- Email: [email protected]
A note on the T+m transformation
(I-
“ae + s) + Pae + s>- (I - ua+ s>- (“2 + s>=
[(uI
+
u)s]“+J
pus + s>(“ae + s) - (I-“ae + SIP + s)
“a~ = ur+ua iCl.Iadold aq~ arleq swal
‘0 luelsuos awes lo3 JolIla aql J! &IO pue J! s = [(UI + u)s]“+J ‘1 VIEIXO~HL
‘qldap alow
LII u0~w1~103sue.11 “+J aql ale8ysaAu!
. (1 - u4 + 4s + cm + 4s - (I - u)s - (4s (1 - 4S(~
MOU ue3 aM
= [cm + u>s,“+J
+ 4s - (1 - ud + w4s
uaA!% uoy~~1.1o3sue.11 “+J. aql 103 uoyaldxa aJay& 9 + s = (u)s se (u)s q3ea swdxa
aql asn aM ‘laqlm3
ue3 aM
‘MON
.(!)D
I?=
.tulal
yrq~ pue dt?lf) U! Jolla UC sluasardar “a stuns IylJed
(u)s
30 I=~{(u)s}
II aDuanbas
aql 30 l!urq aql s! pue
(?)D I?=
s rCq palouap
SF a3uanbas
srql
~JIM
paleposse
I=,’I(.)I DI Ial aM ‘uo!le)ou sno!Aald ~I!M lualysuo3 sa!las aql uaqL .a~uanbas al!ugu! UIZaq OS ‘ul30 saD!oq3 lallaq aq$ alylua~a33!p lou saop uop!puo~ s,l;ralls ‘JSRI~UO:,UI ~d~~w~tueulcp payorlu! aq uw put? aIduris s! a.InpaDold aqL ‘zu 30 satyeA ,,lallaq,, ay$ %uysooy~ SMOG poq$aur .n’Q .aDuanbas IIXI$~O aql wql dlp!dw alotu lyq aql 01 sa%Jaauo3 aDuanbas “+L aql qD!qM lapun uoypuoD t2 sayi9 J!alls ‘swlal aA!l!sod ql!M sa!las alguy lua8raauoD e .IO~ ‘swns IeyJed 30 aDuanbas [email protected] aql ueql ICIp!dw alow lyg aql 01 sa%aAuo:, u1.103sue~1 “‘+J aql icq palEIaUa% a3uanbas aql araqM asw aql ql!M pauIaXIo3 SE?Ml!asS .[z] l!aIlS Aq passrwp &ua3al SI?M pua [I] y.1~13 put Ae~f) dq paluasald SCM 1.11~03sue1~“+J aqL .pa+ap s! u1.103sue~l aql103 UI 30 arqelz a$erldoIddc ue %!sooqD 30 poqlaur e ‘@adold Jolla sty1 30 aflpalmouy E LI.IO.IJ.pagap s! aIqeydde s! uoy~11ro3sue_1l aql qD!qM 103 acwanbas e 30 ICl.Iadold IoIla aql .sa!las algu! 30 awa%aAuoD aql aleJala33e 01 pasn so qXqM ‘u~.103sue~1 “‘1 aql se ufnouy ‘uoy~11~03sw~~ aguanbas-ol-aauanbas .reauguou aql sassnwp z.L.0~ SIHJ
NOILVtWOdSNVXL
“‘J, BH,.L NO WON
V
B. JONES
304
Next,
suppose
T+,[S(n
+m)]
= S, then
(S + G) (S + err +m I) - (S + fn +m) (S + err I) (S + e,) - (S + e,, I) - (.S + e,, +A + (S + e,, +lllm I> = ” which readily
yields
Since these error terms are real numbers. there must exist a 01 such thate,, r,,i =Oie,,. Similarly there must exist a real 02 such that err+ ,,1 1 = t&e,,~,. However, from the equation just derived we have
so that & = 8i and the conclusion
follows.
THEOREM 2. If the error
e,,.,,, = He,, where
property
(1)
h
(4
1
1-
This is clearly
then
S(n + m) = as(n) + h Ly
Proof.
6, # 1 holds.
0’
true since s(n + m) - HS(n) = S + He,, - HS - He,, = S(1 - H),
and taking b = S(1 - O), (1) is true; 8 and h are unique so (2) is true. is applied to sequences which are approximate in the In practice the T+, transformation above error property. That is, e,,+,,, = F)e,, or S(n + m) = OS(n) + h. The purpose of these two theorems is to pinpoint a means of selecting an m for the T,,, transformation. We see from them that when the T,,,, transformation is applicable, we have .S(n + m) = fX(n) + h. Thus, an obvious test for applicability is the lagged serial correlation coefficient. This coefficient is tried over different values (lags) of m, and the m value with most significant correlation is chosen. This coefficient always lies between - 1 and 1, and for values close to * 1 (e.g. ~0.99) the fit is very good. As a matter of interest, Theorem 2 gives another technique for computing S. This is done by estimating H and h and forming h/(1 - 0). As an example, consider
ncos( 1.OSi)
w> =,T, i?
S(SO0) = 0.27416.
very slowly to the Table 1 gives s(n), T+,, T+l, and T+3. S(n), T+, and T+z are converging limit; however, T+3 obtains the limit rather quickly. We use the formula given in the appendix to compute the serial correlation for lags of one, two. and three. Using S(1). S(2). S(3), S(4). S(5), and S(6), we find the serial correlation for a lag of 1 (m = 1) is 0.85, the serial correlation for a lag of 2 is -0.27, and the serial correlation for a lag of 3 (m = 3) is -0.99. Other samples using more values of S(n) or starting the samples later in the S(n) sequence yield similar results. Actually, as we move out in the S(n) sequence with our samples, the correlation for
.
305
A note on the T+, transformation
Table
. n
5 10 15 20 25 30 35 40 45 50
S(n) 0.28042 0.26960 0.27055 0.27324 0.27494 0.27516 0.27450 0.27385 0.27370 0.27398
1. Extrapolation
applied
Tt, 0.24484 0.26206 0.28294 0.27610 0.27637 0.27225 0.27353 0.27327 0.27497 0.27446
= cos( 1.05i) to 7 7
T+z 0.25436 0.23462 0.27670 0.27546 0.28967 0.27356 0.27374 0.26456 0.27442 0.27436
T+, 0.26952 0.27374 0.27408 0.27415 0.27416 0.27416 0.27416 0.27416 0.27416 0.274 16
a lag of 3 goes to -1, whereas the correlation for the other two lags is erratic and low. This demonstrates our analysis very well; the m value to use in the T+, transform should be chosen based on a simple dynamic preliminary correlation analysis. Of course, here we computed T,, and T+* merely for demonstration purposes. APPENDIX The lagged
serial correlation
coefficient
(lag of VI) using the first n + m values of {S(i)} can be written
REFERENCES 1. GRAY H. L. & CLARK W. D., On a class of nonlinear transformations and their applications infinite series, J. Res. natn. Bur. Stand. 73B, (3), 251 (1969). 2. STREIT R., The T+, transformation, Mathematics of Computation, (July 1976).
to the evaluation
of
“ae + s) + Pae + s>- (I - ua+ s>- (“2 + s>=
[(uI
+
u)s]“+J
pus + s>(“ae + s) - (I-“ae + SIP + s)
“a~ = ur+ua iCl.Iadold aq~ arleq swal
‘0 luelsuos awes lo3 JolIla aql J! &IO pue J! s = [(UI + u)s]“+J ‘1 VIEIXO~HL
‘qldap alow
LII u0~w1~103sue.11 “+J aql ale8ysaAu!
. (1 - u4 + 4s + cm + 4s - (I - u)s - (4s (1 - 4S(~
MOU ue3 aM
= [cm + u>s,“+J
+ 4s - (1 - ud + w4s
uaA!% uoy~~1.1o3sue.11 “+J. aql 103 uoyaldxa aJay& 9 + s = (u)s se (u)s q3ea swdxa
aql asn aM ‘laqlm3
ue3 aM
‘MON
.(!)D
I?=
.tulal
yrq~ pue dt?lf) U! Jolla UC sluasardar “a stuns IylJed
(u)s
30 I=~{(u)s}
II aDuanbas
aql 30 l!urq aql s! pue
(?)D I?=
s rCq palouap
SF a3uanbas
srql
~JIM
paleposse
I=,’I(.)I DI Ial aM ‘uo!le)ou sno!Aald ~I!M lualysuo3 sa!las aql uaqL .a~uanbas al!ugu! UIZaq OS ‘ul30 saD!oq3 lallaq aq$ alylua~a33!p lou saop uop!puo~ s,l;ralls ‘JSRI~UO:,UI ~d~~w~tueulcp payorlu! aq uw put? aIduris s! a.InpaDold aqL ‘zu 30 satyeA ,,lallaq,, ay$ %uysooy~ SMOG poq$aur .n’Q .aDuanbas IIXI$~O aql wql dlp!dw alotu lyq aql 01 sa%Jaauo3 aDuanbas “+L aql qD!qM lapun uoypuoD t2 sayi9 J!alls ‘swlal aA!l!sod ql!M sa!las alguy lua8raauoD e .IO~ ‘swns IeyJed 30 aDuanbas [email protected] aql ueql ICIp!dw alow lyg aql 01 sa%aAuo:, u1.103sue~1 “‘+J aql icq palEIaUa% a3uanbas aql araqM asw aql ql!M pauIaXIo3 SE?Ml!asS .[z] l!aIlS Aq passrwp &ua3al SI?M pua [I] y.1~13 put Ae~f) dq paluasald SCM 1.11~03sue1~“+J aqL .pa+ap s! u1.103sue~l aql103 UI 30 arqelz a$erldoIddc ue %!sooqD 30 poqlaur e ‘@adold Jolla sty1 30 aflpalmouy E LI.IO.IJ.pagap s! aIqeydde s! uoy~11ro3sue_1l aql qD!qM 103 acwanbas e 30 ICl.Iadold IoIla aql .sa!las algu! 30 awa%aAuoD aql aleJala33e 01 pasn so qXqM ‘u~.103sue~1 “‘1 aql se ufnouy ‘uoy~11~03sw~~ aguanbas-ol-aauanbas .reauguou aql sassnwp z.L.0~ SIHJ
NOILVtWOdSNVXL
“‘J, BH,.L NO WON
V
B. JONES
304
Next,
suppose
T+,[S(n
+m)]
= S, then
(S + G) (S + err +m I) - (S + fn +m) (S + err I) (S + e,) - (S + e,, I) - (.S + e,, +A + (S + e,, +lllm I> = ” which readily
yields
Since these error terms are real numbers. there must exist a 01 such thate,, r,,i =Oie,,. Similarly there must exist a real 02 such that err+ ,,1 1 = t&e,,~,. However, from the equation just derived we have
so that & = 8i and the conclusion
follows.
THEOREM 2. If the error
e,,.,,, = He,, where
property
(1)
h
(4
1
1-
This is clearly
then
S(n + m) = as(n) + h Ly
Proof.
6, # 1 holds.
0’
true since s(n + m) - HS(n) = S + He,, - HS - He,, = S(1 - H),
and taking b = S(1 - O), (1) is true; 8 and h are unique so (2) is true. is applied to sequences which are approximate in the In practice the T+, transformation above error property. That is, e,,+,,, = F)e,, or S(n + m) = OS(n) + h. The purpose of these two theorems is to pinpoint a means of selecting an m for the T,,, transformation. We see from them that when the T,,,, transformation is applicable, we have .S(n + m) = fX(n) + h. Thus, an obvious test for applicability is the lagged serial correlation coefficient. This coefficient is tried over different values (lags) of m, and the m value with most significant correlation is chosen. This coefficient always lies between - 1 and 1, and for values close to * 1 (e.g. ~0.99) the fit is very good. As a matter of interest, Theorem 2 gives another technique for computing S. This is done by estimating H and h and forming h/(1 - 0). As an example, consider
ncos( 1.OSi)
w> =,T, i?
S(SO0) = 0.27416.
very slowly to the Table 1 gives s(n), T+,, T+l, and T+3. S(n), T+, and T+z are converging limit; however, T+3 obtains the limit rather quickly. We use the formula given in the appendix to compute the serial correlation for lags of one, two. and three. Using S(1). S(2). S(3), S(4). S(5), and S(6), we find the serial correlation for a lag of 1 (m = 1) is 0.85, the serial correlation for a lag of 2 is -0.27, and the serial correlation for a lag of 3 (m = 3) is -0.99. Other samples using more values of S(n) or starting the samples later in the S(n) sequence yield similar results. Actually, as we move out in the S(n) sequence with our samples, the correlation for
.
305
A note on the T+, transformation
Table
. n
5 10 15 20 25 30 35 40 45 50
S(n) 0.28042 0.26960 0.27055 0.27324 0.27494 0.27516 0.27450 0.27385 0.27370 0.27398
1. Extrapolation
applied
Tt, 0.24484 0.26206 0.28294 0.27610 0.27637 0.27225 0.27353 0.27327 0.27497 0.27446
= cos( 1.05i) to 7 7
T+z 0.25436 0.23462 0.27670 0.27546 0.28967 0.27356 0.27374 0.26456 0.27442 0.27436
T+, 0.26952 0.27374 0.27408 0.27415 0.27416 0.27416 0.27416 0.27416 0.27416 0.274 16
a lag of 3 goes to -1, whereas the correlation for the other two lags is erratic and low. This demonstrates our analysis very well; the m value to use in the T+, transform should be chosen based on a simple dynamic preliminary correlation analysis. Of course, here we computed T,, and T+* merely for demonstration purposes. APPENDIX The lagged
serial correlation
coefficient
(lag of VI) using the first n + m values of {S(i)} can be written
REFERENCES 1. GRAY H. L. & CLARK W. D., On a class of nonlinear transformations and their applications infinite series, J. Res. natn. Bur. Stand. 73B, (3), 251 (1969). 2. STREIT R., The T+, transformation, Mathematics of Computation, (July 1976).
to the evaluation
of