Compactly Supported Radial Basis Functions

Transcription

1 Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically ensues that Φ is stictly positive definite Anothe obsevation was that compactly suppoted adial functions can be stictly positive definite on IR s only fo a fixed maximal s-value It is not possible fo a function to be stictly positive definite and adial on IR s fo all s and also have a compact suppot Theefoe we focus ou attention on the chaacteization and constuction of functions that ae compactly suppoted, stictly positive definite and adial on IR s fo some fixed s Accoding to ou ealie wok (Bochne s Theoem and genealizations theeof), a function is stictly positive definite and adial on IR s if its s-vaiate Fouie tansfom is non-negative Theoem 22 gives the Fouie tansfom of Φ ϕ( ) as ˆΦ(x) F s ϕ() (s 2)/2 ϕ(t)t s/2 J (s 2)/2 (t)dt 4 Opeatos fo Radial Functions and Dimension Walks Schaback and Wu [564] defined an integal opeato and its invese diffeential opeato, and discussed an entie calculus fo how these opeatos act on adial functions These opeatos will facilitate the constuction of compactly suppoted adial functions Definition 4 Let ϕ be such that t tϕ(t) L [, ), then we define (Iϕ)() 2 Fo even ϕ C 2 (IR) we define tϕ(t)dt, (Dϕ)() ϕ (), In both cases the esulting functions ae to be intepeted as even functions using even extension 37

2 Remak: Note that the opeato I diffes fom the opeato I intoduced ealie by a facto t in the integand Howeve, the two opeatos ae elated In fact, we have Iϕ( 2/2) Iϕ( ), ie, tϕ(t 2 /2)dt 2 /2 ϕ(t)dt The most impotant popeties of these opeatos ae (see, eg, [564] o [627]): Theoem 42 Both D and I peseve compact suppot, ie, if ϕ has compact suppot, then so do Dϕ and Iϕ 2 If ϕ C ( IR) and t tφ(t) L [, ), then DIϕ ϕ 3 If ϕ C 2 (IR) is even and ϕ L [, ), then IDϕ ϕ 4 If t t s ϕ(t) L [, ) and s 3, then F s (ϕ) F s 2 (Iϕ) 5 If ϕ C 2 (IR) is even and t t s ϕ (t) L [, ), then F s (ϕ) F s+2 (Dϕ) The opeatos I and D allow us to expess s-vaiate Fouie tansfoms as (s 2)- o (s + 2)-vaiate Fouie tansfoms, espectively In paticula, a diect consequence of the above popeties and the chaacteization of stictly positive definite adial functions (Theoem 24) is Theoem 43 Suppose ϕ C(IR) If t t s ϕ(t) L [, ) and s 3, then ϕ is stictly positive definite and adial on IR s if and only if Iϕ is stictly positive definite and adial on IR s 2 2 If ϕ C 2 (IR) is even and t t s ϕ (t) L [, ), then ϕ is stictly positive definite and adial on IR s if and only if Dϕ is stictly positive definite and adial on IR s+2 This allows us to constuct new stictly positive definite adial functions fom given ones by a dimension-walk technique that steps though multivaiate Euclidean space in even incements 42 Wendland s Compactly Suppoted Functions In [627] Wendland constucted a popula family of compactly suppoted adial functions by stating with the tuncated powe function (which we know to be stictly positive definite and adial on IR s fo s 2l ), and then walking though dimensions by epeatedly applying the opeato I Definition 42 With ϕ l () ( ) l + we define ϕ s,k I k ϕ s/2 +k+ It tuns out that the functions ϕ s,k ae all suppoted on [, ] and have a polynomial epesentation thee Moe pecisely, 38

3 Theoem 422 The functions ϕ s,k ae stictly positive definite and adial on IR s and ae of the fom { ps,k (), [, ], ϕ s,k (), >, with a univaiate polynomial p s,k of degee s/2 +3k + Moeove, ϕ s,k C 2k (IR) ae unique up to a constant facto, and the polynomial degee is minimal fo given space dimension s and smoothness 2k Wendland gave ecusive fomulas fo the functions ϕ s,k fo all s, k We instead list the explicit fomulas of [95] Theoem 423 The functions ϕ s,k, k,, 2, 3, have the fom ϕ s, () ( ) l +, ϕ s, () ( ) l+ + [(l + ) + ], ϕ s,2 () ( ) l+2 [ + (l 2 + 4l + 3) 2 + (3l + 6) + 3 ], ϕ s,3 () ( ) l+3 [ (l 3 + 9l l + 5) 3 + (6l l + 45) 2 + (5l + 45) + 5 ], + whee l s/2 + k +, and the symbol positive constant denotes equality up to a multiplicative Poof: The case k follows diectly fom the definition Application of the definition fo the case k yields ϕ s, () (Iϕ l )() t( t) l +dt t( t) l dt tϕ l (t)dt (l + )(l + 2) ( )l+ [(l + ) + ], whee the compact suppot of ϕ l educes the impope integal to a definite integal which can be evaluated using integation by pats The othe two cases ae obtained similaly by epeated application of I Examples: Fo s 3 we get some of the most commonly used functions as ϕ 3, () ( ) 2 +, C SP D(IR 3 ) ϕ 3, () ( ) 4 + (4 + ), C 2 SP D(IR 3 ) ϕ 3,2 () ( ) 6 ( ), C 4 SP D(IR 3 ) ϕ 3,3 () ( ) 8 ( ), C 6 SP D(IR 3 ) 39

4 43 Wu s Compactly Suppoted Functions In [656] Wu pesents anothe way to constuct stictly positive definite adial functions with compact suppot He stats with the function ψ() ( 2 ) l +, l IN, which is stictly positive definite and adial since we know that the tuncated powe function ψ( ) is multiply monotone Wu then constucts anothe function that is stictly positive definite and adial on IR by convolution, ie, ψ l () (ψ ψ)(2) ( t 2 ) l +( (2 t) 2 ) l +dt ( t 2 ) l ( (2 t) 2 ) l dt This function is stictly positive definite since its Fouie tansfom is essentially the squae of the Fouie tansfom of ψ Just like the Wendland functions, this function is a polynomial on its suppot In fact, the degee of the polynomial is 4l +, and ψ l C 2l (IR) Now, a family of stictly positive definite adial functions is constucted by a dimension walk using the D opeato, ie, ψ k,l D k ψ l The functions ψ k,l ae stictly positive definite and adial in IR s fo s 2k +, ae polynomials of degee 4l 2k + on thei suppot and in C 2(l k) in the inteio of the suppot On the bounday the smoothness inceases to C 2l k Example: Fo l 3 we can compute the thee functions ψ k,3 () D k ψ 3 () D k (( 2) 3 + ( 2) 3 +)(2), k,, 2, 3 This esults in ψ,3 () ( ) + ( ) 7 +( ) C 6 SP D(IR) ψ,3 () ( ) + ( ) 6 +( ) C 4 SP D(IR 3 ) ψ 2,3 () ( ) + ( ) 5 +( ) C 2 SP D(IR 5 ) ψ 3,3 () ( ) + ( ) 4 +( ) C SP D(IR 7 ) Remaks: Fo a pescibed smoothness the polynomial degee of Wendland s functions is lowe than that of Wu s functions Fo example, both Wendland s function ϕ 3,2 and Wu s function ψ,3 ae C 4 smooth and stictly positive definite and adial in IR 3 Howeve, the polynomial degee of Wendland s function is 8, wheeas that of Wu s function is 4

5 Figue 4: Plot of Wendland s functions (left), Wu s functions (cente), and Buhmann s function (ight) listed as examples 2 While both families of stictly positive definite compactly suppoted functions ae constucted via dimension walk, Wendland uses integation (and thus obtains a family of inceasingly smoothe functions), wheeas Wu needs to stat with a function of sufficient smoothness, and then obtains successively less smooth functions (via diffeentiation) 44 Buhmann s Compactly Suppoted Functions A thid family of compactly suppoted stictly positive definite adial functions that has appeaed in the liteatue is due to Buhmann (see [84]) Buhmann s functions contain a logaithmic tem in addition to a polynomial His functions have the geneal fom φ() ( 2 /t) λ +t α ( t δ ) ρ + dt Hee < δ 2, ρ, and in ode to obtain functions that ae stictly positive definite and adial on IR s fo s 3 the constaints fo the emaining paametes ae λ, and < α λ 2 Example: An example with α δ 2, ρ and λ 2 is listed in [85]: φ() 2 4 log ,, C 2 SP D(IR 3 ) Remaks: While Buhmann [85] claims that his constuction encompasses both Wendland s and Wu s functions, Wendland [634] gives an even moe geneal theoem that shows that integation of a positive function f L [, ) against a stictly positive definite (compactly suppoted) kenel K esults in a (compactly suppoted) stictly positive definite function, ie, ϕ() 4 K(t, )f(t)dt

6 is stictly positive definite Buhmann s constuction then coesponds to choosing f(t) t α ( t δ ) ρ + and K(t, ) ( 2 /t) λ + 2 Multiply monotone functions ae coveed by this geneal theoem by taking K(t, ) ( t) k + and f an abitay positive function in L so that dµ(t) f(t)dt in Williamson s chaacteization Theoem 262 Also, functions that ae stictly positive definite and adial in IR s fo all s (o equivalently completely monotone functions) ae coveed by choosing K(t, ) e t 42