Comparing Measures of Line Jaggedness

Mandelbrot-Weierstrass function data generation parameters

fractal dimension d

1.5

terms in series T

15

phase ϕ

0.786

frequency multiplier b

2.5

scale factor k

1.

resolution and position parameters

number of points

251

starting point

1

fit MW(t) data with a polynomial of degree

3

4

jaggedness measures

j

2

(σ) and

j

3

(σ)

range factor c

5.

This Demonstration compares three statistical measures of line jaggedness visually and numerically. It creates and plots rugged curves with the Mandelbrot–Weierstrass function,

MW(t)=

T

∑

n=1

1/

n(2-d)

b

(1-kcos(

n

b

t+ϕ))

, of a chosen fractal dimension

d

, starting point, range, and resolution, and curve-fits the data with a third- or fourth-degree polynomial. The fitted ("smoothed") curve is superimposed on the generated

MW(t)

curve with the residuals plotted below. The residuals' standard deviation,

σ

, and two additional jaggedness indices,

j

2

(σ)

and

j

3

(σ)

, having a range from 0 (perfectly smooth) to 1, are calculated and their values displayed. Also included is a plot showing these two indices' dependence on

σ

for a chosen range factor

c

.