Least squares cubic splines without B-splines S.K. Lucas

Transcription

1 Least squares cubc splnes wthout B-splnes S.K. Lucas School of Mathematcs and Statstcs, Unversty of South Australa, Mawson Lakes SA 595 e-mal: Submtted to the Gazette of the Australan Mathematcal Socety May 3, Accepted July 3. Justfcaton Readers of the Gazette wll be famlar wth the use of least squares to fnd the polynomal that best fts a set of data ponts. As wth all curve fttng problems, there are many stuatons where one low order polynomal s nadequate and a hgh order polynomal ft s napproprate. A least squares cubc splne would provde a better ft. There are thousands of artcles publshed on varants of splnes, ncludng least squares cubc splnes. One of the frst least squares artcles was de Boor and Rce [], and a comprehensve explanatory textbook s Derckx []. Unfortunately, every example n the lterature and on the web of a least squares cubc splne makes use of B-splnes. Whle B-splnes have a certan elegance, they are suffcently complex to be beyond the typcal undergraduate level, and have the dsadvantage of beng more expensve to evaluate than tradtonal cubc splnes. For example, Schumacker [4] ponts out that t s more effcent to convert a cubc B-splne to a tradtonal cubc splne and then evaluate f you requre two or more functon evaluatons per nterval. The only excepton to usng B-splnes s Ferguson and Staley [3], where they fnd a least squares cubc ft to data that enforces contnuty of functon and frst dervatve, but not second dervatve. Thus, ther formulaton does not lead to a cubc splne. My colleague, Basl Benjamn, had been usng what s essentally the same cubc ft as [3] when fttng smooth curves to tran lne data. Ths motvated me to seek the alternatve that also enforces contnuty of the second dervatve. The am of ths note, then, s to show how least squares cubc splnes can be formulated n a straghtforward manner usng a tradtonal cubc splne defnton. Whle we shall show that the system of lnear equatons to be solved s larger than when usng B-splnes, (N +) (N +) gven N ntervals, the theory nvolved wll be much more accessble to readers wthout advanced numercal analyss experence.. Dervaton Assume that we are gven a set of ponts {(x, y )} n = for whch we requre a least squares ft. Let t < t < < t N+ be N + nodes, where t x and t N+ x n. The rest of the t s do not need to be placed at data ponts. A pecewse cubc functon f(x) s defned on the doman [t, t N+ ] such that f (x) = f(x) on [t, t + ] s a polynomal of degree at most three. We wsh to fnd the coeffcents of the varous cubcs such that we mnmse the sum of the squares of the errors at the data ponts, wth the constrants that the cubcs have contnuty of functon, frst and second dervatve, so that f(x) s n fact a cubc splne. Our approach wll be the standard one of Lagrange multplers: to mnmse f(x) wth constrants g (x) =, we mnmse f(x) + λ g (x). Followng Ferguson and Staley [3], we defne our pecewse cubc as f(x) = (u (x) + )v (x)z + u (x)( v (x))z + + h u (x)v (x)z + h u v (x)z +, ()

2 for x [t, t + ], =,,..., N, where h = t + t, u (x) = (x x )/h, and v = u (x). () Ths ensures that f(t ) = z and f (t ) = z for =, 3,..., N. In other words, the functon and ts frst dervatve are already contnuous. Whle not a standard way of wrtng a cubc, ths form wll make the followng analyss far more straghtforward. If you prefer a standard cubc for effcent nested evaluaton, () can be rewrtten on [t, t + ] as [ f(x) = z + z (x t ) + 3 (z h z + ) ] (z h + z + ) (x t ) + [ (z h 3 z + ) + ] (z h + z + ) (x t ) 3. The constrants that ensure f(x) s a cubc splne are contnuty of the second dervatve at t, =, 3,..., N. Ths reduces to h z + (h + h )z + h z + 3h (z + z ) 3 h (z z ) = (4) h h for =, 3,..., N. Now let us assume that the data {(x, y )} n = s ordered on x so that we can easly dentfy how many ponts (n ) are n each nterval and where n the lst (p ) they begn. Under these condtons, the constraned mnmsaton problem s reduced to mnmsng S = where N p +n = [ N λ = j=p (y j α j z β j z + γ j z δ j z +) + h z + (h + h )z + h z + 3 h (z + z ) 3 h (z z ) h h α j = (u (x j ) + )v (x j), β j = u (x j)( v (x j )), γ j = h u (x j )v (x j), δ j = h u (x j)v (x j ). The {λ } N = are the Lagrange multplers, and combned wth the unknowns {z, z }N+ = there are 3N + unknowns. As wth any least squares problem, we can take the partal dervatves of S wth respect to each unknown, beng careful to recognse that each unknown occurs n several terms on the rght hand sde of (5), and equate to zero. Ths leads to the followng lnear system of equatons: A B T C T B E F T C F where (wth the abbrevaton k for p k +n k j=p k ) z z λ = D G ], (3) (5) (6), (7) A s the (N + ) (N + ) symmetrc trdagonal matrx wth man dagonal α j, ( α j + β j), ( 3 α 3j + β j),..., ( N α Nj + N β N,j), N β Nj, and codagonal α j β j, α j β j,..., N α Nj β Nj, B s the (N +) (N +) trdagonal matrx wth man dagonal α j γ j, ( α j γ j + β j δ j ), ( 3 α 3j γ 3j + β j δ j ),..., ( N α Nj γ Nj + N β N,j δ N,j ), N β Nj δ Nj, upper dagonal β j γ j, β j γ j,..., N β Nj γ Nj, and lower dagonal α j δ j, α j δ j,..., N α Nj δ Nj,

3 C s the (N ) (N + ) upper dagonal matrx wth c = 3h + /h, c,+ = 3(h /h + h + /h ), and c,+ = 3h /h + for =,,..., N, all other terms zero, D s the (N + ) vector wth terms y j α j, ( y j α j + y j β j ), ( 3 y j α 3j + y j β j ),..., ( N y j α Nj + N y j β N,j ), N y j β Nj, E s the (N + ) (N + ) symmetrc trdagonal matrx wth man dagonal γj, ( γj + δj ), ( 3 γ3j + δj ),..., ( N γnj + N δn,j ), N δnj, and codagonal γ j δ j, γ j δ j,..., N γ Nj δ Nj, F s the (N ) (N + ) upper dagonal matrx wth f = h +, f,+ = (h + h + ), and f,+ = h for =,,..., N, all other terms zero, G s the (N + ) vector y j γ j, ( y j γ j + y j δ j ), ( 3 y j γ 3j + y j δ j ),..., ( N y j γ Nj + N y j δ N,j ), N y j δ nj, z = [z, z,..., z N+ ] T, z = [z, z,..., z N+ ]T, λ = [λ, λ 3,..., λ N ] T, and the zero matrx on the left hand sde s (N ) (N ), and the zero vector on the rght hand sde s (N ). 3. Implementaton Settng up and solvng (7) by Gaussan elmnaton s not dffcult, partcularly f approprate ntermedate varables are used. Equaton (3) can then be used to output the soluton as standard pecewse cubcs. For N pecewse cubcs we requre at least 3N + data ponts, but there s no constrant on the poston of these ponts. In fact t s perfectly acceptable (f unusual) to have ntervals wth no data ponts whatsoever. Pseudocode for a functon that mplements ths formulaton of a least squares cubc splne s lsted n fgure. Fgure shows two examples of least squares cubc splne fts to data usng ths algorthm. The frst s the classc ttanum heat data used to test curve fttng routnes, where splne ntervals have been chosen for a good ft. The approprate postonng of nterval endponts contnues to be an area of actve research. The second s a smple ft to snusodal data wth small Gaussan nose, where data s not avalable n each nterval. As one would expect, the solutons are exactly those obtaned usng a B-splne approach. References [] C. de Boor and J.R. Rce, Least squares cubc splne approxmaton I fxed knots, Techncal Report CSD-TR, Computer Scences, Purdue Unversty (968). Also avalable at ftp://ftp.cs.wsc.edu/approx/tr.pdf [] P. Derckx, Curve and surface fttng wth splnes, Oxford Unversty Press, 993. [3] J. Ferguson and P.A. Staley, Least squares pecewse cubc curve fttng, Comm. ACM (973). [4] L.L. Schumaker, Splne functons: basc theory, John Wley and Sons, New York, 98. 3

4 Algorthm to fnd a least squares cubc splne ft to data (x, y ) n = on ntervals [t, t + ] N =. Input: Data (x, y ) n = and nterval endponts {t } N+ =, number of data ponts n and ntervals N. Output: Functon {z } N+ = and dervatve {z }N+ = data at nterval endponts. procedure least squares splne(x, y, n, t, N, z, z ) sort(x, y) // sort data ponts on x coordnates for =, N h = t + t p =, pn =, j = for =, n // dentfy whch data s n whch nterval f x t j+ pn j = pn j + else j = j +, p j =, pn j = whle x > t j+ // deal wth ntervals wth no ponts j = j +, p j =, pn j = endwhle pn j = endf Set taa, tab, tag, tad, tbb, tbg, tbd, tgg, tgd, tdd to zero vectors of length N Set D and G to zero vectors of length N + for =, N // Set up ntermedate sums and vectors for j = p, p + pn u = (x j t )/h, v = u, α = (u + )v, β = u ( v), γ = h uv, δ = h u v taa = taa +α, tab = tab +αβ, tag = tag +αγ, tad = tad +αδ, tbb = tbb +β tbg = tbg + βγ, tbd = tbd + βδ, tgg = tgg + γ, tgd = tgd + γδ, tdd = tdd + δ D = D + y j α, D + = D + + y j β, G = G + y j γ, G + = G + + y j δ Set A, B, E to zero matrces of sze (N + ) (N + ) Set C, F to zero matrces of sze (N ) (N + ) for =, N // Set up the ntermedate matrces A = A + taa, A +,+ = tbb, A,+ = tab, A +, = A,+, B = B + tag, B +,+ = tbd, B,+ = tbg, B +, = tad E = E + tgg, E +,+ = tdd, E,+ = tgd, E +, = E,+ for =, N C = 3h + /h, C,+ = 3h /h +, C,+ = C C,+ F = h +, F,+ = (h + h + ), F,+ = h A B T C T D Set up and solve B T E F T x = G C F Extract z = {x } N+ =, z = {x } =N+ N+ end(least squares splne) Fgure : Pseudocode for the least squares cubc splne algorthm. 4

5 Fgure : Examples of fttng least squares cubc splnes to data, open squares are data ponts, closed crcles are splne nterval endponts. 5