Acta Polytechnica


Acta Polytechnica 53(2):75–78, 2013 © Czech Technical University in Prague, 2013
available online at http://ctn.cvut.cz/ap/

FRACTAL DIMENSION ESTIMATION IN DIAGNOSING
ALZHEIMER’S DISEASE

Václav Hubata-Vaceka,∗, Jaromír Kukala, Robert Rusinab,
Marie Buncovác

a CTU, Faculty of Nuclear Sciences and Physical Engineering, Department of Software Engineering
in Economics, Břehová 7, 115 19 Praha 1, Czech Republic

b Thomayer’s Hospital, Vídeňská 800, 140 59 Praha 4–Krč, Czech Republic
c Institute for Clinical and Experimental Medicine, Vídeňská 1958/9, 140 21 Praha 4–Krč, Czech Republic
∗ corresponding author: hubatvac@fjfi.cvut.cz

Abstract. Estimated entropies from a limited data set are always biased. Consequently, it is not a
trivial task to calculate the entropy in real tasks. In this paper, we used a generalized definition of
entropy to evaluate the Hartley, Shannon, and Collision entropies. Moreover, we applied the Miller
and Harris estimations of Shannon entropy, which are well known bias approaches based on Taylor
series. Finally, these estimates were improved by Bayesian estimation of individual probabilities. These
methods were tested and used for recognizing Alzheimer’s disease, using the relationship between
entropy and the fractal dimension to obtain fractal dimensions of 3D brain scans.

Keywords: entropy, fractal dimension, Alzheimer’s disease, boxcounting, Rényi entopy.

1. Introduction
Before explaning the relationship between entropy and
dimension, we have to introduce the term of dimension.
Let d ∈ N be a dimension of Euclidean space where a
d-dimensional unit hypercube is placed. Let m ∈ N be
resolution and a = 1/m be edge the length of covering
hypercubes of the same dimension d. The number of
covering elements is given by

N = N(a) = a−D.

Knowledge of N for fixed a enables direct calculation
of the hypercube dimension according to

ln N(a) = −D ln a

D =
ln N(a)

ln 1
a

. (1)

The very popular boxcounting method [1] is based on
the generalization of (1) to the form

ln N(a) = A0 −D0 ln a

and its application to the boundary of any set F ⊂ Rd.
As will be shown in the next section, the quantity
ln N(a) is an estimate of the Hartley entropy.

2. Rényi Entropy
Using a natural logarithm instead of a binary loga-
rithm, we can follow in the definition of Rényi entropy.
Let k ∈ N be number of events, pj > 0 be their prob-
abilities for j = 1, . . . ,k satisfying

∑k
j=1 pj = 1, and

q ∈ R. We can define Rényi entropy [2] as

Hq =
ln
∑k

j=1 p
q
j

1 −q
,

which is a generalization of Shannon entropy. In
respect of q, we obtain the specific entropies:
• Hartley entropy [3] for q = 0 as

H0 = ln
∑
pj >0

1 = ln
k∑

j>0
1 = ln k = ln N(a);

• Shannon entropy [4] for q → 1 as

H1 = lim
q→1

Hq = −
∑
j=1

pj ln pj ;

• Collision entropy [2] for q = 2 as

H2 = − ln
∑
pj >0

p2j ;

The resulting theoretical entropies can be used for
defining the Rényi dimension [2] as

Dq = lim
a→0+

Hq

ln 1
a

,

which corresponds to the relationship

Hq ≈ Aq −Dq ln a (2)

for small covering size a > 0.

3. Entropy Estimates
There are several approaches to entropy estimation
from experimental data sets. Assuming that the num-
ber of experiments n ∈ N is finite, we can count the

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V. Hubata-Vacek, J. Kukal, R. Rusina, M. Buncová Acta Polytechnica

events and obtain nj ∈ N0 as the event frequencies for
j = 1, . . . ,k. The first approach to entropy estimation
is naive estimation. We directly estimate k and pj as

kN =
∑

nj >0
1 ≤ k,

pj,N =
nj
n
.

These biased estimates also produce biased entropy
estimates

H0,N = ln kN,

H1,N = −
∑

nj >0
pj,N ln pj,N,

H2,N = − ln
∑

nj >0
p2j,N.

The second approach is based on Bayesian estimation
of probabilities pj as

pj,B =
nj + 1
n + kN

.

This technique is called here semi-Bayesian estimation.
We obtain other, but also biased, entropy estimates

H1,S = −
∑

nj >0
pj,B ln pj,B,

H2,S = − ln
∑

nj >0
p2j,B.

The estimate H2,S can be improved as

H2,S2 = − ln
∑

nj >0
uj,

where uj =
(nj +2)(nj +1)

(n+kN+1)(n+kN)
is a Bayesian estimate

of p2j . A direct Bayesian estimate of H1 was also
calculated as

H1,B = −
kN∑
i=1

ni + 1
n + kN

(
ψ(ni + 2) −ψ(n + kN + 1)

)
,

where ψ is the digamma function.

4. Bias Reduction
Miller [5] modified the naive estimate H1,N using first
order Taylor expansion, which produces

H1,M = H1,N +
kN − 1

2n
.

Lately, Harris [5] improved the formula to

H1,H = H1,N +
kN − 1

2n
+

1
12n2

(
1 −

∑
pj >0

1
pj

)
From the theoretical point of view, it is prohibited to
estimate pj by its estimates. However we are trying

to investigate biased estimates of H1 in the forms

H1,HN = H1,N +
kN − 1

2n
+

1
12n2

(
1 −

∑
nj >0

1
pj,N

)
,

H1,HS = H1,N +
kN − 1

2n
+

1
12n2

(
1 −

∑
nj >0

1
pj,B

)
,

H1,HB = H1,N +
kN − 1

2n
+

1
12n2

(
1 −

∑
nj >0

rj

)
,

where rj = n+kN−1nj is Bayesian estimate of
1
pj
.

5. Estimation Methodology
Naive, semi-Bayesian, Bayesian and corrected entropy
estimates were subjected of testing on 2D and 3D
structures with known Hausdorff dimension. The
list of involved estimates is included in Tab. 1. A
Sierpinski carpet with Dq = 1.8928 for any q ≥ 0 of
size 81×81 is a typical 2D fractal set model. Using the
estimates from Tab. 1 and a linear regression model
(2), we estimated the Rényi dimensions D̂q and then
evaluated its zscore as a relative measure of bias

zscore =
D̂q −Dq
SDq

.

The results are included in Tab. 2. The best esti-
mations with |zscore| ≤ 1.960 are H1,M followed by
Harris estimations H1,HN,H1,HS,H1,HB. A struc-
ture of Dq = 2.3219 and size 128 × 128 × 128
was then used for 3D testing and the results are
also included in Tab. 2. The best estimators are
H1,HS,H1,HN,H1,HB,H1,M,H2,S.

6. Alzheimer’s Disease Diagnosis
from Fractal Dimension
Estimates

Alzheimer’s disease (AD) is the most common form of
dementia, and is characterised by loss of neurons and
their synapses. This loss is caused by an accumulation
of amyloid plaques between nerve cells in the brain.
Morphologically, the affected areas produce rounded
clusters of destroyed brain cells, which are visible on
brain scans. On the other hand, Amyotrophic lateral
sclerosis (ALS) is a disease of the motor neurons, and
it is not visible on brain scans. In this sense, brain
scans of ALS patien’s look like brain scans of healthy
patients.

These entropy estimators were used for diagnosing
Alzheimer’s disease. We tried to separate two different
groups of samples of human brains. In the first group,
there were brain scans of patients with Alzheimer’s
disease (AD) and in the second group brain scans of
patients with amyotrophic lateral sclerosis (ALS). We
carried out tests on 21 samples (11 for AD and 10
for ALS), represented by 128 × 128 × 128 matrices
of thresholded images (θ = 40 %). We used a two-
sample t-test for null hypotheses, and the alternative

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vol. 53 no. 2/2013 Fractal Dimension Estimation in Diagnosing Alzheimer’s Disease

Estimate Sierpinski carpet Dq = 1.8928 Five Box Fractal Dq = 2.3219

D̂q SDq zscore D̂q SDq zscore

H0,N 1.8158 0.0064 −12.0577 2.0897 0.0284 −8.1757
H1,N 1.8472 0.0059 −7.7116 2.1853 0.0320 −4.2690
H2,N 1.8578 0.0076 −4.6212 2.1949 0.0298 −4.2568
H1,S 1.8515 0.0058 −7.0853 2.2367 0.0315 −2.7012
H2,S 1.8657 0.0072 −3.7494 2.2927 0.0298 −0.9798
H2,S2 1.7898 0.0077 −13.4269 2.1189 0.0268 −7.5904
H1,B 1.8170 0.0060 −12.6863 2.1654 0.0297 −5.2638
H1,M 1.8930 0.0059 0.0306 2.3315 0.0349 0.2730
H1,HN 1.8921 0.0059 −0.1203 2.3208 0.0347 −0.0332
H1,HS 1.8921 0.0059 −0.1164 2.3226 0.0347 0.0196
H1,HB 1.8920 0.0059 −0.1328 2.3182 0.0346 −0.1084

Table 2. Dimension estimates via various entropy estimates.

Method H0 H1 H2
Naive H0,N H1,N H2,N
semibayesian (pj) * H1,S H2,S
semibayesian (p2j ) * * H2,S2
bayesian * H1,B *
Miller * H1,M *
Harris * H1,HN *
Harris semibayesian (pj) * H1,HS *
Harris bayesian (1/pj) * H1,HB *

Table 1. Entropy estimates.

hypotheses were

H0 : ED̂q (AD) = ED̂q (ALS),
HA : ED̂q (AD) 6= ED̂q (ALS).

The results are included in Tab. 3. The most signifi-
cant differences between AD and ALS were observed
for H0,N,H1,S,H1,B.

7. Conclusion
In this paper we tested estimates for Hartley, Shan-
non and Collision entropy. These estimates were im-
proved by Bayesian estimation and tested on fractals
with known fractal dimensions. Finally, these esti-
mates were used on two groups of samples of brain
scans, in order to obtain the best separator. The
best separators, with regard to the experiment, are
H0,N,H1,S,H1,B, and they have a 2 % level of signif-
icance. The rest of the estimates also have results
under a 5 % level of significance, except for H2,N,

Estimate ED̂q (AD) ED̂q (ALS) pvalue
H0,N 1.9745 2.0315 0.017486
H1,N 2.0649 2.1096 0.025128
H2,N 2.0687 2.1034 0.067814
H1,S 2.0968 2.1471 0.018828
H2,S 2.1458 2.1903 0.031375
H2,S2 1.9274 1.9666 0.036419
H1,B 2.0011 2.0506 0.018873
H1,M 2.2607 2.3115 0.041142
H1,HN 2.2428 2.2931 0.037608
H1,HS 2.2452 2.2957 0.037729
H1,HB 2.2366 2.2868 0.035800

Table 3. Diagnostic power.

which was worst. On hte basis of these results, en-
tropy can be used for diagnosing Alzheimer’s disease
in the future, considering that methods can be still
improved, especially by estimating kN or by image
filtering.

Acknowledgements
This paper was created with support from CTU in Prague
grant SGS11/165/OHK4/3T/14.

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	Acta Polytechnica 53(2):75–78, 2013
	1 Introduction
	2 Rényi Entropy
	3 Entropy Estimates
	4 Bias Reduction
	5 Estimation Methodology
	6 Alzheimer's Disease Diagnosis from Fractal Dimension Estimates
	7 Conclusion
	Acknowledgements
	References