моделирование устойчивых случайных величин в случае альфа

ÓÄÊ 519.2
ÌÎÄÅËÈÐÎÂÀÍÈÅ ÓÑÒÎÉ×ÈÂÛÕ ÑËÓ×ÀÉÍÛÕ ÂÅËÈ×ÈÍ
 ÑËÓ×ÀÅ ÀËÜÔÀ ÐÀÂÍÎÌ ÅÄÈÍÈÖÅ
Áàãðîâà È.À.
Êàôåäðà ìàòåìàòè÷åñêîé ñòàòèñòèêè è ñèñòåìíîãî àíàëèçà
Ïîñòóïèëà â ðåäàêöèþ 08.09.2011, ïîñëå ïåðåðàáîòêè 28.12.2011.
 ñòàòüå ðàçðàáîòàí äàò÷èê, ãåíåðèðóþùèé óñòîé÷èâûå ñëó÷àéíûå ÷èñëà ïðè α = 1 è ðàçëè÷íûõ σ , β .  îñíîâó ìîäåëèðîâàíèÿ ïîëîæåíà
îáîáùåííàÿ ÖÏÒ.
Generator for stable random variables with α = 1 and various values of σ ,
β is developed in this article. Modelling is based on GCLT.
Êëþ÷åâûå ñëîâà: ìîäåëèðîâàíèå ñëó÷àéíîé âåëè÷èíû, óñòîé÷èâûå
ðàñïðåäåëåíèÿ, ðàñïðåäåëåíèå Ïàðåòî.
Keywords: modelling random variable, stable distributions, Pareto
distribution.
Ââåäåíèå
 ðàáîòàõ [1], [2], [6] áûëî ðàññìîòðåíî ìîäåëèðîâàíèå óñòîé÷èâûõ ñëó÷àéíûõ
÷èñåë ïðè α ∈ (0, 1). Íàñòîÿùàÿ ñòàòüÿ ïîñâÿùåíà îñîáîìó ñëó÷àþ óñòîé÷èâûõ
ðàñïðåäåëåíèé ñ õàðàêòåðèñòè÷åñêèì ïîêàçàòåëåì óñòîé÷èâîñòè α = 1. Äëÿ ìîäåëèðîâàíèÿ â ýòîé ñèòóàöèè òàêæå áûëà èñïîëüçîâàíà îáîáù¼ííàÿ ïðåäåëüíàÿ òåîðåìà, ñîãëàñíî êîòîðîé óñòîé÷èâûå ðàñïðåäåëåíèÿ ÿâëÿþòñÿ ïðåäåëüíûìè ðàñïðåäåëåíèÿìè äëÿ íîðìèðîâàííîé è öåíòðèðîâàííîé (ïðè α > 1) ñóììû íåçàâèñèìûõ
îäèíàêîâî ðàñïðåäåëåííûõ ñëó÷àéíûõ âåëè÷èí, âõîäÿùèõ â îáëàñòü ïðèòÿæåíèÿ
óñòîé÷èâûõ çàêîíîâ:
Sn =
X1 + ... + Xn − an d
=⇒ Y ïðè n → ∞.
b · n1/α
(1)
Òàêèì îáðàçîì, íåîáõîäèìî ïîäîáðàòü òàêèå çíà÷åíèÿ ïàðàìåòðîâ an è b, ÷òîáû õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ ñóììû íåçàâèñèìûõ îäèíàêîâî ðàñïðåäåëåííûõ
ñëó÷àéíûõ âåëè÷èí ñõîäèëàñü ê õàðàêòåðèñòè÷åñêîé ôóíêöèè óñòîé÷èâîãî ðàñïðåäåëåíèÿ â ôîðìå (À) ïðè α = 1 (ñì. [4], [5], [7], [9]):
½
¾
2
fY (t) = exp −σ|t|(1 + iβ · ln |t|sign(t)) .
(2)
π
Ðàíåå [1], [2] äëÿ ìîäåëèðîâàíèÿ óñòîé÷èâûõ ñëó÷àéíûõ âåëè÷èí â êà÷åñòâå
Xj , j = 1, n èñïîëüçîâàëèñü ñëó÷àéíûå ÷èñëà ñ ðàñïðåäåëåíèåì Ïàðåòî, íîñèòåëåì êîòîðîãî ÿâëÿëîñü ìíîæåñòâî [r; ∞) â îäíîñòîðîííåì ñëó÷àå è ìíîæåñòâî
51
52
ÁÀÃÐÎÂÀ È.À.
(−∞; −l] ∪ [r; ∞) â äâóõñòîðîííåì ñëó÷àå.  ïðåäñòàâëåííîé ñòàòüå, êðîìå òîãî,
ðàññìîòðåíà âòîðàÿ ñèòóàöèÿ, êîãäà ñ öåëüþ èçó÷åíèÿ âëèÿíèÿ íà ñõîäèìîñòü ê
Y ñëó÷àéíàÿ âåëè÷èíà Xj â ðàéîíå íóëÿ èìååò ðàâíîìåðíîå ðàñïðåäåëåíèå.
1. Ìîäåëèðîâàíèå îäíîñòîðîííèõ óñòîé÷èâûõ âåëè÷èí ñ ïîìîùüþ ðàñïðåäåëåíèÿ Ïàðåòî
Ðàññìîòðèì ìîäåëèðîâàíèå îäíîñòîðîííåãî óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè
α = 1 è β = 1, ãäå â êà÷åñòâå Xj âîçüìåì ñëó÷àéíûå âåëè÷èíû, ðàñïðåäåëåííûå
ïî Ïàðåòî.
Ôóíêöèÿ ðàñïðåäåëåíèÿ Ïàðåòî ïðè α = 1 èìååò âèä:
½
1 − xε , x > ε,
F (x) =
0,
x < ε,
åé ñîîòâåòñòâóåò ôóíêöèÿ ïëîòíîñòè:
½
f (x) =
ε
x2 ,
0,
x > ε,
x < ε.
 [3] áûëî ïîëó÷åíî ñëåäóþùåå àñèìïòîòè÷åñêîå ñîîòíîøåíèå äëÿ õàðàêòåðèñòè÷åñêîé ôóíêöèè ðàñïðåäåëåíèÿ Ïàðåòî ïðè α = 1 è |t| → 0:
Z∞
eitx
fXj (t) =
²
ε
π
dx = 1 + itε − |t|ε − itεγ − itε ln(|t|ε) + O(|t|2 ),
x2
2
(3)
ãäå γ ïîñòîÿííàÿ Ýéëåðà.
Ïðè n → ∞ ïðåäåëüíîå âûðàæåíèå äëÿ õàðàêòåðèñòè÷åñêîé ôóíêöèè ñóììû
(1) ïàðåòîâñêèõ ñëó÷àéíûõ âåëè÷èí ìîæíî ïðåäñòàâèòü êàê
µ
¶
µ ¶
n itε π |t|ε itεγ
itε
|t|ε o
1
fSn (t) = exp
−
−
−
ln
+O
.
(4)
b
2 b
b
b
bn
n
Ñðàâíåíèå (4) è (2) ïîêàçûâàåò, ÷òî íåîáõîäèìî öåíòðèðîâàíèå ñóììû Sn . Ïîýòîìó áûë ââåäåí ñäâèã an = a · n, â ðåçóëüòàòå ÷åãî õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ
ñóììû ïàðåòîâñêèõ ñëó÷àéíûõ âåëè÷èí èìååò âèä:
n
o
¡ ¢
itεγ
π |t|ε
itε
itε
ita
fSn (t) = exp itε
−
−
−
ln(|t|ε)
+
ln(bn)
−
+ O n1 =
b
2 b
b
b
b
b
o
n
¡ ¢
= exp itε
(1 − γ) + itε
ln(bn) − ita
− π2 |t|ε
− itε
ln |t| − itε
ln(ε) + O n1 =
b
b
b
b
b
b
n ³
´
³
´o
¡ ¢
2
= exp itb ε(1 − γ) + ε ln(bn) − ε ln(ε) − a − π2 |t|ε
1
+
i
ln
|t|sign(t)
+ O n1 .
b
π
Òàêèì îáðàçîì, åñëè ïîëîæèòü
ε = σ,
a = ε(1 − γ + ln(bn)) − ε ln(ε),
b = π2 ,
òî ïðè n → ∞ ïîëó÷èì õàðàêòåðèñòè÷åñêóþ ôóíêöèþ óñòîé÷èâîãî çàêîíà (2).
(5)
53
ÌÎÄÅËÈÐÎÂÀÍÈÅ ÓÑÒÎÉ×ÈÂÛÕ ÑËÓ×ÀÉÍÛÕ ÂÅËÈ×ÈÍ Â ÑËÓ×ÀÅ...
Äëÿ ìîäåëèðîâàíèÿ óñòîé÷èâûõ ñëó÷àéíûõ ÷èñåë â ýòîì ñëó÷àå áóäåì çàäàâàòü
ïàðàìåòðû: β = 1, σ > 0. Ïðîäåìîíñòðèðóåì ðàáîòó äàò÷èêà äëÿ σ = {1, 2, 0.5, 4}
íà ðèñóíêå 1, ãäå K êîëè÷åñòâî ñãåíåðèðîâàííûõ óñòîé÷èâûõ ÷èñåë. Äëÿ ñðàâíåíèÿ òàêæå ïîñòðîåíà òåîðåòè÷åñêàÿ ôóíêöèÿ ïëîòíîñòè, ïîëó÷åííàÿ ñ ïîìîùüþ
ïðîãðàììû Stable.exe, ðàçðàáîòàííîé J.P. Nolan'îì (ñì. [8]).
Density function
Density function
0.16
0.3
Random generator
Theoretical
Random generator
Theoretical
0.14
0.25
0.12
0.2
0.1
0.08
0.15
0.06
0.1
0.04
0.05
0.02
0
−3
−2
−1
0
1
2
3
4
5
6
7
8
0
−2
0
2
4
(à)
6
8
10
12
14
(b)
Density function
Density function
0.6
Random generator
Theoretical
0.07
0.5
Random generator
Theoretical
0.06
0.4
0.05
0.04
0.3
0.03
0.2
0.02
0.1
0
0.01
−1
0
1
2
3
4
5
0
−5
0
5
10
15
20
25
(c)
(d)
Ðèñ. 1: Ãðàôèê ôóíêöèé ïëîòíîñòè ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîé ñëó÷àéíîé
âåëè÷èíû â ôîðìå (A) ïðè n = 103 , K = 105 , a) σ = 1, b) σ = 2, c) σ = 0.5, d) σ = 4
2. Ìîäåëèðîâàíèå äâóõñòîðîííèõ óñòîé÷èâûõ âåëè÷èí ñ ïîìîùüþ ñìåñè
ðàñïðåäåëåíèé Ïàðåòî
Äëÿ ìîäåëèðîâàíèÿ äâóõñòîðîííåãî óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè α = 1 áóäåì èñïîëüçîâàòü ñìåñü ðàñïðåäåëåíèé Ïàðåòî ñ íîñèòåëÿìè íà ïîëîæèòåëüíîé è
îòðèöàòåëüíîé ïîëóîñÿõ:
Xj = p · Xj+ + q · Xj− ,
p + q = 1.
Ïðè ýòîì Xj+ è Xj− èìåþò ñëåäóþùèå ôóíêöèè ðàñïðåäåëåíèÿ ïðè r, l > 0:
FX + (x) = 1 − xr ,
j
l
FX − (x) = − |x|
,
j
x ≥ r,
x ≤ −l.
54
ÁÀÃÐÎÂÀ È.À.
Õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ ðàñïðåäåëåíèÿ äëÿ Xj+ èìååò âèä (3), à
Xj− èìååò êîìïëåêñíî-ñîïðÿæåííóþ åé õàðàêòåðèñòè÷åñêóþ ôóíêöèþ. Òîãäà õàðàêòåðèñòè÷åñêàÿ
ôóíêöèÿ
íîðìèðîâàííîé è öåíòðèðîâàííîé ñóììû
³P
´
n
Sn =
j=1 (Xj − a)/(b · n) áóäåò âûãëÿäåòü ñëåäóþùèì îáðàçîì:
(
µ ¶´
³
|t|r
itrγ
π |t|r
itr
fSn (t) = exp p · itr
−
−
−
ln
+
b
2 b
b
b
bn
)
(6)
µ ¶´
³
|t|l
−itlγ
π |t|l
−itl
1
−itl
ita
+q · b − 2 b − b − b ln bn
− b + O( n ).
Åñëè ñãðóïïèðîâàòü ñëàãàåìûå â (6), òî ïîëó÷èì:
(
¡
¡
¢
¢
fSn (t) = exp itb (pr − ql) 1 − γ + ln(bn) − pr ln(r) + ql ln(l) − a −
µ
µ
¶
¶)
− π2 |t|
b (pr + ql) 1 + i
pr−ql
pr+ql
2
π
(7)
+ O( n1 ).
ln |t|sign(t)
Äëÿ òîãî, ÷òîáû õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ ñóììû Sn èìåëà â ïðåäåëå âèä
õàðàêòåðèñòè÷åñêîé ôóíêöèè óñòîé÷èâîãî ðàñïðåäåëåíèÿ, íåîáõîäèìî îïðåäåëèòü
ïàðàìåòðû a è b ñëåäóþùèìè âûðàæåíèÿìè:
¡
¢
a = (pr − ql) 1 − γ + ln(bn) − pr ln(r) + ql ln(l),
(8)
b = π2 .
Èç (7) è (8) ïîëó÷àåì ñîîòíîøåíèÿ ìåæäó ïàðàìåòðàìè ñëàãàåìûõ äâóõ ñìåñåé ðàñïðåäåëåíèé Ïàðåòî è ïàðàìåòðàìè ïðåäåëüíîãî óñòîé÷èâîãî ðàñïðåäåëåíèÿ
ïðè α = 1 â ôîðìå (À):

 p + q = 1,
pr + ql = σ,
(9)
 pr−ql
=
β.
pr+ql
Äëÿ ìîäåëèðîâàíèÿ óñòîé÷èâûõ ñëó÷àéíûõ âåëè÷èí â ýòîì ñëó÷àå áóäåì çàäàâàòü ñëåäóþùèå ïàðàìåòðû: β ∈ (−1; 1), σ > 0, p < 1. Ðàáîòà äàò÷èêà ïðè σ = 1
è β = {0.5; −0.5} ïðåäñòàâëåíà íà ðèñ. 2, äëÿ β = 0.2 è σ = {0.5; 2} íà ðèñ. 3.
Density function
Density function
0.3
Random generator
Theoretical
0.3
Random generator
Theoretical
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
−6
−4
−2
0
2
4
6
0
−6
−4
−2
0
2
4
6
(à)
(b)
Ðèñ. 2: Ãðàôèê ôóíêöèé ïëîòíîñòè ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîé ñëó÷àéíîé
âåëè÷èíû â ôîðìå (A) ïðè n = 103 , K = 105 , σ = 1, a) β = 0.5, b) β = −0.5
55
ÌÎÄÅËÈÐÎÂÀÍÈÅ ÓÑÒÎÉ×ÈÂÛÕ ÑËÓ×ÀÉÍÛÕ ÂÅËÈ×ÈÍ Â ÑËÓ×ÀÅ...
Density function
Density function
0.16
0.6
Random generator
Theoretical
0.5
Random generator
Theoretical
0.14
0.12
0.4
0.1
0.08
0.3
0.06
0.2
0.04
0.1
0.02
0
−4
−3
−2
−1
0
1
2
3
4
0
−10
−8
−6
−4
−2
0
2
4
6
8
10
(à)
(b)
Ðèñ. 3: Ãðàôèê ôóíêöèé ïëîòíîñòè ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîé ñëó÷àéíîé
âåëè÷èíû â ôîðìå (A) ïðè n = 103 , K = 105 , β = 0.2, a) σ = 0.5, b) σ = 2
Ïðè β = 0 èìååì äåëî ñ ðàñïðåäåëåíèåì Êîøè, äëÿ êîòîðîãî ñóùåñòâóåò ÿâíîå
âûðàæåíèå ïëîòíîñòè ôóíêöèè ðàñïðåäåëåíèÿ [5]:
σ
g(x, 1, 0, µ, σ) =
.
(10)
2
π(σ + (x − µ)2 )
Îãðàíè÷èìñÿ ñëó÷àåì, êîãäà ïàðàìåòð ïîëîæåíèÿ µ = 0. Ðàáîòà äàò÷èêà äëÿ
σ = {1; 0.8; 0.5; 2} ïðîäåìîíñòðèðîâàíà íà ðèñ. 4:
Density function
Density function
0.4
0.3
Random generator
Cauchy
Random generator
Cauchy
0.35
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
−6
−4
−2
0
2
4
6
0
−6
−4
−2
0
(à)
2
4
6
(b)
Density function
Density function
0.16
Random generator
Cauchy
0.6
Random generator
Cauchy
0.14
0.5
0.12
0.1
0.4
0.08
0.3
0.06
0.2
0.04
0.1
0
−4
0.02
−3
−2
−1
0
1
2
3
4
0
−10
−8
−6
−4
−2
0
2
4
6
8
10
(c)
(d)
Ðèñ. 4: Ãðàôèê ôóíêöèé ïëîòíîñòè ñãåíåðèðîâàííûõ ÷èñåë è ðàñïðåäåëåíèÿ Êîøè ïðè
n = 103 , K = 105 , a) σ = 1, b) σ = 0.8, c) σ = 0.5, d) σ = 2
56
ÁÀÃÐÎÂÀ È.À.
3. Ìîäåëèðîâàíèå îäíîñòîðîííèõ óñòîé÷èâûõ âåëè÷èí ïðè ïîìîùè ñìåñè ðàñïðåäåëåíèé Ïàðåòî è ðàâíîìåðíîãî
Ñ öåëüþ ïðîâåðêè âëèÿíèÿ ðàñïðåäåëåíèÿ â ðàéîíå íóëÿ íà èíòåðâàëå (0; r),
ãäå îòñóòñòâóåò íîñèòåëü ðàñïðåäåëåíèÿ Ïàðåòî, äëÿ ìîäåëèðîâàíèÿ îäíîñòîðîííåãî óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè α = 1 áûëà èñïîëüçîâàíà ñìåñü ðàñïðåäåëåíèé Ïàðåòî è ðàâíîìåðíîãî íà ïîëîæèòåëüíîé ïîëóîñè
Xj = p1 · g1 (x) + p2 · g2 (x),
ãäå p1 +p2 = 1, g1 (x) ïëîòíîñòü ðàñïðåäåëåíèÿ ðàâíîìåðíîãî çàêîíà íà èíòåðâàëå
(0; r), g2 (x) ïëîòíîñòü ðàñïðåäåëåíèÿ Ïàðåòî íà ëó÷å [r; ∞), r > 0.
Ôóíêöèÿ ïëîòíîñòè ðàñïðåäåëåíèÿ ñëó÷àéíîé âåëè÷èíû Xj :
(
g(x) =
p1
åñëè 0 < x < r,
r ,
p2 xr2 , åñëè x ≥ r.
Ïðåîáðàçóåì õàðàêòåðèñòè÷åñêóþ ôóíêöèþ âåëè÷èíû Xj :
fXj (t) =
= p1
Rr
0
R∞
eitx g(x)dx = p1
−∞
eitx 1r dx + p2
R∞
r
Rr
eitx g1 (x)dx + p2
R∞
eitx g2 (x)dx =
r
0
eitx xr2 dx.
Õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ ðàâíîìåðíî ðàñïðåäåëåííîé ñîñòàâëÿþùåé ðàâíà (eitr − 1)/itr, õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ Ïàðåòî ðàñïðåäåëåííîé ñîñòàâëÿþùåé èìååò âèä (3).
Òîãäà
õàðàêòåðèñòè÷åñêàÿ
ôóíêöèÿ
íîðìèðîâàííîé
ñóììû
´
³P
n
Sn =
j=1 Xj /(b · n) áóäåò èìåòü âèä:
"
³
itr
exp bn −1
itr/(bn)
´
³ itε
+ p2 · e bn −
π |t|ε
2 bn
itε
bn γ
itε
bn
´
ln( |t|ε
bn )
#n
O( n12 )
fSn (t) = p1 ·
−
−
+
=
"
#n
³ itr itr 2 1 ´
³
´
1+ bn +( bn ) 2! −1
|t|ε
|t|ε
π
itε
itε
1
= p1 ·
+ p2 · 1 + itε
=
itr/(bn)
bn − 2 bn − bn γ − bn ln( bn ) + O( n2 )
"
#n
³
´
³
´
|t|ε
1
itε
π |t|ε
itε
itε
= p1 · 1 + itr
+
p
·
1
+
−
−
γ
−
ln(
)
+ O( n12 ) =
2
bn 2
bn
2 bn
bn
bn
bn
#n
"
³
´
|t|ε
itr
π |t|ε
itε
itε
1
=
= p1 + p1 2bn
+ p2 + p2 · itε
bn − 2 bn − bn γ − bn ln( bn ) + O( n2 )
"
#n
³
´
|t|ε
itr
π |t|ε
itε
itε
1
= 1 + p1 2bn
+ p2 · itε
.
bn − 2 bn − bn γ − bn ln( bn ) + O( n2 )
(11)
Ïðè n → ∞ ïîëó÷àåì ïðåäñòàâëåíèå õàðàêòåðèñòè÷åñêîé ôóíêöèè íîðìèðîâàííîé ñóììû Sn â âèäå ýêñïîíåíòû, èç êîòîðîãî âèäíà íåîáõîäèìîñòü ââåäåíèÿ
57
ÌÎÄÅËÈÐÎÂÀÍÈÅ ÓÑÒÎÉ×ÈÂÛÕ ÑËÓ×ÀÉÍÛÕ ÂÅËÈ×ÈÍ Â ÑËÓ×ÀÅ...
ñäâèãà an = a · n. Ïîñëå îñóùåñòâëåíèÿ ñîîòâåòñòâóþùèõ èçìåíåíèé õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ íîðìèðîâàííîé è öåíòðèðîâàííîé ñóììû ïðèîáðåòåò âèä:
n
³
´
o
|t|ε
|t|ε2
itε
π |t|ε
tε
tε
ita
fSn (t) = exp p1 itr
+
p
·
−
−
i
γ
−
i
ln(
)
+
O(
−
+ O( n1 ) =
2
2b
b
2 b
b
b
bn
bn ¶
b
µ
n
¡
¢
= exp itb p2 r 1 − γ + ln(bn) − p2 r ln(r) + p1 2r − a −
¶o
µ
2
−p2 π2 |t|r
ln
|t|
·
sign(t)
+ O( n1 ).
1
+
i
b
π
(12)
Åñëè ïîëîæèòü
r = σ/p¡2 ,
¢
a = p2 r 1 − γ + ln(bn) − p2 r ln(r) + p1 2r ,
b = π2 ,
(13)
òî ïîëó÷èì ïðè n → ∞ õàðàêòåðèñòè÷åñêóþ ôóíêöèþ óñòîé÷èâîãî çàêîíà (2).
Äëÿ ìîäåëèðîâàíèÿ óñòîé÷èâûõ ñëó÷àéíûõ ÷èñåë â ýòîì ñëó÷àå áóäåì çàäàâàòü ïàðàìåòðû σ > 0 è p2 < 1. Ðàáîòà äàò÷èêà ïðîäåìîíñòðèðîâàíà íà ðèñóíêàõ
5 è 6, èç êîòîðûõ âèäíî, ÷òî ìîæíî ïîäîáðàòü òàêèå çíà÷åíèÿ a è b, ÷òî ïðè
ïðîèçâîëüíî çàäàííûõ çíà÷åíèÿõ ïàðàìåòðîâ σ, p2 ìîæíî ïîëó÷èòü ñëó÷àéíûå
âåëè÷èíû, èìåþùèå òðåáóåìîå îäíîñòîðîííåå óñòîé÷èâîå ðàñïðåäåëåíèå. Íà ðèñóíêàõ ïðåäñòàâëåíû ãðàôèêè äëÿ ðàçëè÷íûõ ñî÷åòàíèé ïàðàìåòðîâ p1 è p2 ñìåñè
ðàñïðåäåëåíèé ðàâíîìåðíîãî è Ïàðåòî:
Density function X j
g1+ (x) =
0.25
Density function
0.3
p1
r
Random generator
Theoretical
0.25
0.2
← g2+ (x) = p 2
0.15
0.2
r
x2
0.15
0.1
0.1
0.05
0
0.05
r
0
2
4
(a) 6
8
10
12
0
−2
0
2
4
6
8
10
(à)
(b)
Ðèñ. 5: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè n = 104 , K = 105 , β = 1, σ = 1,
p2 = 0.5, r = 2
58
ÁÀÃÐÎÂÀ È.À.
Density function Xj
0.35
Density function
0.15
Random generator
Theoretical
0.3
0.25
← g2+ (x) = p 2
r
x2
0.1
0.2
0.15
0.05
0.1
g1+ (x) =
p1
r
0.05
0
0
2
r
0
4
6
(a) 8
10
12
14
16
−2
0
2
4
6
8
10
12
14
(à)
(b)
Ðèñ. 6: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè n = 104 , K = 105 , β = 1, σ = 2,
p2 = 0.8, r = 2.5
Density function X j
Density function
0.6
0.35
g1+ (x)
p1
=
r
Random generator
Theoretical
0.3
0.5
0.25
0.4
0.2
0.3
0.15
0.2
0.1
← g2+ (x) = p 2
0.05
0
0
1
2
r
3
4 (a)
r
x2
5
0.1
6
7
8
0
−1
0
1
2
3
4
5
6
(à)
(b)
Ðèñ. 7: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè n = 104 , K = 105 , β = 1,
σ = 0.5, p2 = 0.2, r = 2.5
4. Ìîäåëèðîâàíèå äâóõñòîðîííèõ óñòîé÷èâûõ âåëè÷èí ïðè ïîìîùè ñìåñè ðàñïðåäåëåíèé Ïàðåòî è ðàâíîìåðíîãî
Äëÿ ìîäåëèðîâàíèÿ äâóõñòîðîííåãî óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè α = 1
ìîæíî òàêæå èñïîëüçîâàòü ñìåñè ðàñïðåäåëåíèé Ïàðåòî è ðàâíîìåðíîãî íà ïîëîæèòåëüíîé è îòðèöàòåëüíîé ïîëóîñÿõ:
−
−
+
+
Xj = q1 · Xj1
(x) + q2 · Xj2
+ p1 · Xj1
+ p2 · Xj2
,
ãäå p1 + p2 = p, q1 + q2 = q , p + q = 1, g1− (x), g1+ (x) ïëîòíîñòè ðàñïðåäåëåíèÿ ðàâíîìåðíîãî çàêîíà ñîîòâåòñòâåííî íà èíòåðâàëàõ (−l; 0) è (0; r), g2− (x), g2+ (x) ïëîòíîñòè ðàñïðåäåëåíèÿ Ïàðåòî ñîîòâåòñòâåííî íà ëó÷àõ (−∞; −l] è [r; ∞), r, l > 0.
ÌÎÄÅËÈÐÎÂÀÍÈÅ ÓÑÒÎÉ×ÈÂÛÕ ÑËÓ×ÀÉÍÛÕ ÂÅËÈ×ÈÍ Â ÑËÓ×ÀÅ...
59
Ôóíêöèÿ ïëîòíîñòè ðàñïðåäåëåíèÿ ñëó÷àéíîé âåëè÷èíû Xj èìååò âèä:

q l , åñëè x ≤ −l,


 q21 x2
åñëè −l ≥ x < 0,
l ,
g(x) =
p1
,

 r r åñëè 0 < x < r,

p2 x2 , åñëè x ≥ r.
Õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ áóäåò ïðåäñòàâëåíà ñëåäóþùèì îáðàçîì:
fXj (t) = q2
−l
R
−∞
eitx g2− (x)dx + q1
R0
−l
eitx g1− (x)dx + p1
Rr
0
eitx g1+ (x)dx + p2
R∞
r
eitx g2+ (x)dx.
Òîãäà
³ P õàðàêòåðèñòè÷åñêàÿ
´ ôóíêöèÿ íîðìèðîâàííîé è öåíòðèðîâàííîé ñóììû
n
Sn =
j=1 (Xj − a)/(b · n) áóäåò èìåòü âèä ïðè n → ∞:
¶
n µ
¡
¢
r
l
it
fSn (t)= exp b (p2 r − q2 l) 1 − γ + ln(bn) − p2 r ln(r) + q2 l ln(l) + p1 2 − q1 2 − a −
µ
¶o
³
´
p2 r−q2 l 2
π |t|
− 2 b (p2 r + q2 l) 1 + i p2 r+q2 l π ln |t| · sign(t)
+ O( n1 ).
(14)
Âèäíî, ÷òî â ýòîì ñëó÷àå, ÷òîáû ïîëó÷èòü ïðè n → ∞ óñòîé÷èâóþ ñëó÷àéíóþ
âåëè÷èíó, íåîáõîäèìî ïîëîæèòü:
¡
¢
a = (p2 r − q2 l) 1 − γ + ln(bn) − p2 r ln(r) + q2 l ln(l) + p1 2r − q1 2l ,
b = π2 .
Òîãäà ïàðàìåòðû ñìåñè ðàñïðåäåëåíèé Ïàðåòî è ðàâíîìåðíîãî è ïàðàìåòðû
óñòîé÷èâîãî ðàñïðåäåëåíèÿ ñâÿçàíû ñîîòíîøåíèÿìè:

p + q = 1,




 p1 + p2 = p,
q1 + q2 = q,


p2 r + q2 l = σ,


 p2 r−q2 l
p2 r+q2 l = β.
Îòêóäà ïîëó÷àåì, ÷òî r =
σ(1+β)
2p2 ,
l=
r, l > 0
σ(1−β)
2q2 ,
p1 = p − p2 , q1 = q − q2 , q = 1 − p.
Äëÿ ìîäåëèðîâàíèÿ óñòîé÷èâîãî ðàñïðåäåëåíèÿ ñ òðåáóåìûìè ïàðàìåòðàìè
β, σ ñ ïîìîùüþ ñìåñè ðàñïðåäåëåíèé Ïàðåòî è ðàâíîìåðíîãî áóäåì çàäàâàòü ñëåäóþùèå ïàðàìåòðû: β ∈ (−1; 1), σ > 0, p ∈ (0; 1), q2 ≤ 1 − p.
Ðàáîòà äàò÷èêà ïðè ðàçëè÷íûõ ñî÷åòàíèÿõ âõîäíûõ ïàðàìåòðîâ ïðîäåìîíñòðèðîâàíà íà ðèñóíêàõ:
60
ÁÀÃÐÎÂÀ È.À.
Density function
Density function X
0.35
g1− (x) =
0.35
j
Random generator
Theoretical
q1
→
l
0.3
0.3
0.25
0.25
0.2
0.2
g2− (x) = q2
0.15
l
→
x2
0.15
0.1
0.1
g1+ (x)
0.05
0
−10
−8
−6
−4
-l
−2
p1
=
r
0 (a)
r
← g2+ (x) = p 2 2
x
r4
2
6
8
10
0.05
0
−6
−4
−2
0
2
4
6
(à)
(b)
Ðèñ. 8: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè n = 104 , K = 105 , β = 0.3,
σ = 1, p = 0.3, p2 = 0.2, q2 = 0.3
Density function X j
Density function
0.6
0.3
l
= q2 2 →
x
g2− (x)
0.5
0.25
0.4
0.2
0.3
←
g1− (x)
q1
=
l
0.15
0.2
g1+ (x) =
0.1
0
−4
Random generator
Theoretical
−3
−1 -l
−2
0
(a)
← g2+ (x) = p 2
p1
r
1
0.1
r
x2
0.05
r
2
3
4
5
6
0
−4
−2
0
2
4
6
8
(à)
(b)
Ðèñ. 9: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè n = 104 , K = 105 , β = 0.8,
σ = 1, p = 0.6, p2 = 0.4, q2 = 0.3
Density function X
Density function
j
← g1+ (x) =
0.3
0.15
p1
r
Random generator
Theoretical
0.25
0.1
0.2
0.15
← g2+ (x) = p 2
r
x2
0.05
0.1
g2− (x) = q2
0.05
0
−12
l
→
x2
g1− (x) =
−10
−8
−6
-l −4
−2
q1
l
(a)
0
r
2
4
6
8
0
−14
−12
−10
−8
−6
−4
−2
0
2
4
6
(à)
(b)
Ðèñ. 10: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè n = 104 , K = 105 , β = −0.8,
σ = 2, p = 0.5, p2 = 0.2, q2 = 0.4
ÌÎÄÅËÈÐÎÂÀÍÈÅ ÓÑÒÎÉ×ÈÂÛÕ ÑËÓ×ÀÉÍÛÕ ÂÅËÈ×ÈÍ Â ÑËÓ×ÀÅ...
61
Îòäåëüíî ðàññìîòðèì ñëó÷àé, ñîîòâåòñòâóþùèé ðàñïðåäåëåíèþ Êîøè, êîãäà
2l
α = 1, β = 0, ïðè ýòîì ïàðàìåòð ïîëîæåíèÿ µ = 0.  ýòîì ñëó÷àå β = 0, pp22 r−q
r+q2 l = 0,
p2 r = q2 l, r = σ/2p2 , l = σ/2q2 , p1 = p − p2 , q1 = q − q2 .
Density function X
0.18
Density function
j
Random generator
Cauchy
0.3
0.16
0.14
0.25
g1− (x) =
0.12
q1
→
l
← g1+ (x) =
p1
r
0.2
0.1
0.15
0.08
g2− (x) = q2
0.06
l
→
x2
← g2+ (x) = p 2
r
x2
0.1
0.04
0.05
0.02
0
−10
−8
−6
−2 -l
−4
0 (a)
r2
4
6
8
10
0
−6
−4
−2
0
2
4
6
(à)
(b)
Ðèñ. 11: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è ðàñïðåäåëåíèÿ Êîøè ïðè n = 104 , K = 105 , β = 0, σ = 1,
p = 0.5, p2 = 0.3, q2 = 0.3
Density function X j
q1
→
g1− (x) =
l
0.06
Density function
← g1+ (x) =
p1
r
0.16
Random generator
Cauchy
0.14
0.05
0.12
0.1
0.04
0.03
g2− (x) = q2
l
→
x2
←
g2+ (x)
r
= p2 2
x
0.08
0.06
0.02
0.04
0.01
0
−20
0.02
−15
−10
−5
-l
r
0 (a)
5
10
15
20
0
−10
−5
0
5
10
(à)
(b)
Ðèñ. 12: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è ðàñïðåäåëåíèÿ Êîøè ïðè n = 104 , K = 104 , β = 0, σ = 2,
p = 0.5, p2 = 0.25, q2 = 0.25
Density function X j
Density function
1
Random generator
Cauchy
0.6
0.9
← g2+ (x) = p 2
0.8
r
x2
0.5
0.7
0.4
0.6
0.5
0.3
0.4
0.3
g2− (x) = q2
l
→
x2
0.2
← g1+ (x) =
0.2
g1− (x) =
0.1
0
−4
−3
−2
q1
→
l
−1
-l
0 (a)
r
1
p1
r
2
0.1
3
4
0
−5
−4
−3
−2
−1
0
1
2
3
4
5
(à)
(b)
Ðèñ. 13: a) âèä ïëîòíîñòè ðàñïðåäåëåíèÿ Xj , b) ãðàôèê ôóíêöèé ïëîòíîñòè
ñãåíåðèðîâàííûõ ÷èñåë è ðàñïðåäåëåíèÿ Êîøè ïðè n = 104 , K = 104 , β = 0, σ = 0.5,
p = 0.6, p2 = 0.5, q2 = 0.3
62
ÁÀÃÐÎÂÀ È.À.
Çàêëþ÷åíèå
 ñòàòüå áûëè ïîëó÷åíû âûðàæåíèÿ äëÿ ïàðàìåòðîâ a è b èç (1), íåîáõîäèìûå
äëÿ ìîäåëèðîâàíèÿ óñòîé÷èâûõ ñëó÷àéíûõ âåëè÷èí ïðè α = 1. Áûëè îïðåäåëåíû
ñîîòíîøåíèÿ ìåæäó ïàðàìåòðàìè ñìåñè ðàñïðåäåëåíèé Ïàðåòî è ðàâíîìåðíîãî
è ïàðàìåòðàìè óñòîé÷èâîãî çàêîíà. Ìîäåëèðîâàíèå ïîêàçàëî, ÷òî è äëÿ ñëó÷àÿ
α = 1 ïîëó÷åíèå óñòîé÷èâûõ ñëó÷àéíûõ ÷èñåë âîçìîæíî ñ ïîìîùüþ äàò÷èêà, îñíîâàííîãî íà îáîáùåííîé öåíòðàëüíîé ïðåäåëüíîé òåîðåìå.
Ñïèñîê ëèòåðàòóðû
[1] Àðõèïîâ Ñ.Â., Áàãðîâà È.À. Î ìîäåëèðîâàíèè îäíîñòîðîííèõ óñòîé÷èâûõ ñëó÷àéíûõ âåëè÷èí // Âåñòíèê Òâåðñêîãî ãîñóíèâåðñèòåòà. Ñåðèÿ: Ïðèêëàäíàÿ
ìàòåìàòèêà, âûïóñê 4(15). Òâåðü: èçä-âî Òâåðñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà, 2009. Ñ. 53-62.
[2] Àðõèïîâ Ñ.Â., Áàãðîâà È.À. Î ìîäåëèðîâàíèè óñòîé÷èâûõ ñëó÷àéíûõ âåëè÷è ïðè α áëèçêèõ ê åäèíèöå// Âåñòíèê Òâåðñêîãî ãîñóíèâåðñèòåòà. Ñåðèÿ:
Ïðèêëàäíàÿ ìàòåìàòèêà, âûïóñê 3(18). Òâåðü: èçä-âî Òâåðñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà, 2010. Ñ. 5-14.
[3] Àðõèïîâ Ñ. Πïðåäñòàâëåíèè õàðàêòåðèñòè÷åñêîé ôóíêöèè óñòîé÷èâîãî ðàñïðåäåëåíèÿ ïðè α = 1 (â ïå÷àòè)
[4] Ëåâè Ï. Ñòîõàñòè÷åñêèå ïðîöåññû è áðîóíîâñêîå äâèæåíèå. Ì.: Íàóêà,
1972. 375 ñ.
[5] Çîëîòàðåâ Â.Ì. Îäíîìåðíûå óñòîé÷èâûå ðàñïðåäåëåíèÿ. Ì.: Íàóêà, 1983.
304 ñ.
[6] Janicki A., Weron A. Simulation and Chaotic Behavior of α-Stable Stochastic
Processes. New York: Marcel Dekker, 1994. 355 p.
[7] Samorodnitsky G., Taqqu M. Stable Non-Gaussian Random Processes.
York: Chapman and Hall, 1994. 632 p.
New
[8] J. P. Nolan. http://academic2.american.edu/ jpnolan/stable/stable.html
[9] Uchaikin V.V., Zolotarev V.M. Chance and Stability. Stable Distributions and
their Applications. Utrecht: VSP, 1999. 594 p.