Computational Statistics and Data Analysis. Robust smoothing of gridded data in one and higher dimensions with missing values

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1 Computatonal Statstcs and Data Analyss 54 () Contents lsts avalable at ScenceDrect Computatonal Statstcs and Data Analyss journal homepage: Robust smoothng of grdded data n one and hgher dmensons wth mssng values Damen Garca CRCHUM Research Centre, Unversty of Montreal Hosptal, Montreal, Canada artcle nfo abstract Artcle hstory: Receved 8 March 9 Receved n revsed form 5 September 9 Accepted 6 September 9 Avalable onlne 3 September 9 A fully automated smoothng procedure for unformly sampled datasets s descrbed. The algorthm, based on a penalzed least squares method, allows fast smoothng of data n one and hgher dmensons by means of the dscrete cosne transform. Automatc choce of the amount of smoothng s carred out by mnmzng the generalzed cross-valdaton score. An teratvely weghted robust verson of the algorthm s proposed to deal wth occurrences of mssng and outlyng values. Smplfed Matlab codes wth typcal examples n one to three dmensons are provded. A complete user-frly Matlab program s also suppled. The proposed algorthm, whch s very fast, automatc, robust and requrng low storage, provdes an effcent smoother for numerous applcatons n the area of data analyss. 9 Elsever B.V. All rghts reserved.. Introducton In ths paper, a fast robust verson of a dscretzed smoothng splne s ntroduced. The proposed method, based on the dscrete cosne transform (DCT), allows robust smoothng of equally spaced data n one and hgher dmensons. The followng paragraphs of the present secton brefly descrbe the underlyng penalzed least squares approach, n the partcular case of one-dmensonal data. Secton provdes a revew of the generalzed cross-valdaton whch s used to estmate the smoothng parameter. When data are evenly spaced, t s shown (Secton 3) that the lnear system can be greatly smplfed and solved by means of the DCT. An teratve robust scheme of ths DCT-based penalzed regresson s then proposed n order to deal wth weghted, mssng (Secton 4) and outlyng (Secton 5) values. It s fnally explaned, n Secton 6, how the smoothng technque proposed n ths paper can be appled to multdmensonal data. To llustrate the effectveness of the smoother, several examples related to one-, two- and three-dmensonal data are presented (Secton 7). Two smplfed Matlab functons and one complete optmzed Matlab code are also provded. In statstcs and data analyss, smoothng s used to reduce expermental nose or small-scale nformaton whle keepng the most mportant mprnts of a dataset. Consder the followng model for the one-dmensonal nosy sgnal y: y = ŷ + ", where " represents a Gaussan nose wth mean zero and unknown varance, and ŷ s supposed to be smooth,.e. has contnuous dervatves up to some order (typcally ) over the whole doman. Smoothng y reles upon fndng the best estmate of ŷ. Data smoothng s generally carred out by means of parametrc or nonparametrc regresson. Parametrc regresson requres some a pror knowledge of the so-called regresson equaton that can represent the data well. The majorty of observed values, however, cannot be parameterzed n terms of predetermned analytcal functons, so nonparametrc regresson s usually the best opton for smoothng of data (Takezawa, 5). To cte a few nstances, some of the most common approaches to nonparametrc regresson used n data processng nclude kernel regresson (Haste () Correspondng address: LBUM, CRCHUM, Pavlon J.A. de Sève (room Y-69), 99 Alexandre de Sève, Montreal, QC, HL W5, Canada. Tel.: (475). E-mal address: Damen.Garca@crchum.qc.ca /$ see front matter 9 Elsever B.V. All rghts reserved. do:.6/j.csda.9.9.

2 68 D. Garca / Computatonal Statstcs and Data Analyss 54 () and Loader, 993) lke movng average flterng, local polynomal regresson (Watson, 964) and the Savtzky Golay flter (Savtzky and Golay, 964). Another classcal approach to smoothng s the penalzed least squares regresson. Ths method was frst ntroduced n the early 9 s by Whttaker (93) and t has been extensvely studed ever snce (Wahba, 99a). Ths technque conssts n mnmzng a crteron that balances the fdelty to the data, measured by the resdual sum-ofsquares (RSS), and a penalty term (P) that reflects the roughness of the smooth data. One thus seeks to mnmze F(ŷ) = RSS + s P ŷ = ŷ y + s P ŷ, () where kk denotes the Eucldean norm. The parameter s s a real postve scalar that controls the degree of smoothng: as the smoothng parameter ncreases, the smoothng of ŷ also ncreases. When the roughness penalty s wrtten n terms of the square ntegral of the pth dervatve of ŷ, the penalzed regresson s known as a smoothng splne (Schoenberg, 964; Takezawa, 5; Wahba, 99a). Another smple and straghtforward approach to express the roughness s by usng a second-order dvded dfference (Wenert, 7; Whttaker, 93) whch yelds, for a one-dmensonal data array, P ŷ = D ŷ, where D s a trdagonal square matrx defned by D, = h (h + h ), D, = h h, D, = h (h + h ), for apple apple n, where n s the number of elements n ŷ, and h represents the step between ŷ and ŷ +. Assumng repeatng border elements (y = y and y n+ = y n ) gves D, = D, = and D h n,n = D n,n =. h n Ths procedure s smlar to the Whttaker smoothng (Elers, 3; Wenert, 7; Whttaker, 93), apart from the repeatng boundares. Such boundary condtons lead to further numercal smplfcatons wth evenly spaced data (Buckley, 994), as explaned further n Secton 3. Now, usng Eqs. () and (3), mnmzaton of F(ŷ) gves the followng lnear system that allows the determnaton of the smoothed data: I n + sd T D ŷ = y, where I n s the n n dentty matrx and D T stands for the transpose of D. Because (I n + sd T D) s a symmetrc pentadagonal matrx, the latter equaton can be numercally solved very effcently wth techncal computng software (Elers, 3; Wenert, 7). Computaton of ŷ usng Matlab (The MathWorks, Natck, MA, USA) and Eq. (4) has recently been ntroduced by Elers (3) and Wenert (7). In the case of evenly spaced data, Wenert has proposed a relatvely fast algorthm based on Cholesky decomposton (Wenert, 7). Ths algorthm, however, loses effcency when an estmaton of the smoothng parameter s requred, as explaned n Secton.. Estmaton of the smoothng parameter As ndcated by Eq. (4), the output ŷ s strongly nfluenced by the smoothng parameter s. Assumng the model gven by Eq. (), t s approprate to use the smoothng parameter that yelds the best estmate of the orgnal data and thus avods over- or under-smoothng as much as possble. Such a correct value can be estmated by the method of generalzed crossvaldaton (GCV). The GCV method was ntroduced by Craven and Wahba (978) and Wahba (99b) n the context of smoothng splnes. Assumng that one wants to solve the smoothng lnear system ŷ = H(s)y, where H s the so-called hat matrx, the GCV method chooses the parameter s that mnmzes the GCV score (Craven and Wahba, 978; Golub et al., 979): s = argmn (GCV) wth GCV (s) RSS/n ( Tr(H)/n), where Tr denotes the matrx trace. Fndng s whch mnmzes the GCV score thus requres Tr(H) to be known. Here, accordng to Eq. (4), H s gven by H = I n + sd T D. To calculate the GCV score, Elers (3) and Wenert (7) have both proposed algorthms that nvolve the determnaton of H at each teratve step of the mnmzaton. Such a process s very tme-consumng and can be avoded. Indeed, Tr(H) can be smply reduced to nx Tr(H) =, (6) + s = (3) (4) (5)

3 D. Garca / Computatonal Statstcs and Data Analyss 54 () where ( ) =...n are the egenvalues of D T D. The GCV score thus reduces to! nx nx GCV (s) = n ŷ y n + s. (7) = = Fndng the s value that mnmzes the GCV score yelded by Eq. (7) makes the smoothng algorthm fully automated. Because the components of ŷ appear n the expresson of the GCV score, ŷ has to be calculated at each step of the mnmzaton process. Ths can be avoded for the partcular case of equally spaced data, as demonstrated n Secton. 3. Smoothng of evenly spaced data Eq. (4) can be effcently solved wth a Matlab code usng the so-called left matrx dvson (see Matlab documentaton) appled to sparse matrces. Solvng such a lnear system, however, may rapdly become tme-consumng wth an ncreasng number of data. Ths algorthm can be greatly smplfed and sped up f the data are evenly spaced. Indeed, n numerous condtons, expermental acquston and data measurement lead to equally spaced or evenly grdded datasets. An effcent algorthm for smoothng of unformly sampled data s descrbed n the present secton. A generalzaton for multdmensonal data wll be descrbed n Secton 6. Assumng now that the data are equally spaced wth h =, 8, the dvded dfference matrx D (n Eq. (4)) can be rewrtten as the smple dfference matrx D = An egecomposton of D yelds D = U U, C A. where s the dagonal matrx contanng the egenvalues of D defned by Yueh (5): = dag (,..., n) wth = + cos (( ) /n). (8) Because U s a untary matrx (.e. U = U T and UU T = I n ), Eq. (4) leads to ŷ = U I n + s U T y U U T y, where the components of the dagonal matrx, accordng to Eq. (8), are gven by (9), = + s ( cos (( ) /n)) and,j = f 6= j. () It s worth notng that U T and U are actually n-by-n type- dscrete cosne transform (DCT) and nverse DCT matrces, respectvely (Strang, 999). Thus, the smooth output ŷ can also be expressed as ŷ = U DCT(y) = IDCT( DCT(y)), where DCT and IDCT refer to the dscrete cosne transform and the nverse dscrete cosne transform, respectvely. In the case of equally spaced data, the GCV score can also be smplfed consderably. The trace of the hat matrx [see Eqs. (6) and (8)] ndeed reduces to nx Tr(H) = + s ( cos (( ) /n)). = Note that, when n becomes large, one has Tr(H) Z n + s ( cos x) dx = p + p + 6s p p + 6s. () Moreover, usng Eq. (9), the resdual sum of squares (RSS) can be wrtten as RSS = ŷ y = I n + s nx = + s = I n DCT (y) DCT (y), ()

4 7 D. Garca / Computatonal Statstcs and Data Analyss 54 () where DCT refers to the th component of the dscrete cosne transform. The GCV score gven by Eq. (7) thus becomes np n +s DCT (y) = GCV (s) = np, (3) n = +s wth gven by Eq. (8). It s much more convenent to use an algorthm based on Eq. () rather than Eq. (4) when the data are equspaced. The DCT ndeed has a computatonal complexty of O(n log(n)), whereas Eq. (4) requres a Cholesky factorzaton whose computatonal complexty s O(n 3 ). Note also that the computaton of the GCV score from Eq. (3) s straghtforward and does not requre any matrx operaton and manpulaton, whch makes the automated smoothng very fast. A smplfed Matlab code for fully automated smoothng s gven n Appx A. The above-mentoned smoothng procedure requrng the DCT s smlar to that ntroduced by Buckley (994). It s shown n Secton 4 how ths DCT-based smoother can be adapted to weghted data and mssng values. 4. Dealng wth weghted data and occurrence of mssng values The occurrence of mssng data due to measurement nfeasblty or nstrumentaton falure s frequent n practce. It can also be convenent to gve outlers a low weght or, on the contrary, allocate a relatvely hgh weght to hgh-qualty data. Let W be the dagonal matrx dag(w ) that contans the weghts w [, ] correspondng to the data y. In the presence of weghted data, the RSS becomes wrss = W / ŷ y. The mnmzaton of F(ŷ) (Eq. ()) thus yelds W + sd T D ŷ = Wy, whch can be rewrtten as I n + sd T D ŷ = (I n W) ŷ + Wy. Lettng A sd T D + W and usng the above-mentoned hat matrx H, ths equaton can also be expressed as H ŷ = H A ŷ + Wy. Ths mplct formula can be solved usng an teratve procedure: H ŷ {k+} = H A ŷ {k} + Wy wth an arbtrary ŷ {}, (4) where ŷ {k} refers to ŷ calculated at the kth teraton step. It can be shown that ths equaton converges, for any ŷ {}, snce the lnear system s postve defnte. Indeed, the matrx D s real nonsngular; then sd T D s postve defnte f s >. In addton, W s postve semdefnte (because w ); A s thus postve defnte and, accordng to the Theorem 3 of Keller (965), convergence of ŷ {k} towards the desred soluton ŷ s ensured for any s >. Note that, n the presence of mssng values, W s smply defned by w = f y s mssng, whle an arbtrary fnte value s assgned to y. In that case, the algorthm performs both smoothng and nterpolaton. Contrarly to classcal methods whch usually operate wth lnear or cubc local nterpolatons, mssng data are assgned to values that are estmated usng the entre dataset. In the case of evenly spaced data, one can take advantage of the DCT by rewrtng Eq. (4) as ŷ {k+} = I n + sd T D W(y ŷ {k} ) + ŷ {k}, whch, smlarly to Eq. (), becomes ŷ {k+} = IDCT DCT W(y ŷ {k} ) + ŷ {k}, (5) wth gven by Eq. (). The teratve convergent process defned by Eq. (5), wth the addtonal determnaton of the GCV score, allows automatc smoothng wth weghted and/or mssng data. When weghted and/or mssng data occur, Eq. (7) descrbng the GCV score, however, becomes nvald snce the weghted resduals (wrss) must be accounted for. Moreover, the number of mssng data must be taken nto account n the expresson of the average square error. In the presence of weghted data, the GCV score s therefore gven by GCV (s) = wrss/(n n mss) = kw / ŷ y k /(n n mss ), ( Tr(H)/n) ( Tr(H)/n) where n mss represents the number of mssng data and n s the number of elements n y. Note that the GCV score gven by the latter equaton, n comparson wth Eq. (3), requres ŷ to be known, so ŷ must be calculated at each teraton of the GCV mnmzaton. Smoothng of weghted data wll thus nvolve addtonal computaton tme.

5 D. Garca / Computatonal Statstcs and Data Analyss 54 () Robust smoothng In regresson analyss, t s habtually assumed that the resduals follow a normal dstrbuton wth mean zero and constant varance, usually unknown. In spte of that, faulty data, erroneous measurements or nstrumentaton malfuncton may lead to observatons that le abnormally far from the others. The man drawback of the penalzed least squares methods s ther senstvty to these outlers. Besdes outlyng ponts, hgh leverage ponts also adversely affect the outcome of regresson models. In statstcs, the leverage denotes a measure of the nfluence (between and ) of a gven pont on a fttng model due to ts locaton n the space of the nputs. More precsely, the ponts whch are far removed from the man body of ponts wll have hgh leverage (Chatterjee and Had, 986). To mnmze or cancel the sde effects of hgh leverage ponts and outlers, one can assgn a low weght to them usng an teratvely reweghted process, as often used n robust local regresson. Ths method conssts n constructng weghts wth a specfed weghtng functon by usng the current resduals and updatng them, from teraton to teraton, untl the resduals reman unchanged (Rousseeuw and Leroy, 987). In practce, fve teratve steps are suffcent. Several weghtng functons are avalable for robust regresson. The most common one, the so-called bsquare weght functon, wll be used herenafter, although other weght functons would be sutable as well (Heberger and Becker, 99). The bsquare weghts are gven by 8 >< u u f W = < >: u f, where u s the Studentzed resdual whch s adjusted for standard devaton and leverage and s defned as (Rousseeuw and Leroy, 987) u = r ˆ p h. In Eq. (6), r = y ŷ s the resdual of the th observaton, h s ts correspondng leverage and ˆ s a robust estmate for the standard devaton of the resduals gven by.486 MAD, where MAD denotes the medan absolute devaton (Rousseeuw and Croux, 993). The leverage values h are all gven by the dagonal elements of the hat matrx H (Hoagln and Welsch, 978). However, a faster and economcal alternatve for robust smoothng can be obtaned usng an average leverage: 8, h = h = X H = Tr(H)/n, n where an approxmated value for Tr(H)/n s gven by Eq. (). The approxmated Studentzed resduals fnally reduce to p u = r MAD (r) s p 3 + 6s p p s (6) The use of the bsquare weghts n combnaton wth these approxmated Studentzed resduals provdes a robust verson of the above-mentoned smoother. Note that f y represents a mssng data, r s not nvolved n the calculaton of the MAD(r). Appx B contans a smplfed Matlab code for robust smoothng usng the method descrbed n ths paper. 6. Multdmensonal smoothng of evenly grdded data Because a multdmensonal DCT s bascally a composton of one-dmensonal DCTs along each dmenson (Strang, 999), Eq. () can be mmedately exted to two-dmensonal or N-dmensonal regularly grdded data. A detaled descrpton of the extenson of Eq. () to hgher dmensons s gven n Buckley (994). In the case of N-dmensonal data, Eq. () becomes ŷ = IDCTN N DCTN (y), where DCTN and IDCTN refer to the N-dmensonal dscrete cosne transform and nverse dscrete cosne transform, respectvely, and stands for the Schur (elementwse) product. As an extenson of Eq. (), N s a tensor of rank N defned by N = N N + s N N. Here, the operator symbolzes the element-by-element dvson and N s an N-rank tensor of ones. N, smlarly to Eq. (8), s the followng N-rank tensor (Buckley, 994): NX N,..., N = + cos ( j ), (9) n j j= (7) (8)

6 7 D. Garca / Computatonal Statstcs and Data Analyss 54 () Fg.. Pseudocode of the automatc robust smoothng algorthm. The varable names and acronyms are descrbed n the text. The complete Matlab code s suppled n the supplemental materal (smoothn.m). When robust opton s requred, mnmzaton of the GCV score s performed at the frst robust teratve step only (see 4.c.) and the last estmated smoothness parameter (s) s used durng the successve steps. Ths makes the algorthm faster wthout alterng the fnal results sgnfcantly. where n j denotes the sze of N along the jth dmenson. Automated determnaton of s requres the GCV score whch, n the case of multdmensonal data, can be wrtten as GCVs = ŷ y F /n N /n, where kk F denotes the Frobenus norm, kk s the -norm, and n = Q N j= n j s the number of elements n y. Smlarly to Eq. (7), the GCV score wth no weghted and no mssng data can be rewrtten as follows: GCVs = n N n N DCTN (y) F N. The same procedure as descrbed n Secton 5 can also be drectly appled to weghted or mssng data. The teratve process llustrated by Eq. (5) thus becomes ŷ {k+} = IDCTN N DCTN W (y ŷ {k} ) + ŷ {k}. () A robust verson can also be mmedately obtaned usng the Studentzed resduals gven by (6) wth the followng average leverage: p + p! N + 6s h = p p. + 6s 7. Matlab codes and examples Eqs. () and (5) wth optmal smoothng by means of the GCV method are relatvely easy to mplement n Matlab. Smplfed Matlab codes for automatc smoothng (smooth) and robust smoothng (rsmooth) of one-dmensonal and twodmensonal datasets are gven n Appces A and B. These two programs have a smlar syntax and both requre the two-dmensonal (nverse) dscrete cosne transform (dct and dct) provded by the Matlab mage processng toolbox. The nput represents the dataset to be smoothed, whle the output represents the smoothed data. A complete documented Matlab functon for smoothng of one-dmensonal to N-dmensonal data (smoothn), wth automated and robust optons, and whch can deal wth weghted and/or mssng data, s also suppled n the supplemental materal. The pseudocode for smoothn s gven n Fg.. No Matlab toolbox s needed for the use of smoothn. Updated versons of ths functon are downloadable from the author s personal webste (Garca, 9). The frst four examples llustrated herenafter can be run n Matlab usng the functon smoothndemo, also avalable n the supplemental materal.

7 D. Garca / Computatonal Statstcs and Data Analyss 54 () A B C - D -4-6 Fg.. Automatc smoothng of two-dmensonal data wth mssng values. (A). Nosy data. (B). Corrupted data wth mssng values. (C). Smoothed data restored from B. (D). Absolute errors between the restored and orgnal data. Two-dmensonal smoothng wth mssng values. To llustrate the effectveness of the DCT-based smoothng algorthm, hghly corrupted two-dmensonal data have been smoothed and the result has been compared wth the orgnal data. The orgnal matrx data (3 3) conssted of a functon of two varables obtaned by translatng and scalng Gaussan dstrbutons created by the Matlab peaks functon. The corrupted data were obtaned by frst addng a Gaussan nose wth mean zero (Fg. A), then randomly removng half of the data, and fnally creatng a 5 5 square of mssng data (Fg. B). The automatc smoother was able satsfactorly to recover the orgnal surface (Fg. C) wth an average relatve error (defned by ksmoothed orgnalk/korgnalk) smaller than 5% (see also the absolute error mappng on Fg. D). Robust smoothng. Penalzed least squares methods are very senstve to outlyng data. Fg. 3A demonstrates how outlers may adversely b the smoothed curve (see also the one-dmensonal example n Appx B). Clearly, the smoothed values do not reflect the behavor of the bulk of the data ponts. To overcome the dstorton due to a small fracton of outlers, the data can be smoothed usng the robust procedure descrbed n Secton 5. Fg. 3B shows that the robust smoothng s resstant to outlers. An example for two-dmensonal robust smoothng s also gven n Appx B. Three-dmensonal smoothng. Fg. 4 now llustrates three-dmensonal automated smoothng by means of the smoothn program suppled n the supplementary materal. Gaussan nose wth mean zero and standard devaton of.6 has been added to a three-dmensonal functon defned by f (x, y, z) = x exp( x y z ) n the [ ; ] 3 doman (Fg. 4A). Fg. 4B shows that smoothn sgnfcantly reduced the addtve nose. Note that smoothn could also be appled to hgher dmensons. Smoothng of parametrc curves. The dscrete cosne transform (DCT) and nverse DCT are lnear operators and thus work wth complex numbers. Ths property can be used to smooth parametrc curves defned by (x(t), y(t)), where t s equspaced. By way of example, Fg. 5 depcts a nosy cardod (dots) whose parametrc equatons are x(t) = cos(t)[ cos(t)]+n (,.) and y(t) = sn(t)[ cos(t)]+n (,.), where N refers to the normal dstrbuton and t s equally spaced between and. Let us now defne the complex functon z(t) = x(t) + y(t), where s the unt complex number, and ts smoothed

8 74 D. Garca / Computatonal Statstcs and Data Analyss 54 () A 6 B Fg. 3. Non-robust versus robust smoothng. Outlers may b the smoothed curve (A). A robust smoothng may get rd of ths drawback (B). A B Fg. 4. Automatc smoothng of three-dmensonal data Fg. 5. Automatc smoothng of a nosy cardod. output ẑ(t). The parametrc curve defned by Re ẑ(t), Im ẑ(t) respectvely, provdes a smooth cardod (sold lne; see Fg. 5)., where Re and Im refer to the real and magnary parts, Applcatons to surface temperature anomales. Automatc smoothng wth the mnmzaton of the GCV score has fnally been tested on temperature data avalable n the Met Offce Hadley Centre s webste (Brohan et al., 6; Kennedy, 7). Fg. 6 llustrates the evoluton of global average land temperature anomaly (n C) wth respect to The smooth

9 D. Garca / Computatonal Statstcs and Data Analyss 54 () Temperature anomaly ( C) Fg. 6. Global average land temperature anomaly ( C) wth respect to 96 99: smoothed versus orgnal year-averaged data. Year-averaged data are avalable n Fg. 7. August 3 surface temperature dfference ( C) wth respect to 96 99: smoothed (bottom panel) versus orgnal year-averaged (top panel) data. The year-averaged dataset s avalable n The smoothed output has been upsampled for a better vsualzaton of the surface temperatures. curve clearly depcts the contnung rse of global temperature that has occurred snce the 97 s. Fg. 7 shows the Earth s surface temperature dfference of August 3 wth respect to The top panel represents a mappng of the best estmates of temperature anomales as provded by the Met Offce Hadley Centre. Note that numerous temperature data are mssng due to nonexstent staton measurements. A smooth map of Earth temperature anomales wth flled gaps can be automatcally obtaned usng the DCT-based algorthm descrbed n the present paper (Fg. 7, bottom panel). The resoluton of Fg. 7, bottom panel, has been ncreased to obtan a better vsualzaton of surface temperatures. A basc upsamplng of the smoothed data was performed by paddng zeros after the last element of DCTN ŷ along each dmenson before usng the nverse DCT operator, as follows: ŷ upsampled = r nupsampled n IDCTN ZeroPad DCTN ŷ,

10 76 D. Garca / Computatonal Statstcs and Data Analyss 54 () where n upsampled represents the number of elements after upsamplng. It s worth recallng that the DCT assumes repeatng boundary condtons whereas perodc boundares would have been more approprate for the east west edges. Such condtons caused east west dscordances at the level of New Zealand and Antarctca. 8. Dscusson The best choce of the smoothng algorthm to be used n data analyss deps specfcally upon the orgnal data and on the propertes of the addtve nose. As a consequence, the search for a perfect unversal smoother remans llusory (Elers, 3). In ths paper, a robust, fast and fully automated smoothng procedure has been proposed, and ts effcacy has been llustrated on a few cases n the prevous secton. The user, however, should be aware of some lmtatons of the algorthm. Better results wll be obtaned f ŷ tself s suffcently smooth,.e. f t has contnuous dervatves up to some desred order over the whole doman of nterest. Moreover, because the algorthm process nvolves a dscrete cosne transform, the nherent repeatng condtons may slghtly dstort the smoothed output on the boundares, especally when the total number of data s small. The GCV crteron s used to make the smoothng procedure fully automated and good results are expected f the addtve nose " n Eq. () follows a Gaussan dstrbuton wth nl mean and constant varance. It has been reported, however, that the GCV remans farly adapted even wth nonhomogenous varances and non-gaussan errors (Wahba, 99b). Alternatvely, f the measurement errors n the data present varable standard devatons and f the varances are known ( ), the response varances can be transformed to a constant value usng weghts gven by w = /. In addton, f outlyng data occur, t could be convenent to perform a robust smoothng. Fnally, the GCV crteron may cause problems when the sample sze s small, whch could make the automated verson unsatsfactory wth a small number of data. If ths occurs, the smoothng parameter s must be tuned manually untl a vsually acceptable result s obtaned. Appx A A smplfed Matlab code (smooth) for one-dmensonal (-D) and two-dmensonal (-D) smoothng of equally grdded data, and two examples are gven below. The nput represents the dataset to be smoothed and the output represents the smoothed data. Ths code has been wrtten wth Matlab R7b. functon z = smooth(y) [n,n] = sze(y); n = n*n; Lambda = bsxfun(@plus,repmat(-+*cos((:n-)*p/n),n,),... -+*cos((:n-). *p/n)); DCTy = dct(y); fmnbnd(@gcvscore,-5,38); z = dct(gamma.*dcty); functon GCVs = GCVscore(p) s = ^p; Gamma =./(+s*lambda.^); RSS = norm((dcty(:).*(gamma(:)-)))^; TrH = sum(gamma(:)); GCVs = RSS/n/(-TrH/n)^; -D Example: x = lnspace(,,^8); y = cos(x/)+(x/5).^ + randn(sze(x))/; z = smooth(y); plot(x,y,.,x,z) -D Example: xp = lnspace(,,^6); [x,y] = meshgrd(xp); f = exp(x+y)+sn((x-*y)*3) + randn(sze(x))/; g = smooth(f); fgure, subplot(), surf(xp,xp,f), zlm([ 8]) subplot(), surf(xp,xp,g), zlm([ 8]) Appx B A smplfed Matlab code (rsmooth) for robust smoothng of one-dmensonal (-D) and two-dmensonal (-D) equally sampled data, and two examples are gven below. The syntax s smlar to that of smooth (see Appx A). A complete

11 D. Garca / Computatonal Statstcs and Data Analyss 54 () optmzed code (smoothn) allowng smoothng n one or more dmensons s also avalable n the supplemental materal and n the author s personal webste (Garca, 9). functon z = rsmooth(y) [n,n] = sze(y); n = n*n; N = sum([n,n]~=); Lambda = bsxfun(@plus,repmat(-+*cos((:n-)*p/n),n,),... -+*cos((:n-). *p/n)); W = ones(n,n); zz = y; for k = :6 tol = Inf; whle tol>e-5 DCTy = dct(w.*(y-zz)+zz); fmnbnd(@gcvscore,-5,38); tol = norm(zz(:)-z(:))/norm(z(:)); zz = z; tmp = sqrt(+6*s); h = (sqrt(+tmp)/sqrt()/tmp)^n; W = bsquare(y-z,h); functon GCVs = GCVscore(p) s = ^p; Gamma =./(+s*lambda.^); z = dct(gamma.*dcty); RSS = norm(sqrt(w(:)).*(y(:)-z(:)))^; TrH = sum(gamma(:)); GCVs = RSS/n/(-TrH/n)^; functon W = bsquare(r,h) MAD = medan(abs(r(:)-medan(r(:)))); u = abs(r/(.486*mad)/sqrt(-h)); W = (-(u/4.685).^).^.*((u/4.685)<); -D Example: x = lnspace(,,^8); y = cos(x/)+(x/5).^ + randn(sze(x))/; y([7 75 8]) = [ ]; z = smooth(y); zr = rsmooth(y); subplot(), plot(x,y,.,x,z), ttle( Non robust ) subplot(), plot(x,y,.,x,zr), ttle( Robust ) -D Example: xp = lnspace(,,^6); [x,y] = meshgrd(xp); f = exp(x+y)+sn((x-*y)*3) + randn(sze(x))/; f(:3:35,:3:35) = randn(6,6)*+; g = smooth(f); gr = rsmooth(f); subplot(), surf(xp,xp,g), zlm([ 8]), ttle( Non robust ) subplot(), surf(xp,xp,gr), zlm([ 8]), ttle( Robust ) Appx C. Supplemental data The supplemental materal contans four Matlab programs (smoothn, smoothndemo, dctn and dctn). The functon smoothn ncludes all the propertes descrbed n ths paper. It carres out manual or automatc smoothng of one-dmensonal

12 78 D. Garca / Computatonal Statstcs and Data Analyss 54 () to N-dmensonal unformly sampled data, and can deal wth weghted and/or mssng values. A robust opton s also offered. Enter help smoothn n the Matlab command wndow to obtan a detaled descrpton and the syntax for smoothn. Updated versons of smoothn are also downloadable from the author s personal webste (Garca, 9). The second program smoothndemo llustrates the frst four examples from Secton 7. Enter smoothndemo n the Matlab command wndow to create four Matlab fgures correspondng to the Fgs. 5 of ths paper. The scrpt of smoothndemo can be of help for a better understandng of smoothn. Fnally, dctn and dctn allow the computaton of the dscrete cosne transform (DCT) and nverse DCT of N-dmensonal arrays. These two functons are necessary for the use of smoothn. The functons smoothn, smoothndemo, dctn and dctn have been wrtten wth Matlab R7b. Supplementary data assocated wth ths artcle can be found, n the onlne verson, at do:.6/j.csda References Brohan, P., Kennedy, J.J., Harrs, I., Tett, S.F.B., Jones, P.D., 6. Uncertanty estmates n regonal and global observed temperature changes: A new data set from 85. Journal of Geophyscal Research, D6. do:.9/5jd6548. Buckley, M.J., 994. Fast computaton of a dscretzed thn-plate smoothng splne for mage data. Bometrka 8, Chatterjee, S., Had, A.S., 986. Influental observatons, hgh leverage ponts, and outlers n lnear regresson. Statstcal Scence, Craven, P., Wahba, G., 978. Smoothng nosy data wth splne functons. Estmatng the correct degree of smoothng by the method of generalzed crossvaldaton. Numersche Mathematk 3, Elers, P.H., 3. A perfect smoother. Anal. Chem. 75, Garca, D., Matlab functons n BoméCardo Ref Type: Electronc Ctaton. Golub, G., Heath, M., Wahba, G., 979. Generalzed cross-valdaton as a method for choosng a good rdge parameter. Technometrcs, 5 3. Haste, T., Loader, C., 993. Local regresson: Automatc kernel carpentry. Statstcal Scence 8, 9. Heberger, R.M., Becker, R.A., 99. Desgn of an S functon for robust regresson usng teratvely reweghted least squares. Journal of Computatonal and Graphcal Statstcs, Hoagln, D.C., Welsch, R.E., 978. The hat matrx n regresson and ANOVA. The Amercan Statstcan 3, 7. Keller, H.B., 965. On the soluton of sngular and semdefnte lnear systems by teraton. Journal of the Socety for Industral and Appled Mathematcs: Seres B, Numercal Analyss, 8 9. Kennedy, J., Met Offce Hadley Centre observatons datasets Ref Type: Electronc Ctaton. Rousseeuw, P.J., Leroy, A.M., 987. Robust Regresson and Outler Detecton. John Wley & Sons, Inc., New York, NY, USA. Rousseeuw, P.J., Croux, C., 993. Alternatves to the medan absolute devaton. Journal of the Amercan Statstcal Assocaton 88, Savtzky, A., Golay, M.J.E., 964. Smoothng and dfferentaton of data by smplfed least squares procedures. Anal. Chem. 36, Schoenberg, I.J., 964. Splne functons and the problem of graduaton. Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca 5, Strang, G., 999. The dscrete cosne transform. SIAM Revew 4, Takezawa, K., 5. Introducton to Nonparametrc Regresson. John Wley & Sons, Inc., Hoboken, NJ. Wahba, G., 99a. Splne Models for Observatonal Data. Socety for Industral Mathematcs, Phladelpha. Wahba, G., 99b. Estmatng the smoothng parameter. In: Splne Models for Observatonal Data. Socety for Industral Mathematcs, Phladelpha, pp Watson, G.S., 964. Smooth regresson analyss. The Indan Journal of Statstcs, Seres A 6, Wenert, H.L., 7. Effcent computaton for Whttaker Herson smoothng. Computatonal Statstcs & Data Analyss 5, Whttaker, E.T., 93. On a new method of graduaton. Proceedngs of the Ednburgh Mathematcal Socety 4, Yueh, W.C., 5. Egenvalues of several trdagonal matrces. Appled Mathematcs E-Notes 5,