The information-theoretical approach to monitoring network design is not the only option, and other objective functions have also been used for this problem. Examples are cost, geographical spread, and squared-error-based metrics. Some approaches use models describing spatial variability with certain assumptions, e.g., Kriging . In the case of network expansion to new locations, models are always needed to describe what could be measured in those locations. These could vary from simple linear models to full hydrodynamic transport models, such as those used in , who compared performance of the sensor placement in a polder network based on a simple low-order PCA (principal component analysis) model and a detailed hydrodynamic and salt transport model.

In this paper, our main focus is to discuss the formulation of information-theoretical objective functions and previous literature. Therefore, we restrict our scope to those information-theory-based objective functions based on observed data on one single variable in multiple locations. Keeping this limited scope allows us to discuss the interpretation of these objective functions for monitoring network design, which formalize what we actually want from a network. Furthermore, we investigate whether the desired optimum in the objective function can be found by greedy approaches or whether exhaustive search is needed to prevent a loss of optimality.

Only after it is agreed on what is wanted from a network and this is captured as an optimization problem do other issues such as the solution or approximation of the solution to the problem become relevant. The numerical approach to this solution and calculation of information measures warrants independent discussion, which is outside our current scope and will be presented in a future paper. Our discussion is numerically illustrated by a case study using data from Brazos River in Texas, as presented in , to allow for comparisons. However, as we argue, the case study can only serve as an illustration and not be used for normative arguments for use of a particular objective function. Such an argument would be circular, as the performance metric will be one of the objective functions.

In this paper, since we are discussing the appropriate choice of objective function, there is no experimental setup that could be used to provide evidence for one objective version vs. the other. Rather, we must make use of normative theoretical reasoning and shine light on the interpretation of the objectives used and their possible justifications. The practical case studies in this paper therefore serve as an illustration but not as evidence for all the conclusions advocated in this paper, some of which are arrived at through interpretation and argumentation in the Results and discussion section.

The paper is organized as follows. In the following Methodology section, we introduce the methods used to investigate and illustrate the role of objective functions. In Sect. 3, we discuss the case study on the streamflow monitoring network of Brazos River. Section 4 introduces the results for the various methods and then discusses the need for multiple objectives, the interpretation of trade-offs between redundancy and total information, and the feasibility of greedy algorithms reaching the optimum. The article concludes with summarizing the key messages and raising important questions about the calculation of the measures to be addressed in future research.

developed information theory (IT) based on entropy, the concept that explains a system's uncertainty reduction as a function of added information. To understand how, consider a set of N events for which possible outcomes are categorized into m classes. Uncertainty is a measure of our knowledge about which outcome will occur. Once an event is observed, and it is identified which of the m classes it belongs to, our uncertainty about the outcome decreases to 0. Therefore, information can be characterized as the decrease of an observer's uncertainty about the outcome . For monitoring networks, we are interested in the information content of the observations from all stations. The information content is equal to the uncertainty about the observations before measuring. The uncertainty is quantified through probability distributions that describe the possible observations, based on the data.

In monitoring network design, IT has been applied in the literature to evaluate data collection networks that serve a variety of purposes, including rainfall measurement, water quality monitoring, and streamflow monitoring. These evaluations are then used to optimize the placement of sensors. In the monitoring network optimization literature, three expressions from IT are often used in monitoring network design: (1) entropy (H), to estimate the expected information content of observations of random variables; (2) mutual information, often called transinformation (T), to measure redundant information or dependency between two variables; and (3) total correlation (C), a multivariate analogue to mutual information, to measure the total nonlinear dependency among multiple random variables.

The Shannon entropy H(X) is a nonparametric measure, directly on the discrete probabilities, with no prior assumptions on data distribution. It is also referred to as discrete marginal entropy, to distinguish it both from continuous entropy and from conditional entropy. Discrete marginal entropy (Eq. ), defined as the average information content of observations of a single discrete random variable X, is given by 1H(X)=-∑x∈Xp(x)log⁡2p(x), where p(x)(0⩽p(x)⩽1) is the probability of occurrence of outcome x of random variable X. Equation () gives the entropy in the unit of “bits” since it uses a logarithm of base 2. The choice of the logarithm's base for entropy calculation is determined by the desired unit – other information units are “nats” and “Hartley” for the natural and base 10 logarithms, respectively. For monitoring network design, the logarithm of base 2 is common in the literature since it can be interpreted as the necessary number of answers to a series of binary questions and allows for comparisons with file sizes in bits; see, e.g., . Joint entropy (Eq. ), as an extension of entropy beyond a single variable, measures the number of questions needed to determine the outcome of a multivariate system. 2H(X1,X2)=-∑x1∈X1∑x2∈X2p(x1,x2)log⁡2p(x1,x2), where p(x1,x2) is the joint probability distribution of random variable X1 and X2. For a bivariate case, if two random variables are independent, then their joint entropy, H(X1,X2), is equal to the sum of marginal entropies H(X1)+H(X2). Conditional entropy (Eq. ), H(X1|X2), explains the amount of information X1 delivers that X2 can not explain. 3H(X1|X2)=-∑x1∈X1∑x2∈X2p(x1,x2)log⁡2p(x1,x2)p(x2)40⩽H(X1|X2)⩽H(X1) H(X1|X2) can have a range (Eq. ) between zero when both variables are completely dependent and marginal entropy H(X1) when they are independent. Mutual information, in this field often referred to as transinformation (Eq. ), T(X1,X2), explains the level of dependency and shared information between two variables by considering their joint distribution. 5T(X1;X2)=-∑x1∈X1∑x2∈X2p(x1,x2)log⁡2p(x1,x2)p(x1)*p(x2), where p(x1) and p(x2) constitute the marginal probability distribution of random variable X1 and X2, respectively; and p(x1,x2) form their joint probability distribution. The assessment of the dependencies beyond three variables can be estimated by the concept of total correlation (Eq. ) (proposed by and named by ). 6C(X1,X2,…,Xn)=∑i=1nH(Xi)-H(X1,X2,…,Xn) Total correlation (C) gives the amount of information shared between all variables by taking into account their nonlinear dependencies. C can only be non-negative since the sum of all marginal entropies cannot be smaller than their multivariate joint entropy, though in the special case of independent variables, C would become zero.

In this paper, we argue for the maximum joint entropy (maxJE) objective for maximizing the total information collected by a monitoring network. This is equivalent to the GR3 objective proposed by , as part of six other objectives (see Appendix  for more detail) proposed in the same paper, which did not provide preference for its use. In the discussion, we argue that a single-objective optimization of the joint entropy of all selected sensors leads to a maximally informative sensor network, which minimizes total remaining uncertainty about the outcomes at all potential locations. Also, it should be noted that the maxJE objective function already penalizes redundant information through its network selection process, which aims to find a new station that produces maximum joint entropy when it is combined with already selected stations in each iteration. When applied in a greedy search, adding one new station at a time, this approach ranks stations based on growing joint information as quickly as possible. This is mathematically equivalent to adding to the selection, in each iteration, the new station FC that provides maximum conditional entropy HFC|S on top of an already selected set (S) of stations (see Fig.  for visual illustration).

Our research compares and contrasts a variety of objective functions from literature. Information-theory-based multi-objective optimization methods for monitoring networks have gained significant attention recently. Maximizing network information content, through either the sum of marginal entropy or joint entropy, is the common theme among existing methods . However, there is no consensus on whether to use total correlation or transinformation measures to minimize redundant information. Table  gives an overview of the large number of objectives and combinations of objectives used in the last decade. On the one hand, the water monitoring in polders (WMP) method and joint permutation entropy (JPE) method used normalized transinformation to minimize redundant information, while, on the other hand, the multi-objective optimization problem (MOOP) method , combined regionalization and dual entropy multi-objective optimization (CRDEMO) method , multivariable hydrometric networks (MHN) method , and greedy-rank-based optimization (GR 5 and 6) method adopted total correlation to achieve minimum redundancy. Interestingly, both C and T were used as competing objectives in the maximum information minimum redundancy (MIMR) method proposed by . They argued that transinformation between selected stations in the optimal set and non-selected stations should be maximized to account for the information transfer ability of a network. Meanwhile, recently proposed methods in the literature have attempted to improve monitoring network design by introducing yet more other additional objectives . These additional objectives are further discussed in Appendix .

Apart from the objective function, the optimization of monitoring networks is also characterized by constraints. These constraints can either be implemented for numerical reasons or for the reflection of practical aspects of the real-world problem. The majority of existing literature listed in Table  often implicitly imposed a constraint by treating stations' selection as greedy optimization. Greedy optimization adds one station to the selected stations each time, without reconsidering the set's already selected stations. A practical reason for this is numerical efficiency; an exhaustive search of all subsets of k stations out of n possible stations will need to consider a large number of combinations, since the search space grows exponentially with the size n of the full set of sensors (2n combinations of sensors need to be considered).

In this paper, for the maximization of joint entropy that we advocate, we consider and compare three cases for constraints with a large influence on computational cost, with the purpose of investigating whether these influence the results. We also interpret the constraints as reflections of placement strategies. Firstly, the “greedy add” strategy is the commonly applied constraint that each time the network expands, the most favorable additional station is chosen, while leaving the already chosen network intact. The optimal network for k stations is found by expanding one station at a time. This approach can for example be useful in Alpine terrain, where relocating a sensor requires significant effort . Secondly, “greedy drop” is the reverse strategy, not previously discussed in the literature, where the starting point is the full network with all n stations. The optimal network for k stations is found by reducing the full network one step at a time, each step dropping the least informative station. Since all of the discussed monitoring design strategies use recorded data and hence discuss networks whose stations are already established, network reduction is perhaps the more realistic application scenario for information-based design methods. Thirdly, “exhaustive search” is the strategy in which the optimal network of k stations is found by considering all subsets of k stations out of n. This unconstrained search is far more computationally expensive and may not be feasible in larger networks for computational reasons. It can therefore be seen as an optimality benchmark. Because all options are considered, this is guaranteed to find the optimal combination of selected sensors for each network size, given the objective function.

In this comparison, we investigate whether the exhaustive optimization yields a series of networks where an increase in network size may also involve relocating stations. This may not always be practically feasible or desired in actual placement strategies, where networks are slowly expanded one station at a time. Occurrence of relocation in the sequence of growing subsets would also show that no greedy algorithm could exist that guarantees optimality.

In this paper, we argue that, due to the additive relations of information measures (Eq. ), the proposed objective functions in the literature are unnecessarily complicated, and a single-objective optimization of the joint entropy of all selected sensors will lead to a maximally informative sensor network. The additive relations between some of the information measures discussed in this paper are illustrated in Fig. . In this figure and later is this paper, we use a shorthand notation: we use the sets of stations directly in the information measures, as a compact notation for the multivariate random variable measured by that set of stations. Various types of information interactions for three variables are conceptually understandable using a Venn diagram (Fig. a. Although a Venn diagram can be used to illustrate information of more than three variables when they are grouped in three sets (Fig. ), it cannot be used to illustrate pairwise information interactions beyond three variables. A chord diagram, on the other hand, can be useful to better understand pairwise information interaction beyond three variables. Figure  provides a simple template to interpret and compare Venn and chord diagrams.

There are two important caveats with these visualizations. In the general Venn diagram of three-variate interactions, the “interaction information”, represented by the area where three circles overlap, can become negative. Hence, the Venn diagram ceases to be an adequate visualization. For similar reasons, in the chord diagram, the sector size of outer arc lengths should not be interpreted as total information transferred . Information that can contribute to interactions is a combination of unique, redundant, and synergistic components . Their information entanglement is an active area of research in three or more dimensions. In this paper, the total size of the outer arc lengths is set to represent the sum of pairwise information interactions (used in ) and conditional entropy of each variable. This size may be larger than the total entropy of the variable and does not have any natural or fundamental interpretation.

In this paper, we use Venn diagrams to illustrate information relations between three groups of variables. Group one is the set of all sensors that are currently selected as being part of the monitoring network, which we denote as S. Group two is the set of all sensors that are currently not selected, denoted as F, and group 3 is the single candidate sensor that is currently considered for addition to the network, FC; see Appendix  for an overview of notation. Since group 3 is a subset of group 2, one Venn circle is contained in the other, and there are only five distinct areas vs. seven in a general three-set Venn diagram. In this particular case, there is no issue arising from negative interaction information.

For the purpose of illustrating the main arguments of this study, we compare maxJE objective function (Eq. ) with three other (sets of) objective functions from previously proposed methods: MIMR (Eq. ), WMP (Eq. ), and minT (Eq. ). These methods were chosen since they are highly cited methods in this field, and more importantly, recent new approaches in the literature have mostly been built on one of these methods with additional objectives . Objective function (maxJE):=7maximize HXS1,XS2,…,XSk,XFC

8Objective function (MIMR):=maximize HXS1,XS2,…,XSk,XFCmaximize ∑i=1mTXS1,XS2,…,XSk,XFiminimize CXS1,XS2,…,XSk,XFC▾ integrated format ▾maximize λ1(HXS1,XS2,…,XSk,XFC+∑i=1mTXS1,XS2,…,XSk,XFi)-(1-λ1)CXS1,XS2,…,XSk,XFC Objective function (WMP):=9maximize H(FC)subject to ∑i∈STSi;FCH(Si)<SBMObjective function(minT):=10maximize first H(FC)minimize TXS1,XS2,…,XSk,XFC, where XS1,XS2,…,XSk refers to selected stations in the previous iterations. XFC and H(FC) denote the variable at the current candidate station and its marginal entropy, respectively. FC is the station considered for addition to the current set in a greedy add approach. This formulation was chosen to allow for a uniform presentation between methods. The objectives for methods maxJE and MIMR can easily be modified to consider FC as part of S, so that the objective function evaluates the entire network rather than one candidate station for addition. This allows for greedy add, greedy drop and exhaustive search methods. λ1 is information-redundancy trade-off weight . SBM (“select below median”) stands for the constraint whereby only stations are considered that are below the median score of all potential stations for that objective. m is equal to the number of non-selected stations in each iteration (m+k=n total number of stations). For the multi-objective approaches used in the case study, we used the same weights as the original authors to identify a single solution. It can be seen that a large number of different combinations of information-theoretical measures are used as objectives.

As indicated in the Introduction, we should not attempt to gauge the merits of the objective functions by the intuitive optimality of the resulting network. Rather, the merits of the networks should be gauged by the objective functions. Still, the case study can provide insight into some behaviors resulting from the objective functions.

To assess and illustrate the workings of the different objectives in retrieving information from the water system, we compared three existing methods with a direct maximization of the joint entropy of selected sensors, HS,FC, indicated with maxJE in the results, such as Tables  and . The joint entropy results in Table  indicate that maxJE is able to find a combination of 8 stations that contains joint information of all 12 stations ranked by other existing methods. Figure  displays spatial distribution of the top eight stations chosen by different methods. Before any interpretation of the placement, we must note that the choices made in quantization and the availability of data play an important role in the optimal networks identified. Whether the saturation that occurs with eight stations has meaning for the real-world case study depends on whether the joint probability distribution can be reliably estimated. This is highly debatable and merits a separate detailed discussion which is out of the scope of this paper. We present this case study solely to illustrate behavior of the various objectives.

The most notable difference between maxJE and the other methods is the selection of all three of the stations located most downstream. While other methods would not select these together due to high redundancy between them, maxJE still selects all stations because despite the redundancy, there is still found to be enough new information in the second-most and third-most downstream station. This can be in part attributed to the quantization choice of equally sized bins throughout the network, leading to higher information contents downstream. While this quantization choice is debatable, it is important, in our opinion, to not compensate artifacts from quantization by modifying the objective function, even if the resulting network may seem more reasonable, but rather to address those artifacts in the quantization choices themselves. To repeat the key point: an objective function should not be chosen based on whether it yields a “reasonable network” but rather based on whether the principles that define it are reasonable.

Though already necessarily true from the formulation of the objective functions, we use the case study to illustrate how other methods with a separate minimum redundancy objective lead to the selection of stations with lower new information content (green area in Fig. ). Reduction of the yellow area in each iteration (i.e., the information loss compared to the full network) in Fig.  corresponds to the growth of joint entropy values in Table  for each method. maxJE (by definition) has the fastest and minT the slowest rate of reduction of information loss. Methods' preference for reaching minimum redundancy or growing joint information (red area in Fig. ) governs the reduction rate of information loss. Also, Fig.  provides auxiliary information about the evolution of pairwise information interaction between already selected stations X1,X2,…,Xi-1 in the previous iterations and new proposed station Xi. Figure  illustrates the contrast between the choice of the proposed stations in the first six iterations by different methods. For instance, the minT method aims to find a station that has minimum mutual information (red links in Fig. ) with already selected stations. In contrast, the maxJE method tries to grow joint entropy, which translates to finding a station that has maximum conditional entropy (green segments in Fig. ). Other methods opt to combine two approaches by either imposing a constraint (WMP) or having a trade-off between them (MIMR).

The existing approaches considered above have in common that they all involve some form of dependence criterion to be minimized. For example, the total correlation gives a measure of total redundant information within the selected set. This is information that is duplicated and therefore does not contribute to the total information content of the sensors, which is given by the joint entropy. Focusing fully on minimizing dependence, such as what is done in the minT objective optimization, makes the optimization insensitive to the amount of non-duplicated information added. This results in many low-entropy sensors being selected. It is important to note that the joint entropy already accounts for duplicated information and only quantifies the non-redundant information. This is exactly the reason why it is smaller than the sum of individual entropies. In terms of joint entropy, two completely dependent stations are considered to be exactly as informative as one of them. This means that the negative effect that dependency has on total information content is already accounted for by maximizing joint entropy only.

stated that “the fundamental basis in designing monitoring networks based on the entropy approach is that the stations should have as little transinformation as possible, meaning that the stations must be independent of each other”. However, no underlying argument for this fundamental basis is given in the paper. The question is then whether there is another reason, apart from information maximization, why the total correlation should be minimized. In three of the early papers introducing the approaches that employed or evaluated total correlation, no such reason was given other than the one by . Also in later citing research , no such arguments have been found, except for effectiveness, which we argue is covered by looking at the total non-redundant information the network delivers. Traditional reasons for minimizing redundancy are reducing the burden of data storage and transmission, but these are not very relevant in monitoring network design, since those costs are often negligible compared to the costs of the sensor installation and maintenance (see ). Moreover, information theory tells us that, if needed, redundant information can be removed before transmission and storage by employing data compression . The counter-side of minimal redundancy is less reliability, a far more relevant criterion for monitoring network design. Given that sensors often fail or give erroneous values, one could argue that redundancy (total correlation) should actually be maximized, given a maximum value of joint entropy. We might even want to gain more robustness at the cost of losing some information. One could for example imagine placing a new sensor directly next to another to gain confidence in the values and increase reliability, instead of using it to collect more informative data in other locations.

The Pareto front that would be interesting to explore in this context is the trade-off between maximum total correlation (robustness) vs. joint entropy (expected information gained from the network), indicated by the red line in Fig. . Different points on this Pareto front reflect different levels of trust in the sensors' reliability. Less trust requires more robustness and leads to a network design yielding more redundant information. Previous approaches, such as the MOOP approach proposed by , explore the Pareto front given by the dashed black line, where minimum total correlation is conflicting with maximizing joint entropy. As argued in this section, this trade-off is not fundamental in information-theoretical terms. Still, it results from the fact that there is usually redundant information as a by-product of new information, so highly informative stations also carry more redundant information. This redundant information does not reduce the utility of the new information so does not need to be included as a minimization objective in the optimization.

In summary, the maximization of joint entropy while minimizing redundancy is akin to maximizing effectiveness while maximizing a form of efficiency (i.e., bits of unique information/bits collected). However, bits do not have any significant associated cost. If installing and maintaining a monitoring location has a fixed cost, then efficiency should be expressed as unique information gathered per sensor installed, which could be found by maximizing joint entropy (effectiveness) for a given number of stations, as we suggest in this paper.

Different search strategies have been adopted in the literature for monitoring network design. The most commonly used greedy algorithms impose a constraint on exhaustive search space to reduce computational effort. We investigated three different search strategies to obtain the optimal network in the context of using maxJE as an objective function. We discuss the advantages and limitations of each search strategy in terms of optimality of the solution and computational effort.

The exhaustive optimization tests all possible new combinations, not restricted to those combinations containing the already selected set in a smaller network. Since the joint entropy of a set of locations does not depend on the order in which they are added, the number of possible combinations is nk (i.e., n choose k), where n is the number of potential stations in the pool, and k is the number of selected stations. The computational burden is therefore greatest when about half of the stations are selected. For a number of potential sensors under 20, this is still quite tractable (4 min on normal PC, implemented – by a hydrologist – in MATLAB), but for larger numbers, the computation time increases very rapidly. When considering all sub-network sizes, the number of combinations to consider is 2n, so an exponential growth. We could make an optimistic estimate, only considering the scaling from station combinations to evaluate but not considering the dimensionality of the information measures. For 40 stations, this estimate would yield a calculation time of more than 5 years, unless a more efficient algorithm can be found. Regardless of potential improvements in implementation, the exponential scaling will cause problems for larger systems.

Greedy approaches might be candidates for efficient algorithms. For the proposed joint entropy objective, we tested the optimality of greedy approaches against the benchmark of exhaustive optimization of all possible station combinations. For the Brazos River case study, both the “add” and “drop” greedy selection strategies resulted in the global optimum sets, i.e., the same gauge order and resulting joint entropy as was found by the exhaustive optimization. These results can be read from the last row of Tables  and . Therefore, for this case, the greedy approaches did not result in any loss of optimality. For the last few sensors, multiple different optimal sets could be identified, which are detailed in Table . Results in Table  show that multiple network layouts with equal network size and joint information exist. For this case, network robustness could be an argument to prefer the network with maximum redundancy.

In a further test, using artificially generated data, we experimentally falsified the hypothesis that greedy approaches can guarantee optimality. For this test, we generated a correlated random Gaussian dataset for 12 stations, based on the covariance matrix of the data from the case study. We increased the number of generated observations to 860 time steps, to get a more reliable multidimensional probability distribution. Table  shows the resulting orders for 12 stations for the three different approaches. Note how for the exhaustive optimization in this example, in some instances, one or two previously selected gauges are dropped in favor of selecting new stations. The resulting joint entropies for the selected sets are shown in Table . This means no greedy approach can exist that finds results equivalent to the exhaustive approach.

Based on our limited case study, the questions remain open: (1) whether faster algorithms can be formulated that yield guaranteed optimal solutions, and (2) in which cases the greedy algorithm provides a close approximation. It is also possible to formulate modified greedy methods with the ability of replacing a limited number of stations instead of just adding stations. This leads to a significantly reduced computational burden while reaching the optimum more often than when adding stations one at a time. In Table , it can be seen that allowing a maximum number of two relocated stations would already reach the optimal configurations for this specific case. Another limitation of this comparison is that we did not consider metaheuristic search approaches , which fall in between greedy and exhaustive approaches in terms of computational complexity and could serve to further explore the optimality vs. computational complexity trade-off. It would be interesting to further investigate what properties in the data drive the suboptimality of greedy algorithms. Synergistic interactions are a possible explanation, although our generated data example shows that even when moving from one to two selected stations, a replacement occurs. Since there are only pairs of variables involved, synergy is not needed in the explanation of this behavior. Rather, the pair with maximum joint entropy does not always include the station with maximum entropy, which could perhaps be too highly correlated with other high entropy variables.

In this paper, we focused on the theoretical arguments for justifying the use of various objective functions and compared a maximization of joint entropy to other methods while using the same dataset and quantization scheme. Since the majority of previous research used greedy search tools to find optimal network configurations, we compared greedy and exhaustive search approaches to raise awareness in the scientific community that greedy optimization might fall into a local optimum, though its application can be justified considering the computation cost of the exhaustive approach. compared computation cost for greedy and metaheuristic optimization (Non-dominated Sorting Genetic Algorithm II). They reported that the greedy approach resulted in drastic reduction of the computational time for the same set of objective functions (metaheuristic computation cost was higher 58 times in one trial and 476 times in another). We recommend further investigation of these three search tools in terms of both optimality (for the maxJE objective) and computation cost.

Another important question that needs to be addressed in future research is to investigate how the choices and assumptions made (i.e., data quantization which influences probability distribution) in the numerical calculation of objective functions would affect network ranking. What many of these objective functions have in common is that they rely on multivariate probability distributions. For example, in our case study, the joint entropy is calculated from a 12-dimensional probability distribution. These probability distributions are hard to reliably estimate from limited data, especially in higher dimensions, since data requirements grow exponentially. Also, these probability distributions and the resulting information measures are influenced by multiple factors, including choices about the data's temporal scale and quantization. To have an unbiased comparison framework of objective functions, we kept data and quantization choices from a case study previously described in the literature. It is worth acknowledging that these assumptions, as well as data availability, can greatly influence station selection and require more attention in future research.

Numerically, the limited data size in the case study presents a problem for the calculation of multivariate information measures. Estimating multivariate discrete joint distributions exclusively from data requires quantities of data that exponentially grow with the number of variables, i.e., potential locations. When these data requirements are not met, and joint distributions are still estimated directly based on frequencies, independent data will be falsely qualified as dependent and joint information content severely underestimated. This can also lead to apparent earlier saturation of joint entropy, at a relatively low number of stations. For the case study presented here, we do not recommend interpreting this saturation as reaching the number of needed stations, since it could be a numerical artifact. This problem applies to all methods discussed in this paper. Before numerics can be discussed, clarity is needed on the interpretation and choice of the objective function. In other words, before thinking about how to optimize, we should be clear on what to optimize. We hope that this paper helped illuminate this.