Cultural Heritage Applications: Archaeological Ceramics and Building Restoration - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 137

Table 6.4 Porosity and density statistics of the prior study

Open porosity ( %) Apparent density (gr/cm3)

Period i Mean l Standard deviation r i Mean l Standard deviation r i

1. Bronze Age 28.20 3.7794 1.80 0.0676

2. Iberian 22.70 3.3320 1.85 0.0663

3. Roman 31.06 8.3532 1.79 0.1607

4. Middle Ages 22.69 5.3441 1.84 0.0949

the sample number in the stratum i(3, 15, 15, 12 for Bronze Age, Iberian, Roman,
and Middle Ages pieces, respectively), and r i is the standard deviation for the
stratum i(estimates of Table 6.4 were applied). The results for the strata i ¼ 1...4
were:

n 1
n ¼ 6 : 85 % ;

n 2
n ¼ 19 : 90 % ;

n 3
n ¼ 44 : 42 % ;

n 4
n ¼ 28 : 83 % for

and open

n 1
n ¼ 6 : 5 % ;

n 2
n ¼ 21 : 01 % ;

n 3
n ¼ 45 : 34 % ; and

n 4
n ¼ 27 : 16 % for

porosity; and

apparent density, respectively.
We specified that the estimate of the sample mean should lie between B units of
the population mean, with probability equal to 0.95. This is equivalent to impose
that the mean estimate should lie in the interval l 2 r, i.e., B ¼ 2 r. From
the analysis of the variable means of Table 6.4 , we chose B ¼ 1 : 1% and
B ¼ 0 : 02 gr/cm 3 as the bounds on the error of estimation of the population mean
for open porosity and apparent density, respectively. These bounds allowed the
stratum mean of the sampling to be separated adequately.

2

P

L

N i r i

The total number of samples was estimated using 0, n ¼

i ¼ 1

; where

N 2 D þ P

L

N i r i

i ¼ 1

D ¼ B 2 = 4 : Thus, we obtained the total number of samples n ¼ 79 and n ¼ 83 for
open porosity and apparent density, respectively. These were the number of pieces
that the morphological and physiochemical characterization analyses were applied
to. Using the estimated fractions n i

n for the strata and the total number of samples n ;
we obtained the sampling population for each stratum. The final results of the
stratified random sampling for an error margin of 0.05 are in Table 6.5 . The
estimate of the population mean for open porosity and apparent density for each
stratum are shown with an approximate 2 standard deviation bound on the error of
estimation.
Table 6.5 shows that the samples of the different strata (chronological periods)
can be clearly separated by open porosity, since the bounds of the distributions
define the most part of the densities to be disjoint. The separation of the samples by
apparent density is more difficult because there is a degree of overlapping between
densities of Roman and Bronze Age pieces, and a higher overlapping between
densities of Iberian and Middle Ages pieces. However the joint densities of these
two collections of pieces are well-separated between them. In conclusion, physical