Chemistry Reference
In-Depth Information
Fig. 4 Ideal parabolic shape for the fractal dimension distribution (RDA-plot) in the presence of
Gaussian noise and absence of model errors. For a discussion of the distributions and the
horizontal
and
vertical red bars
, see the text.
Black square
:
p
1
¼
0.000;
red circles
:
p
1
¼
0.222;
green
triangles
:
p
1
¼
0.444;
blue rhombuses
:
p
1
¼
0.888
Table 1 Residual density descriptors applied to simulated data on S(N
t
Bu)
3
for differing levels of
Gaussian noise as quantified by the noise control parameter
p
1
p
1
d
f
(0)
e
gross
r
0,min
r
0,max
Dr
0
[e
˚
3
]
[e
˚
3
]
[e
˚
3
]
[e]
0.000
2.7956
0.34
0.00
0.00
0.00
0.222
2.7693
8.38
0.12
0.12
0.24
0.444
2.7678
16.41
0.22
0.23
0.45
0.888
2.7647
30.92
0.41
0.46
0.87
Of course, there is an implicit dependency on the data processing, too, which is
ignored for this discussion. For a fixed experimental resolution and residual density
grid, and a perfect model, the baseline is in proportion to the noise. This is exactly
what Fig.
4
shows: the more noise, the broader the distribution. The figure also
shows deviations from the ideal parabolic shape in the periphery. This just
expresses the usual statistical fluctuations, which do not play a role for a very
large number of data points (like in the center at
r
0
¼
0), but become increasingly
dominant with a decreasing number of data points.
The vertical red line, which shows the difference between the actual value of the
zero residual density value
d
f
(0) and the ideal value
d
f
(0)
¼
3.0 depends on the
