Page  00000345 Applications of Principal Differential Analysis, to Data Reduction and Extraction of Musical Features of Sound SUZANNE WINSBERG'AND PHILLIPPE DEPALLEt "winsberg@ircam.fr, IRCAM,1 place Igor-Stravinsky, 75004 PARIS, FRANCE tdepallep@ere.umontreal.ca, Universite de Montreal, Montreal, CANADA Abstract In sound signal analysis and modeling, one often wants to reduce large sets of functions into smaller sets, thus obtaining an efficient parametrization of a sound or a useful extraction of a musical feature. Given a set of observed functions, {z n}n=l,N, data reduction may be obtained by determining the linear differential operator minimizing the homogenous differential equation Lz, = 0, and expressing each function as a linear combination of the set of functions spanning the null space of L. This paper presents principal differential analysis and its application to the estimation of vibrato and to the segmentation of sounds. 1 Introduction In sound signal analysis and modeling, one often wants to reduce large sets of functions into much smaller sets of functions, obtaining an efficient parametrization of a sound or a useful extraction of a musical feature. The problem is then, given a set of N functional observations {z,(t)}n=1......N we wish to define a much smaller set of m functions, {uj}j=1,...,J, in order to obtain efficient approximations of the observed functions. Although we use the continuous index, t, in practice the data consist of a finite number, T, of sampled values on t. We consider a method, principal differential anaysis, PDA, (Ramsay,1996), which like principal components analysis, PCA, obtains an efficient approximation, but in addition explicitly takes into account the smoothness of the data, and may reveal the underlying processes that generated the data. We identify the linear differential operator, LDO, L =woI+ lD +...+ wm-lDm + m, where I is the identity operator, and D" is the differential operator of order m, that is, the mth derivative operator, that comes as close as possible to satisfying the homogenous linear differential equation, HLDE, Lz, = 0, for each observed function,,,. A least squares approach yields the following fitting criterion: SSEpDAO(L) = [Lxn()]2dt n=lJ which is minimized over L to find an estimate of the LDO. L is defined by the m-vector w of functions (wo,...,,m-l)', and so SSEpDA(L) depends on those functions, and estimating L is equivalent to estimating the m weight functions, wj. Having determined L, we can then generally define m linearly independent functions, u(t), that span the null space of L. Finally, any function z(t), satisfying the HLDE can be expressed as a linear combination of the u(t). Since L,, has been minimized, a good aproximation of,, can thus be obtained. This procedure is reminiscient of PCA where the first m components are a good m-dimensional set for approximating the data. The nature of the approximation is different, however. In PCA we also find a linear operator, Q, such that Qxn = 0. The difference between PDA and PCA lies in the difference between the operators L and Q. Since ICMC Proceedings 1999 -345 -

Page  00000346 Qz is in the same space as x, we measure the performance of Q in the same space as the functions x to which it is applied. However, L is a roughening operator, since Lx has fewer derivatives and is generally more variable. PDA is useful because it explicitly takes into account the smoothness of the data, while PCA does not. An interesting way to look at the residual error is to interpret it as the forcing functions, f, defined as Lx, = f,. These functions should be small if the homogenous equation is valid, and their size is a measure of the fit to the model. 2 Estimation One way to estimate the weight functions wj is a pointwise approach. Define the pointwise criterion PSSEL(t) = N-' (Lxn)2(t) = n j=O where wm (t) 1 for all t. If t is regarded as fixed, the following argument demonstrates that the above is a least squares fitting criterion. Define: the m-dimensional vector, w(t)= (wo(t),...,,im-(t)), the N x m pointwise design matrix, Z(t) = {(1'n)(t)}n=l,...,Nj=0...,m-l, and the N-dimensional dependent vector, The existence of these pointwise values of w(t) depends on the determinant of Z(t)'Z(t) being bounded away from zero for all values of t. 3 An Example PDA has been applied to the data reduction of the time-evolution of the frequency of the twenty first partials of a tenor voice. Using a second order differential operator L to give the best overall fit to the data, the time-dependent vibrato frequency is obtained. The tenor data consists of an excerpt of duration 1.77 seconds sampled each 10 milliseconds. The data was centered and normed to remove the effect of the base frequency, leaving only the frequency about the central frequency of the partial. Prior to analysis, the data was warped using a data-based monotone transformation of the time, t to make the functions line up with each other as well as possible in a least squares sense. A second order differential operator was found to give the best overall fit. The recovered function for wo yielded the time-dependent vibrato frequency. The system described by the optimal HLDE was found to be underdamped. Furthermore, the forcing functions are generally very close to zero, except at the instant of time at which the tenor changes a note. This moment is clearly demarcated, both in the forcing functions and in the solution for the resonant frequency. Using the model, the frequencies of the partials are all well recovered by the linear combinations of the basis functions, that span the space of the resultant differential operator.. The functions are smoother than the original data, with noisy parts removed. 4 Conclusion This technique is thus useful for uncovering aspects of the data, as well as for data reduction, smoothing, and segmentation of a sound. References Ramsay,J.O. (1996), Principal differential analysis: data reduction by differential operators.Journal of the Royal Statistical Society, Series B,58, 495-508. y(t) = {-DmZ,(t)}n=1,.N. Then in matrix terms, PSSEL(t) = [ N-'[y(t) - Z(t)w(t)]'[y(t) - Z(t)w()]. Holding t fixed, the least squares solution minimizing PSSEL (t) with respect to wj is w(t)= [Z= )Z(t'Z7)- )'y( ). -346 - ICMC Proceedings 1999