PE&RS August 2015 - page 662

Proposal of a Statistical Model for Spatial Data Adequate For ISO
2859-1 and ISO 2859-2
In order to apply ISO 2859-1 or ISO 2859-2, we need a mecha-
nism to convert a continuous variable, such as positional
errors (1 D, 2D, 3D, or nD), into a counting variable and, at the
same time, allow for the use of the framework proposed by
this standard. Here, we propose applying the method pre-
sented by Ariza-López and Rodríguez-Avi (2014a), where a
complete statistical formulation and demonstration of the
behavior of this method is presented when working with
commonly assumed parametric models for positional errors
(Normal, Chi2, and Gamma distributions). In Ariza-López and
Rodríguez-Avi (2014b), there are examples of its application
to line strings in 2D and 3D with non-parametric models.
This method is based on two statistical models, a base
model and a binomial model, and has been developed to be
applied to isolated lots. The first is the Base Model (BaM),
which can be any parametric or non-parametric model,
but with the sole condition of adequately representing the
population of errors under control. BaMs can be derived for
any dimension and for any kind of error measure based on
any distance definition. For example, Figure 4 shows the
observed planimetric error distribution of the Base Model for
a road dataset measured by means of the Hausdorff distance.
The BaM plots the cumulative frequency distribution of
positional errors on the Y-axis and the size in meters of such
errors on the X-axis. This BaM comes from the control of the
road dataset of the product, which is called MTA10v (from
“Mapa Topográfico de Andalucía escala 1:10,000 vectorial ”).
The MTA10v is the official cartography of the Autonomous Re-
gion of Andalusia (Spain). It is a topographic-vector database
which includes the geometric axis of all paved roads. The
BaM is derived from a large control of more than 1,200 km of
the road axis included in this product and a GPS field survey.
Table 1 summarizes other main facts of the data sources of the
BaM (see Ariza-López et al., 2011 for more details).
The second model is the Binomial Model (BiM) and is
applied over the former. By means of the BiM the control is
T able 1. Principal Characteristics of the Data S ources Originating from the B ase M odel Presented in Figure 4
Characteristics MTA10v Product subset GPS Field Survey
Total length
Total cases
Mean length
Standard deviation of the length
Total points involved
Mean points per road segment
Mean distance between points
Standard deviation of points distance
Positional accuracy
1,210 Km
1,254 road segments
965 m
1671 m
28,823 points
22.98 points/road segment
41.98 m
28.49 m
10.65 m (95%)
1,210 Km
1,254 road segments
965 m
1671 m
122,467 points
97.66 points/road segment
9.88 m
3.19 m
1.41 m (95%)
Figure 4. Base Model of 2D positional errors measured by Hausdorff distance of a line-string data set representing paved roads of the
“Mapa Topográfico de Andalucía.”
Figure 5. General idea of the basis of the proposed method.
662 August 2015 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING