POWER TRANSFORMATIONS WHEN THEORETICAL MODELS TO. Raymond J. Carroll and. David Ruppert
|
|
- Gervase Kelley
- 6 years ago
- Views:
Transcription
1 POWER TRANSFORMATIONS WHEN FITTING THEORETICAL MODELS TO DATA Raymnd J. Carrll and 2 David Ruppert 1 Running Title: Fitting Theretical Mdels 1 Natinal Heart, Lung and Bld Institute and the University f Nrth Carlina. Grant AFOSR Supprted by the Air Frce Office f Scientific Research 2 University f Nrth Carlina. Supprted by Natinal Science Fundatin Grant MCS Sme key wrds: Transfrmatins, Bx-Cx mdels, theretical mdels, rbustness. ~ AMS 1970 Subject Classificatins: Primary 62F20; Secndary 62G35.
2 -2- A B S T R ACT We investigate pwer transfrmatins in nn-linear regressin prblems when there is a physical mdel fr the respnse but little understanding f the underlying errr structure. In such circumstances and unlike the rdinary pwer transfrmatin mdel, bth the respnse and the mdel must be transfrmed simultaneusly and in the same way. We shw by an asympttic thery and a small Mnte-Carl study that fr estimating the mdel parameters there is little cst fr nt knwing the crrect transfrm a priri; this is in dramatic cntrast t the results fr the usual case that nly the ~ respnse is transfrmed.
3 -3-1: INTRODUCTION t Often in scientific wrk, ne bserves data y and x = (xl.. x p ) and pstulates that these data fllw a mdel (1.1) y. = f(x.~ 8 0 ), i = l~..., N~ t. t. where 8 0 is a k-parameter vectr. The functin f may be derived, fr example, frm differential equatins believed t gvern the physical system which gave rise t the data. The deterministic mdel (1.1) is ften inadequate since the data exhibit randm variatin, but whereas f was derived frm theretical cnsideratins, there is really n firm understanding f the mechanism prducing the randmness. In this case, ne typically assumes that (1. 2) where the {Ei} are i.i.d. N{O,a~). In thse cases in which the data suggest that mdel (1.2) is als unsatisfactry, ne might then assume that the errrs are multiplicative and lg-nrmal, s that (1. 3) The pint here is that mdel (l.l) is equivalent t the mdel h{y.) = h{f{x., 8 0 )) t. t. whenever h{ ) is a mntnic transfrmatin. Therefre (1.2) and (1.3) are based n the same theretical mdel, but they allw variability int the mdel in different fashins.
4 -4- A mre flexible apprach is t take a sufficiently rich family f strictly mntnic transfrmatins h(y~a), indexed by the m-vectr parameter A, and t assume that fr sme value A (1.4) The mdel (1.4) is in the spirit f Bx and Cx (1964), wh suggested the family f pwer transfrmatins with m = 1 and (1.4b) = lg (y) if A= O. Hwever, as we will make clear, ur prpsed mdel (1.4) has greatly different ramificatins than usually assciated with the pwer family. Bx and Cx (1964) used their family in a study f the transfrmatin mdel (1.5) h(y~a ) = x~ e + E. 0 Ntice here that, unlike (1.4), the regressin functin in (1.5) is nt transfrmed. Bx and Cx sught a transfrmatin which achieves 1) a simple, additive r linear mdel, 2) hmscedastic errrs and 3) nrmally distributed errrs. Our mdel is different. Theretical cnsideratins already prvide a regressin functin. We hpe t transfrm the respnse and the regressin functin simultaneusly t btain hmscedasticity and nrmality. There are tw reasns fr using mdel (1.4) instead f simply fitting (1.1) by least squares r sme ther methd. First, estimatin f e based n mdel (1.4) shuld be mre efficient than ther methds. Secnd, it may
5 -5- be necessary t estimate the entire cnditinal distributin f y given x; if the data clearly suggest that the distributins f {y.-f(x.,e )} are nt 1" 1" 0 cnstant, ne must g beynd standard regressin methdlgy. An example, which partly mtivated the research f this paper, cncerns the relatinship between egg prductin in a fish stck and subsequent recruitment int the stck. At least fr sme species, as egg prductin increases, the change in the skewness and variance f recruitment is as large as the change in the median recruitment, and this change in distributinal shape may have imprtant implicatins fr management f the fishery. The utline f the paper is as fllws. Sectin 2 discusses a current cntrversy cncerning the mdel f Bx and Cx. Bickel and Dksum (1981) have shwn that, in mdel (1.5), the ML estimate f e can be much mre variable when A is estimated cmpared t when A is knwn. In Sectin 3, 0 4It we demnstrate fr ur mdel (1.5) an entirely different result: the ML estimate f e in mdel when A is unknwn cmpared t when A is knwn. 0 cnsiderably strnger result. (1.4) turns ut t be nly slightly mre variable In Sectin 4 we prve a By examining a weighted least abslute deviatins estimatr, we prvide a lwer bund f 2/n n the asympttic relative efficiency f the ML estimatr f e in mdel (1.4) when A is unknwn cmpared 0 t the MLE when A is knwn. 2: RECENT STUDIES OF THE BOX AND COX MODEL In Sectin 7 f Bx and Cx's riginal paper they discuss the analysis e f effects after transfrmatin. A They state that, after finding A, ne shuld A -estimate effects (regressin parameters) n the scale A which has been chsen fr analysis and nt n the true but unknwn A scale. 0 Hwever, in discuss- ing interactins, they g n t state that lithe general cnclusin will be A that t allw fr the effect f analysing in terms f A rather than A, the 0
6 -6- residual degrees f freedm need nly be reduced by... the number f cmpnent parameters in A". Bx and Tia (1968) agree, stating that the nly practical effect between using A in the psterir distributin f 8, , is an adjustment in the degrees f freedm. Bickel and Dksum (1981) disagree with this cnclusin. rather than the true Fllwing calculatins fr the lcatin prblem dne by Hinkley (1975) and suggestive Mnte-Carl results f Spitzer (1978) and Carrll (1980), they calculated 2 fr general regressin the large sample infrmatin matrix f A, a and They fund that the large sample variance f 8 is larger, ften much larger, when A is estimated cmpared t when is knwn. They als state that the cnclusin f Bx and Tia is nt crrect. On a technical level, part f the (A) (1..)/ (0)1..-1 z = y y, where y is the gemetric mean f the {Yi}' Hwever, Hinkley and Runger (1982) fund z(a) unsatisfactry in several respects. The differences may als be cntextual; at the null hypthesis f n interactin effects, ne can act as if were knwn, with an apprpriate change in the degrees f freedm. See Carrll (1982) and Dksum and Wng (1981). Since pwer transfrmatins have been used ften and with real satisfactin by applied statisticians, the findings f Bickel and Dksum were surprising and led t further research. Hinkley and Runger argue that the parameter 8 0 in (1.5) is nt physically meaningful; it is defined in an unknwn scale s that a unit change in x is nt easily interpreted by 8 0 alne. Instead, ~ they argue that in practice, the relevant distributin is the cnditinal A distributin f e given A. As N ~ 00, the cnditinal variance f e given A
7 -7- and the variance f ewhen A is knwn cnverge t the same matrix. They then argue that, when analyzing 6 0 ' n adjustment need be made fr the fact that A was estimated. This appealing behavir is smewhat cunter-balanced by difficulties with the cnditinal mean in hypthesis testing in unbalanced designs, as pinted ut by Carrll (1982). Carrll and Ruppert (1981) als nticed the difficulty with interpreting 6 0 and studied predicting the median f Y n the riginaz data scale by t'" backtransfrming x 6. the prblems f definitin inherent with 6 0 data dependent scale. effect f nt knwing A This idea f lking at the respnse surface avids being defined in an unknwn r They fund that when predicting the median f Y, the can be large but is in general similar t the effect f adding ne mre regressin parameter, and it is certainly much less severe than the effect when estimating 6. 0 The abve discussin establishes the extent t the cntrversy surrunding the Bx and Cx mdel applied (1.5). We believe (1.4) entirely avids this cntrversy. First, the parameter 6 0 has physical meaning even if A is unknwn, since f{x ij 6 ) is the median f Yi n matter what the 0 true scale. Secndly, the large sample analysis t fllw indicates that '" 6 is nly slightly mre variable when A is estimated than when A is knwn. 3: LIKELIHOOD ANALYSIS The likelihd analysis prceeds as fllws: define z. = dh{f.{6 ),A )ld6 '/- '/-000 f (6) = f{x. J6), f = f (8 0 )' '/- '/- '/- '/- h (y) = h (YJ A) = dh{yj A)ldYJ and h{y) = h{yj A). Y Y Let h A (y) and h AA (y) be the gradi ent vectr and Hessi an f h{yj A) with respect t A. By simple algebra we find the jint infrmatin matrix f (6 0,O,A )
8 -8- as (all summatins are frm 1 t N) (3.1) s/ L 0 C 1 / N- 1 I = C /a 2 lf 1/(20 1f 0 ) 0 C /0 2 J 0 where (3.2) C 2 = -N-1ELEi[hA(Yi) - ha(fi)]t C J = N-1EL{IhA(Yi) - ha(f i )] [ha(yi) - ha(fi)t t + Ei[hAA(Yi) - haa(f i )] + (a/aa}(a/aa) lg[hy(yi)]}. Using the wrk f Hadley (1971), it is straightfrward, thugh perhaps smewhat tedius, t establish cnditins sufficient that (8, 0 2, ~) is cnsistent and asympttically nrmal. We will nt pursue this matter further, but rather we will assume that ( 8, 0, A ;s apprximately N 8 0 A), I -1) and we will study I -1. A "2 ") t ( ( 2 t 000 In general, C 1 and C 2 are nt zer and the asympttic distributin f (~, 0 2 ) when A ;s estimated differs frm when A is knwn. At least t this 0 pint then, the analysis ;s similar t thse dne in the usual Bx-Cx mdel (1.5). The key questin, f curse, is whether r nt C 1 and C 2 are " sufficiently different frm zer t seriusly affect the distributin f A. The expressins C 1, C 2 and C J are cmplex even when f i (8 0 ) has a nice frm such as simple linear regressin. T simplify matters sufficiently that we can gain sme insight abut the difference between knwing and estima-
9 -9- ting A ' we fllw Bickel and Dksum and thers and let 0 0 ~. While Bickel and Dksum let N ~ 00 and simultaneusly, we let N + 00 and then 0 +. There is n essential difference between the tw appraches. Our is very suitable fr heuristic arguments. It shuld be emphasized that we are nt cncerned nly, r even primarily, with small 0 0 In fact, the need fr transfrmatin is greater when 0 0 is large. The small 0 asympttics d, hwever, lead t majr simplificatins, and the Mnte-Carl results presented later agree with them. (3.3) Taylr expansins shw that under mild regularity cnditins Standard calculatins shw that when A is knwn, (3.4) N~ Cvariance [(8 - e )/0, ( )/0 2 1A knwn] A -1 = [-1 S 0] 02. Let D = Diag(, 0 2, 1). Then, t find this limiting cvariance matrix when 0 A is unknwn, we must find the upper left (k + 1) x (k + 1) crner f DID = s C 1 /O C / C /0 2 ;) 0 which by standard results n inverting partitined matrices is A- 1 + FE- 1 F t where A- 1 is given in (3.4), t E = C3/0~ - B A B, -1 F = A B,
10 -10- and B = Clearly, F= and In rder t btain simple asympttics, we will assume that fr 00 fixed, c1/a~, C2/0~, and CJ/~ cnverge as N + 00, and that these, in turn, have limits D1~ D2~ and D J respectively as We als assume that t S + S (psitive definite) as N If D J - 2D 2 D 2 is nnsingular, then Urn Urn N + 00 Urn + 0 Urn N+ S-l 0 ] [ = W THEOREM 1. Assume that the limits D 1 ~ D 2 ~ D J ~ S mentined abve exist and that D J _ 2D 2 D 2 is nnsingular. As N + 00 and then + O~ the limit A distribu~in f 6 is the same whether A is knwn r unknwn. The limit distributin f 0 depends n whether A is knwn r unknwn. As an example cnsider multiple linear regressin and the pwer transfnnatin family, i.e., h(y~a) is given by (l.4b) and h(y.~a) t =x. e + E '/; '/; 0
11 -11- where xl'.., x n are knwn k x 1 vectrs. Als, suppse that A = 0, A -1. i.e., the lg transfrmatin is needed. Then hy{y) = Y -, ha{y) = (lg y)2/2, and haa(y) = (lg y)3/3. We find that and -1 t t 2 A = N \ x.x./{x.e ) L 1,1, 1,0 C 1 = -(2N)-lE\[x./{x~e )J{[lg(x~e ) + E.J2_[lg{x~e )]2} L 1, 1, 0 1, 0 1, 1, 0 = -002{2N)-1\ x./{x~e ), L 1, 1, 0 1 t 2 t 2 C 2 = -(2N)- ELEi{[Zg{xie) + Ei] - [Zg{x i 80)] } = _N- 1 \ lg{x~e ) 0 2 L 1, 0 0' ė -1 t 3 t 3 + (3N) E\ E.{[lg(x.e ) + E.] - [lg(x.e )] } L 1, 1,0 1, 1,0 t 2 =?/ /N \[lg{x.e)] 0 L 1, 0 Therefre, D 2 = lim N+ N- 1 I Zg(x~e)' and prvided the abve limits exists. Thus, the 1 x 1 matrix D 3-2D;V 2 is twice the 1imi t f the vari ance f lg(x;e),..., lg (x;e ), and wi 11 be nnsi ngu 1ar except in degenerate situatins. There is thus a fundamental difference between the mdels (1.4) and (1.5). A small simulatin study is utlined in Sectin 6 and helps back up Therem 1. This result can be extended t nn-nrmal errr distributins as
12 -12- well as the rbust methds f Carrll (1980) and Bickel and Dksum (1981). The details are nt instructive. ė 4: A LOWER BOUND ON THE EFFICIENCY OF THE MLE. Let e(~) and 8(~) dente the ML estimatr with A estimated and knwn respectively. Let ARE(81~82) be the asympttic relative efficiency f 8 1 t 8 2. Fr fixed 00' it is difficult t find ARE(8(A),e(~)) and, in fact, this may depend n e, A ' the {x.} and the crdinate f e being estimated. 1-- that can be said fr certain is that this ARE All is at least ne and cnverges t ne as In this sectin we will define a weighted L 1 r least abslute deviatin estimatr 8(w)and shw that ARE(8(A )' e(w)) ~ rr/2. Under reasnable regularity cnditins, thi's means that ARE(8(A ), 8(~)) is bunded between ne and rr/2~ in vivid cntrast t the Bx and Cx mdel (1.5) in which this last ARE can apprach infinity. We first lk at general weighted L 1 estimatrs. wn right. The results stated here seem t be new and are f interest in their Let w1~... ~ w N be psitive numbers and let e(l) be any pint which minimizes the expressin I w. IY -!.(8(L))\ Under (1.4),!.(e ) is the unique median f Y.~ s we can expect 8(L) t be cnsistent. The unweighted L 1 estimate fr linear mdels was studied by Ruppert and Carrll (1980). Thse results suggest that
13 -13- (4.1) 0;' LW. sign (y. - f.(8(l))s... ~ ~ ~ ~ s. = df.(e )/de. ~ ~ 0 Define r. = y. - f.(e ) and let m. be the density f r.. ~ ~ ~ 0 ~ ~ By a generalizatin f the strng law, fr example Therem 7.1 f Carrll and Ruppert (1982) which itself generalizes Lemma 4.2 f Bickel (1975), (4.2) ~ lw. {sign(y. - f.(8(l))) -sign(r.)}s. ~ ~ ~ ~ ~ -(ELw.{sign(y. - f.(e))-sign(r.)}s.)! e=8(l)). ~ ~ ~ ~ ~ Nw, as E ~ 0.. we btain that ė (4.3) E(sign(r. + E) - sign(r.)) -2Em.(O) ~. ~ ~ ~ Cmbining (4.1)-(4.3) we get t rder (N- Yz ), (4.4) Nw, since fr mdel (1.4) E. = h(f.(e )+ r... A ) - h(f.(e ), A ), ~ ~ ~ 0 ~ 0 we then have (4.5) m. (0) ~
14 -14- Thus, if we chse (4.6) w. = h (f. (e ), A ), ~ y ~ 0 0 we have by (4.4)-(4.6) and the Central Limit Therem that Nw 8(L) is nt a bna fide estimatr ~ALl N 2 (e(l) - e )/ 0 -> N(O,(n/2) S- ). 0 since w. in (4.6) requires A, e t be knwn. Hwever, if in (4.6) ne plugs in any N Yz cnsistent estimatrs A f e and A and calls the L 1 estimate based n these new weights e(w), then 0 using Therem 7.1 f Carrll and Ruppert (1982), ne can als shw that. ~ Nw, because it then fllws that ~ A L -1) N 2 (e(a )- e )/0 -> N(O, S (4.8) A A ARE(e(A ), e(w)) = n/2, ARE(8(A ), e(~)) ~ n/2. Therem 1 and the Mnte-Carl results t fllw indicate that the upper bund in (4.8) is quite cnservative. it is a bund that des nt depend n 0 0, estimatr f e prvided that Ei is nt needed. The beauty f (4.8) is that The weighted L 1 estimatr may well be useful fr example if in (1.4) ne suspected that the errrs {E.} are nt nrmal. 1- is the unique median f E It is a cnsistent 1- Symmetry f
15 -15-5: THE K-SAMPLE PROBLEM Our mdel (1.4) and Therem 1 prvide sme useful insight int the k-samp1e prblem under the frmulatin (1.5) f Bx and Cx. In their mdel, fr each f k ppulatins we have (5.l) h{y. " A ) = fl. + E.. j = 1, 1-J 0 J 1-J... ~ k; i = 1,...,N. 1- The equivalent frmulatin frm ur viewpint is (5.2) e Here ~. is the median f y. n the riginal scale and fl. is the expected J 1-J J value f y in the A scale. The results f Carrll and Ruppert imply 1-J 0 that f~ estimating the ~IS, there is little cst in nt knwing A, while fr estimating the fl's, Bickel and Dksum shw that the cst f nt knwing A can be enrmus. Since there shuld be little cst in testing fr equality f means when 1. 0 is unknwn. These heuristics are frmally prven by Carrll (1982) and Dksum and Wng (1981). 6: MONTE-CARLO. " T study 8 when N is finite and a tk a small simulatin f the mdel is nt necessarily small, we under (6.1 ) h{y.,a ) = h{ x.,a ) + a E.,
16 -16- where h( ) is the Bx and Cx pwer family (l.4b). In ur simulatins, were nrmally distributed with mean zer and variance ne and 8 1 = 7, 8 2 = 2. We cnsidered three estimatrs: 1) 2) 3) ML estimatr, A knwn (KNOWN) ML estimatr, A unknwn (MLE) The rdinary least squares estimatr (LSE) withut any transfrmatin. The median f y is x, s that LSE frms an especially plausible estimatr f the slpe 8 2 (fr which it is cnsistent). We chse three values f 0 0 : = 0.05, 0.10, and We present results in Tables 1 and 2 fr A = 0 (lg-nrmal data) and A = There were 600 replicatins f the experiment fr each (A,E ) 0 0 and each estimatr, all generated frm a cmmn set f randm numbers. The nrmal randm deviates were generated frm the IMSL ruti:ne GGNPM. N = 50, the design pints {x.} were equally spaced n [-1, 1], the errrs 'Z.- Estimatin f (8 1, 8 2 ) fr each A was dne by the IMSL rutine ZXSSQ while ZXGSN was used t estimate A. The results fr the ML estimatr with A unknwn (dented MLE) are very encuraging. The mean square errrs fr MLE are quite clse t thse fr KNOWN, the ML estimatr with A knwn, especially fr the slpe 8 2. These results agree with ur small 0 thery and indicate the minimal cst fr nt knwing A. The relative efficiencies f MLE t KNOWN are always well abve ~ the lwer bund f 2/n. T appreciate hw well MLE des relative t KNOWN
17 -17~ (line 2 f Tables 1 and 2), it is enlightening t study Table 5 f Bickel and Dksum (1981); in their mdel which we call (1.5), they have ratis MLE(A estimated)/known(a knwn) always at least 1.5 and as large as 211, 0 while urs never exceed 1.2. The ther valuable pint learned frm Table 2 is that when estimating the slpe 8 2, the ML estimatr MLE with A unknwn tends t dminate the LSE, especially fr larger values f a. In ther wrds, fr ur mdel (1.4), there is real value t transfrmatin when it is apprpriate. ė
18 ..18- REF ERE N C E S BICKEL, PETER J. (1975). One step Huber estimates in the linear mdel. J. Amer. Statist. Assc.?O~ BICKEL, P.J., and DOKSUM, K.A. (1981). An analysis f transfrmatins revisited. J. Am. Statist. Assc.?6~ BOX, GEORGE E.P. and COX, DAVID R. (1964). An analysis f transfrmatins. J. Ry. Statist. Sc. Sere B 26~ BOX, G.E.P., and TIAO, G.C. (1973). Bayesian Inference in Statistical Analysis. Reading, Mass. Addisn-Wesley. CARROLL, R.J. (1980). A rbust methd fr testing transfrmatins t achieve apprximate nrmality. J. Ry. Statist. Sc. Series B 42~ CARROLL, R.J. (1982). Tests fr regressin parameters in pwer transfrmatin mdels. Tentatively accepted by the Scand. J. Statist. CARROLL, R.J., and RUPPERT, D. (1981). Predictin and the pwer transfrmatin family. Bi,metrika 68~ CARROLL, R.J. and RUPPERT, DAVID (1982). Rbust estimatin in heterscedastic linear mdels. T appear in Ann. Statist. DOKSUM, K.A., and WONG, C.W. (1981). Statistical tests after transf~atins. Manuscript. HINKLEY, D.V., and RUNGER, G. (1981). Analysis f transfrmed data. T appear in J. Am. Statist. Assc. HOADLEY, B.A. (1971). Asympttic prperties f maximum likelihd estimatrs fr the independent nt identically distributed case. Ann. Math. Statist. 42~ RUPPERT, D. and CARROLL, R.J. (1980). Trimmed least squares estimatin in the linear mdel. J. Am. Statist. Assc.?5~ SPITZER, J.J. (1978). A Mnte-Carl investigatin f the Bx-Cx transfrmatin in small samples. J. Am. statist. Assc.?3~
19 TABLE #1,. Results f the Mnte-Carl study described in the text. These results are fr the INTERCEPT. The median respnse is linear with intercept = 7 and slpe = 2. KNOWN ML estimate with A knwn. MLE = ML estimate with A unknwn. LSE = rdinary least squares estimate cr BIAS OF KNOWN MSE OF KNmJN BIAS OF MLE MSE MSE OF MLE OF KNOWN MSE OF MLE MSE OF KNOWN S.E. OF ABOVE DIFF BIAS OF LSE MSE MSE OF MLE OF LSE t1se OF MLE MSE OF LSE S.E. OF ABOVE DIFF In these calculatins, the mean square errr (MSE) and S.E. f difference terms are multiplied by T**2. Here T = 10 if cr ~ 0.10, T = 1 if cr = 0.50.
20 TABLE #2 Results f the Mnte-Carl study described in the text. These results are fr the SLOPE. The median respnse is linear with intercept = 7 and slpe = 2. KNOWN = ML estimate with A knwn. MLE ML estimate with A unknwn. LSE = rdinary least squares estimate. A a BIAS OF KNOWN MSE OF KNOWN BIAS OF MLE MSE OF MLE MSE OF KNOWN e.. MSE OF MLE - MSE OF KNOWN S.E. OF DIFF , BIAS OF LSE MSE OF MLE MSE OF LSE MSE OF MLE - MSE OF LSE S.E. OF DIFF ~ In these calculatins, the mean square errrs (MSE) and S.E. f di fference terms are multiplied by T**2. Here, T = 10 if a s 0.10, T = 1 if e a = 0.50.
Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);
More informationCHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the
More informationA Matrix Representation of Panel Data
web Extensin 6 Appendix 6.A A Matrix Representatin f Panel Data Panel data mdels cme in tw brad varieties, distinct intercept DGPs and errr cmpnent DGPs. his appendix presents matrix algebra representatins
More informationOn Huntsberger Type Shrinkage Estimator for the Mean of Normal Distribution ABSTRACT INTRODUCTION
Malaysian Jurnal f Mathematical Sciences 4(): 7-4 () On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin Department f Mathematical and Physical Sciences, University f Nizwa, Sultanate
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationDistributions, spatial statistics and a Bayesian perspective
Distributins, spatial statistics and a Bayesian perspective Dug Nychka Natinal Center fr Atmspheric Research Distributins and densities Cnditinal distributins and Bayes Thm Bivariate nrmal Spatial statistics
More informationInternal vs. external validity. External validity. This section is based on Stock and Watson s Chapter 9.
Sectin 7 Mdel Assessment This sectin is based n Stck and Watsn s Chapter 9. Internal vs. external validity Internal validity refers t whether the analysis is valid fr the ppulatin and sample being studied.
More informationMath Foundations 20 Work Plan
Math Fundatins 20 Wrk Plan Units / Tpics 20.8 Demnstrate understanding f systems f linear inequalities in tw variables. Time Frame December 1-3 weeks 6-10 Majr Learning Indicatrs Identify situatins relevant
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More informationAdmissibility Conditions and Asymptotic Behavior of Strongly Regular Graphs
Admissibility Cnditins and Asympttic Behavir f Strngly Regular Graphs VASCO MOÇO MANO Department f Mathematics University f Prt Oprt PORTUGAL vascmcman@gmailcm LUÍS ANTÓNIO DE ALMEIDA VIEIRA Department
More informationUNIV1"'RSITY OF NORTH CAROLINA Department of Statistics Chapel Hill, N. C. CUMULATIVE SUM CONTROL CHARTS FOR THE FOLDED NORMAL DISTRIBUTION
UNIV1"'RSITY OF NORTH CAROLINA Department f Statistics Chapel Hill, N. C. CUMULATIVE SUM CONTROL CHARTS FOR THE FOLDED NORMAL DISTRIBUTION by N. L. Jlmsn December 1962 Grant N. AFOSR -62..148 Methds f
More information4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression
4th Indian Institute f Astrphysics - PennState Astrstatistics Schl July, 2013 Vainu Bappu Observatry, Kavalur Crrelatin and Regressin Rahul Ry Indian Statistical Institute, Delhi. Crrelatin Cnsider a tw
More informationSUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical model for microarray data analysis
SUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical mdel fr micrarray data analysis David Rssell Department f Bistatistics M.D. Andersn Cancer Center, Hustn, TX 77030, USA rsselldavid@gmail.cm
More informationPSU GISPOPSCI June 2011 Ordinary Least Squares & Spatial Linear Regression in GeoDa
There are tw parts t this lab. The first is intended t demnstrate hw t request and interpret the spatial diagnstics f a standard OLS regressin mdel using GeDa. The diagnstics prvide infrmatin abut the
More informationDetermining the Accuracy of Modal Parameter Estimation Methods
Determining the Accuracy f Mdal Parameter Estimatin Methds by Michael Lee Ph.D., P.E. & Mar Richardsn Ph.D. Structural Measurement Systems Milpitas, CA Abstract The mst cmmn type f mdal testing system
More informationLecture 2: Supervised vs. unsupervised learning, bias-variance tradeoff
Lecture 2: Supervised vs. unsupervised learning, bias-variance tradeff Reading: Chapter 2 STATS 202: Data mining and analysis September 27, 2017 1 / 20 Supervised vs. unsupervised learning In unsupervised
More informationLecture 2: Supervised vs. unsupervised learning, bias-variance tradeoff
Lecture 2: Supervised vs. unsupervised learning, bias-variance tradeff Reading: Chapter 2 STATS 202: Data mining and analysis September 27, 2017 1 / 20 Supervised vs. unsupervised learning In unsupervised
More informationSIZE BIAS IN LINE TRANSECT SAMPLING: A FIELD TEST. Mark C. Otto Statistics Research Division, Bureau of the Census Washington, D.C , U.S.A.
SIZE BIAS IN LINE TRANSECT SAMPLING: A FIELD TEST Mark C. Ott Statistics Research Divisin, Bureau f the Census Washingtn, D.C. 20233, U.S.A. and Kenneth H. Pllck Department f Statistics, Nrth Carlina State
More informationA NOTE ON THE EQUIVAImCE OF SOME TEST CRITERIA. v. P. Bhapkar. University of Horth Carolina. and
~ A NOTE ON THE EQUVAmCE OF SOME TEST CRTERA by v. P. Bhapkar University f Hrth Carlina University f Pna nstitute f Statistics Mime Series N. 421 February 1965 This research was supprted by the Mathematics
More informationCHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS
CHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS 1 Influential bservatins are bservatins whse presence in the data can have a distrting effect n the parameter estimates and pssibly the entire analysis,
More informationRevision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax
.7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical
More informationECEN 4872/5827 Lecture Notes
ECEN 4872/5827 Lecture Ntes Lecture #5 Objectives fr lecture #5: 1. Analysis f precisin current reference 2. Appraches fr evaluating tlerances 3. Temperature Cefficients evaluatin technique 4. Fundamentals
More informationModule 4: General Formulation of Electric Circuit Theory
Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated
More informationThermodynamics and Equilibrium
Thermdynamics and Equilibrium Thermdynamics Thermdynamics is the study f the relatinship between heat and ther frms f energy in a chemical r physical prcess. We intrduced the thermdynamic prperty f enthalpy,
More informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More informationCS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007
CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is
More informationA New Evaluation Measure. J. Joiner and L. Werner. The problems of evaluation and the needed criteria of evaluation
III-l III. A New Evaluatin Measure J. Jiner and L. Werner Abstract The prblems f evaluatin and the needed criteria f evaluatin measures in the SMART system f infrmatin retrieval are reviewed and discussed.
More informationResampling Methods. Chapter 5. Chapter 5 1 / 52
Resampling Methds Chapter 5 Chapter 5 1 / 52 1 51 Validatin set apprach 2 52 Crss validatin 3 53 Btstrap Chapter 5 2 / 52 Abut Resampling An imprtant statistical tl Pretending the data as ppulatin and
More informationIN a recent article, Geary [1972] discussed the merit of taking first differences
The Efficiency f Taking First Differences in Regressin Analysis: A Nte J. A. TILLMAN IN a recent article, Geary [1972] discussed the merit f taking first differences t deal with the prblems that trends
More informationSimple Linear Regression (single variable)
Simple Linear Regressin (single variable) Intrductin t Machine Learning Marek Petrik January 31, 2017 Sme f the figures in this presentatin are taken frm An Intrductin t Statistical Learning, with applicatins
More informationKinetic Model Completeness
5.68J/10.652J Spring 2003 Lecture Ntes Tuesday April 15, 2003 Kinetic Mdel Cmpleteness We say a chemical kinetic mdel is cmplete fr a particular reactin cnditin when it cntains all the species and reactins
More informationLead/Lag Compensator Frequency Domain Properties and Design Methods
Lectures 6 and 7 Lead/Lag Cmpensatr Frequency Dmain Prperties and Design Methds Definitin Cnsider the cmpensatr (ie cntrller Fr, it is called a lag cmpensatr s K Fr s, it is called a lead cmpensatr Ntatin
More informationThe blessing of dimensionality for kernel methods
fr kernel methds Building classifiers in high dimensinal space Pierre Dupnt Pierre.Dupnt@ucluvain.be Classifiers define decisin surfaces in sme feature space where the data is either initially represented
More informationANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels
ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1
More informationHomology groups of disks with holes
Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.
More informationKinematic transformation of mechanical behavior Neville Hogan
inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More information7 TH GRADE MATH STANDARDS
ALGEBRA STANDARDS Gal 1: Students will use the language f algebra t explre, describe, represent, and analyze number expressins and relatins 7 TH GRADE MATH STANDARDS 7.M.1.1: (Cmprehensin) Select, use,
More informationENSC Discrete Time Systems. Project Outline. Semester
ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding
More informationEquilibrium of Stress
Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small
More information1996 Engineering Systems Design and Analysis Conference, Montpellier, France, July 1-4, 1996, Vol. 7, pp
THE POWER AND LIMIT OF NEURAL NETWORKS T. Y. Lin Department f Mathematics and Cmputer Science San Jse State University San Jse, Califrnia 959-003 tylin@cs.ssu.edu and Bereley Initiative in Sft Cmputing*
More informationAP Statistics Practice Test Unit Three Exploring Relationships Between Variables. Name Period Date
AP Statistics Practice Test Unit Three Explring Relatinships Between Variables Name Perid Date True r False: 1. Crrelatin and regressin require explanatry and respnse variables. 1. 2. Every least squares
More informationOn Out-of-Sample Statistics for Financial Time-Series
On Out-f-Sample Statistics fr Financial Time-Series Françis Gingras Yshua Bengi Claude Nadeau CRM-2585 January 1999 Département de physique, Université de Mntréal Labratire d infrmatique des systèmes adaptatifs,
More informationCOMP 551 Applied Machine Learning Lecture 5: Generative models for linear classification
COMP 551 Applied Machine Learning Lecture 5: Generative mdels fr linear classificatin Instructr: Herke van Hf (herke.vanhf@mail.mcgill.ca) Slides mstly by: Jelle Pineau Class web page: www.cs.mcgill.ca/~hvanh2/cmp551
More informationEric Klein and Ning Sa
Week 12. Statistical Appraches t Netwrks: p1 and p* Wasserman and Faust Chapter 15: Statistical Analysis f Single Relatinal Netwrks There are fur tasks in psitinal analysis: 1) Define Equivalence 2) Measure
More information3.4 Shrinkage Methods Prostate Cancer Data Example (Continued) Ridge Regression
3.3.4 Prstate Cancer Data Example (Cntinued) 3.4 Shrinkage Methds 61 Table 3.3 shws the cefficients frm a number f different selectin and shrinkage methds. They are best-subset selectin using an all-subsets
More information[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )
(Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well
More informationx 1 Outline IAML: Logistic Regression Decision Boundaries Example Data
Outline IAML: Lgistic Regressin Charles Suttn and Victr Lavrenk Schl f Infrmatics Semester Lgistic functin Lgistic regressin Learning lgistic regressin Optimizatin The pwer f nn-linear basis functins Least-squares
More informationComparing Several Means: ANOVA. Group Means and Grand Mean
STAT 511 ANOVA and Regressin 1 Cmparing Several Means: ANOVA Slide 1 Blue Lake snap beans were grwn in 12 pen-tp chambers which are subject t 4 treatments 3 each with O 3 and SO 2 present/absent. The ttal
More informationModeling the Nonlinear Rheological Behavior of Materials with a Hyper-Exponential Type Function
www.ccsenet.rg/mer Mechanical Engineering Research Vl. 1, N. 1; December 011 Mdeling the Nnlinear Rhelgical Behavir f Materials with a Hyper-Expnential Type Functin Marc Delphin Mnsia Département de Physique,
More informationThe general linear model and Statistical Parametric Mapping I: Introduction to the GLM
The general linear mdel and Statistical Parametric Mapping I: Intrductin t the GLM Alexa Mrcm and Stefan Kiebel, Rik Hensn, Andrew Hlmes & J-B J Pline Overview Intrductin Essential cncepts Mdelling Design
More informationNOTE ON APPELL POLYNOMIALS
NOTE ON APPELL POLYNOMIALS I. M. SHEFFER An interesting characterizatin f Appell plynmials by means f a Stieltjes integral has recently been given by Thrne. 1 We prpse t give a secnd such representatin,
More informationInference in the Multiple-Regression
Sectin 5 Mdel Inference in the Multiple-Regressin Kinds f hypthesis tests in a multiple regressin There are several distinct kinds f hypthesis tests we can run in a multiple regressin. Suppse that amng
More informationChapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms
Chapter 5 1 Chapter Summary Mathematical Inductin Strng Inductin Recursive Definitins Structural Inductin Recursive Algrithms Sectin 5.1 3 Sectin Summary Mathematical Inductin Examples f Prf by Mathematical
More informationModelling of Clock Behaviour. Don Percival. Applied Physics Laboratory University of Washington Seattle, Washington, USA
Mdelling f Clck Behaviur Dn Percival Applied Physics Labratry University f Washingtn Seattle, Washingtn, USA verheads and paper fr talk available at http://faculty.washingtn.edu/dbp/talks.html 1 Overview
More informationFIELD QUALITY IN ACCELERATOR MAGNETS
FIELD QUALITY IN ACCELERATOR MAGNETS S. Russenschuck CERN, 1211 Geneva 23, Switzerland Abstract The field quality in the supercnducting magnets is expressed in terms f the cefficients f the Furier series
More informationThis section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving.
Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus
More informationFebruary 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA
February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal
More informationCAUSAL INFERENCE. Technical Track Session I. Phillippe Leite. The World Bank
CAUSAL INFERENCE Technical Track Sessin I Phillippe Leite The Wrld Bank These slides were develped by Christel Vermeersch and mdified by Phillippe Leite fr the purpse f this wrkshp Plicy questins are causal
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationEnd of Course Algebra I ~ Practice Test #2
End f Curse Algebra I ~ Practice Test #2 Name: Perid: Date: 1: Order the fllwing frm greatest t least., 3, 8.9, 8,, 9.3 A. 8, 8.9,, 9.3, 3 B., 3, 8, 8.9,, 9.3 C. 9.3, 3,,, 8.9, 8 D. 3, 9.3,,, 8.9, 8 2:
More informationResampling Methods. Cross-validation, Bootstrapping. Marek Petrik 2/21/2017
Resampling Methds Crss-validatin, Btstrapping Marek Petrik 2/21/2017 Sme f the figures in this presentatin are taken frm An Intrductin t Statistical Learning, with applicatins in R (Springer, 2013) with
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical
More informationCOMP 551 Applied Machine Learning Lecture 9: Support Vector Machines (cont d)
COMP 551 Applied Machine Learning Lecture 9: Supprt Vectr Machines (cnt d) Instructr: Herke van Hf (herke.vanhf@mail.mcgill.ca) Slides mstly by: Class web page: www.cs.mcgill.ca/~hvanh2/cmp551 Unless therwise
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationPattern Recognition 2014 Support Vector Machines
Pattern Recgnitin 2014 Supprt Vectr Machines Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Pattern Recgnitin 1 / 55 Overview 1 Separable Case 2 Kernel Functins 3 Allwing Errrs (Sft
More informationChE 471: LECTURE 4 Fall 2003
ChE 47: LECTURE 4 Fall 003 IDEL RECTORS One f the key gals f chemical reactin engineering is t quantify the relatinship between prductin rate, reactr size, reactin kinetics and selected perating cnditins.
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More informationarxiv:hep-ph/ v1 2 Jun 1995
WIS-95//May-PH The rati F n /F p frm the analysis f data using a new scaling variable S. A. Gurvitz arxiv:hep-ph/95063v1 Jun 1995 Department f Particle Physics, Weizmann Institute f Science, Rehvt 76100,
More informationSmoothing, penalized least squares and splines
Smthing, penalized least squares and splines Duglas Nychka, www.image.ucar.edu/~nychka Lcally weighted averages Penalized least squares smthers Prperties f smthers Splines and Reprducing Kernels The interplatin
More informationSAMPLING DYNAMICAL SYSTEMS
SAMPLING DYNAMICAL SYSTEMS Melvin J. Hinich Applied Research Labratries The University f Texas at Austin Austin, TX 78713-8029, USA (512) 835-3278 (Vice) 835-3259 (Fax) hinich@mail.la.utexas.edu ABSTRACT
More informationALE 21. Gibbs Free Energy. At what temperature does the spontaneity of a reaction change?
Name Chem 163 Sectin: Team Number: ALE 21. Gibbs Free Energy (Reference: 20.3 Silberberg 5 th editin) At what temperature des the spntaneity f a reactin change? The Mdel: The Definitin f Free Energy S
More informationPerfrmance f Sensitizing Rules n Shewhart Cntrl Charts with Autcrrelated Data Key Wrds: Autregressive, Mving Average, Runs Tests, Shewhart Cntrl Chart
Perfrmance f Sensitizing Rules n Shewhart Cntrl Charts with Autcrrelated Data Sandy D. Balkin Dennis K. J. Lin y Pennsylvania State University, University Park, PA 16802 Sandy Balkin is a graduate student
More informationHubble s Law PHYS 1301
1 PHYS 1301 Hubble s Law Why: The lab will verify Hubble s law fr the expansin f the universe which is ne f the imprtant cnsequences f general relativity. What: Frm measurements f the angular size and
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationCHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India
CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce
More informationWhat is Statistical Learning?
What is Statistical Learning? Sales 5 10 15 20 25 Sales 5 10 15 20 25 Sales 5 10 15 20 25 0 50 100 200 300 TV 0 10 20 30 40 50 Radi 0 20 40 60 80 100 Newspaper Shwn are Sales vs TV, Radi and Newspaper,
More informationENGI 4430 Parametric Vector Functions Page 2-01
ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr
More informationGeneral Chemistry II, Unit I: Study Guide (part I)
1 General Chemistry II, Unit I: Study Guide (part I) CDS Chapter 14: Physical Prperties f Gases Observatin 1: Pressure- Vlume Measurements n Gases The spring f air is measured as pressure, defined as the
More informationPressure And Entropy Variations Across The Weak Shock Wave Due To Viscosity Effects
Pressure And Entrpy Variatins Acrss The Weak Shck Wave Due T Viscsity Effects OSTAFA A. A. AHOUD Department f athematics Faculty f Science Benha University 13518 Benha EGYPT Abstract:-The nnlinear differential
More informationRevisiting the Socrates Example
Sectin 1.6 Sectin Summary Valid Arguments Inference Rules fr Prpsitinal Lgic Using Rules f Inference t Build Arguments Rules f Inference fr Quantified Statements Building Arguments fr Quantified Statements
More informationBOUNDED UNCERTAINTY AND CLIMATE CHANGE ECONOMICS. Christopher Costello, Andrew Solow, Michael Neubert, and Stephen Polasky
BOUNDED UNCERTAINTY AND CLIMATE CHANGE ECONOMICS Christpher Cstell, Andrew Slw, Michael Neubert, and Stephen Plasky Intrductin The central questin in the ecnmic analysis f climate change plicy cncerns
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion
.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References -- J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (Addisn-Wesley, Reading, 966) T study neutrn diffusin
More informationChapters 29 and 35 Thermochemistry and Chemical Thermodynamics
Chapters 9 and 35 Thermchemistry and Chemical Thermdynamics 1 Cpyright (c) 011 by Michael A. Janusa, PhD. All rights reserved. Thermchemistry Thermchemistry is the study f the energy effects that accmpany
More informationDispersion Ref Feynman Vol-I, Ch-31
Dispersin Ref Feynman Vl-I, Ch-31 n () = 1 + q N q /m 2 2 2 0 i ( b/m) We have learned that the index f refractin is nt just a simple number, but a quantity that varies with the frequency f the light.
More informationCOMP 551 Applied Machine Learning Lecture 11: Support Vector Machines
COMP 551 Applied Machine Learning Lecture 11: Supprt Vectr Machines Instructr: (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/cmp551 Unless therwise nted, all material psted fr this curse
More informationWe can see from the graph above that the intersection is, i.e., [ ).
MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Functin Arithmetic: With nt t much difficulty, we ntice that inputs f functins are numbers, and utputs f functins are numbers. S whatever we can d with
More informationBASD HIGH SCHOOL FORMAL LAB REPORT
BASD HIGH SCHOOL FORMAL LAB REPORT *WARNING: After an explanatin f what t include in each sectin, there is an example f hw the sectin might lk using a sample experiment Keep in mind, the sample lab used
More informationHypothesis Tests for One Population Mean
Hypthesis Tests fr One Ppulatin Mean Chapter 9 Ala Abdelbaki Objective Objective: T estimate the value f ne ppulatin mean Inferential statistics using statistics in rder t estimate parameters We will be
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationTree Structured Classifier
Tree Structured Classifier Reference: Classificatin and Regressin Trees by L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stne, Chapman & Hall, 98. A Medical Eample (CART): Predict high risk patients
More informationLHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More informationWe say that y is a linear function of x if. Chapter 13: The Correlation Coefficient and the Regression Line
Chapter 13: The Crrelatin Cefficient and the Regressin Line We begin with a sme useful facts abut straight lines. Recall the x, y crdinate system, as pictured belw. 3 2 1 y = 2.5 y = 0.5x 3 2 1 1 2 3 1
More informationI. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is
Length L>>a,b,c Phys 232 Lab 4 Ch 17 Electric Ptential Difference Materials: whitebards & pens, cmputers with VPythn, pwer supply & cables, multimeter, crkbard, thumbtacks, individual prbes and jined prbes,
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationSupport-Vector Machines
Supprt-Vectr Machines Intrductin Supprt vectr machine is a linear machine with sme very nice prperties. Haykin chapter 6. See Alpaydin chapter 13 fr similar cntent. Nte: Part f this lecture drew material
More informationA.H. Helou Ph.D.~P.E.
1 EVALUATION OF THE STIFFNESS MATRIX OF AN INDETERMINATE TRUSS USING MINIMIZATION TECHNIQUES A.H. Helu Ph.D.~P.E. :\.!.\STRAC'l' Fr an existing structure the evaluatin f the Sti"ffness matrix may be hampered
More informationCONSTRUCTING STATECHART DIAGRAMS
CONSTRUCTING STATECHART DIAGRAMS The fllwing checklist shws the necessary steps fr cnstructing the statechart diagrams f a class. Subsequently, we will explain the individual steps further. Checklist 4.6
More informationA Correlation of. to the. South Carolina Academic Standards for Mathematics Precalculus
A Crrelatin f Suth Carlina Academic Standards fr Mathematics Precalculus INTRODUCTION This dcument demnstrates hw Precalculus (Blitzer), 4 th Editin 010, meets the indicatrs f the. Crrelatin page references
More information