MATHEMATICAL METHODS OF ANALYSIS AND EXPERIMENTAL DATA PROCESSING (Or Methods of Curve Fitting)

Transcription

1 MATHEMATICAL METHODS OF ANALYSIS AND EXPERIMENTAL DATA PROCESSING (Or Methods of Curve Fittig) I this chapter, we will eamie some methods of aalysis ad data processig; data obtaied as a result of a give eperimet. The epositio will begi with the study of simple aalytic procedures of costructio of empirical fuctioal depedeces usig the Method of Least Squares.. THE METHOD OF LEAST SQUARES. Let s suppose that as a result of a eecutio of a series of measuremets i a give eperimet, the followig data table was obtaied. Table f() y y y 3... y It is required to fid a formula that aalytically epresses this fuctioal depedece. Of course, we ca use Iterpolatio Methods, for istace, Lagrage s Iterpolatio Polyomial or Newto s Divided Differece Iterpolatio Formula, whose values at the iterpolatio odes,,...,, will match with Table s correspodig values of f(). Nevertheless, occasioally, this fuctio-ad-iterpolatio-value matchig at the give odes could ot mea a matchig of characters of behavior betwee f() ad the iterpolatio fuctio. The demad that these values should match at the odes is much more ujustified, if the values of f(), obtaied as a result of the give measuremets, are doubtful. We will formulate the problem so that from the begiig ad, for sure, the character of the give fuctio is actually cosidered, that is, we will fid the fuctio of type y = F() () at the poits,,...,, that best fits Table s y, y,..., y values. The type of fuctio F that best fits these data ca be defied as follows: usig Table, plot the poits of f o a coordiate plae ad sketch a smooth curve which reflects better the distributio character of the give poits (see Fig. ). The obtaied curve establishes the type of fuctio that best fits these data, that is, the best curve fittig for the give data. (It is frequetly oe of the simple aalytic fuctios) y 0 Fig..

2 It is fair to metio that a rigorous fuctioal depedece for Table s eperimetal data is seldom observed, sice each of the ivolved magitudes ca deped o may radom factors. Nevertheless, formula () (which is called empirical formula or regressio equatio of y o ) is of such a poit of iterest that allows to fid the values of f for values of that are ot i Table, fittig i that way the results of the measuremets for the correspodig y-magitude. This approach is justified due to the practical utility of the obtaied formula. We will eamie oe of the most widespread methods to fid formula (). We will assume that the approimatio fuctio F at,,...,, has values y, y,..., y. () The demad of proimity of Table s y, y,..., y values with values () is justified as follows: let s eamie the group of Table s values ad group () as the coordiates of two poits i a -dimesioal space. Uder this assumptio, the problem of fuctioal approimatio ca be formulated as follows: fid a fuctio F such that the distace betwee the poits M(y, y,..., y ) ad M( y is miimum. Usig the metric, y,..., y ) of a Euclidea Space, it is requir ed that the magitude ( y y ) + ( y y ) ( y y ) is miimum, which is equivalet to say that the sum of squares ( y y ) + ( y y ) ( y y ) (3) (4) is miimum. Thus, the problem of approimatig fuctio f ca ow be formulated as follows: for a fuctio f give i Table, fid a fuctio F of a certai type such that the sum of squares (4) is miimum. This problem is called Approimatio of f usig The Method of Least squares. As fuctios of approimatio, depedig o the graphical character of the poits of fuctio f, are usually used the followig oes:. y = a + b (Liear Regressio).. y = a + b + c (Regressio of Degree Two ). 3. y = a m (Power Fuctio or Geometric Regressio). 4. y = a.e m (Epoetial Fuctio). 5. y = /(a + b) (Liear Ratioal Fuctio). 6. y = a.l + b (Logarithmic Fuctio ). 7. y = (a/) + b (Hyperbola). 8. y = /(a + b) (Ratioal Fuctio). Where a, b, c, m are parameters. Whe the type of approimatio fuctio is established, the problem is reduced to fid ig the values of these parameters.

3 3 Let s eamie the method of fidig the parameters of a approimatio fuctio i the geeral case of a approimatio fuctio with three parameters, y i y = F(, a, b, c). (5) Thus F( i, a, b, c) =, i =,,...,. The sum of the square differeces for the correspodig values of f ad F ca be writte as i i i= [ y F(, a, b, c)] = Φ ( a, b, c). This sum is a fuctio Φ(a, b, c) of three variables: the parameters a, b, c. Thus, the problem is reduced to searchig a miimum. We will use the ecessary coditio for the eistece of a optimum: that is, i= i= i = Φ/ a = 0, Φ/ b = 0, Φ/ c = 0, ' [ y F(, a, b, c)] F (, a, b, c) = 0, i i a i ' [ y F(, a, b, c)] F (, a, b, c) = 0, (6) i i b i ' [ y F(, a, b, c)] F (, a, b, c) = 0. i i c i Resolvig this system of three equatio s with three ukows; the parameters a, b, c, we will obtai a cocrete type of the search fuctio F(, a, b, c). Note that the chage of the umber of parameters does ot alter the essece of approachig the problem, ad it is epressed oly i the chage of the umber of equatios of system (6). It is atural to epect that the values of the foud fuctio F(, a, b, c) at the poits,,..., will be differet of Table s y, y,..., y values. The values of the differeces y i - F( i, a, b, c) = ε i (i =,,...,) (7) are called deflectios (or deviatios) of the measured y-values i relatio to those oes calculated usig formula (5). For the foud empirical formula (5) ad i accordace with Table, the sum of the square deflectios ca be epressed as σ = ε i, (8) i= which, accordig to The Method of Least Squares for the give type of approimatio fuctio ad the values foud for parameters a, b, c, must be miimum.

4 4 Of two differet approimatios of the same tabulated fuctio usig The Method of Least Squares, it is cosidered better that oe for which the sum (8) has a miimum value.. FINDING THE TYPE OF APPROXIMATION FUNCTIONS CALLED LINEAR REGRESSION AND REGRESSION OF DEGREE TWO. Let s determie the approimatio fuctio called Liear Regressio, The partial derivatives with respect to the parameters are The system of type (6) ca be writte as F(, a, b) = a + b. (9) F/ a =, F/ b =. (y i - a i - b) i = 0, (y i - a i - b) = 0. (I the sum, ide i varies from to ). Thus we get: ad dividig each equatio by, Keepig i mid the followig desigatios: i y i - a i - b i = 0, (0) y i - a i - b = 0, i a + i b i yi =, i a + b = yi. i = M, y = M i y, i yi = My i = M,. () We coclude that the last system ca be writte as M a + M b M = y, () M a + b = M. y

5 5 The coefficiets M, M y, M, M y of system () are umbers that i each cocrete problem of approimatio ca easily be calculated usig formulae (), where i, y i are the values i Table. Let s determie the Secod Degree Approimatio Fuctio or Regressio of Degree Two, Its partial derivatives are Thus, the system of type (6) ca be writte as F(, a, b, c) = a + b + c. (3) F/ a =, F/ b =, F/ c =. (y i - a i - b i - c) i = 0, (y i - a i - b i - c) i = 0, (y i - a i - b i - c) = 0. After a series of trasformatios, we get a system of three liear equatios with ukows a, b, c. The system s coefficiets, as i the case of a liear regressio, are calculated usig Table s kow data, M a + M b + M c = M, 4 3 y 3 + y + + = y. M a + M b M c = M, (4) M a M b c M Here, we have used () ad also the followig desigatios: 4 3 i = M 4, M 3 y M i =, i i =. (5) y 3. FINDING THE OTHER TYPES OF APPROXIMATION FUNCTIONS. We will show that the problem of fidig approimatio fuctios F(, a, b) with two parameters of the type of the eight oes listed above ca be reduced to fidig the parameters of a liear fuctio. 3.. Power Fuctio (Geometric Regressio). Fid a approimatio fuctio of type F(, a, m) = a m. (6)

6 6 Suppose that i Table, both -values ad y-values are positive. Etractig the atural logarithm to (6), assumig a > 0, we get: l F = l a + m l. (7) Sice fuctio F is a approimatio fuctio for fuctio f, fuctio l F will be a approimatio for l f. Let s make the substitutio u = l. Thus, as it follows from (7), l F will be a fuctio of u, say Φ(u). Makig equatio (7) ca be epressed as m = A, l a = B, (8) Φ(u, A, B) = Au + B, (9) that is, the problem is reduced to fidig a approimatio fuctio of liear type. Keepig i mid the above coditios, i order to fid the required approimatio fuctio, we will proceed as follows:. I accordace with Table, costruct a ew table which cotais the atural logarithms of both ad y.. Usig this ew table, fid the parameters A ad B (see Paragraph ) of the approimatio fuctio of type (9). 3. Usig formulae (8), fid the values of parameters a, m, ad substitute them i (6). 3.. Epoetial Fuctio. Suppose that Table is such that its approimatio fuctio is of epoetial type: Etractig the atural logarithm to (0), we get: F(, a, m) = a ep(m), a > 0. (0) Accordig to (8), equatio () ca be epressed as l F = l a + m. () l F = A + B. () Cosequetly, to fid the approimatio fuctio of type (0), it is required to etract the atural logarithms to the values of f i Table ad cosiderig them alog with its -values, costruct for this ew table of values a fuctio of type (). After that ad

7 7 accordig to desigatios (8), it oly remais to obtai the values of the search parameters a, b, ad to substitute them i formula (0) Liear Ratioal Fuctio. Fid a approimatio fuctio of type Formula (3) ca be epressed as F(, a, b) = /(a + b). (3) /[F(, a, b)] = a + b. Keepig i mid the last epressio, to fid the values of parameters a, b ad i accordace with Table, it is required to costruct a ew table i which its -values remai uchaged, ad those oes of f are substituted by their reciprocals. After that ad i accordace with the ew obtaied table, oe ca fid the approimatio fuctio a + b. The search parameters a, b are the substituted i formula (3) Logarithmic Fuctio. Suppose that we wat to fid a approimatio fuctio of type F(, a, b) = a l + b. (4) I order to trasform (4) ito a liear fuctio, it is eough to make the substitutio l = u. So, to determie the values of a ad b, it is ecessary to etract the atural logarithm to Table s -values ad eamiig the obtaied results alog with those oes of fuctio f, fid, for the ew obtaied table, the approimatio fuctio of liear type. The coefficiets a, b of the search fuctio should be substituted i formula (4) Hyperbola. If the graph of poits plotted accordig to Table turs out to be the brach of a hyperbola, the approimatio fuctio has the form F(, a, b) = (a/)+ b. (5) For its trasformatio ito a liear fuctio, we make the substitutio u = /. Φ(u, a, b) = a u + b. (6) So, to determie the approimatio fuctio of type (6), Table s -values should be substituted by their reciprocals ad the, for the ew table, fid the approimatio (6) of liear type. The coefficiets a, b of the obtaied fuctio are the substituted i formula (5).

8 Ratioal Fuctio. Suppose that the approimatio fuctio has the form F(, a, b) = /(a + b). (7) Thus, we get: /[F(, a, b)] = a + (b/), so, the problem tur s out to be oe of the same form of the previous case. I fact, if i Table we chage the values of both ad y by their reciprocals the, accordig to the formulae z = /, u =/y, we fid for the ew table of values a approimatio fuctio of type u = bz + a. The values of a, b are the substituted i formula (7). Eample. Costruct a approimatio fuctio usig the Method of Least Squares for the fuctioal depedece give by the followig table: Table y The graph of poits is show i Fig.. To compare the quality of this approimatio, cosider, at the same time, two approimatio curves for the give fuctio: the liear regressio y = a + b ad the power fittig y = c m. After fidig the values for parameters a, b, c, ad m, the sum of the square deflectios (8) ca be foud to establish which of these two approimatios is better. y Fig.. I the case of the liear regressio, the values of parameters a, b are foud usig system (), whose coefficiets are calculated accordig to Table s data ad i accordace with desigatios (). I order to compute the coefficiets of the give

9 9 system, keep i mid Table ad costruct a auiliary table (Table 3) whose last row correspods to the values of the sums of the terms of the correspodig colum. Table 3. y y Dividig the obtaied sums by the umber of elemets = 8, i accordace with formulae (), we get: M M Let s form ow the system of type (): y = 3. 4, M = 75., = , M = y 4.056a + 3.4b = a + b =.75. Its solutio, applyig Cramer s Rule, is a = 0.9, b = Thus, the approimatio fuctio y = F(, a, b) ca be writte as y = (8) To determie the values of the parameters c, m of the power fittig, as it follows from Paragraph 3., a ew table with the atural logarithms of both the -values ad y-values must be costructed. Deote the values of the ew correspodig variables u ad z by u = l, z = l y. Accordig to the ew table s umerical data, form the system of equatios of type (): M A + M B M u u = uz (9) M A + B = M, u whose coefficiets are the umb ers calculated usig the ew table ad formulae (), while ukows A ad B are related to the search parameters by the correlatios A = m, B = l c. z

10 0 Table 4. u z uz u To fid the coefficiets of system (9), costruct a auiliary table (Table 4). Dividig the values of the last row of Table 4 by 8, we obtai: Let s form the system of type (9): Its solutio is M =., M = , M = , M = u z uz u.498a +.B = A + B = A =.656, B = Let s fid ow the values of parameters c ad m: m =.656, c = ep(-.359) = Therefore, the power fittig ca be epressed as y = (30) Table 5. y ε ε ε ε

11 To compare the quality of the approimatios (8) ad (30), calculate the sum of the square deflectios (see Table 5). It follows from Table 5 that the sum of the square deflectios for the liear regressio is σ = 0.0 ad for the power fittig is σ = Comparig the quality of these two approimatios, (i the sese of The Method of Least Squares) we ca coclude that the power fittig is better. The previous eample shows that to solve the approimatio problem of a fuctio usig The Method of Least Squares, it is required to carry out a sigificat amout of calculatios alog with the use of a large amout of umerical values, which are essetially origiated i the ow process of calculatio. It is evidet that, i this case, the most appropriate istrumet of calculatio should be a computer. Below, for the iterested reader, we have listed a Pascal program for simulatig the values of the parameters of a regressio equatio for each of the followig three types: y = a + b, y = a m, y = ae m. Pascal was chose as the programmig laguage because of its clarity ad readability. I Pascal, the iformatio occurrig betwee brackets {} is commetary ad is ot read by the computer. Oce the program is eecuted, you could eter, for eample, Table s data. We kow that with certai data trasformatios, the calculatio of the parameters of the power ad epoetial fittigs is reduced to calculatig the parameters of the liear regressio, that is, to solvig the system of type (). For this reaso, the fudametal steps of carryig this algorithm out ca be described as follows:. Eter the give data.. Choose the type of curve fittig. 3. Trasform the data ito the fuctioal depedece of liear type. 4. Obtai the parameters of the regressio equatio. 5. Trasform coversely the data ad fid the square deflectios of the foud values of the fuctio related to the data. 6. Prit the output results. {Curve Fittig} uses crt; var A,B,D,s,s,s,s3,s4 N,i,k X,Y,U,V : real; : iteger; : array[0..50] of real; begi clrscr; writel('eter the umber of ordered pairs i the table:'); readl(n); writel('eter the values of X:'); begi for i:= to N do readl(x[i]); ed; writel('eter the values of Y:'); begi for i:= to N do readl(y[i]); ed;

12 writel('choose the type of curve fittig:'); writel(': y=a*+b'); writel(': y=a*^m'); writel('3: y=a*ep(m*)'); writel('eter the umber of the required equatio:'); readl(k); s:=0; s:=0; s3:=0; for i:= to N do begi if k= the begi U[i]:=X[i]; V[i]:=Y[i]; ed; if k= the begi U[i]:=L(X[i]); V[i]:=L(Y[i]); ed; if k=3 the begi U[i]:=X[i]; V[i]:=L(Y[i]); ed; s:=s+u[i]; s:=s+v[i]; s3:=s3+u[i]*v[i]; s4:=s4+u[i]*u[i]; ed; D:=N*s4-s*s; A:=(N*s3-s*s)/D; B:=(s4*s-s*s3)/D; s:=0; for i:= to N do begi V[i]:=A*U[i]+B; if (k=) or (k=3) the V[i]:=Ep(V[i]); writel(v[i]); s:=s+sqr(y[i]-v[i]); ed; writel('equatio of the regressio curve:'); if k= the begi writel('y = ',A,'*X + ',B); writel('s = ',s); ed; if k= the begi writel('y = ',Ep(B),'*X^',A); writel('s = ',s); ed; if k=3 the begi writel('y = ',Ep(B),'*ep(',A,'*X)'); writel('s = ',s); ed; ed. I the previous Pascal program, to save the data, we use the -dimesioal arrays X ad Y. The trasformed data are the saved, respectively, i the arrays U ad V. Eecutig the calculatios for Table s data whe usig this program, to the request Choose the type of curve fittig ad, keepig i mid that, the uderlied commad i the program allows the output of the values of the calculated fuctio i accordace with the obtaied regressio equatio, we get:

13 Equatio of the regressio curve: Y = *X S = For k =, the program gives: Equatio of the regressio curve: Y = *X^ S = , ad for k = 3: Equatio of the regressio curve: Y = *ep( *X) S = Eercise: Fit Table 6 s data usig the approimatio fuctios of liear, power, epoetial, ad logarithmic types. Determie which fittig is better ad why. Table y Solutio: k = : Y = *X S = k = : Y = *X^ S = k = 3: Y = *ep( *X) S =