Costraied liear discrimiat rule for -groups via the Studetized classificatio statistic W for large dimesio Takayuki Yamada Istitute for Comprehesive Educatio Ceter of Geeral Educatio Kagoshima Uiversity --30 Korimoto Kagoshima 890-0065 Japa Abstract This paper is cocered with -group liear discrimiat aalysis for multivariate ormal populatios with ukow mea vectors ad ukow commo covariace matrix for the case i which the sample sizes N N ad the dimesio p are large. We give Studetized versio of the W statistic uder the high-dimesioal asymptotic framework A that N N ad p ted to ifiity together uder the coditio that p/n + N coverges to a costat i 0 ad N /N coverges to a costat i 0. Asymptotic expasio of the distributio for the coditioal probability of misclassificatiocpmc of the Studetized W is derived uder A. By usig this asymptotic expasio we give the cut-off poit such that the oe of two CPMCs is less tha the presettig value. Such the costraied discrimiat rule is studied by Aderso 973 ad McLachla 977. Mote Carlo simulatio revealed that the proposed method is more accurate tha McLachla 977 s method for the case i which p is relatively large. AMS 000 subject classificatio: primary 6H30; secodary 6H Key Words ad Phrases: Discrimiat aalysis W classificatio statistic locatio ad scale mixtured expressio CPMC High-dimesioal asymptotic results Upper cofidece bouds o CPMC. Itroductio Let x ij j =... N i i = be the jth sample observatio j =... N i from the ith populatio Π i i = with mea µ i ad commo covariace matrix Σ. We cosider the problem to allocate a observatio vector x which is accordig to either Π or Π. A commoly used rule is that W = x x S {x } x + x < c > c allocate x as Π Π which is called the liear discrimiat rule where c is a cut-off poit x x ad S are the sample mea vectors ad the pooled sample covariace matrix defied by x i = N i N i j= x ij i = S = = N = N + N. N i x ij x i x ij x i i= j= There are two types of probability of misclassificatio. Oe is the probability of allocatig x ito Π eve though it is actually belogig to Π. The other is the probability that x is classified as Π although it is actually belogig to Π. These two types of expected probabilities of misclassificatios EPMCs for W-rule are expressed as e c = P W < c x Π ad e c = P W > c x Π. E-mail address:yamada@gm.kagoshima-u.ac.jp
I geeral it is hard to evaluate these expected probabilities of misclassificatio EPMC explicitly but some asymptotic results icludig asymptotic expasios have bee obtaied. Aderso [] derived a asymptotic expasio for Studetized W ad applied it to idetify c such that e c = ε + O where ε 0 is a presettig level which is give by experimeter. This discrimiat rule is used to cotrol oe of EPMCs for the case i which oe type of errors is geerally regarded as more serious tha the others such as medical applicatios associated with the diagosis of diseases. Aderso [] s asymptotic expasio is obtaied uder the asymptotic framework A0: A0 : N N N /N γ 0 p is fixed. For achievig the same aim with Aderso [] McLachla [0] proposed the cut-off poit c such that where P c c < Ξ L = ε + O c c = P W < c x Π ; x x S ε 0 is a presettig level which is set by experimeter ad Ξ L is a upper boud which is set by experimeter. These ad some other asymptotic results were reviewed by Siotai [3] by McLachla [] ad by Aderso []. Geerally the precisio of asymptotic approximatios uder A0 gets worth as the dimesio p becomes large. As a alterative approach to overcome this shortcomig it has bee cosidered to derive asymptotic distributios of discrimiat fuctios i a high-dimesioal situatio where ad p ted to ifiity together. Yamada et al. [4] derived a asymptotic expasio of Studetized W uder the high-dimesioal asymptotic framework A ad the assumptio C such that A : N N N /N γ 0 p p/ γ 0 [0 ; C : 0 0 where is the squared Mahalaobis distace defied as = µ µ Σ µ µ. Usig the asymptotic expasio they proposed a cut-off poit c such that e c = ε + O AC where the symbol O AC stads for the term such that O AC coverges to a costat as uder A ad C. The usefuless such the high-dimesioal asymptotic framework is metioed i Fujikoshi et al. [6]. The aim of this paper is to obtai a cut-off poit c h which satisfies that P [ c c h < Ξ H ] = ε + OAC for the presettig values ε ad Ξ H. I order to derive this we show a asymptotic expasio of the distributio for the statistic c c h uder A ad C. Sice the distributio for c c is the same as the oe for c c with iterchagig N ad N we oly derive asymptotic expasio of the distributio for c c. This paper is orgaized as the followig: Sectio presets Studetizatio for W uder A. I Sectio 3 we derive the asymptotic distributio of c c via the Studetized statistic W uder A. Asymptotic expasio for the Studetized c c is derived by makig use of a powerful method kow as the method by the differetial operator which was used by James [8] Okamoto [] etc. I Sectio 4 we propose costraied liear discrimiat rule for CPMC for large dimesioal case. Simulatio results are writte i Sectio 5. We revealed that the proposed methods performs well for the case i which Ξ H is ot so small. I Sectio 6 cocludig remarks are writte. Some proofs ad techical results are give i Appedix.
Studetizatio for W uder A For x Π i it follows from Lachebruch [9] that W = x x S {x } x + x = V / Z i U i i = where V = x x S ΣS x x Z i = V / x x S x µ i U i = x x S x µ i D ad D is the squared sample Mahalaobis distace defied by D = x x S x x. The we fid that V is a positive radom variable ad U i V are joitly idepedet of Z i. Further Z i is distributed as N0. This ormality follows by cosiderig the coditioal distributio of Z i whe x x ad S are give. I this case W is called a locatio ad scale mixture of the stadard ormal distributio. It ca be expressed that E[U i ] = i E[V ] = m p m N N + Np N N + m m + where m = p. The aalytic expressios for VarU i ad for VarV which are provided by Fujikoshi [3] show that VarU i 0 ad VarV 0 uder A ad C. It implies that the limitig distributio of W uder A ad C is ormal with mea u 0i = lim A E[U i ] ad variace v 0 = lim A {E[V ]}. The atural estimate for E[U i ] E[V ] is obtaied by replacig with the followig ubiased estimator : = m D Np N N ad we write it as Ê[U i] Ê[V ]. The ubiasedess for Ê[U i] Ê[V ] also holds. We ca show that Ê[U i] Ê[V ] has cosistecy uder A ad C. From Slutsky s theorem W + Ê[U i] Ê[V ] D N0 for x Π i D where the symbol stads for the covergece i distributio. From the techical reaso istead of usig Ê[U i] Ê[V ] we use U 0i V 0 for the Studetizatio of W i this paper which is defied as follows. U 0i = Ê[U i] + i m + m N i = i m p + i m N N m + m V 0 = m m + Ê[V ] = + m m + m + + Np N N It is oted that U 0i V 0 is ot the ubiased estimator of E[U i ] E[V ] but has has the cosistecy uder A ad C. We ca also show that W i. = W + U 0i D N0 for x Πi. V0 N i 3
3 Asymptotic distributio for the Studetized CPMC Let c j be the cut-off poit for Wj ad let C i j deote the coditioal probability of misclassificatio of Wj misallocatig a observatio from Π j where i j. The C i j is give by C i j = c i j V0 c j U 0j = P i W < i V0 c j U 0j x Π j x x S = Φ i V0 c j U 0j + U j V V0 Φ V c V {U 0 U } i = j = = V0 Φ V c V {U 0 U } i = j =. From Lemma ad 3 i Appedix A the distributio for C is idetical to C if c = c. From that reaso we oly deal with C. Asymptotic expasio of the distributio for C ca be obtaied by the oe for C by replacig N N c with N N c. Hereafter we set U 0 = U 0 U = U ad c = c uless makig cofusio. 3. Stochastic expressio for CPMC Now we cosider to express C as the fuctio of simple variables. Let u = + / Σ / x x N N u = Σ / N x + N x N µ N µ N B = Σ / SΣ /. The u u ad B are idepedet. I additio u N p /N + /N / δ I p ad u N p 0 I p where δ = Σ / µ µ. It also holds that B is distributed as a Wishart distributio with degrees of freedom ad covariace matrix I p which is deoted as W p I p. Substitutig them we have U = u B u u + B u N δ B u N N N N NN It is also described that V = N u B u. N N Usig this expressio we ca write = Nm u B u Np. N N N N U 0 = N u B u + p N N m + N + N u V 0 = B u. m + m + N N The followig lemma which is give by Yamada et al. [4] eables to see the fuctios of the idepedet stadard ormal ad chi-squared variables for U V U 0 ad V 0. 4
Lemma. Let v N p a I p v N p 0 I p W W p I p ad v v ad W are idepedet. The a a Z + X a a W X a Z X v 3 v W v v W v = D X + X {Z + a X a + Z + X 4}Z 3 3 v W v {Z + a X a + Z + X 4 } + X {Z + a X a + Z + X 4 } 3 X where X i χ f i i = 3 4; Z i N0 i = 3; all the seve variables X X X 3. X 4 Z Z Z 3 are idepedet; f = p + f = p = p + f 4 = p. Similar results to Lemma was treated i Fujikoshi ad Seo [5] Fujikoshi [4] ad Hyodo ad Kubokawa [7]. Put where b = b w w w 3 z z = N N z + f + w N b = b w w w 3 w 4 z z z 3 = + f t sz 3 f + w f tz q = q w w 4 z z = s f + w q = q w w w 3 w 4 z z = + f t s f + w s = sw 4 z z = z + t = tw w 3 = + w + w 3 ; N N N + z + f 4 + w 4 f f f 4 are defied i Lemma. The we have B δ B u B Q def = u B u u B u Q u B u b /f W /f W / W 3 Z / Z / D = b /f W /f W / W 3 /f 4 W 4 Z / Z / Z 3 / q /f W /f 4 W 4 Z / Z / q /f W /f W / W 3 /f 4 W 4 Z / Z / where W i = f i /X i /f i i =... 4. This implies that B U U 0 α 4 α 3 α 0 α α V 0 = D 0 0 β 0 B Q V 0 0 0 β 0 0 Q 0 5
where Lettig α = { N } = α = p N N N N N m + α N 3 = α 4 = N N NN + N β = β = N. m + m + N N N N β q c + α q α + α 3 b α 4 b Ψw w w 3 w 4 z z z 3 = Φ β q we ca express that V0 c U 0 + U C = Φ V D = Ψ / W /f W / W 3 /f 4 W 4 Z / Z / Z 3 /. 3 Hereafter we set v as the variable vector ad y as the radom variable vector which are defied by v = v v v 3 v 4 v 5 v 6 v 7 = w w w 3 w 4 z z z 3 y = Y Y Y 3 Y 4 Y 5 Y 6 Y 7 = W W W 3 W 4 Z Z Z 3 ad uless makig cofusio. Ψv = Ψw w w 3 w 4 z z z 3 4 3. Studetizatio for CPMC uder A It ca be expressed that b 0 0 0 0 0 = N N f N = b 0 b 0 0 0 0 0 0 0 = 0 q 0 0 0 0 = N N f N + f 4 q 0 0 0 0 0 0 = f + f = q 0 N N N + f 4 = q 0. So we have Ψ0 = Φ β q 0 c + α q 0 α α 4 b 0 β q 0 = Φc. It is oted that Ψv is the smooth fuctio o R 3. We will expad Ψ / W /f W / W 3 /f 4 W 4 Z / Z / Z 3 / at /f W /f W / W 3 /f 4 W 4 Z / Z / Z 3 / = 0 0 0 0 0 0 0. Let ψ = ψ c be the vector valued fuctio i R 7 defied by ψ = D v Ψv 0 5 6
where D = diag /f /f / /f 4 Ψv is defied as 4 ad the otatio 0 stads for the value at the poit that v = 0. We ca express that ψ = ϕcp where p = p c δ = c f f f f + f + f 4 f q0 β f α 0 0 f + 0 0 0 0 f 4 α f 4 f α β f f f α 4 α 3 q0. 6 I additio let Ψ = Ψ c be 7 7 matrix valued fuctio defied by We ca show that Ψ = D v v ΨvD 0. DΨ D = ϕcp where P = P c is the 7 7 matrix valued fuctio defied as P = P 0 + cp + c P + c 3 P 3 with P i = P i i = 0 3 beig the 7 7 symmetric matrix valued fuctio. The aalytic form for P i i = 0 3 are give i Appedix C. It ca be expressed that Ψ / W /f W / W 3 /f 4 W 4 Z / Z / Z 3 / = Φc + ψ y + y Ψ y + 3/ R 7 where R is a remaider term cosistig of a homogeeous polyomial of order 3 i the elemets of y of which the coefficiets are O as uder A ad C plus / times a homogeeous polyomial of order 4 plus a remaider term that is O uder A ad C for fixed y. By virtue of 7 with combied the use of the formula 3 we have C Φc ψ ψ / D = ψ ψ y + y Ψ y + ψ R. 8 It follows from the defiitio of y that y D N 7 0 I 7 uder A ad C which implies the followig theorem. Theorem. Uder the high-dimesioal asymptotic framework A ad the assumptio C where C Φc ϕc ρc / ρc = ψ ψ /{ϕc} = c + f f f + { + + β q 0 f f 4 f c f f 4 D N0 α f β α + q0 + f } α + α β f 3q 0. 9 7
Sice ρc is ukow parameter it is eeded to estimate for Studetizatio. The atural estimate ρc ca ot be used. The reaso is that {q 0 } / which is icluded i ρc ca ot by defied for the case i which 0 < D < {/m }N/N N sice q 0 takes egative value. Istead of usig the ubiased estimator we use which is ot the ubiased estimate of. It ca be expressed that ρc = f + f A = m + Np D 0 N N + β q 0 f c f 4 α β f + q 0 f f f + f 3 f + f ατ + α 3 β where τ = f + f f + f + + + + f. f f 4 β It is sufficiet to show the positiveess for τ A to esure that ρc A > 0. We ca express that τ A = = f = f + f f + f + + f f 4 + + f + f D + + β f f 4 f + f f f + [ q 0 A + + f { f + + f f } f f 4 β + f ] D f 4 β f f 4 f f + f + f f + where the last equality follows from the fact that q 0 A = + f D. 3 f β The o-egativeess for τ A follows from that f f 4 + f { f + + f f } = p 3 + 5 + p p + + p p + p + 7 > 0. Note that A p uder A ad C. From this rate cosistecy we obtai that By Theorem ad Slutsky s theorem ρc A ρc p. C Φc D N0. 4 ϕc ρc A / 8
3.3 Asymptotic expasio for the distributio of the proposed Studetized statistic for CPMC uder A I this sectio we derive a asymptotic expasio for the distributio for the Studetized C to improve the covergece rate i 4. Firstly we give a geeral result for the cumulative distributio fuctio of the radom variable T which has the form: T = h h h y + h h y Hy + R 5 for h R 7 ad the symmetric matrix H where R is the term cosistig of a homogeeous polyomial of order 3 i the elemets of y of which the coefficiets are O uder A ad C plus / times a homogeeous polyomial of order 4 plus a remaider term that is O uder A ad C for fixed y. Theorem. The cumulative distributio fuctio of T which is described as 5 ca be expressed as P T x = Φx s H 0 x + s H xϕx + O where Φ deotes the cumulative distributio fuctio of the stadard ormal distributio ϕ is the derivative of Φ ad H k x deotes the Hermite polyomial of degree k especially H 0 x = H x = x. Here for h h 7 = h. { s = tr H h h s = h h 3/ 3 4 k= } h 3 k + h Hh f k The proof of Theorem is give i Appedix B. Now we cosider to express the proposed Studetized statistic as the form 5. By virtue o combied with the use of the formula ad the fact that Q = {N/N N }D we have q 0 = D + f q /f W /f 4 W 4 Z / Z /. 6 f Put Ωw w 4 z z = where q = q w w 4 z z α 5 = + f f f + α 6 = f 4 α α 5 f f β α 5 c α 6 + α 7 + f + f q α 8 f + f q α 7 = α + α3 β [ α 8 = + f ] 3 f { f + + f f } f α f f 4 3f. 3f + α 5 β Without makig cofusio we express Ωv = Ωw w 4 z z. / 9
for v = w w 4 z z. From the expressios ad 6 we have Ω /f W /f 4 W 4 Z / Z / D = ρc A. By takig ito cosideratio that q 0 = /f + f / q 0 0 0 0 it is easy to see that Ω0 = ρc. Sice Ωv is the smooth fuctio o R Taylor series expasio at /f W /f 4 W 4 Z / Z / = 0 0 0 0 gives Ω /f W /f 4 W 4 Z / Z / = ρc + ω y + R where y = W W 4 Z Z ω = ω c is the vector valued fuctio i R 4 defied by ω = D Ωv 0 7 v with beig that D = diag /f /f 4 ad R is the residue term of which the property is similar to R. We ca express that where ω = {ρc } p 3/ p = p c α5 α 6 = c α 8 + α 5 α 6 q0 q f 0 Let ω be the extesio for ω defied as f f 4 q 0 N N f N q 0 0. 8 for ω = ω ω ω 3 ω 4. The we have ω = ω 0 0 ω ω 3 ω 4 0 ρc D ρc = + ρc ω A y + R 3 9 where R 3 is the residue term of which the property is similar to R. Combiig 8 ad 9 we have C Φc D = ϕc ρc p A / p y + } y {P + p p p ρc p p + p p y + R 4 where p is the extesio for p of which the defiitio is the same as ω ad R 4 is the residue term of which the property is similar to R. Summarizig the above results asymptotic expasio for the coditioal probability of misclassificatio is obtaied which is give as the followig theorem. 0
Theorem 3. P c V0 c U 0 Φc ϕc ρc x = Φx s H 0 x + s H x ϕx + O AC A / where s = s c = s = s c = { ρc tr P + { {ρc } 3/ 3 } ρc p p + p p 4 p 3 k + p f P p + p p k k= }. Aalytic forms for s ad s are complicated so we omit to describe them. We otice that s ad s cotai oly the term i which the power of is the eve umber. 4 Costraied liear discrimiat rule for CPMC I this sectio we give a costraied liear discrimiat rule for -groups of which oe of the two coditioal misclassificatio probabilities does ot exceed the presettig value Ξ H with the cofidece level α. Suppose that c H = ξ H ρξ H A z ε 0 where ξ H = Φ Ξ H for Ξ H 0. By virtue of 4 combied with Slutsky s theorem we have lim P c V0 c H U 0 < Ξ H = ε A uder the assumptio C. As a extesio of 0 we obtai the followig result. Theorem 4. Let c H = ξ H h h where h = ρξ H A z ε ξ H h = ρξ H A + f 4 + f α z β q 0 A ε ρξ H A b ξ H A H 0z ε + b ξ H A H z ε with beig that ξ H = Φ Ξ H for Ξ H 0. The P c V0 c H U 0 < Ξ H = ε + O AC. The proof of Theorem 4 is similar to the oe of Theorem i McLachra [0] ad so we omit to describe it.
5 Simulatio result Simulatio experimets were performed to cofirm the asymptotic result of Theorem 4. We also compared the accuracies with the asymptotic result of Theorem i McLachla [0] for the case i which N = N = 50 p = 0 30 50 70 = 3 ε = 0.05 Ξ = Φ/ where the settigs of ad Ξ are followed to McLachla [0]. Whe we treat the distributios of W -rule without loss of geerality from ivariat property of the distributio for the orthogoal trasformatio of observatio vector we may assume that two give ormal populatios with the same covariace matrix are Π : N p /e I p Π : N p /e I p where e = 0... 0. To compute misclassificatio probability geerate 0 4 traiig samples. For each traiig samples we geerate 0 4 test samples i which observatio vectors are i.i.d. as N p /e I p. The value of the coditioal misclassificatio probability was calculated by sim k = umber of misclassificatio 0 4 k =... 0 4 i each traiig samples. We took the average of Isim < Ξ... Isim 0 4 < Ξ where I. deotes the idicator fuctio ad wrote it as the value for the actual level i row Y i Tables. The same value for McLachla [0] s approximatio was writte i row Mc. Table : Actual levels of cofideces that the coditioal error probabilities are less tha Ξ whe the omial level is ε = 0.95. p = 0 p = 30 p = 50 p = 70 = Y 0.95 0.95 0.96 0.95 Mc 0.93 0.9 0.85 0.78 = Y 0.95 0.95 0.95 0.95 Mc 0.93 0.90 0.84 0.75 = 3 Y 0.94 0.94 0.93 0.93 Mc 0.9 0.87 0.79 0.67 From Tables we ca see that our proposed asymptotic approximatio has good accuracy whe =. The actual level of cofidece becomes small as the dimesio gets large for the case i which = 3. We ca check that McLachla [0] s result does ot work well for our settigs. Extra simulatio results which does ot writte i this paper reveals that the actual cofidece level gets small from the omial level as the dimesio becomes close to sample size for the case i which Ξ is small. 6 Cocludig remarks I this paper we derived Studetized statistic for the coditioal probability of misclassificatio for the Studetized W ad derived its asymptotic expasio for distributio up to the term of O / uder the high-dimesioal asymptotic framework A. It may be oted that the order of its error is O. Based o the derived asymptotic expasio we gave the cut-off poit for the liear discrimiat rule that the oe of two coditioal error probabilities is less tha the presettig value. Simulatio results revealed that our proposed rule is superior tha McLachla [0] s result. Ufortuately our proposed rule did ot work well for the case i which Ξ is small. The modificatio should be cosidered ad is beig a future problem. A Equality i distributios for proposed statistics I this sectio firstly we metio the equality i distributios for U 0 U ad U 0 U which is give as the followig lemma
Lemma. The distributio for U 0 U is the same as the oe for U 0 U with exchagig N for N. Proof. Set S N N = U 0 U ad set S N N = U 0 U. I additio put x i D = µi + Ni z i i = S D = W δ = Σ/ µ µ where z z W are idepedet; z ad z are distributed as N p 0 I p ; W is distributed as W p I p. The we have { S N N = D z z δ W z } z δ Np m N N N N N N p m N N m m + N z z δ W z + z δ N N N N { S N N = D z z δ W z } z δ Np m N N N N N N + p m N N m m + N + z z δ W z + z + δ. N N N N By iterchagig N ad N { S N N = D m + p m N N + z z δ W z z δ N N N N z z δ N N { = m p m N N m m + N W z + z + δ N N z z δ W z z δ N N N N N z N z δ D = S N N where z = z ad z = z. m m + N W z + z δ N N Np N N Np N N Next we show the equality i distributio for V 0 ad for V which is give as the followig lemma. Lemma 3. Each of the distributios for V 0 ad for V is the same as the oe with exchagig N for N. We omit to write the proof of Lemma 3 sice it is similar to Lemma. } } 3
B Proof of Theorem I this sectio we gave the proof of Theorem. Firstly we gave the followig two lemmas which is to be used. Lemma 4. Suppose that a R ad g. is a polyomial fuctio. Let Z ad Y are radom variables; Z is distributed as the stadard ormal distributio; Y is distributed as the chi-square distributio with f degrees of freedom. The E [ gze itaz] a = exp it E[gZ + ita] E [ gw e itaw ] f/ [ ] f W + ita = ita exp ita E g f ita /a where i = ad W = f/y/f. It is easy to prove Lemma 4 so we omit to write the proof. Proof of Theorem. From the assumptio for T give i 5 the characteristic fuctio ca be expaded as [ E[expitT ] = E T 0 + it ] h h T + O where T 0 = exp it h h h y T = y Hy exp it h h h y. From Lemma 4 we have 4 h k E[T 0 ] = it k= h h f k [ { = + 4 3 it3 k= k= fk / exp f k h 3 k h h 3/ uder A. It ca be expressed that 7 [ ] E[T ] = h kk E Yk h k exp it h h Y k E exp it + 7 k= l= l k h k it h h }] fk 7 k=5 e it / + O 7 h k h h Y k l= l k 7 [ ] h k h kl E Y k Y l exp it h h Y h l k exp it h h Y l E exp it exp it h k h h 7 α= α kα l h α h h Y α 4
for h kl = H. From Lemma agai we have [ 4 ] + it h k E[T ] = E[T 0 ] h /h h 7 kk k= { ith k / h h /f k } + h kk { + it h k/h h} k=5 4 4 ith k / h h + E[T 0 ] h kl k= l= ith k / h h ith l / h h /f k ith l / h h /f k l k 4 7 7 7 + = E[T 0 ] k= l=5 h kl ith k / h h ith k / h h h it l /f k h h + { tr H + it h Hh h h + O/ k=5 l=5 l k h kl it h kh l h h uder A. The desired result ow follows by formally ivertig the expasio for the characteristic fuctio. } C Aalytic forms for P 0 P P ad P 3 I this sectio we give the aalytic form for P i i = 0 3. The derivatio is straightforward ad so is omitted. 5
P 0 = P = q 0β f 0 f f+f3 f α f f+f3 f3 α 0 0 0 α q0 f f+f3 f f+f3 f f α f3 α α q0 3 f f+f3 f f+f3 3/ α β q0 f3 f α β α β α β q0 f f+f3 f f+f3 0 0 f3 α 4 0 f f3 f α 4 f3 q0 f α4 0 α 0 q0 f3 α 0 0 { } α 3/ α f3 f f3 f αα4 β β + q0 4 3 f f f+f3 f f+f3 f { 3 f f+f3 { 3 4 f f+f3 f f+f3 + f f3 f+f3 f f+f3 f } } α β q0 ff3 0 0 f3 0 0 q0 0 0 0 0 f3 f f3 αα4 β αα4 β 3/ α 4 β q0 q0 q0 f f3 αα3 β αα3 β 3/ αα3 β α3α4 β α 3 β q0 q0 q0 q0 α β q0 α4 β q0 α β 3/ α β β 3/ q0 q0 0 q0 6
P = P 3 = β q0 f α f f+f3 f α f f+f3 0 0 f3 α 0 f f+f3 f f+f3 3/ α α β f α f3 α f f+f3 f f+f3 0 0 0 3/ f f f+f3 f f+f3 f f+f3 f f+f3 f3 α 4 f3 α 4 f f3 α4 f f+f3 f f+f3 α 3 f q 0 α 3 q 0 0 0 0 0 0 0 0 f f f+f3 f f f+f3 f f+f3 f3 α 3 q 0 0 0 0 0 f3 ff3 0 0 0 0 f3 0 0 0 0 0 0 0 0 0 0 0 0 0 0. f f3 α β α β 7
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