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Geometrical correlation indices using homological
constructions on manifolds
Alberto J. Hernández · Maikol Soĺıs ·
Ronald A. Zúñiga-Rojas
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Abstract The course of dimensionality is a common problem in statistics and data
analysis. Variable sensitivity analysis methods are a well studied and established
set of tools designed to overcome these sorts of problems. However, as this work
shows, these methods fail to capture relevant features and patterns hidden within
the geometry of the enveloping manifold projected into a variable. We propose an
index that captures, reflects and correlates the relevance of distinct variables within
a model by focusing on the geometry of their projections. The analysis was made
with an original R-package called TopSA, short for Topological Sensitivity Anal-
ysis. The TopSA R-package is available on the site https://github.com/maikol-
solis/TopSA.
Keywords Betti numbers · Data analysis · Homology · Topological manifolds ·
Sensitivity Analysis · Simplexes
Mathematics Subject Classification (2010) 49Q12 · 52C35 · 57T30 · 55N35
1 Introduction
Let (X1, X2, . . . Xp) ∈ Rp for p > 1 and Y ∈ R two random variables. Define the
non-linear regression model as
Y = m(X1, X2, . . . Xp) + ε. (1)
Here ε is a random noise independent of (X1, X2, . . . Xp). The unknown function
m : Rp 7→ R describes the conditional expectation of Y given (X1, X2, . . . Xp).
Alberto J. Hernández
Centro de Matemática Pura y Aplicada (CIMPA), Escuela de Matemática, Universidad de
Costa Rica, San José, Costa Rica.
Maikol Soĺıs
Centro de Matemática Pura y Aplicada (CIMPA), Escuela de Matemática, Universidad de
Costa Rica, San José, Costa Rica. Corresponding author: E-mail: maikol.solis@ucr.ac.cr
Ronald A. Zúñiga-Rojas
Centro de Investigaciones Matemáticas y Meta-Matemáticas (CIMM), Escuela de Matemática,
Universidad de Costa Rica, San José, Costa Rica.
2 Alberto J. Hernández et al.
Suppose as well that (Xi1, Xi2 . . . Xip, Yi) for i = 1, . . . , n is a size n sample for the
random vector(X1, X2 . . . Xp, Y ).
If p n, the model (1) suffers from the “curse of dimensionality”, a term intro-
duced by Bellman (1957) and Bellman (1961), where he showed that the sample
size n required to fit a model increases with the number of variables p. In a sta-
tistical context, the model selection techniques solve the problem using indicators
as the AIC or BIC or more advanced techniques as Ridge or Lasso regression. The
interested reader can find a comprehensive survey of these methodologies in Hastie
et al. (2009).
Professionals on computational modelling deal with these problems all the
time; the choosing of relevant variables is a recurring task for them. Another
way to approach this problem is through the variable importance measures or
indices. These indices rely on indicators to find the relevance or sensitivity of the
variables in a model. The works of Saltelli et al. (2002), Saltelli et al. (2007), Saltelli
et al. (2009) and Wei et al. (2015) compile different approaches to estimate those
indicators. They present techniques based on differences methods; parametric and
non-parametric settings, variance-based measures, moment independent measures,
and graphical techniques.
These techniques overlook the geometric arrangement of the data to build the
complete relation between X and Y and then present that information in the
form of an indicator. Depending on this simplification they do not consider the
geometric properties of the data. For example, most indices will fail to recognize
structure when the input variable is of zero-sum, treating it as random noise.
The analysis of topological data is a recent field of research that aims to over-
come these shortcomings. Given a set of points generated in space, it tries to
reconstruct the model through an embedded manifold that covers the data set.
With this manifold, we can study the characteristics of the model using topologi-
cal and geometrical tools instead of using classic statistical tools.
Two classic tools used to discover the intrinsic geometry of the data are the
Principal Components Analysis (PCA) and the Multidimensional Scaling (MDS).
The PCA transforms the data in a smaller linear space preserving the statistical
variance. The other approach, the MDS, performs the same task but preserving
the distances between points. Recent methods like the isomap algorithm developed
by Tenenbaum (2000) and expanded by Bernstein et al. (2000); Balasubramanian
(2002) unify these two concepts to allow the reconstruction of a low-dimensional
variety for non-linear functions. Using the geodesic distance it identifies the cor-
responding manifolds and search lower dimension spaces to project it.
In recent years, new theoretical developments use tools such as persistent ho-
mology, simplicial complexes, and Betti numbers to reconstruct manifolds, the
reconstruction works for clouds of random data and functional data, see Ghrist
(2008), Carlsson (2009, 2014). In Gallón et al. (2013) and in Dimeglio et al. (2014)
some examples are presented. This approach allows handling “ Big Data” quickly
and efficiently, for example, see Snášel et al. (2017).
In this work, we aim to connect the concepts of sensitivity analysis with the
analysis of topological data through a geometrical correlation index. By doing this,
it will be possible to create relevance indicators for statistical models using the
geometric information extracted from the data.
The outline of this paper is: Section 2 deals with basic notions, both in sensi-
tivity analysis and in topology. In Subsection 2.1 some of the most used and well-
Geometrical correlation indices using homological constructions on manifolds 3
known statistics methods are reviewed and commented. We finish this subsection
with an example that motivates this work. Subsection 2.2 deals with preliminaries
in homology. Section 3 explains the method used to create our sensitivity index;
Subsection 3.1 describes the construction of the neighborhood graph, and deals
with different topics such as the importance of scale, the Ishigami Model and presents
a strip of code to determine the radius of proximity. Subsection 3.2 describes the
algorithm used to construct the homology complex through the Vietoris-Rips com-
plex and Subsection 3.3 explains our proposed sensitivity index. Section 4 contains
a description of our results, it describes the software and packages used to run our
theoretical examples. Subsection 4.1 is a full description of each theoretical exam-
ple together with a visual aid, such as graphics and tables describing the results.
Subsection (4.2) is an application of our algorithm to a well known hydrology
model. Finally, Section 5 contains our conclusions and explores scenarios for fu-
ture research.
2 Preliminary Aspects
2.1 Sensitivity Analysis
According to Saltelli et al. (2009), the sensitivity analysis (SA) is “the study of how
uncertainty in the output of a model can be apportioned to different sources of uncer-
tainty in the model input”. The process of validation could be seen as an optional
step. But, a complete analysis requires it to find hidden patterns and uncertainties
in the data and to identify relevant factors and validating simplifications of the
problem.
We could classify sensitivity analysis methods into one-at-the-time and global
sensitivity analysis. In the one-at-the-time methods, one variable is sampled at a
time while keeping the rest constant. The methods reveal the pattern of the model,
however; they work if the problem is linear. Otherwise, features of the data can
be missed and misleading conclusions can arise. The correlations and regression
sensitivity analysis share this problematic behavior. We refer the reader to Saltelli
et al. (2007) for a further review on the flaws and shortcomings of linear sensitivity
analysis models.
The standarized regression coefficients are a basic tool to detect linear sensitiv-
ities between the inputs and the outputs. It relies in the multiple linear regression,
Y = β0 + β1X1 + · · ·+ βpXp + ε.
Set X = [1, X1, . . . , Xp] where 1 is a vector of ones and the variables Xi’s are
disposed as columns. We know that the least-square estimator for the β’s are
> −1
(β̂0, . . . , β̂p) = (X
>X) X>Y
>
Y = X(β̂0, . . . , β̂p) . (2)
The regression coefficients β̂i (i = 1, . . . , p) measure how much sensitive are the
Xi’s with respect Y . However, they fail in describe the relative importance between
4 Alberto J. Hernández et al.
the variables because their values are still in the units of the inputs variables. To
solve this, we can rewrite the fitted re(gressio)n model 2 as
Ŷ − ∑nȲ β̂iŝi (Xi − X̄ )= i ,
ŝ ŝ ŝi
i=1
where √√√∑ √n √√∑n
ŝ = √ (Yi − Ȳ )/(n− 1), ŝi = √ (Xij − X̄j)/(n− 1)
i=1 i=1
are the standard deviations for the output samples and the input samples respec-
tively. The values ρSRCi = β̂iŝi/ŝ are called the standard regression coefficients
(SRC) of Xi with respect Y . It reflects the sensitiviy of Xi removing any units in
the data. So the larger |ρSRCi | the most influential is Xi to the model.
To discover the nonlinear patterns of the data one could use other techniques.
Some popular and well-documented ones are screening, Morris method, global
variance or the moment independent measures. For instance, the Morris method
creates an approximation of the variation over the variable we want to study.
This method can detect no linearity, but due to its nature assumes monotonicity.
This characteristic is not always present in complex models Hamby (1994). The
higher-order effects are difficult to estimate due to their demanding computational
requirements Saltelli et al. (2007).
One popular method to asses relevant variables is the variance-based global
sensitivity analysis. The method was proposed by Sobol’ (1993) based on the
ANOVA decomposition. He proved that if f is an squared integrable function,
then he decomposed it in the unit∑ary cube∑as:
Y = f = f0 + fi + fij + · · ·+ f12···p (3)
i ij
i=6 j
where each term is also square integrable over the domain. Also, each function is
defined by fi = fi(Xi), fij = fij(Xi, Xj) and so on. This decomposition has 2
p
terms and the first one, f0, is constant. The remaining are non-constant functions.
Sobol also proved that this representation is unique if each term has zero mean
and the functions are pairwise orthogonal. Equation (3) could have been seen as
the decomposition of the output variable Y into its effects due to the interaction of
none, one or multiple variables. Taking expectation in Equation (3) and simplifying
the expression we get:
f0 = E[Y ]
fi(Xi) = E[Y |Xi]− E[Y ]
fij(Xi, Xj) = E[Y |Xi, Xj ]− fi − fj − f0
and so on for all combinations of variables.
Once with this orthogonal decomposition, we measure the variance of each el-
ement. The global-variance method estimates the regression curves (surfaces) for
each dimension (or multiple dimensions) removing the effects due to variables in
Geometrical correlation indices using homological constructions on manifolds 5
lower dimensions. Then, it gauges the variance of each curve (surface) normal-
ized by the total variance in the model. For the first and second-order effects the
formulas remain as:
Cov(f (X ), Y ) Var(E[Y |X ])
Si =
i i = i (4)
Var(Y ) Var(Y )
Cov(fij(Xi, Xj), Y ) Var(E[Y |Xi, Xj ])
Sij = = − S − S . (5)Var(Y ) Var( i jY )
and so on for the higher order terms.
To quantify how much we could apportion of each variable to the whole model,
the total effect is estimated by
− Var(E[Y |X ])ST = 1 ∼ii Var(Y )
The moment independent indices estimates relevant inputs in other way. These
estimate the average of the distance between the random variable Y and Y |Xi = x
for any x ∈ R. The initial ideas come from Borgonovo et al. (2014) which used
monotonic invariant transformations to gauge the distance between two probability
measures. Therefore, if the variable Xi is irrelevant or independent to the model,
the random variables Y and Y |Xi = x will be almost identical and the impact
will be small. Otherwise the impact will be significant. The following are popular
measures using this technique:
– Kolmogorov-Smirnov∫: E[sup |FY (y)− FY |X (y)|],iy∈R
1
– Radon-Nykodym: ∫ |fY (y)−(fY |X (y)|dy,2 i )
1 f (y)
– Kullback-Leiber: fY (y) log
Y dy
2 fY |X (y)i
where FY and FY |X represent the distribution function for Y and Y |Xi re-i
spectively. The density functions are presented as fY and fY |X .i
All those methods try to compact all the features of the space in a single
measure and then average over the set of values into the sample.
However, the sample could have particularities not detected by the method.
For example, Figure 1 bellow presents a toy example where the data points are
arranged into a circle with a hole in the middle (further details in Section 4). The
first variable presents all the information about the model while the second one is
a noisy arrangement.
We estimated the SRC for both cases. Here, both results agree that the vari-
ables X1 and X2 are irrelevant to the model. In this paper we propose an al-
gorithm that handles this examples and notices such geometric and topological
details within the data.
2.2 Homology
In this subsection we recall some of the topological definitions needed to under-
stand the frame of our work.
6 Alberto J. Hernández et al.
X1 X2
1.0
0.5
0.0
−0.5
−1.0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
Independent Variables
X1 X2
ρSRC 0.00 -0.01
Fig. 1: Estimation of correlation index for the example of a Circle with a hole (see
Section 4.1. In this case both variables are irrelevant to the model.
n—topological manifold is a Hausdorff space M with a countable basis such
that each point x ∈ M has an open neighborhood that is homeomorphic with an
open subset of Rn.
We recall here that Hausdorff means that any two distinct points lie in disjoint
open subsets. A countable basis means countable family of open subsets such that
each open subset is the union of a subfamily. The reader may see Barden and
Thomas (2003) for details.
In terms of differential manifolds, a chart on a topological manifold M is a pair
(U, φ) where U ⊆M is an open subset and φ : M → Rn is a homeomorphism that
sends U to an open subset V = φ(U) ⊆ Rn. The set U ⊆M is so-called a coordinate
neighborhood and the set V ⊆ Rn is so-called its coordinate space. An atlas for a
topological manifold M is a collection of charts A = {(Ui, φi)|i ∈ I} such that the
coordinate neighborhoods of A cove⋃r the whole manifold M :
Ui = M.
i∈I
To avoid ambiguity, we introduce the coordinate transformations: if two dis-
tinct coordinate neighborhoods are not disjoint then, we can define the following
functions from the images of the overlaping open subset:
θ12 = φ ◦ φ−11 2 : φ2(U1 ∩ U2)→ φ1(U1 ∩ U2)
and
θ −121 = φ2 ◦ φ1 : φ1(U1 ∩ U2)→ φ2(U1 ∩ U2)
From Barden & Thomas Barden and Thomas (2003) An n—differential man-
ifold is a topological manifold M of dimension n together with a maximal smooth
atlas on it. The reader may consult Lang (1963) for differential manifolds details.
Dependent Variable
Geometrical correlation indices using homological constructions on manifolds 7
To define a simplex, let consider the unit interval I = [0, 1] and two topological
spaces X and Y . Consider the triple product X×Y × I and the quotient (X×Y ×
I)/∼ where the relation ∼ is given by the identifications
(x, y0, 0) ∼ (x, y1, 0) and (x0, y, 1) ∼ (x1, y, 1).
In other words, we are collapsing the subspaces X×Y ×{0} and X×Y ×{1} to
X and Y respectively. To visualize it, take for instance X = Y = I two real closed
intervals. Hence, we are collapsing two opposite faces of a cube onto line segments
so that, the cube becomes a tetrahedron:
X
Y
I
According to Hatcher (2000) (may see also Munkres (1984)), a 4-complex is a
generalization of a simplex. Recall that 0-simplices are points, called vertices. As
well as 0-simplices, we have line segments as 1-simplices, called edges, 2-simplices
called faces, 3-simplices called tetrahedron, and a generalization of dimension n
will be a convex set in Rm{ co}ntaining {v0, v1, . . . , vn} a subset of n + 1 distinctpoints that do not lie in the same hyper plane of dimension n or, equivalently,
that the vectors wj = vj − v0 are linearly independent. In such a case, we are
denoting the points {v0, v1, . . . , vn} as vertices, and the usual notation would be
[v0, v1] for edges, [v0, v1, v2] for faces, [v0, . . . , v4] for tetrahedron, and [v0, . . . , vn]
for n-simplicies.
A 4-complex is a qoutient topological space of a collection of disjoint simplices
identifying some of their faces via certain homeomorphisms {α}α∈A that preserve
the order of the vertices. Those homeomorphisms are linear, and give us a better
notation for n-simplices: enα.
Now, we may define the simplicial homology groups of a 4-complex X as
follows. Let consider the free abelian group 4n(X) with open n-simplices enα ⊆ X
as basis elements. It means that the elements of this group 4n(X), known as
chains, look like linear combinations of∑the form
c = n nαeα (6)
α
with integer coeficients nα ∈ Z. We als∑o could wirte chains as linear combinationsof characteristic maps
c = nασα (7)
α
where every σ : 4nα → X is the corresponding characteristic map of each enα, with
image the closure of enα. So, c ∈ 4n(X) is a finite collection of n-simplices in X
with integer multiplicities nα. The boundary of the n-simplex [v0, . . . , vn] consists
8 Alberto J. Hernández et al.
of the various (n−1)-simplices [v0, . . . , v̂j , . . . , vn] = [v0, . . . , vj−1, vj+1, . . . , vn]. For
chains, the boundary of c = [v∑0, . . . , vn] is an oriented (n− 1)-chain of the formn
= (−1)j∂c [v0, . . . , v̂j , . . . , vn] (8)
j=0
which is a linear combination of faces. This allows us to define the boundary ho-
momorphisms for a general 4-complex X, ∂n : 4n (X) → 4n−1(X) as follows:
∑n
j
∂n(σα) = (−1) σα|[v0, . . . , v̂j , . . . , vn]. (9)
j=0
Hence, we get a sequence of homomorphisms of abelian groups
∂
· · · → 4 ∂n(X) −−→n 4n−1(X) −−
n−−→1 4n−2(X)→ · · · → 4 −
∂1 ∂0
1(X) →40(X) −→ 0
where ∂ ◦ ∂ = ∂n ◦ ∂n−1 = 0 for all n. This is usually known as a chain complex.
Since ∂n ◦ ∂n−1 = 0, the Im(∂n) ⊆ ker(∂n−1), and so, we define the nth simplicial
homology group of X as the quotient
4 ker(∂n)hn (X) = . (10)Im(∂n+1)
The elements of the kernel are known as cycles and the elements of the image are
known as boundaries.
Easy computations of sequences give us the simplicial homology of some ex-
amples: the circle X = S1:
H40 (S
1) 4 1 4 1=∼ Z, H (S ) =∼1 Z, H (S ) =∼n 0 for n > 2,
and the torus X = T =∼ S1 × S1:
H40 (T ) =
∼ Z, H4(T ) =∼1 Z⊕ Z, H
4
2 (T )
4
=∼ Z, Hn (T ) =∼ 0 for n > 3.
In a very natural way, one can extend this process to define singular homol-
ogy groups Hn(X). This process, nevertheless, is not trivial but natural. If X is
a 4-complex with finitely many n-simplices, then Hn(X) (and of course H4n (X))
is finitely generated. The nth-Betti number of X is the number of summands iso-
morphic to the additive group Z. The reader may see Hatcher (2000) or Munkres
(1984) for details.
Definition 1 (VR neigboorhood graph) Given S ⊆ Rp and scale ε ∈ R, the VR
neighborhood graph is a graph where Gε(V ) = (V,Eε(V )) and
Eε(V ) = {{u, v} | d(u, v) 6 ε, u 6= v ∈ V }.
Definition 2 (VR expansion) Given a neigborhood graph G, their Vietoris-Rips
complex V(G) is defined as all the edges of a simplex σ that are in G. In this case
σ belongs to V(G). For G = (V,E), we h{ave ( ) }
V ∪ ∪ | σ(G) = V E σ ⊆ E .
2
where σ is a simplex of G.
Geometrical correlation indices using homological constructions on manifolds 9
3 Methodology
Recall the model (1). Here the random variables (X1, . . . , Xp) are distorted by the
function m and its topology. Our aim is to measure how much each of the Xi
influenced this distortion, i.e, we want to determine which variables influence the
model the most.
In Section 2.1 we reviewed the most common statistical indices to estimate
the sensitivity or correlation of Xi for i = 1, . . . , p with respect to Y . In our case,
we want to consider the geometry of the point-cloud, its enveloping manifold and
create an index that will reveal information about the model.
The first step is to create a neighborhood graph for the point-cloud formed
by (Xi, Y ) where an edge is set if a pair of nodes are within an ε of euclidean
distance. In this way, we connect only the nodes nearby by a fixed distance. With
this neighborhood graph, we construct the persistent homology using the method
of Zomorodian (2010) for the Vietoris-Rips (VR) complex.
The algorithm of Zomorodian (2010) works in two-phases: First it creates a VR
neighborhood graph (Definition 1) and then builds, step by step, the VR complex
(Definition 2).
These definitions state the procedure to construct the two-phase scheme for
the Vietoris-Rips complex with resolution ε:
1. Using Definition 1 compute the neighborhood graph Gε(V ) with parameter ε.
2. Using Definition 2 compute V(Gε(V )). From now on set Vε(V ) = V(Gε(V )).
This scheme provides us with a geometrical skeleton for the data cloud points
with resolution ε. If ε is large there will be more edges connecting points. Here,
we could have a large interconnected graph with little information. Otherwise, if ε
is small, there would be fewer edges connecting points, resulting in a sparse graph
and missing relevant topological features within the data cloud.
In the second step, we unveil the topological structure of the neighborhood
graph through the Vietoris-Rips complex. The expansion builds the cofaces related
to our simplicial complex. In Section 3.2 we will discuss more about the algorithm
used for this purpose.
3.1 Neigborhood graph
The neighborhood graph collects the vertices V , and for each vertex v ∈ V it adds
all the edges v − −u within the set u ∈ V , satisfying d(v, u) 6 ε. This brute-force
operation works in O(n2) time.After considering a variety of theoretical examples
it becomes clear that the scale factor in the data set is relevant. The scale in one
variable may differ with the scale of the output by many orders of magnitude.
Thus, proximity is relative to the scale of the axis in which the information is
presented and the proximity neighborhood could be misjudged.
The Ishigami model, presented in Figure 2 bellow shows how the proximity
neigborhood becomes distorted when scale is taken into consideration.
We conclude that the aspect ratio between both variables define how the al-
gorithm construct the neighborhood graph. Therefore, to use circles to build the
neighborhood graph, we need to set both variables into the same scale. The Algo-
rithm 1 constructs the VR-neigborhood graph for a cloud of points with arbitrary
scales.
10 Alberto J. Hernández et al.
Rescalated to [0,1] True Scale
1.00 15
10
0.75
5
0.50
0
0.25
−5
0.00
−10
0.00 0.25 0.50 0.75 1.00 −2 0 2
Range of X2 = 6.28
Fig. 2: The Second variable of the Ishigami model escalated to [0, 1] with a circle
centered in (0.5, 0.5) and radius 1 (left). The same circle draw in the true scale of
the data (right).
Data: A set of points X = (X1, . . . , Xn) and Y = (Y1, . . . , Yn)
A value 0 < α < 1.
Result: The Neigborhood Graph.
1 Function Create-VR-Neigborhood((X,Y ), α):
2 Xr ←
X −minX
maxX −minX
← Y −minY3 Yr
maxY −minY
4 n← length(Y )
5 DistanceMatrix ← Matrix(n× n)
6 for i = 1:n do
7 for j = 1:n do √
8 DistanceMatrix[i,j] ← (Xr[i]−Xr[j])2 + (Yr[i]− Yr[j])2
9 end
10 end
11 ε ← Quantile (DistanceMatrix, α)
12 AdjacencyMatrix ← DistanceMatrix 6 ε
13 NeigborhoodGraph ← CreateGraph (AdjacencyMatrix, xCoordinates =
X, y Coordinates = Y )
14 return (NeigborhoodGraph)
15 end
Algorithm 1: Procedure to estimate the neigborhood graph from a set of
points in the plane.
Range of Y = 24.87
Geometrical correlation indices using homological constructions on manifolds 11
1. Rescale the points (Xi, Y ) i = 1, . . . , n onto the square [0, 1]× [0, 1].
2. Estimate the distance matrix between points.
3. With the distance chart, estimate the α quantile of the distances. Declare the
radius εi as this quantile.
4. Using Definition 1, build the VR-neigborhood graph with ε changed by εi for
each projection.
5. Rescale the data points to their original scale.
3.2 VR expansion
In his work Zomorodian (2010), Zomorodian describes three methods to build the
Vietoris-Rips complex. The first approach builds the complex adding the vertices,
the edges and then increasing the dimension to create triangles, tetrahedrons, etc.
The second method starts from an empty complex, and adds, step by step, all the
simplices stopping in the desired dimension. In the third one, one takes advantage
from the fact that the VR-complex is the combinations of cliques in the graph in
the desired dimension.
Due to its simplicity, we adopt the third approach and detect the cliques in
the graph. We use the algorithm in Eppstein et al. (2010) which is a variant of
the classic algorithm from Bron and Kerbosch (1973). This algorithm orders the
graph G and then computes the cliques using the Bron-Kerbosch method without
pivoting. This procedure reduced the worst-case scenario from time O(3n/3) to
O(dn3d/3) where n is the number of vertices and d is the smallest value such that
every nonempty subgraph of G contains a vertex of degree at most d.
Constructing the manifold via the VR-complex is not efficient in the sense that
the co-faces may overlap, increasing the computational time required., One can
overcome this issue by creating an ordered neighborhood graph.
3.3 Geometrical correlation construction
One of the main objectives in a sensitivity analysis is to discover patterns and
relationships between the inputs against the output and determine which of those
patterns are more influential for the model in consideration. For our case, the
patterns are described through empty spaces in the projection space generated by
each individual variable. If the point-cloud fills all the domain then the unknown
function m applied to Xi produces erratic values of Y. Otherwise, the function
sheds an structural pattern which could be recognized geometrically.
The VR-complex V(G) estimates the geometric structure of the data by filling
the voids between close points. Then, we estimate the area of the created object.
This number will not give much information about the influence that the variable
has within the model. We estimate the area of the minimum rectangle containing
all the object. If some input variable presents a weak correlation with the output
variable, its behavior will be almost random with uniform distributed points into
the rectangular box. In other case, if it has some relevant correlation, it will create
a pattern causing empty spaces to appear across the box.
To clarify the notation we will denote as Gε,i the neighborhood graph generated
by the pair of variables (Xi, Y ) and the radius ε. Denote as V(Gε,i) the VR-complex
12 Alberto J. Hernández et al.
generated by Gε,i. We also denote as Area(V(Gε,i)) as the geometrical area of the
object formed by the VR-complex V(Gε,i).
We define[ the rectangular box for the]pro[jection the data (Xi, Y ) as ]
Bi = min(V(Gε,i)),max(V(Gε,i)) × min(V(Gε,i)),max(V(Gε,i)) .
Xi Xi Y Y
The geometrical area of Bi will be denote as Area(Bi).
Therefore, we can create the measure,
Geom Area(V(G− ε,i
))
ρi = 1 .Area(Bi)
Notice that if the areas of the object and the box are similar, then the index
ρAreai is close to zero. Otherwise, if there is a lot of empty spaces and both areas
differ and the index would approach 1.
4 Results
To asses the quality of the estimated index described before, we performed nu-
merical examples. The software used was R (R Core Team (2017)), along with
the packages igraph (Csárdi and Nepusz (2006)) for all the graph manipulations,
and the packages rgeos and sp (Bivand and Rundel (2018); Pebesma and Bivand
(2005); Bivand et al. (2013)) for all the geometric estimations. A package contain-
ing all these algorithms will be available soon in CRAN.
For all the settings, we sample n = 1000 points with the distribution specified
in each case. Due to the number of points, we choose the quantile 5% for each
case to determine the radius of the neigborhood graph. Further insights about
this choosing will be presented in the conclusions section
We will consider five settings, each one with different topological features. The
cases are not exhaustive and there are other settings with interesting features
as well. However, through this sample we show how the method captures the
geometrical correlation of the variables where other classic methods have failed,
as well as a case for which our method fails to retrieve the desired information.
4.1 Theoretical examples
The examples considered are the following:
Linear This is a simple setting with
Y = 2X1 +X2
and X3, X4 and X5 independent random variables. We set Xi ∼ Uniform(−1, 1)
for i = 1, . . . , 5.
Quartic: This is another simple scheme with
Y = X1 +X
4
2
with Xi ∼ Uniform(−1, 1) for i = 1, 2.
Geometrical correlation indices using homological constructions on manifolds 13
Circle with hole: The model in th{is case is
X1 = r cos(θ)
Y = r sin(θ)
with θ ∼ Uniform(0, 2π) and r ∼ Uniform(0.5, 1). This form creates circle with
with a hole in the middle.
Connected circles with holes: The model is set in two sections, where in both parts
we set θ ∼ Uniform(0, 2π):
1. Circle centered in (0, 0) with r{adius between 1 and 2:
X1 = r1 cos(θ)
Y = r1 sin(θ)
where r1 ∼ Uniform(1, 2).
2. Circle centered in (1.5, 1.5){with radius between 0.5 and 1:
X1 − 1.5 = r2 cos(θ)
Y − 1.5 = r2 sin(θ)
where r2 ∼ Uniform(0.5, 1).
Ishigami: The final model is
Y = sinX1 + 7 sin
2X 42 + 0.1 X3 sinX1
where Xi ∼ Uniform(−π, π) for i = 1, 2, 3, a = 7 and b = 0.1.
4.1.1 Numerical results
The figures presented in this section represent the estimated manifold for each
input variable Xi with respect to the output variable Y . The lower table presents
the radius used to build the neighborhood graph, the estimated areas for the
manifold object, the reference square and the proposed index.
The linear model in Figure 3 is simple enough to allow us to declare the variable
X1 has the double of relevance compared to variable X2. The other variables will
have less relevant indices. As we expected, the index for X1 almost doubles the
counterpart for X2. The examples show us how the empty spaces are present
according to the relevance level of the variable.
For the Quartic model in Figure 4, we could estimate the theoretical Sobol
indices according to Equation (4). The indices are S1 = 0.82, S2 = 0.18 and
S3 = 0. We observe in Figure 4 how the Sobol indices match with our algorithm,
ranking the variables in order of relevance as X1, X2 and X3.
The model of Circle with a hole in Figure 5 was discussed in the preliminaries.
Recall that in this case, both variables were irrelevant to the model, even if the
geometric shape showed the contrary. Figure 5 present the results. Observe how
the first variable has index equal to 0.48 and the second one with 0.06. These
14 Alberto J. Hernández et al.
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−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
X1 X2 X3
Variable ε Area(V(G)) Area(B) ρGeom ρSRC
X1 0.08 3.79 11.38 0.67 0.90
X2 0.11 7.69 11.39 0.32 0.46
X3 0.12 9.78 11.39 0.14 0.00
X4 0.12 10.18 11.38 0.10 0.00
X5 0.12 10.16 11.39 0.11 0.00
Fig. 3: Results for the Linear case.
indices allow us to say the X1 has an impact into the model much more relevant
than X2.
To test our algorithm further, we present the model of Connected circles with
holes in Figure 6. Here we created two circles with different scales and positions.
Even if the choice is an arguable point for the method, we could capture the most
relevant features for each projection. Again, it is possible to rank the variables as
X1 first with index equal to 0.58 and X2 second with index 0.08.
The final model is produced by the Ishigami function, Figure 7. This is a
popular model in sensitivity analysis because presents a strong nonlinearity and
non-monotonicity with interactions in X3. With other sensitivity estimators, the
variables X1 and X2 have great relevance to the model, while the third one X3
has almost zero. For a further explanation of this function, we refer the reader
to Sobol’ and Levitan (1999). In our case, all the variables result into a index near
to 0.5. This is relevant because our index has captured the presence of structure,
as well as with the model Circle with hole whereas other tools treat them as noise.
Y
−3 −1 1 2 3
Y
−3 −1 1 2 3
Y
−3 −1 1 2 3
Geometrical correlation indices using homological constructions on manifolds 15
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X1 X2
Variable ε Area(V(G)) Area(B) ρGeom ρSRC
X1 0.06 1.50 5.88 0.75 0.91
X2 0.11 3.83 5.89 0.35 0.01
Fig. 4: Results for the Quartic case.
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−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
X1 X2
Variable ε Area(V(G)) Area(B) ρGeom ρSRC
X1 0.11 2.10 3.92 0.46 0.00
X2 0.13 3.67 3.94 0.07 0.01
Fig. 5: Results for the Circle with 1 hole case.
Y Y
−1.0 −0.5 0.0 0.5 1.0 −1.0 0.0 1.0 2.0
Y Y
−1.0 −0.5 0.0 0.5 1.0 −1.0 0.0 1.0 2.0
16 Alberto J. Hernández et al.
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−2 −1 0 1 2 −1.0 −0.5 0.0 0.5 1.0
X1 X2
Variable ε Area(V(G)) Area(B) ρGeom ρSRC
X1 0.08 9.44 20.07 0.53 0.36
X2 0.12 8.60 8.96 0.04 0.00
Fig. 6: Results for the Circle with 2 holes case.
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−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
X1 X2 X3
Variable ε Area(V(G)) Area(B) ρGeom ρSRC
X1 0.08 64.83 158.65 0.59 0.44
X2 0.07 56.78 152.50 0.63 0.02
X3 0.09 70.96 152.59 0.53 0.01
Fig. 7: Results for the Ishigami case.
Y
−10 −5 0 5 10 15 Y
−2 −1 0 1 2
Y
−10 −5 0 5 10 15
Y
−2 −1 0 1 2
Y
−10 −5 0 5 10 15
Geometrical correlation indices using homological constructions on manifolds 17
4.2 Application: A Hydrology model
One academic real case model to test the performance in sensitivity analysis is
the dyke model. This model simplifies the 1D hydro-dynamical equations of Saint
Venant under the assumptions of uniform and constant flow rate and large rect-
angular sections.
The following equations recreate the variable S which measures the maximal
annual overflow of the river (in meters) and the variable Cp which is the associated
cost (in millions of euros) of the dyke.
S = Zv +H −Hd − Cb (11)
with  √QH = 
[ BK ( Zm−Zvs L )]
−1000
Cp = 1S>0 + 0.2 + 0.8 1− exp 1S4 S60
1
+ (Hd1H >8 + 81H 68) (12)20 d d
Table 1 shows the inputs (p = 8). Here 1A(x) is equal to 1 for x ∈ A and 0
otherwise. The variable Hd in Equation (11) is a design parameter for the Dyke’s
height set as a Uniform(7, 9).
In Equation (4.2), the first term mean a cost of 1 million euros due to a flooding
(S > 0). The second term corresponds to the cost of the dyke maintenance (S 6 0)
and the third term is the construction cost related to the dyke. The latter cost
is constant for a height of dyke less than 8m and is growing like the dyke height
otherwise.
Input Description Unit Probability Distribution
Q Maximal annual flowrate m3/s Gumbel(1013, 558) truncated on [500, 3000]
Ks Strickler coefficient — N (30, 8) truncated on [15,∞)
Zv River downstream level m Triangular(49, 50, 51)
Zm River upstream level m Triangular(54, 55, 56)
Hd Dyke height m Uniform(7, 9)
Cb Bank level m Triangular(55, 55.5, 56)
L Length of the river stretch m Triangular(4990, 5000, 5010)
B River width m Triangular(295, 300, 305)
Table 1: Input variables and their probability distributions.
For a complete discussion about the model, parameters definition and meaning
the reader can review (Iooss and Lemâıtre, 2015), (de Rocquigny, 2006) and their
references.
The work of Iooss and Lemâıtre (2015) detects the most influential variables for
models (4.2) and (11). They use a combination of a Morris screening method, the
18 Alberto J. Hernández et al.
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Zv Zm
Variable ε Area(V(G)) Area(B) ρGeom ρSRC
Q 0.07 96.81 311.84 0.69 0.63
Ks 0.07 1.58 4.39 0.64 0.38
Zv 0.08 0.09 0.23 0.61 0.42
Zm 0.08 0.10 0.21 0.53 0.12
Hd 0.09 0.06 0.13 0.55 0.27
Cb 0.08 0.05 0.09 0.51 0.20
L 0.08 1.13 2.26 0.50 0.01
B 0.08 0.45 1.06 0.57 0.03
Fig. 8: Results for the Dyke Cp case.
standarized regression coefficient and sobol indices, The variables more correlated
and influential to the outputs Cp and S are: Q, Hd, Zv, Ks and Cb.
Figures 8 and 9 present the results of this model for the variables Q, Ks, Zv
and Zm. For the output Cp, the variables the three first variables present a clearer
geometric pattern than the other inputs. In the case of the output S we recover as
the most correlated variables as Q and Zv. The other ones has not a clever pattern
in which we could discriminate their influence.
Cp Cp
0.64 0.68 0.72 0.64 0.70 0.76
Cp Cp
0.64 0.68 0.72 0.64 0.68 0.72 0.76
Geometrical correlation indices using homological constructions on manifolds 19
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Variable ε Area(V(G)) Area(B) ρGeom ρSRC
Q 0.08 4289.10 11554.51 0.63 0.65
Ks 0.09 80.24 163.61 0.51 0.41
Zv 0.09 3.89 9.98 0.61 0.52
Zm 0.09 4.77 9.46 0.50 0.08
Hd 0.11 3.01 5.13 0.41 0.32
Cb 0.09 2.35 4.71 0.50 0.22
L 0.09 46.66 83.16 0.44 0.01
B 0.09 23.11 45.34 0.49 0.03
Fig. 9: Results for the Dyke S case.
Also, the variable B has a higher value of ρGeom even if it is not correlated
using the classic models. Figure 10 presents a zoom for this case. The reason comes
from the bounding box which enclosed the manifold. The isolated points near to
the box frontier creates 2-simplexes (triangles) far away of the concentrated data.
Therefore, those simplexes rises artificially the area of the bounding box B. There
are still improvements to be researched in order to create a robust version of the
algorithm.
S S
−13 −11 −9 −12 −10 −8
S S
−12 −10 −8 −12 −10 −8
20 Alberto J. Hernández et al.
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296 298 300 302 304
B
Fig. 10: Scaling issue with irrelevant variables. The VR-complex spreadout causes
bounding boxes larger than the geometrical structure of the estimated manifold.
5 Conclusions and further research
As it was mentioned above, the aim of this paper was to build a sensitivity index
that relied solely on the topological and geometrical features of a given data-
cloud, since it is clear that purely analytic or statistic methods fail to recognize
the structure within the projection of certain variables, primarily when the input
is of zero sum, that is, what might be considered artificial noise. In such cases,
those projections, or the variables in question, have positive conditional variance
that might contribute to the model in ways that hadn’t been explored so far.
Our index proved to be reliable in detecting this conditional variance when
the variable is of zero sum, differentiating between pure random noise and well
structured inputs -even of zero sum-. In the cases where the model presents pure
noise our index coincides fully with other method’s indexes in detecting relevant
structured inputs, in the other cases our index reflects the presence of structure
in all the variables, which was the case we wanted to explore.
As of sensitivity, we can not fully declare that our index measures it accurately,
at least for the moment, and in a conventional way. To achieve the confidence for
such a claim, we have identified a series of research problems to be dealt with
in near future, namely: Improving the algorithm for the construction of the base
graph, or change it completely to make the process more efficient and hence allow
ourselves to run more sophisticated examples, both theoretical as well as real data
examples from well known models. One of the central points to be discussed and
Cp
0.64 0.68 0.72
Geometrical correlation indices using homological constructions on manifolds 21
studied further is the determination of the radius of proximity, which we believe
must be given by the data set itself, probably by a more detailed application of
persistent homology. Finally we will be looking forward to extend our method to
more than one variable at the time, to be able to check for crossed relevance.
We do not claim this to be an exhaustive list of problems related to the im-
provement of our method, but are confident that they would help us run more
examples and more sophisticated ones, as well as will help us get more data to
compare our results with other methods.
Acknowledgements
We acknowledge Santiago Gallón for enlightining discussions about the subject.
His help has been very valuable.
First and second authors acknowledge the financial support from CIMPA, Cen-
tro de Investigaciones en Matemática Pura y Aplicada through the projects 821–
B7–254 and 821–B8–221 respectively.
Third author acknowledges the financial support from CIMM, Centro de In-
vestigaciones Matemáticas y Metamatemáticas through the project 820–B8–224.
The three authors also acknowledge Escuela de Matemática, Universidad de
Costa Rica for their support.
6 Supplementary Material
R-package for TopSA routine: R-package “TopSA” estimates sensitivity indices
reconstructing the embedding manifold of the data. The reconstruction is done
via a Vietoris Rips with a fixed radius. Then the homology of order 2 and the
indices are estimated.
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