDUH�VLQWHWL]HD] �QXPDL�LQIOXHQ D�FDUDFWHULVWLFLL�IDFWRULDOH��t); ti - UHSUH]LQW � YDORULOH� FDUDFWHULVWLFLL� IDFWRULDOH� FDUH�� vQ� FD]XO� VHULLORU� FURQRORJLFH�� HVWH�

timpul. Parametrii a� úL� b� VH� GHWHUPLQ � SULQ� UH]ROYDUHD� VLVWHPXOXL� GH� HFXD LL� QRUPDOH� RE LQXW� SULQ�

PHWRGD�FHORU�PDL�PLFL�S WUDWH�� [ ]∑ =+− min)( 2ii btay ):

=+

=+

∑∑∑∑∑

iiii

ii

yttbta

ytbna2

Pentru ti∑ ����VLVWHPXO�GH�HFXD LL�QRUPDOH�GHYLQH�

=

=

∑∑∑

iii

i

yttb

yna2

, de unde

=

=

∑∑

∑

2i

ii

i

t

ytb

n

ya

ÌQORFXLQG�YDORULOH�FDOFXODWH�DOH�FHORU�GRL�SDUDPHWUL� vQ�HFXD LD�GH� UHJUHVLH�úL�DSRL� vQORFXLQG�

VXFFHVLY�YDORULOH�YDULDELOHL�WLPS�VH�RE LQ�YDORULOH�DMXVWDWH�DOH�FDUDFWHULVWLFLL�UH]XOWDWLYH� Verificarea coUHFWLWXGLQLL�FDOFXO ULL�HFXD LLORU�GH�UHJUHVLH�VH�IDFH�SH�ED]D�UHOD LHL�

∑∑ = it yYi

.

Trend parabolic

2iit ctbtaY

i++=

3XQkQG� DFHHDúL� FRQGL LH�� FD� VXPD� S WUDWHORU� DEDWHULORU� WHUPHQLORU� VHULHL� GH� OD� YDORULOH�

WHRUHWLFH�V �ILH�PLQLP ��VH�RE LQH�

minim))(( 22∑ =++− iii ctbtay

LDU�VLVWHPXO�GH�HFXD LL�QRUPDOH�

=++

=++

=++

∑∑∑∑∑∑ ∑∑

∑ ∑ ∑

2432

32

2

iiiii

iiiii

iii

yttctbta

yttctbta

ytctbna

Pentru ∑ it ���FD]�vQ�FDUH�úL�∑ 3it ���VLVWHPXO�GH�HFXD LL�QRUPDOH�GHYLQH�

=+

=

=+

∑∑∑∑∑

∑ ∑

242

2

2

iiii

iii

ii

yttcta

yttb

ytcna

Rezolvând sistemul de HFXD LL� QRUPDOH�� VH� FDOFXOHD] � YDORDUHD� FHORU� WUHL� SDUDPHWUL� úL�� vQ�

IXQF LH�GH�YDORULOH�LQGLYLGXDOH�DOH�YDULDELOHL�t��VH�DMXVWHD] �YDORULOH�FDUDFWHULVWLFLL�UH]XOWDWLYH� Trend hiperbolic

bt

aYi

ti

1+=

6LVWHPXO�GH�HFXD LL�HVWH�

=+

=+

∑ ∑ ∑

∑ ∑

iiii

ii

ytt

bt

a

yt

bna

111

1

2

7UHQG�H[SRQHQ LDO

i

i

tt baY ⋅=

3ULQ�ORJDULWPDUH��PRGHOXO�VH�WUDQVIRUP �vQWU-un model liniar de forma:

btaY itilglglg +=

6LVWHPXO�GH�HFXD LL�QRUPDOH�YD�IL�

⋅=⋅+⋅

=⋅+

∑∑∑∑∑

)lg(lglg

lglglg

2iiii

ii

yttbta

ytban

2SHUD LD� GH� DMXVWDUH� vQ� DFHVW� FD]� VH� YD� IDFH� GXS � FH� VH� YRU� FDOFXOD� ORJDULWPLL� HFXD LLORU�

LQGLYLGXDOH� GH� DMXVWDUH�� $MXVWDUHD� GXS � R� IXQF LH� H[SRQHQ LDO � VH� IDFH� SULQ� DQWLORJDULWPDUHD�HFXD LLORU�GH�DMXVWDUH�FDOFXODWH�vQ�IXQF LH�GH� it .

Criterii de alegere a procedeelor de ajustare

a) 6H� FDOFXOHD] � VXPD�DEDWHULORU�� OXDWH� vQ� YDORDUH� DEVROXW ��GLQWUH� GDWHOH� HPSLULFH� úL� FHOH�

ajustate tt Yy − �� 6H� FRQVLGHU � FHO� PDL� SRWULYLW� SURFHGHXO� OD� FDUH� VH� RE LQH�

∑ =− .mintt Yy

b) 6H�FDOFXOHD] �FRHILFLHQWXO�GH�YDULD LH�

100// ⋅=′

y

dV ty

ty

în care tyd / � UHSUH]LQW � DEDWHUHD�PHGLH� OLQLDU � D� YDORULORU� UHDOH� GH� OD� YDORULOH� DMXVWDWH�

FDOFXODW �GXS �IRUPXOD�

n

Yyd itt

ty

∑ −=/

EXTRAPOLAREA SERIILOR CRONOLOGICE

Extrapolarea datelor unei serii statistiFH� DUH� OD� ED] � PHWRGHOH� úL� SURFHGHHOH� IRORVLWH� OD�ajustare.

3HQWUX�D�IDFH�GLVWLQF LH�vQWUH�WHUPHQLL�DMXVWD L��itY ��úL�FHL�H[WUDSROD L�-�FDUH�VXQW�FRQVLGHUD L�

tot termenii teoretici -�VH�YRU�QRWD�WHUPHQLL�H[WUDSROD L�FX� ,it

Y ′ iar variabila de timp cu ti’.

Deci, formulele de calcul vor fi: • pentru extrapolarea pe baza sporului mediu:

∆+=′ ’0 it tyY

i

• SHQWUX�H[WUDSRODUHD�SH�ED]D�LQGLFHOXL�PHGLX�GH�FUHúWHUH�

’

0i

i

tt IyY ⋅=′

$FHVWH�IRUPXOH�VH�DSOLF �DWXQci când se folosesc valorile parametrilor ( )I,∆ din perioada H[SLUDW ��ÌQ�FD]XO�FkQG�DFHúWLD�VH�PRGLILF ��SDUDPHWULL�VH�PRGLILF �FX�XQ�FRHILFLHQW�k, astfel:

∆′+=′ ’0 it tyY

i , în care: ∆⋅=∆ k’

′

′

⋅=′i

i

t

t IyY 0 , în care: IkI =′

Coeficientul k�SRDWH�V �ILH�PDL�PDUH�VDX�PDL�PLF�GHFkW��� 'DF �k<1, DWXQFL� vQVHDPQ �F �VH�UHGXFH�YDULD LD�PHGLH�DEVROXW �VDX�UHODWLY ��GXS �FXP�VH�

DSOLF �OD�SULPXO�VDX�OD�DO�GRLOHD�SURFHGHX� 'DF � k>1, atunci înseaPQ � F � YDORDUHD� SDUDPHWULORU� IRORVL L� vQ� H[WUDSRODUH� HVWH�PDL�PDUH�

decât în perioada de analizat. 3HQWUX�H[WUDSRODUHD�SH�ED]D�PHWRGHORU�DQDOLWLFH�GH�FDOFXO�VH�SXQH��vQ�SULPXO�UkQG��FRQGL LD�FD�

GDWHOH� V � VH�GHWHUPLQH�DVWIHO� vQFkW� V �QX�PRGLILFH�RULJLQHD�YDULD LHL�GH� WLPS�FDUH�HVWH� vQ�PLMORFXO�VHULHL�FURQRORJLFH�úL�SHQWUX�FDUH�Σti = 0. 'HFL��YDULD LD�GH�WLPS�VH�H[WLQGH�vQ�DPEHOH�VHQVXUL��GHúL�LQWHUHVHD] �QXPDL�WHQGLQ D�RE LQXW �SULQ�H[WLQGHUHD�VHULHL�SHQWUX�SHULRDGD�XUP WRDUH�

ùL� vQ� DFHVW� FD]� VH�YRU� IRORVL� DFHOHDúL� QRWD LL�� DGLF � WHUPHQLL� H[WUDSROD L� VH�YRU�QRWD� FX�it

Y ′ ,

astfel: ’it btaY

i+=′

i

i

tt abY ′=′

2

iit tctbaYi

′+′+=′

'DF �IHQRPHQHOH�VH�DQDOL]HD] �vQ�LQWHUGHSHQGHQ �FX�YDULD LD�IDFWRULORU�FDUH�OH�GHWHUPLQ �în GLQDPLF ��DWXQFL�VH�DOHJ�FDUDFWHULVWLFLOH�IDFWRULDOH�FDUH�VXQW�SUHYL]LELOH�QXPDL�vQ�IXQF LH�GH�WLPS�úL�DSRL�� DSUHFLLQG� F � V-DU� S VWUD� DFHHDúL� WHQGLQ � GH� UHDOL]DUH� D� OHJ WXULL�� VH� FDOFXOHD] � YDORULOH� GH�SHUVSHFWLY � úL� SHQWUX� FDUDFWHULVWLFD� GHSHQGHQW �� 'H� H[HPSOX�� SHQWUX� R� FDUDFWHULVWLF � UH]XOWDWLY ��

DQDOL]DW �vQ�IXQF LH�GH�GRL�IDFWRUL�FX�FDUH�HVWH�OHJDW �OLQLDU��VH�FDOFXOHD] ��PDL�vQWkL��HFXD LD�PHGLH�GH�WHQGLQ �SH�SHULRDGD�H[SLUDW ��DGLF �

Yt = a0 + a1x1 + a2x2

în care x1, x2 sunt variabile factoriale luate în timp.

6HSDUDW� VH� DQDOL]HD] � WHQGLQ D� GH� GH]YROWDUH� D� ILHF UHL� YDULDELOH� LQGHSHQGHQWH� úL� VH�H[WUDSROHD] �FHOH�GRX �VHULL�

tbaX xx ′+=′

111

tbaX xx ′+=′

222

$FHVWH� YDORUL� VH� LQWURGXF� vQ� HFXD LD� IXQF LHL� GH� UHJUHVLH� PXOWLSO � calculându-se valorile

H[WUDSRODWH�DOH�YDULDELOHL�<��GXS �YDORULOH�PRGLILFDWH�DOH�FHORU�GRL�IDFWRUL�

2211021xaxaaY xx ′+′+=′

PROBLEME REZOLVATE

1. $QDOL]D�VWDWLVWLF �D�XQHL�VHULL�FURQRORJLFH�GH�LQWHUYDOH 3URGXF LD�LQWHUQ �GH�DOXPLQLX�D�vQUHJLVWrat în perioada 1992-�����XUP WRDUHOH�YDORUL�

Tabelul 6.1.

Anul 1992 1993 1994 1995 1996 1997 1998 1999 2000 3URGXF LD�GH�

aluminiu (mii tone)

120 116 122 144 145 164 175 176 178

Sursa: Revista „Capital” nr. 23 din 7 iunie 2001.

Se cere: 1. V �VH�FDUDFWHUL]H]H�HYROX LD�SURGXF LHL�GH�DOXPLQLX�IRORVLQG�LQGLFDWRULL�

• DEVROX L� • relativi; • medii;

2. V �VH�UHSUH]LQWH�JUDILF�VHULD�YDORULORU�DEVROXWH� 3. V �VH�GHWHUPLQH�WUHQGXO�GH�HYROX LH�SH�ED]D�PHWRGHORU�PHFDQLFH�úL�DQDOLWLFH� 4. V �VH�DOHDJ �FHD�PDL�DGHFYDW �PHWRG �FDUH�DMXVWHD] �IHQRPHQXO�úL�V �VH�H[WUDSROH]H�VHULD�

SHQWUX�DQLL������úL������

Rezolvare: 1. &DOFXOXO�LQGLFDWRULORU�DEVROX L:

Nivelul absolut ( )ty este reprezentat de termenii seriei (vezi tabelul 6.1.).

Nivelul totalizat: ∑=

n

tty

1

0RGLILFDUHD�DEVROXW � ( )∆ : ¾�FX�ED] �IL[ � ( )1t∆ :

11 yy tt −=∆ (tabelul 6.2., coloana 2)

41201161212 −=−=−=∆ yy (mii tone)

� 581201781919 =−=−=∆ yy (mii tone)

¾�FX�ED] �vQ�ODQ � ( )1−∆ tt :

11 −− −=∆ tttt yy (tabelul 6.2., coloana 3)

41201161212 −=−=−=∆ yy (mii tone)

� =−=−=∆ 1761788919 yy 2 (mii tone)

5HOD LLOH�vQWUH� 1−∆ tt �úL 1t∆ :

¾� 11 ttt ∆=∆∑ −

5821111912264

19892312

=+++++++−

∆=∆++∆+∆ �

¾� 1111 −− ∆=∆−∆ tttt

25658

891819

=−

∆=∆−∆

Tabelul 6.2.

0RGLILFDUHD�DEVROXW ��PLL�WRQH�

Anul 3URGXF LD� (mii tone)

FX�ED] �IL[

11 yy tt −=∆ FX�ED] �vQ�ODQ

11 −− −=∆ tttt yy

A 1 2 3 1992 120 - - 1993 116 - 4 - 4 1994 122 2 6 1995 144 24 22 1996 145 25 1 1997 164 44 19 1998 175 55 11 1999 176 56 1 2000 178 58 2 Total 1340 - 58

Calculul indicatorilor relativi:

,QGLFHOH�GH�GLQDPLF ��FUHúWHUH�GHVFUHúWHUH� ( )I :

¾�FX�ED] �IL[ � ( )1tI (vezi tabelul 6.3., coloana 2)

( ) 1001

%1 ⋅=y

yI t

t

( )

( ) %33,148100120

178

%66,96100120

116

%19

%12

=⋅=

=⋅=

I

I

�

¾�FX�ED] �vQ�ODQ � ( )1−ttI (vezi tabelul 6.3., coloana 3)

( )

( )

( ) %14,101100176

178

%66,96100120

116

100

%89

%12

1%1

=⋅=

=⋅=

⋅=−

−

I

I

y

yI

t

ttt

�

5HOD LLOH�vQWUH� 1tI �úL� 1−ttI :

¾� 11 ttt II =∏ −

4833,10114,10517,19666,0 ≅⋅⋅⋅ �

¾� 1tI : 111 −− = ttt II

19I : 8918 II =

1,4833 : 1,4666 0114,1≅ Tabelul 6.3.

Indicele de dinDPLF Ritmul 9DORDUHD�DEVROXW �D�

1% din ritm

Anul 3URGXF LD�(mii tone) FX�ED] �IL[

11

y

ty

tI =

FX�ED] �vQ�ODQ

11

−=−

ty

ty

ttI FX�ED] �IL[

( ) 100111 ⋅−= tItR FX�ED] �vQ�ODQ

( ) 100111 ⋅−−=− ttIttR

FX�ED] �IL[

100

11

y

tA =

FX�ED] �vQ�ODQ

100

11

−=−

ty

ttA

A 1 2 3 4 5 6 7 1992 120 - 1,0000 - - - - 1993 116 0,9666 0,9666 -3,34 -3.34 1,2 1,20 1994 122 1,0166 1,0517 1,66 5,17 1,2 1,16 1995 144 1,2000 1,1803 20,00 18,03 1,2 1,22 1996 145 1,2083 1,0069 20,83 0,69 1,2 1,44 1997 164 1,3666 1,1310 36,66 13,1 1,2 1,45 1998 175 1,4583 1,0670 45,83 6,70 1,2 1,64 1999 176 1,4666 1,0057 46,66 0,57 1,2 1,75 2000 178 1,4833 1,0114 48,33 1,14 1,2 1,76 Total 1340 - - - - - -

Ritmul de modificare (FUHúWHUH�GHVFUHúWHUH���5�: ¾�FX�ED] �IL[ �� 1tR (vezi tabelul 6.3., coloana 4)

( )

( )

( ) %33,4810014833,1

%34,310019666,0

1001100

19

12

11

1

1

=⋅−=

−=⋅−=

⋅−=⋅∆

=

R

R

Iy

R t

t

t

�

¾�FX�ED] �vQ�ODQ ��YH]L�WDEHOXO�������FRORDQD���

( )

( )

( ) %14,110010114,1

%34,310019666,0

1001100

89

12

11

1

1

=⋅−=

−=⋅−=

⋅−=⋅∆

= −−

−−

R

R

Iy

R ttt

tt

tt

�

9DORDUHD�DEVROXW �D�XQXL�SURFHQW�GLQ�ULWPXO�PRGLILF ULL��$��

¾�FX�ED] �IL[ �� 1tA (vezi tabelul 6.3., coloana 6)

1001

1

1

1

y

RA

t

t

t =∆

=

2,1100

12012 ==A mii tone

�

2,1100

12019 ==A mii tone

¾�FX�ED] �vQ�ODQ �� 1−ttA (vezi tabelul 6.3., coloana 7)

100

1

1

1

1

−=

∆=

−

−−

t

tt

tt

tt

y

RA

2,1100

12012 ==A mii tone

�

76,1100

17619 ==A mii tone

Calculul indicatorilor medii

Nivelul mediu ( )y : 6HULD�FURQRORJLF �SUH]HQWDW �HVWH�R�VHULH�GH�LQWHUYDOH�SHQWUX�FDUH�QLYHlul mediu se FDOFXOHD] �DSOLFkQG�IRUPXOD�PHGLHL�DULWPHWLFH�

89,1489

13401 ≅==∑

=

n

y

y

n

tt

mii tone

3URGXF LD�PHGLH�DQXO �GH�DOXPLQLX�D�IRVW�GH��������PLL�WRQH�

0RGLILFDUHD�PHGLH�DEVROXW � ( )∆ :

25,78

58

11

11==

−

∆=

−

∆=∆

∑ −

nn

ntt mii tone

În medie creúWHUHD�DQXDO �vQ�SHULRDGD�����-2000 a fost de 7,25 mii tone. Indicele mediu

0505,14833,1811

11 ==== −−

−∏ nn

ntt III sau 105,05%

Ritmul mediu

( ) ( ) %05,510010505,11001 =⋅−=⋅−= IR 3URGXF LD�V-a modificat în medie cu 5,05% anual.

2. 3HQWUX�UHSUH]HQWDUHD�JUDILF �D�VHULHL�VH�FRQVWUXLHúWH�FURQRJUDPD��KLVWRULRJUDPD��

100110120130140150160170180190

1992 1993 1994 1995 1996 1997 1998 1999 2000

anii

���

� ��

� ��

�� ���

)LJXUD������'LQDPLFD�SURGXF LHL�GH�DOXPLQLX�D�5RPkQLHL�vQ�DQLL�����-2000

3. 'HWHUPLQDUHD�WUHQGXOXL��WHQGLQ HL�JHQHUDOH�

Metode mecanice 0HWRGD�PRGLILF ULL�PHGLL�DEVROXWH Pentru ajustarea termHQLORU�VH�XWLOL]HD] �UHOD LD�

( )11 −⋅∆+= tyY t Metoda indicelui mediu:

( ) 1

1

−⋅=

t

t IyY 5H]XOWDWHOH�RE LQXWH�SULYLQG�DMXVWDUHD�WHUPHQLORU�SULQ�DFHVWH�GRX �PHWRGH�VXQW�SUH]HQWDWH�vQ�WDEHOXO�������FRORDQHOH���úL���

Tabelul 6.4.

Valori ajustate prin:

Anul ty PRGLILFDUHD�PHGLHL�DEVROXW

( )11 −⋅∆+= tyY t

indicele mediu

( )( )1

1

−⋅=

t

t IyY

A 1 2 3

1992 120 120025,71201 =⋅+=Y 1201 =Y

1993 116 25,127125,71202 =⋅+=Y 06,1260505,11202 =⋅=Y

1994 122 5,134225,71203 =⋅+=Y 43,1320505,1120 2

3 =⋅=Y

1995 144 75,141325,71204 =⋅+=Y 1,13930505,11204 =⋅=Y

1996 145 0,149425,71205 =⋅+=Y 14,14640505,11205 =⋅=Y

1997 164 25,156525,71206 =⋅+=Y 52,15350505,11206 =⋅=Y

1998 175 5,163625,71207 =⋅+=Y 27,16160505,11207 =⋅=Y

1999 176 25,170725,71208 =⋅+=Y 42,16970505,11208 =⋅=Y

2000 178 178825,71209 =⋅+=Y 17880505,11209 =⋅=Y

Total 1340 1340,5 1325,95

Metode analitice 'LQ�FURQRJUDPD�SUH]HQWDW ��ILJ���������VH�REVHUY �F �DMXVWDUHD�VH�SRDWH�IDFH�SULQ�PHWRGD�

OLQLDU �úL�SH�ED]D�SDUDEROHL�GH�JUDGXO�GRL� 3HQWUX�DMXVWDUHD�OLQLDU �VH�XWLOL]HD] �UHOD LD�

it tbaY +=

Pentru calculul parametrilor a�úL�b�VH�DSOLF �meWRGD�FHORU�PDL�PLFL�S WUDWH:

( )[ ] min2 =⋅+−∑ it tbaY

6LVWHPXO�GH�HFXD LL�HVWH�

=+

=+

∑ ∑ ∑∑ ∑

iiii

ii

yttbta

ytban

2

'DF �VH�SXQH�FRQGL LD�FD�� 0=∑ it sistemul devine:

∑∑∑

∑ ∑∑

==

=

=

2

2

;i

iii

iii

i

t

ytb

n

ya

yttb

yan

&DOFXOHOH�QHFHVDUH�UH]ROY ULL�VLVWHPXOXL�VXQW�SUH]HQWDWe în tabelul 6.5.

96,8;89,1489

1340

53860

13409

===

=

=

ba

b

a

Deci:

ii tY 96,889,148 += Valorile ajustate sunt prezentate în tabelul 6.5., coloana 7.

Tabelul 6.5. Valori ajustate prin

IXQF LD Anul iy it 2it 4

it ii yt ii yt 2 OLQLDU SDUDEROLF

A 1 2 3 4 5 6 7 8 1992 120 -4 16 256 -480 1920 113,05 111,83 1993 116 -3 9 81 -348 1044 122,00 121,77 1994 122 -2 4 16 -244 488 131,69 132,03 1995 144 -1 1 1 -144 144 139,93 140,66 1996 145 0 0 0 0 0 148,89 149,75 1997 164 1 1 1 164 164 157,85 158,58 1998 175 2 4 16 350 700 166,09 166,43 1999 176 3 9 81 528 1584 175,77 175,53 2000 178 4 16 256 712 2848 188,73 183,51 Total 1340 0=∑ it 602 =∑ it 7084 =∑ it 538=∑ ii yt 88922 =∑ ii yt 1340=∑ iY 1340=∑ iY

Pentru trendul parabolic rela LD�HVWH�

2iii tctbaY ++=

6LVWHPXO�GH�HFXD LL�QRUPDOH�SHQWUX�FDOFXODUHD�SDUDPHWULORU�

=++

=++

=++

∑ ∑ ∑ ∑∑ ∑ ∑ ∑

∑ ∑ ∑

iiiii

iiiii

iii

yttctbta

yttctbta

ytctban

2432

32

2

'DF �� 0=∑ it , sistemul devine:

=+

=

=+

∑ ∑ ∑∑ ∑

∑ ∑

iiii

iii

ii

yttcta

yttb

ytcan

242

2

2

Calculele sunt prezentate în tabelul 6.5. Deci:

−=

=

=

⇔

=+

=

=+

13,0

96,8

75,149

889270860

53860

1340609

c

b

a

ca

b

ca

(FXD LD�SHQWUX�DMXVWDUHD�SULQ�WUHQGXO�SDUDEROLF�HVWH�

213,096,875,149 iit ttYi

−+=

Valorile ajustate sunt prezentate în tabelul 6.5., coloana 8.

4. &ULWHULLOH�IRORVLWH�SHQWUX�DOHJHUHD�PHWRGHL�FDUH�DMXVWHD] �FHO�PDL�ELQH�VHULD�SUH]HQWDW �

sunt:

• compararea volumului total empiric

∑=

n

iiy

1

cu cel teoretic ∑=

n

it i

Y1

;

• FRPSDUDUHD�DEDWHULORU�DEVROXWH�DOH�YDORULORU�WHRUHWLFH��DMXVWDWH��ID �GH�FHOH�HPSLULFH��

∑=

−n

iiti Yy

1

• FRPSDUDUHD�S WUDWHORU�DEDWHULORU�GLQWUH�YDORULOH�HPSLULFH�úL�FHOH�WHRUHWLFH

( )∑=

−n

iti i

Yy1

2

• FRPSDUDUHD�FRHILFLHQ LORU�GH�YDULD LH�FDOFXOD L�GXS �UHOD LLOH�

( )

100100

1

2

⋅

−

=⋅=

∑=

y

n

Yy

yv

n

iti

y

i

iσ

sau

100100

1

⋅

−

=⋅=′

∑=

y

n

Yy

y

dv

n

iti

y

i

i ,

unde n

y

y

n

ii∑

== 1 .

Calculele necesare sunt prezentate în tabelul 6.6..

6H�UH LQH�PHWRGD��PRGHOXO��GH�DMXVWDUH�SHQWUX�FDUH�FRHILFLHQWXO�GH�YDULD LH�HVWH�FHO�PDL�PLF�(tabelul 6.7.).

&DOFXOXO�FRHILFLHQWXOXL�GH�YDULD LH

Tabelul 6.7.

Nr. crt.

Metoda de ajustare prin: ( )∑ − 2iti Yy

tyσ 100⋅=y

V i

t

y

y

σ

1 PRGLILFDUHD�PHGLH�DEVROXW 523,80 7,62 5,12 2 indicele mediu 576,83 8,01 5,37 3 trend liniar 442,40 7,01 4,71 4 trend parabolic 367,76 6,39 4,30

([WUDSRODUHD� �SUHYL]LRQDUHD�� SURGXF LHL� GH� DOXPLQLX� SHQWUX� XUP WRULL� �� DQL� VH� IDFH� LQkQG�FRQW�F �VXQW�DFHOHDúL�FRQGL LL��9DORULOH�H[WUDSRODWH�SULQ�PHWRGHOH�IRORVLWH�OD�DMXVWDUH�VXQW�SUH]HQWDWH�în tabelul 6.8..

Tabelul 6.8.

Anii Nr.

crt. Metoda de

ajustare prin: 2001 2002 A 1 2 3

1 modificarea medie DEVROXW

25,185925,71202001 =⋅+=Y

5,1921025,71202002 =⋅+=Y

2 indicele mediu ( ) 99,1860505,1120 9

2001 =⋅=Y

( ) 43,1960505,1120 10

2002 =⋅=Y

3 IXQF LD�OLQLDU 69,193596,889,1482001 =⋅+=Y

65,202696,889,1482002 =⋅+=Y

4 IXQF LD�SDUDEROLF 5,1892513,0596,875,1492001 =⋅−⋅+=Y

83,1983613,0696,875,1492002 =⋅−⋅+=Y

2. Se cunRVF�XUP WRDUHOH�GDWH�SULYLQG�YDORDUHD�H[SRUWXULORU�OD�SURGXVHOH�GLQ�OHPQ�vQ�SHULRDGD�

1991-1998:

Tabelul 6.9.

Anul 1992 1993 1994 1995 1996 1997 1998 0RGLILF UL�

ID �GH�DQXO�

precendent (mil. $)

40 22 45 36 38 42 47

Se cere:

1. V � VH� UHFRQVWLWXLH� VHULD� GH� YDORUL� DEVROXWH�� úWLLQG� F � YDORDUHD� H[SRUWXOXL� D� FUHVFXW� vQ�perioada 1991-1998, în medie, cu 18,84%;

2. V �VH�FDOFXOH]H�LQGLFDWRULL�PHGLL�úL�V �VH�LQWHUSUHWH]H� 3. V �VH�DMXVWH]H�VHULD�SULQWU-un procedeu analitic; 4. V �VH�FDOFXOH]H�FRHILFLHQWXO�GH�YDULD LH� Rezolvare: 1. 'LQ�GDWHOH�SUH]HQWDWH�UH]XOW �F �

1991199812

1 yyyy t

n

ttt −=−=∆∑

=−

27047423836452240 =++++++

'H�DVHPHQHD��VH�úWLH�F �

( ) %84,11810084,18100%84,18 % =+=+=⇒= RIR

1884,17

91

981

91

98 === −

y

y

y

yI n

Deci:

( )

=

=⇒

==

=−

$ mil.385y

$mil.115

3478,31884,1

270

1998

1991

7

1991

1998

19911998y

y

y

yy

7HUPHQLL�VHULHL�VH�FDOFXOHD] �DVWIHO�

38547338

33842296

15540115

115

1998

1997

1992

1991

=+=

=+=

=+=

=

y

y

y

y

�

6HULD�UHFRQVWLWXLW �HVWH�SUH]HQWDW �vQ�WDEHOXO������

Tabelul 6.10.

Anul 1991 1992 1993 1994 1995 1996 1997 1998 Export produse din lemn (mil. $)

115 155 177 222 258 296 338 385

2. Indicatorii medii

• PRGLILFDUHD�PHGLH�DEVROXW �

57,387

115385

18111991199812

1

=−=−−

=−−

=−

∆

=∆∑=

−yy

n

yy

nn

n

ttt

mil. $

Exportul a crescut în medie cu 47,5 mil. lei.

• nivelul mediu:

25,2438

19461===

∑=

n

y

y

n

tt

mil. $

ÌQ�DFHDVW �SHULRDG �V-au exportat în medie, anual, produse în valoare de 243,25 mil. $. 5LWPXO�PHGLX�D�IRVW�GDW�úL�SH�ED]D�OXL�V-a calculat indicele mediu.

3. 3HQWUX�DMXVWDUHD�SULQ�PHWRGD�DQDOLWLF �VH�FRQVWUXLHúWH�FURQRJUDPD��KLVWRULRJUDPD��

0

50

100

150

200

250

300

350

400

450

1991 1992 1993 1994 1995 1996 1997 1998

anii

expo

rtul

(m

il. $

)

Figura 6.2.

*UDILFXO�VXJHUHD] �XQ�WUHQG�OLQLDU� (FXD LD�GUHSWHL�

it tbaYi

+=

6LVWHPXO�GH�HFXD LL�QRUPDOH:

=+

=+

∑ ∑ ∑∑ ∑

iiii

ii

yttbta

ytban

2

&X�FRQGL LD�FD� 0=∑ it �VLVWHPXO�GH�HFXD LL�GHYLQH�

=

=

∑ ∑∑

iii

i

yttb

yan

2

===

===

∑∑

∑=

04,19168

3198

25,2438

1946

2

1

i

ii

n

ii

t

tyb

n

y

a

&DOFXOHOH�QHFHVDUH�GHWHUPLQ ULL�SDUDPHWULORU�úL�YDORULOH�DMXVWDWH��WHRUHWLFH��VXQW�SUH]HQWDWH�vQ�tabelul 6.11.

Tabelul 6.11.

Anii iy it 2it ii ty it tY

i04,1925,243 += ( )2

iti Yy −

0 1 2 3 4 5 6 1991 115 -7 49 805 109,97 25,3 1992 155 -5 25 775 148,05 48,3 1993 177 -3 9 531 186,13 83,36 1994 222 -1 1 222 224,21 4,88 1995 258 1 1 258 262,29 18,4 1996 296 3 9 888 300,37 19,1 1997 338 5 25 1690 338,45 0,2 1998 385 7 49 2695 376,53 71,74

Total 1946=∑ iy 0=∑ it 1682 =∑ it 3198=∑ ii ty 1946=∑ itY ( ) 28,271=−∑ iti Yy

4. &DOFXOXO�FRHILFLHQWXOXL�GH�FRUHOD LH�VH�IDFH�FX�DMXWRUXO�UHOD LHL�

100⋅=Y

v t

t

Y

y

σ, unde

( )82,5

2

=−

=∑

n

Yyi

t

ti

yσ

3HQWUX�DEDWHUHD��HURDUHD��PHGLH�S WUDWLF ��FDOFXOHOH�VXnt prezentate în tabelul 6.11, coloana 6.

%39,210025,243

82,5=⋅=

tyv

3. 6H� FXQRVF� XUP WRDUHOH� GDWH� FX� SULYLUH� OD� vQ]HVWUDUHD� SRSXOD LHL� FX� EXQXUL� GH� IRORVLQ �

vQGHOXQJDW ��DXWRPRELOH��������ORFXLWRUL��vQ�SHULRDGD������–1998.

Tabelul 6.12.

Anul 1992 1993 1994 1995 1996 1997 1998 1999 0RGLILF UL�

DEVROXWH�ID �GH�

anul precedent (buc.)

6,5 8,3 9,3 7,3 8,2 8,9 8,9 6,7

0RGLILF UL�

UHODWLYH�ID �GH�

anul precedent (%)

10,58 12,22 12,20 8,54 8,84 8,81 8,10 5,64

Se cere: 1. V �VH�UHFRQVWLWXLH�VHULD�GH�YDORUi absolute; 2. V �VH�FDOFXOH]H�WR L�LQGLFDWRULL�DEVROX L�úL�UHODWLYL�

Rezolvare: 1. 3HQWUX� UHFRQVWLWXLUHD�VHULHL� FURQRORJLFH� VH�XWLOL]HD] � UHOD LD� vQWUH�PRGLILFDUHD�DEVROXW ��

PRGLILFDUHD�UHODWLY �úL�YDORDUHD�DEVROXW �D�XQXL�SURFHQW�GLQ�ULWPXO�PRGLILF ULL�

1001001

1

1

1−

−

−− =

⋅

∆= t

tt

tt

tt

y

RA

De exemplu,

10091

9192

9192

9192

y

RA =

∆=

100614,0

58,10

5,6 919192

yA ===

4,61100614,091 =⋅=y buc

Pentru anul 1999 valoarea indicatorului se poate calcula astfel:

5,1257,68,11898999899 =+=∆+= yy

sau

5,1250564,18,11898999899 =⋅=⋅= Iyy

2. 6HULD�UHFRQVWLWXLW �úL�LQGLFDWRULL�DEVROX L�úL�UHODWLYL�VXQW�SUH]HQWD L�vQ�WDEHOXO�������

Tabelul 6.13.

,QGLFDWRUL�DEVROX L Indicatori relativi Anul

ty (auto / 1000

locuitori) 1t∆ 1−∆ tt ( )%1tI ( )%1−ttI 1tR 1−ttR 1tA 1−ttA

A 1 2 3 4 5 6 7 8 9 1991 61,4 - - 100 - - - - - 1992 67,9 6,5 6,5 110,52 110,52 10,52 10,58 0,614 0,614 1993 76,2 14,8 8,3 124,10 112,22 24,10 12,22 0,614 0,679 1994 85,5 24,1 9,3 139,25 112,20 39,25 12,20 0,614 0,762 1995 92,8 31,4 7,3 151,14 108,53 51,14 8,54 0,614 0,855 1996 101,0 39,6 8,2 164,49 108,83 64,49 8,84 0,614 0,928 1997 109,9 48,5 8,9 178,99 108,81 78,99 8,81 0,614 1,010 1998 118,8 57,4 8,9 193,48 108,10 93,48 8,10 0,614 1,099 1999 125,5 64,1 6,7 204,39 105,64 104,39 5,64 0,614 1,188

4. ÌQ�DQLL������úL������V-DX�vQUHJLVWUDW�XUP WRDUHOH�GDWH�SULYLQG�VWRFXULOH�H[LVWHQWH�vQ�GHSR]LWHOH�

XQHL�VRFLHW L�FRPHUFLDOH��WDEHOXO 6.14.).

Tabelul 6.14.

1999 2000 'DWD�vQUHJLVWU ULL Cantitatea (kg) 'DWD�vQUHJLVWU ULL Cantitatea (kg)

1 ianuarie 180 1 ianuarie 240 1 aprilie 200 25 martie 270 1 iulie 250 1 august 240 1 octombrie 270 20 octombrie 260 31 decembrie 240 31 decembrie 280

Se cere: 1. V �VH�UHSUH]LQWH�JUDILF�VHULLOH�FURQRORJLFH� 2. V �VH�FDOFXOH]H�VWRFXO�PHGLX�vQ�FHL�GRL�DQL� Rezolvare: 1. Seria se timp pentru anul 1999 este o serie de momente cu intervale egale între

PRPHQWH��5HSUH]HQWDUHD�JUDILF �VH�IDFH�SULQWU-o cronograP ��ILJ�������� Seria din anul 2000 este tot o serie de momente cu intervale neegale între momente. 3HQWUX�UHSUH]HQWDUHD�JUDILF �VH�IRORVHúWH�XQ�JUDILF�SULQ�FRORDQH��ILJ��������

)LJXUD������9DULD LD�VWRFXULORU�GH�P UIXUL�vQ�DQXO�����

)LJXUD������9DULD LD�VWRFXULORU�GH�P UIXUL�vQ�DQXO�����

0

50

100

150

200

250

300

GDWD�vQUHJLVWU ULL

cant

itat

ea (

kg)

1.01.99 1.04.99 1.07.99 1.10.99 31.12.99

280260

240240270

0

50

100

150

200

250

300

GDWD�vQUHJLVWU ULL

cant

itat

ea (

kg)

1.01.2000 25.03.2000 1.08.2000 20.10.2000 31.12.2000

2. Stocul mediu 3HQWUX�SULPD� VHULH� VH�FDOFXOHD] �R�PHGLH�FURQRORJLF � VLPSO � LDU�SHQWUX�VHULD� D�GRXD�R�PHGLH�FURQRORJLF �SRQGHUDW �

5,23215

2

240200250200

2

180

1225

4321

1999 =−

++++=

−

++++=

n

yyyy

y

y cr kg

∑−

=

++

++

++

+=

1

1

45

434

323

212

11

200022222

n

ii

cr

d

dy

ddy

ddy

ddy

dy

y ,

unde id �VXQW�LQWHUYDOHOH�GH�WLPS��GLVWDQ HOH��GLQWUH�PRPHQWH�

71;79;126;84 4321 ==== dddd .

kg 86,256360

92605

3602

71280

2

7179260

2

79126240

2

12684270

2

84240

2000

==

+++++++=cry

5. Ponderea sectorului privat în PIB a evoluat astfel în perioada 1991 - 1992.

Tabelul 6.15.

Anul 1992 1993 1994 1995 1996 1997 1998 1999 0RGLILF UL�

ID �GH�

anul 1991(%)

2,8 11,2 15,3 21,7 31,3 37 38,4 39,7

Se cere: 1. V �VH�UHFRQVWLWXLH�VHULD�GDF �SRQGHUHD�VHFWRUXOXL�SULYDW�vQ�3,%�D�FUHVFXW�vQ�DQXO������ID �

de anul 1991 de 2,326 ori; 2. V �VH�UHSUH]LQWH�JUDILF�VHULD�UHFRQVWLWXLW � 3. V �VH�H[WUDSROH]H�YDORULOH�SHQWUX�DQLL������úL������

Rezolvare: 1. 'LQ�GDWHOH�SUH]HQWDWH�UH]XOW �F �

=

=⇒

=

=−

9,54

6,23

326,2

3,31

96

91

91

96

9196

y

y

y

y

yy

6HULD�GH�GDWH�UHFRQVWLWXLW �HVWH�SUH]HQWDW �vQ�WDEHOXO�������úL�V-D�RE LQXW�DVWIHO�

3,637,396,23

3,638,26,23

6,23

98

92

91

=+=

=+=

=

y

y

y

�

Tabelul 6.16.

Anii 1991 1992 1993 1994 1995 1996 1997 1998 1999 Ponderea sectorului privat în PIB (%)

23,6 26,4 34,8 38,9 45,3 54,9 60,6 62,0 63,3

2. 5HSUH]HQWDUHD�JUDILF �VH�IDFH�SULQ�FURQRJUDP ��KLVWRJUDP ��

Figura 6.4.

0

10

20

30

40

50

60

70

1991 1992 1993 1994 1995 1996 1997 1998 1999

anii

pond

erea

sec

toru

lui p

riva

t în

PIB

(%

)

3. Pentru extrapolarea valoriORU�SHQWUX�DQLL������úL������IRORVLP�R�PHWRG �PHFDQLF �úL�XQD�DQDOLWLF �GH�DMXVWDUH��FRQVLGHUkQG�F �FHOH�GRX �PHWRGH�DMXVWHD] �FHO�PDL�ELQH�IHQRPHQXO�

0HWRGD�PHFDQLF 0HWRGD�PRGLILF ULL�PHGLL�DEVROXWH� 5HOD LD�GH�DMXVWDUH�HVWH�

( )11 −⋅∆+= tyyit

9625,48

6,233,63

811919911

=−

=−

=−−

=−

∆=∆

∑ − yy

n

yy

nntt

%

'DF � IHQRPHQXO� VH� GHVI úRDU � vQ� DFHOHDúL� FRQGL LL�� YDORULOH� SHQWUX� DQLL� ����� úL� ����� vor fi:

( ) 225,73109625,46,2311112001 =⋅+=−⋅∆+= yy %

( ) 1875,78119625,46,2311212002 =⋅+=−⋅∆+= yy % 0HWRGD�DQDOLWLF *UDILFXO�VXJHUHD] �XQ�WUHQG�OLQLDU�

it tbayi

+=

ApOLFkQG�PHWRGD�FHORU�PDL�PLFL�S WUDWH�úL�DSRL�FRQGL LD� 0=∑ it �� VLVWHPXO�GH�HFXD LL�

normale devine:

===

===

⇒

=

=

∑∑

∑

∑ ∑∑

55,560

332

53,459

8,409

2

2

t

tyb

n

ya

yttb

yan

ii

i

iii

i

Calculele pentru determinarea parametrilor a�úL�b sunt prezentate în tabelul 6.17..

Tabelul 6.17.

Anii iy it 2

it ii ty it tYi

55,553,45 +=

0 1 2 3 4 5 1991 23,6 - 4 16 -94,9 23,32 1992 26,4 - 3 9 -79,2 28,88 1993 34,8 - 2 4 -69,6 34,43 1994 38,9 - 1 1 -38,9 39,98 1995 45,3 0 0 0 45,53 1996 54,9 1 1 54,9 51,08 1997 60,6 2 4 121,2 56,63 1998 62,0 3 9 186,0 62,19 1999 63,3 4 16 253,2 67,74

Total ∑ = 8,409iy 0=∑ it 602 =∑ it 2,333=∑ ii ty 8,409=∑ itY

Pentru anLL������úL������YDORULOH�YRU�IL�

83,78655,553,452001 =⋅+=Y %

38,84755,553,452002 =⋅+=Y % (VWH�GLILFLO�GH�SUHVXSXV�F �IHQRPHQXO�XUPHD] �DFHHDúL�HYROX LH�–�VH�REVHUY �F �vQ�XOWLPLL�3 ani (1997 –� ������ PRGLILF ULOH� DX� IRVW� UHGXVH� �FUHúWHULOH� DX� IRVW� IRDUWH� PLFL� în FRPSDUD LH�FX�DQLL�SUHFHGHQ L��

PROBLEME PROPUSE

1. &RQVXPXO�PHGLX�DQXDO�GH�ODSWH�úL�SURGXVH�GLQ�ODSWH��O���ORFXLWRU��GLQ�5RPkQLD�D�FXQRVFXW�vQ�

perioada 1989 –������XUP WRDUHD�HYROX LH�

Tabelul 6.18.

Anul 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 Consum (l) 135,9 140,1 163,3 163,7 176,9 179,5 188,6 192,7 192,4 194,4

Se cere: 1. V �VH�UHSUH]LQWH�JUDILF�VHULD�FURQRORJLF � 2. V �VH�FDOFXOH]H�LQGLFDWRULL�DEVROX L��UHODWLYL�úL�PHGLL�FH�FDUDFWHUL]HD] �VHULD�FURQRORJLF � 3. V �VH�DMXVWH]H�VHULD�SULQ�PHWRGH�PHFDQLFH�úL�DQDOLWLFH� 4. V �VH�SUHFL]H]H�PHWRGD�FDUH�DMXVWHD] �FHO�PDL�ELQH�WHQGLQ D�GH�HYROX LH�D�IHQRPHQXOXL�

2. Dinamica exporturilor produsului Y în perioada 1989 –������D�IRVW�XUP WRDUHD�

Tabelul 6.19.

Anul 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Dinamica ID �GH�

anul precedent (%)

122,11 107,08 77,94 148,11 205,73 85,34 127,77 97,62 103,23 136,86 92,83

ùWLLQG�F �YDORDUHD�H[SRUWXOXL�vQ�DQXO������D�IRVW�GH�����PLO����VH�FHUH� 1. V �VH�UHFRQVWLWXLH�VHULD�GH�GDWH� 2. V �VH�FDOFXOH]H�WR L�LQGLFDWRULL�DEVROX L��UHODWLYL�úL�PHGLL� 3. V �VH�DMXVWH]H�VHULD�SULQ�PHWRGH�PHFDQLFH�

3. 3RSXOD LD�RFXSDW ��WRWDO��D�5RPkQLHL�vQWUH������–�������OD�VIkUúLWXO�DQXOXL��HVWH�SUH]HQWDW �vQ�

tabelul 6.20..

Tabelul 6.20.

Anul 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 3RSXOD LD�

RFXSDW � (mii persoane)

10,840 10,786 10,458 10,062 10,011 9,493 9,379 9,023 8,813 8,420

Se cere:

1. V �VH�UHSUH]LQWH�JUDILF�VHULD� 2. V �VH�FDOFXOH]H�LQGLFDWRULL�PHGLL� 3. V �VH�DMXVWH]H�VHULD�SULQWU-R�PHWRG �DQDOLWLF �úL�V �VH�FDOFXOH]H�HURDUHD�PHGLH�S WUDWLF �D�

DMXVW ULL�úL�FRHILFLHQWXO�GH�YDULD LH�

4. &UHúWHULOH� vQUHJLVWUDWH� SULYLQG� YDORDUHD� H[SRUWXULORU� 5RPkQLHL� OD� SURGXVHOH� GH� OHPQ� vQ�

perioada 1991 –������DX�IRVW�XUP WRDUHOH�

Tabelul 6.21. Anul 1992 1993 1994 1995 1996 1997 1998 &UHúWHUL�ID �GH�

anul precedent (mil. $)

40 22 45 36 38 42 47

Se cere: 1. V �VH�UHFRQVWLWXLH�VHULD�GH�YDORUL�DEVROXWH��úWLLQG�F �YDORDUHD�H[SRUWXOXL�D�FUHVFXW�vQ�DQXO�

�����ID �GH�DQXO������FX�������� 2. V �VH�FDOFXOH]H�LQGLFDWRULL�PHGLL�úL�V �VH�Lnterpreteze; 3. V �VH�DMXVWH]H�VHULD�SULQWU-XQ�SURFHGHX�PHFDQLF�úL�XQXO�DQDOLWLF�–�DUJXPHQWD L�SURFHGHXO�

ales; 4. SUHFL]D L� FDUH� GLQ� FHOH� GRX � SURFHGHH� DMXVWHD] � FHO� PDL� ELQH� HYROX LD� H[SRUWXULORU� vQ�

DFHDVW �SHULRDG �

5. 6H�FXQRVF�XUP WRDUHOH�GDWH�SULYLQG�Ginamica PIB (1991 = 100).

Tabelul 6.22.

Anii 1992 1993 1994 1995 1996 1997 1998 1999 Dinamica PIB (%)

88,17 82,24 88,56 97,02 92,91 85,16 87,62 85,39

Se cere: 1. V � VH� UHFRQVWLWXLH� VHULD� GH� GDWH� úWLLQG� F � vQ� DQXO� ����� YDORDUHD� 3,%� D� IRVW� GH�

2138,2 mld.�OHL��vQ�SUH XULOH�DQXOXL������� 2. V �VH�UHSUH]LQWH�JUDILF�VHULD�GH�GDWH� 3. V �VH�FDOFXOH]H�LQGLFDWRULL�DEVROX L��UHODWLYL�úL�PHGLL�úL�V �VH�LQWHUSUHWH]H�

6. 6H�FXQRVF�XUP WRDUHOH�GDWH�SULYLQG�HYROX LD�XQXL�IHQRPHQ�vQ�SHULRDGD������– 1998:

Tabelul 6.22.

Anul 1992 1993 1994 1995 1996 1997 1998 Modificarea DEVROXW �ID �GH�

anul precedent 25 32 28 22 31 24 30

Se cere:

1. V �VH�UHFRQVWLWXLH�VHULD�GH�YDORUL�DEVROXWH�FXQRVFkQG�F �IHQRPHQXO�D�FUHVFXW�vQ�DQXO������

ID �GH������FX��������� 2. V �VH�FDOFXOH]H�LQGLFDWRULL�DEVROX L��UHODWLYL�úL�PHGLL� 3. V �VH�DMXVWH]H�VHULD�SULQWU-XQ�SURFHGHX�PHFDQLF�úL�XQXO�DQDOLWLF��DUJXPHQWD L�SURFHGHHOH�

alese; 4. SUHFL]D L� FDUH� GLQ� PHWRGHOH� XWLOL]DWH� DMXVWHD] � FHO� PDL� ELQH� IHQRPHQXO� vQ� DFHDVW �

SHULRDG � 5. H[WUDSROD L�VHULD�SHQWUX�XUP WRULi doi ani.