Definition of empirical dependence parameters at two dimensions using ambiguity ellipse algorithm in a generalized method of uncertainty centre has been considered. The algorithm of optimal parameters definition is offere

Belov, V. M.

Evstigneev, V. V.

Goncharov, S. A.

Ryabova, E. V.

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in nplane Ln1. 1families (23) giving transformations W of2plane L21 into themselves correspond to its generators.If the reference point is canonized then the equations (29) are reduced to the equations (27).3. Let point X=xαAα, belonging to equipping straight line l1=(An+1,An+2) of nsurface Sn1 be given. Reasoning similarly, we obtain transformation x~α=xβωβiωiα of straight lineinto itself which is defined by matrix (Пαβ): Пαβ=ΛiαjΛikβ t kt k.If this is transformation W, then we obtain cone(Λijn+1+Λijn+2)xixj=0; xα=0 in nplane Ln1. 1families (23)giving transformations W of straight line l1 correspondto its generators. If the reference point is canonical thenthe equations take on formIf the reference point is canonized (accurate withinnormalization), then the similar constructions with thesame properties may be also obtained for surfaces S n2and S n3.Conclusion It was shown that the third surface possessing the same properties that the initial surfaces Sn1 and Sn2 may beattached to a pair of nsurfaces. The canonical reference point was constructed provided that the coordinatenet of lines is conjugate on these surfaces. The privateclass of a pair of nsurfaces when the coordinate net isconjugate on all surfaces was noted. A series of invariantgeometric images for each of surfaces of constructed triplet of nsurfaces was found.1 2 2( )( ) 0; 0.n n iii ii x xα+ +Λ + Λ = =Mathematics and mechanics. Physics9REFERENCES1. Luchinin А.А. On a pair msurfaces with the given net in multidimensional projective space // Bulletin of the Tomsk PolytechnicUniversity. – 2006. – V. 309. – № 4. – P. 11–14.2. Luchinin A.A., Knjazeva O.G. Connectivities, associated with a pair of msurfaces // Progressive technologies and economy in mechanical engineering: Collection works of AllRussian scientificpractical conference. – Yurga, 2003. – P. 314–315.3. Basylev V.T. On nets on multidimensional surfaces of projective space // Izvestiya Vuzov. Mathematics. – 1966. – № 2. – P. 9–19.4. Ostianu N.M. On geometry of a multidimensional surface of projective space // Works of a geometrical seminar. – ARISTI of USSRAS. – 1966. – V. 1. – P. 239–264.5. Ivlev Е.Т. on the issue of geometrical interpretation of convolutionoperation of some tensors // Materials of the final scientific conference on mathematics and mechanics for 1970s. – Tomsk. – 1970. –V. 1. – P. 121–123.Received on 17.04.2006 To ascertain the laws of some phenomena the experimental investigations are carried out. During them thevalues of one or another physicochemical magnitudeare measured. At physical experiment processing theempirical models or formula are often used. They include experimentally inaccurately measured magnitudes;inaccuracy is as a rule taken into account in output variables. The case when the output and input variables ofthe model are timed measured, especially in the case ofdeveloping concrete techniques of physical experimentdata analysis, is insufficiently represented in scientificliterature and is urgent. Statement and investigation ofproblem solvability of parameters uncertainty set immersion of twodimensional linear parameterorientedmodel at accurate measurement of input variables andinaccurate measurement of output variables were studied in detail in papers [1–4].Algorithm of parameter uncertainty set immersionof twodimensional linear dependence into ambiguityellipse at interval assignment of input and output variables was examined in [5–8]. In these papers inaccuracieswhich influenced the result of the algorithm operationwere made when ambiguity ellipse parameters being determined by generalized method of uncertainty centre.Let us examine the algorithm [5–8] more detailed.At two dimensions the experimental points should meetthe system of interval equations[y]1=[a]+[b][x]1,[y]2=[a]+[b][x]2.UDC 541.127DETERMINATION OF AMBIGUITY ELLIPSE PARAMETERS AT TWO DIMENSIONS USING GENERALIZED METHOD OF UNCERTAINTY CENTRE V.M. Belov, S.A. Goncharov, E.V. Ryabova, V.V. EvstigneevI.I. Polsunov’s Altai State Technical University, BarnaulEmail: sim_64@mail.ruDefinition of empirical dependence parameters at two dimensions using ambiguity ellipse algorithm in a generalized method of uncertainty centre has been considered. The algorithm of optimal parameters definition is offered.The range of possible dimension of linear functionparameters is of the form of irregular quadrilateral theangular points of which may be determined using the rules of interval arithmetic. In case of increasing function[у]=[а]+[b][х] i. e. at x1+<x2–⇒y1+<y2–, the angular pointsof uncertainty quadrilateral are determined as(1)In case if the function of the form y=[a]+[b][x] is decreasing i.e. at x1+<x2–⇒y1+>y2– then the angular points of uncertainty quadrilateral are determined from the ratios(2)To determine the uncertainty quadrilateral centre ofgravity the leastsquares procedure is used. Then for searching centre of gravity coordinates we obtain the ratiowhere x–1, x–2, y–1, y–2 are the average values of measured input and output variables respectively.The centre of gravity of the given uncertainty quadrilateral may be also determined in different ways. Anycombination of three angular points defined by the ratios (1) and (2) forms the uncertainty triangle. There arefour of such triangles: А1А2А3, А1А2А4, А1А3А4, А2А3А4. It isknown that the centre of gravity of any triangle is on thecrossing of its medians. Thus, for centre of gravity coordinates of triangles there is a ratioAs a result four points В1, В2, В3, В4, which definenew uncertainty quadrilateral are obtained. Let us suppose that А1=В1, А2=В2, А3=В3, А4=В4. Then the procedure of determining the centers of gravity of new uncertainty quadrilaterals is repeated till the moment whilethe required accuracy of defining the centre of gravity ofthe given uncertainty quadrilateral is achieved.Let us denote quadrilateral centre of gravity by а0, b0.At parameter uncertainty set immersion of linear parameteroriented function into the ambiguity ellipse thecentre of gravity of the given quadrilateral is consideredto coincide with the centre of ambiguity ellipse circumscribed around it. Then the angular points of uncertainty quadrilateral are arranged in order of parameter bmagnitude that is b1<b2<b3<b4.Segment А3А4 is a chord of ambiguity ellipse. Thenthe parameters of the line А3А4Let us draw a straight line parallel to А3А4. For thispurpose let us determine the coordinates of bisectingpoint of the segment А3А4:The parameters of the line а=kb+α passing throughthe bisecting point of the segment А3А4 and ellipse centre are found as(3)After that the distance from chord А3А4 midpoint tothe ambiguity ellipse centre is determinedwhere а01=а0–а34; b01=b0–b34.The chord of ellipse A1*A2* is parallel to chord А3А4 andsymmetric to it with respect to ellipse centre [5–8]. Themidpoint of chord A1*A2* is on the line with parameters (3)its coordinates are determined by the ratiosа12=а0+а01; b12=b0+b01.As chords A1*A2* and А3А4 are symmetrical with respectto the ellipse centre then let us calculate the additionalmagnitudes for defining the coordinates of points A1* andA2*Then the coordinates of point A1* are calculated asand coordinates of point A2*Thus, the parallelogram A1*A2*А3А4 circumscribes theuncertainty quadrilateral of the parameters а, b. The* *2 12 12 2 12 12; .a a a b b b= + Δ = + Δ* *1 12 12 1 12 12; ,a a a b b b= − Δ = − Δ4 334 12 4 344 334 12 4 34,2.2a aa a a ab bb b b b−Δ = Δ = − =−Δ = Δ = − =2 201 01 .l a b= +0 34 34 0 0 340 34 0 34; .a a b a bk b b b bαα− −= =− −3 4 3 434 34; .2 2a a b ba b+ += =4 3 3 4 4 334 344 3 4 3; .a a a b a bk b b b bα− −= =− −1 2 3 1 2 311 2 4 1 2 421 3 4 1 3 432 3 4 2 3 44; ;3 3; ;3 3; ;3 3; .3 3a a a b b bBa a a b b bBa a a b b bBa a a b b bB+ + + +⎛ ⎞= ⎜ ⎟⎝ ⎠+ + + +⎛ ⎞= ⎜ ⎟⎝ ⎠+ + + +⎛ ⎞= ⎜ ⎟⎝ ⎠+ + + +⎛ ⎞= ⎜ ⎟⎝ ⎠2 11 22 10 02 1 2 1; ,x y x y y ya bx x x x− −= =− −1 2 2 1 2 11 1 12 1 2 11 2 2 1 2 12 2 22 1 2 11 2 2 1 2 13 3 32 1 2 11 2 2 1 2 14 4 42 1 2( , ) ; ,( , ) ; ,( , ) ; ,( , ) ;y x y x y yA a bx x x xy x y x y yA a bx x x xy x y x y yA a bx x x xy x y x y yA a bx x x− − − − − −− − − −− + + − + −+ − + −+ + + + + ++ + + ++ − − + − +− +⎛ ⎞− −= = ⎜ ⎟− −⎝ ⎠⎛ ⎞− −= = ⎜ ⎟− −⎝ ⎠⎛ ⎞− −= = ⎜ ⎟− −⎝ ⎠− −= = − 1.x− +⎛ ⎞⎜ ⎟−⎝ ⎠1 2 2 1 2 11 1 12 1 2 11 2 2 1 2 12 2 22 1 2 11 2 2 1 2 13 3 32 1 2 11 2 2 1 2 14 4 42 1 2( , ) ; ;( , ) ; ;( , ) ; ;( , ) ;y x y x y yA a bx x x xy x y x y yA a bx x x xy x y x y yA a bx x x xy x y x y yA a bx x x− − + + + −− + − +− + − + − −+ + + ++ − + − + +− − − −+ + − − − ++ −⎛ ⎞− −= = ⎜ ⎟− −⎝ ⎠⎛ ⎞− −= = ⎜ ⎟− −⎝ ⎠⎛ ⎞− −= = ⎜ ⎟− −⎝ ⎠− −= = − 1.x+ −⎛ ⎞⎜ ⎟−⎝ ⎠Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 210centre of gravity of parallelogram A1*A2*А3А4 coincideswith the centre of ellipse containing it.The ambiguity ellipse circumscribed next to the parallelogram A1*A2*А3А4 is searched in the form of where а0, b0 are the coordinates of ambiguity ellipse centre of the parameters.To define the parameters of ambiguity ellipse let usmake a set of equations(4)where the following notations are accepted:System (4) is rewritten in the form(5)In papers [5–8] parameters Q and F were inaccuratelydetermined. Let us solve system (5) with respect to Q and F:(6)whereThe area of ambiguity ellipse with parameters F, Q,D is calculated as(6)To determine the ellipse of minimal area the maximum of function fD is found(7)Subject to (6), the expression (7) may be presentedin the form ofLet us calculate the derivative f 'D(F, Q, D):ThenHaving solved this equation with respect to parameter D we obtain the expressionIn this case using the obtained ratios for parametersF, Q, D the ambiguity ellipse area is minimal..)1(2 110110−+=QFQFQFD.DQFf 'D 0),,( ='0 1 1 0 1 1( , , ) 2 ( 1).Df F Q D F Q F Q D F Q= + + −20 0 0 1 1 0 1 1( , , ) ( ) ( 1).Df F Q D F Q F Q F Q D D F Q= + + + − .DFQDQFfD2),,( −= .2DFQ −= πν2 22 1 1 20 12 2 2 21 2 2 11 2 2 12; .b b b bF F a b a ba b a bΔ − Δ − Δ Δ= = Δ Δ + Δ ΔΔ Δ − Δ Δ2 22 1 1 20 12 2 2 22 1 1 22 1 1 22; ;a a a aQ Q a b a ba b a bΔ − Δ − Δ Δ= = Δ Δ + Δ ΔΔ Δ − Δ Δ ,; 1010 DFFFDQQQ +=+=2 21 1 1 12 22 2 2 21 2 ,1 2 .F a Q b D a bF a Q b D a bΔ + Δ = − Δ ΔΔ + Δ = − Δ Δ**2 31 41 2**3 24 11 2; ;2 2; .2 2a aa aa ab bb bb b−−Δ = Δ =−−Δ = Δ =2 21 1 1 12 22 2 2 22 1,2 1,F a D a b Q bF a D a b Q bΔ + Δ Δ + Δ =Δ + Δ Δ + Δ =2 20 0 0 0( ) 2 ( )( ) ( ) 1,F a a D a a b b Q b b− + − − + − =Mathematics and mechanics. Physics11REFERENCES1. Belov V.M., Suhanov V.A., Unger F.G. Theoretical and appliedaspects of uncertainty centre method. – Novosibirsk: Nauka, 1995.– 144 p.2. Belov V.M., Unger F.G., Karbainov Yu.A., Prolubnikov V.I., Tubalov N.P. Estimation of empirical dependences parameters by the uncertainty centre method. – Novosibirsk: Nauka, 2001. – 175 p.3. Belov V.M., Suhanov V.A., Unger F.G. Approximation by the ellipseof uncertainty set of dependences parameters come to the linear ones. – Tomsk, 1990. – 28 p. Preprint of USSR AS SD TSC; № 45.4. Belov V.M. Estimation of parameters of linear chemicalanalyticaland physicochemical dependences by uncertainty centre method:Thesis ... of a candidate of phys.math. science. – Tomsk, 1992. –166 p.5. Belov V.M., Goncharov S.A., Goncharova N.L. Recurrent estimation algorithm of parameters of linear twoparameter function // Fisicohimicheskie protsessy v neorganicheskih materialah: 8 Mezhdunarodnaya konferentsiya. – V. 2. – Kemerovo, 2001. – P. 134–135.6. Goncharov S.A. Generalized method of uncertainty centre for estimating parameters of linear experimental physical dependences.Thesis ... of a candidate of tech. science. – Barnaul, 2003. – 160 p.7. Goncharov S.A., Belov V.M., Goncharova N.L. Estimating parameters uncertainty range of linear functions by ambiguity ellipse // Valihanovskie chteniya 7: International theoretical and practical conference. – Т. 7. – Kokshetau, 2002. – P. 3–5.8. Goncharov S.A., Dudnik E.A., Sharapov S.V. Estimating linear function parameters by ambiguity ellipse // IV Scientific and technicalconference of students and graduate students. – Rubtsovsk, 2002. –P. 5–9.Received on 14.11.2006

Determination of ambiguity ellipse parameters at two dimensions using generalized method of uncertainty centre

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