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KAH 76, CAW 92, FAL 94, ARB 96, OLS 94, HOL 92, RIE 95a, RIE 97b, PES 97] for which dim(Ka ) and T (a) are equal and have the form of a (see Figure 43 and also 45(e)) These cascades are constructed through a multiplicative iteration scheme such as the binomial cascade, which is presented in detail earlier in this chapter with special emphasis on its wavelet decomposition Having positive increments, this class of processes is, however, sometimes too restrictive FBM, as noted, has the disadvantage of a poor multifractal structure and does not contribute to a larger pool of stochastic processes with multifractal characteristics It is also notable that the rst natural , truly multifractal stochastic process to be identi ed was the L vy motion [JAF 99] This example is particularly appealing since scaling is not injected into the model by an iterative construction (this is what we mean by the term natural) However, its spectrum is degenerative, though it shows a non-trivial range of scaling exponents h(t), in the sense that it is linear 481 Construction and simulation With the formalism presented here, the stage is set for constructing and studying new classes of truly multifractional processes The idea, to speak in Mandelbrot s own words, is inevitable after the fact The ingredients are simple: a multifractal time warp , ie an increasing function or process M(t) for which the multifractal formalism is known to hold, and a function or process V with strong monofractal scaling properties such as fractional Brownian motion (FBM), a Weierstrass process or self-similar martingales such as L vy motion We then form the compound process V(t) := V (M(t)) (491)

To x the ideas, let us recall the method of midpoint displacement which can be used to de ne a simple Brownian motion B1/2 which we will also call the Wiener motion (WM) for a clear distinction from FBM This method constructs B1/2 iteratively at dyadic points Having constructed B1/2 (k2 n ) and B1/2 ((k + 1)2 n ) we de ne B1/2 ((2k + 1)2 n 1 ) as (B1/2 (k2 n ) + B1/2 ((k + 1)2 n ))/2 + Xk,n The offsets Xk,n are independent zero-mean Gaussian variables with variance such as to satisfy (41) with H = 1/2, hence the name of the method One way to obtain Wiener motion in multifractal time WM(MF) is then to keep the offset variables Xk,n as they are but to apply them at the time instances tk,n de ned by tk,n = M 1 (k2 n ), ie M(tk,n ) = k2 n : B1/2 (t2k+1,n+1 ) := B1/2 (tk,n ) + B1/2 (tk+1,n ) + Xk,n 2 (492)

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This amounts to a randomly located random displacement, the location being determined by M Indeed, (491) is nothing but a time warp An alternative construction of warped Wiener motion WM(MF) which yields equally spaced sampling, as opposed to the samples B1/2 (tk,n ) provided by (492), is

The problem with this model goes deeper than its lack of any understanding of the customer While well-understood and relatively easy to measure, these traditional measures also present some serious difficulties for project organizations These difficulties arise as a result of the natural tendency toward operating in ways that are rewarded by this triple constraint model, including:4 1 The tendency to sacrifice external concerns for internal performance In other words, when a company measures their

desirable To this end, note rst that the increments of WM(MF) become independent Gaussians once the path of M(t) is realized To be more precise, x n and let G(k) := B((k + 1)2 n ) B(k2 n ) = B1/2 (M(k + 1)2 n ) B1/2 (M(k2 n )) (493)

.

For a sample path of G we start by producing rst the random variables M(k2 n ) Once this is done, the G(k) are simply independent zero-mean Gaussian variables with variance |M((k + 1)2 n ) M(k2 n )| This procedure has been used in Figure 45 482 Global analysis To calculate the multifractal envelope T (q) we need only to know that V is an H-sssi process, ie that the increment V (t + u) V (t) is equal in distribution to uH V (1) (see (41)) Assuming independence between V and M, a simple calculation reads as

2n 1

2n 1

M k2 n ,

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