To make or use a mathematical model, we must find an effective algorithm and approach to determine its parameter values. The algorithms include the ordinary least squares (OLS), maximum likelihood estimation (MLE), and major axis method (MAM). A number of measurement approaches, as displayed above, are proposed in literature to estimate fractal dimension values (
Table 1,
Table 2 and
Table 3). Generally speaking, different methods are applied to different directions (different aspects or properties). For example, the walking-divider method can be used to estimate the fractal dimension of urban boundary dimension rather than urban area; power spectrum is used to research the urban evolution based on time series rather than urban form based on spatial data; fractional Brownian motion (FBM) is used to estimate self-affine record dimension rather than self-similar trail dimension; the sandbox method, clustering growing, and wave-spectrum are used to calculate the radial dimension for characterizing urban growth; the box-counting method is used to compute fractal dimensions for describing spatial structure and texture of urban morphology, and so on. Sometimes, several different methods can be applied to the same aspect of cities. For example, the box-counting method, area–radius scaling method, sandbox method, and wave spectrum analysis based on density–radius scaling can be employed to estimate the fractal dimension of urbanized area. In theory, a fractal aspect has only one fractal dimension value, but unfortunately, in empirical studies, different methods often result in different fractal dimension estimation values, and in many cases, the numerical differences are statistically significant and cannot be ignored in a spatial analysis. Even for a given method, a fractal dimension value often depends on the size and central location of the study area defined by a researcher. This is involved with the uncertainty of fractal dimension calculation, which puzzles many fractal scientists.
A simple prototype is helpful for understanding complex phenomena in scientific research. In order to study the atomic structure, physicists first explored the structure of the simplest atom, the hydrogen atom; in order to study the structure of viruses, biologists first concentrated on exploring the structure of simple virus, bacteriophages. Simple prototypes often form the beginning of theoretical analysis. To reveal the root of the problem of uncertainty in fractal dimension calculation, we can examine two regular fractals, including monofractal and multifractal patterns. All these regular fractals reflect prefractal structure because we can never look the real fractal patterns. The real fractals in geometry are just like the high-dimensional spaces in linear algebra, which can be imagined but can never be observed. All of the fractal images we encounter in books or articles represent prefractals rather than real fractals [
54]. The difference between real fractals and prefractals is as follows: A real fractal has infinite levels, but a prefractal is a limited hierarchy; therefore, the Lebesgue measure of a real fractal equals 0, but the Lebesgue measure of a prefractal is not equal to 0. For a given aspect (say, area or boundary) of a regular monofractal object, we can apply different methods to its prefractal structure to determine its fractal dimension. Different methods lead to the same result, which represents the real fractal dimension value. However, for a multifractal object, the real fractal dimension cannot be computed by applying some method to its prefractal pattern. We can only obtain comparable parameters rather than real fractal dimension for multifractal systems.
By analyzing the regular fractal objects, we can gain new insight into fractal structure and fractal dimension measurement. First of all, let us see a simple regular growing fractal, which is employed to model urban growth in literature [
11,
21,
39,
42,
55]. This fractal was proposed by Jullien and Botet [
56] and became well known due to the work of Vicsek [
33], and it is also termed Vicsek’s figure or box fractal (
Figure 1). Three approaches can be applied to its prefractal pattern, including the box-counting method, sandbox method, and cluster growing scaling method. The third approached can be divided into two equivalent methods: area (number)–radius scaling and density–radius scaling. According to its regular composition, we can obtain the datasets comprising the first 10 steps (
Table 4). Based on the box-counting method, sandbox method, and area–radius scaling method, the scaling exponent is just its fractal dimension, and the value is
D = ln(5)/ln(3) = 1.465. Based on the density–radius scaling method, the scaling exponent is
a = 2 −
D = 0.535, and thus the fractal dimension is also
D = 2 − 0.535 = 1.465. This value is exactly the real fractal dimension of this fractal object.
Further, let us examine a regular growing multifractal object, which reflects the pattern of spatial heterogeneity. This fractal is presented by Vicsek [
33]. It can be used to model multifractal growth of cities [
57]. The first three steps represent a prefractal process (
Figure 2). The box-counting method can be used to calculate its global dimension. Step 1: fractal dimension
D = 0 (for a point, the fractal dimension can be obtained by L’Hospital’s rule). Step 2: box dimension
D = −ln(17)/ln(1/5) = 1.7604. Step 3: box dimension
D = −ln(289)/ln(1/25) = 1.7604. If we apply the sandbox method to the figure in the third step, the fractal dimension is also
D = 1.7604. However, two problems can be found by careful investigation. First, different fractal units bear different fractal dimension values. One of basic properties of fractals, including monofractals and multifractals, is entropy conservation: different fractal units at a given level has the same Shannon entropy value [
9,
28,
58,
59,
60,
61]. In fact, different fractals, except fat fractals, can be unified into the same framework [
28], and expressed by a transcendental equation as below [
60,
61]
in which
Pi denotes the growth probability of the
ith fractal unit,
ri represents the linear size of the
ith fractal unit,
q refers to the order of moment, and the power exponent
Dq is termed the generalized correlation dimension [
59]. For monofractals, we have,
Dq ≡
D0; for self-affine fractals, different directions have different fractal dimension values, and for a given direction, we have
Dq =
D0; for multifractals, different parts of a multifractal system have different local fractal dimension values, and the global fractal dimension
Dq depends on the moment order
q [
28]. Equation (1) can be employed to identify different fractals from varied complex systems. One of the commonalities of different fractals is the conservation of entropy, which can be derived from Equation (1). However, the fractal dimension does not comply with a conservation law. In fact, for a multifractal system, different parts have different local fractal dimensions. For example, for the second level of the third step, the five parts have two fractal dimension values. The central part, box dimension is
D = ln(1/17)/ln(1/5) = 1.7604; the other four parts, box dimension is
D = ln(4/17)/ln(2/5) = 1.5791. Second, the parameter value estimated by the box-counting method and the sandbox method is not equal to its real dimension value. In theory, the calculated values represent the capacity dimension of this multifractals, i.e.,
D0 = 1.7604. The regular multifractal structure can be modeled by a transcendental equation based on probabilities and the corresponding scales. Where the third step is concerned, the multifractal transcendental equation can be constructed as follows
Using Matlab to find its numerical solutions, we can obtain its multifractal parameter values (
Table 5). The results show that the real capacity dimension is about
D = 1.5995 <
D = 1.7604. The capacity dimension based on the box-counting method and the sandbox method is in fact the maximum dimension, that is
D−∞ = 1.7604. The capacity (
D0) is the maximum value of the local dimension, while the maximum dimension (
D−∞) is the upper range value of global dimension.
Now, a basic judgment can be reached as follows. For a regular monofractal (
Figure 1), the real fractal dimension value can be calculated by the prefractal structure. However, for a regular multifractal (
Figure 2), the real fractal dimension values cannot be obtained by applying some method such as the box-counting method to its prefractal structure. This suggests that the resolution of remote sensing images influences the multifractal parameter estimation of cities, reminding us of the finite size effect in fractal measurements. In fact, for a random multifractal system, we cannot construct its multi-scaling transcendental equation such as in Equation (1). As a result, we will never know the real fractal parameter values. We can only estimate a set of comparable parameter values to replace the real values. In the real world, fractal cities have two properties. First, they are random multifractals rather the regular monofractals or regular multifractals; second, they only develop prefractal structure rather than real fractal structure. What is more, the prefractal structure of multifractals are always mixed up with self-affine processes, fat fractal components, or even fractal complements. In this case, the processes of measurements and analyses become very complicated relative to the regular fractals.