Bayesian multi-resolution spatial analysis with applications to marketing

Abstract

Marketing researchers have become increasingly interested in spatial datasets. A main challenge of analyzing spatial data is that researchers must a priori choose the size and make-up of the areal units, hence the resolution of the analysis. Analyzing the data at a resolution that is too high may mask “macro” patterns, while analyzing the data at a resolution that is too low may result in aggregation bias. Thus, ideally marketing researchers would want a “data-driven” method to determine the “optimal” resolution of analysis, and at the same time automatically explore the same dataset under different resolutions, to obtain a full set of empirical insights to help with managerial decision making. In this paper, we propose a new approach for multi-resolution spatial analysis that is based on Bayesian model selection. We demonstrate our method using two recent marketing datasets from published studies: (i) the Netgrocer spatial sales data in Bell and Song (Quantitative Marketing and Economics 5:361–400, 2007), and (ii) the Pathtracker^® data in Hui et al. (Marketing Science 28:566–572, 2009b; Journal of Consumer Research 36:478–493, 2009c) that track shoppers’ in-store movements. In both cases, our method allows researchers to not only automatically select the resolution of the analysis, but also analyze the data under different resolutions to understand the variation in insights and robustness to the level of aggregation.

Appendices

Appendix

1.1 Finding the configuration G with the highest posterior model probability

We propose a simulated annealing (SA) algorithm (Goffe et al. 1994) to stochastically search for the configuration with the highest posterior model probability, ${G^*} = \mathop {{\arg \,\max }}\limits_G \Pr \left( {G|y} \right)$. The SA algorithm is a Monte Carlo optimization technique (Robert and Casella 2004) that is particularly suited for finding the global optima over a finite and discrete search domain, which is the case for configuration G where the number of potential configurations is finite. Specifically, it has been shown that in a finite search space, the SA algorithm converges to the global optima with probability 1, given that the “cooling schedule” (discussed later) is slow enough (Bertsimas and Tsitsiklis 1993; Hajek 1988). Thus, the SA algorithm is particularly suitable for our problem.

The SA algorithm can be described by the following steps:

• 1. Start with an initial configuration G ₀ and initial “temperature” T ₀.

• 2. For each iteration n (n = 1, …, N):

  • a. Propose a candidate move G’ (discussed below)

  • b. Move to G’ $\left( {{\text{i}}.{\text{e}}.,{G_{n + 1}} = G\prime } \right)$with probability min (1, r), where $r = \exp \left( {\frac{{\ln (\Pr (G\prime |y) - \ln (\Pr ({G_n}|y)}}{{{T_n}}}} \right)$. Otherwise, ${G_{n + 1}} = {G_n}$.

  • c. Update the temperature using a “cooling schedule”, ${T_{n + 1}} = k{T_n}\left( {0 < k < 1} \right)$.

We now discuss how a candidate move G’ in step (2a) is defined. First, we define a “boundary representation” of G that automatically incorporates the contiguity structure in the data. The leftmost panel of Fig. 14 shows the areal units of the raw data for a hypothetical example; originally, there are 4 areal units in total. The upper middle and right panels show two potential configurations for G, denoted as G ₁ and G ₂ , respectively. We denote G ₁ using the notation {(1), (3), (2,4)}, and G ₂ using the notation {(1,3), (2,4)}.

Fig. 14

Illustration of the raw data, two example configurations, and their corresponding boundary representations

The lower middle and right panels of Fig. 14 shows the boundary representations of the configurations. The boundary representation of G ₁ is denoted as B ₁ in the lower middle panel. The boundary representation provides an alternative representation for a configuration, and is interpreted as follows: The absence of a boundary between two areal units indicates that they are grouped into the same region. As can be seen in the figure, boundaries 1—2, 3—4, 1—3 are present in B ₁ , but boundary 2—4 is absent, thus areal units 2 and 4 are grouped into the same region, resulting in the configuration G ₁  = {(1), (3), (2,4)}. Likewise, B ₂ is the boundary representation of G ₂ . Figure 15 shows all 12 possible configurations with this example, along with their corresponding boundary representations.

Fig. 15

All possible configurations and their corresponding boundary representations

From the current configuration G _n (and its corresponding boundary representation B _n ), the proposed candidate move “turns” one of the randomly-selected boundaries in B _n from “on” to “off,” or from “off” to “on.” This gives us a candidate boundary representation B’, and hence a candidate configuration G’ that, because of the definition of boundary representations, already satisfies the contiguity constraint.

Figure 16 summarizes the “neighborhood structure” of our proposed procedure of generating candidate moves. Configurations that are connected can be reached in one move through their boundary representations. As can be seen, our proposed procedure allows us to search through the domain of configurations efficiently through a sequence of local moves, the key idea behind the SA algorithm.

Fig. 16

Summary of the neighborhood structure for the hypothetical example. Two configurations that are connected can be a candidate move from each other

We follow standard guidelines in setting other parameter values for the SA algorithm (Robert and Casella 2004). Specifically, the initial configuration G ₀ is randomly generated. The initial temperature is set to a high value, so that moves from the initial configuration have around 80% probability of being accepted (Robert and Casella 2004). The number of iterations N is set to one million, and the “cooling schedule” k is set so that at the last iteration, the temperature drops to 0.001. To ensure that the SA algorithm converges to the global optimal configuration, we repeat the SA algorithm 100 times, each time from a different random starting point. The algorithm is coded up in C++ with the GNU scientific library. The source code is available upon request.

1.2 Derivation of Eq. 9 (Sales Mapping)

Starting with Eqs. 5 and 8,

$$ \matrix{ {P(Y|G) = \int {P(Y|G,{\theta^G})\pi ({\theta^G}|G)d{\theta^G}} } \hfill \\ { = \left( {\prod\limits_j {\frac{{{n_j}^{{y_j}}}}{{{y_j}!}}} } \right){{\left( {\frac{{{\beta^\alpha }}}{{\Gamma (\alpha )}}} \right)}^{{K^G}}}\prod\limits_k {\int {{{\left( {\lambda_k^G} \right)}^{\left( {y_k^G + \alpha - 1} \right)}}\exp \left( { - \left( {n_k^G + \beta } \right)\lambda_k^G} \right)d\lambda_k^G} } } \hfill \\ { = \left( {\prod\limits_j {\frac{{{n_j}^{{y_j}}}}{{{y_j}!}}} } \right){{\left( {\frac{{{\beta^\alpha }}}{{\Gamma (\alpha )}}} \right)}^{{K^G}}}\prod\limits_k {\frac{{\Gamma (y_k^G + \alpha )}}{{{{\left( {n_k^G + \beta } \right)}^{\left( {y_k^G + \alpha } \right)}}}}} .} \hfill \\ }<!end array> $$

Dropping the constant $\left( {\prod\limits_j {\frac{{{n_j}^{{y_j}}}}{{{y_j}!}}} } \right)$ from consideration because it is the same for all G, we have:

$$ P(G|Y) = \pi (G){\left( {\frac{{{\beta^\alpha }}}{{\Gamma (\alpha )}}} \right)^{{K^G}}}\prod\limits_k {\frac{{\Gamma (y_k^G + \alpha )}}{{{{\left( {n_k^G + \beta } \right)}^{\left( {y_k^G + \alpha } \right)}}}}} . $$

1.3 Derivation of Eq. 15 (Flow Mapping)

Starting with Eqs. 13 and 14,

$$ \matrix{ {P(Y|G) = \int {P(Y|G,{\theta^G})\pi ({\theta^G}|G)d{\theta^G}} } \hfill \\ { = \left[ {\frac{1}{{{{\left( {\left( {{K^G} - 1} \right)!} \right)}^{{K^G}}}}} \cdot \frac{1}{{\prod\limits_k {\left( {|{M_G}(k)| - 1} \right)!} }}} \right]\int {\left[ {\prod\limits_{{k_1}} {\prod\limits_{{k_2}} {{{\left( {\lambda_{{k_1}{k_2}}^G} \right)}^{n_{{k_1}{k_2}}^G}}} } } \right]} \left[ {\prod\limits_k {\prod\limits_{j \in {M_G}(k)} {{{\left( {\varphi_{kj}^H} \right)}^{{m_j}}}} } } \right]d{\theta^G}} \hfill \\ { = \left[ {\frac{1}{{{{\left( {\left( {{K^G} - 1} \right)!} \right)}^{{K^G}}}}} \cdot \frac{1}{{\prod\limits_k {\left( {|{M_G}(k)| - 1} \right)!} }}} \right]\prod\limits_{{k_1}} {\left( {\int {\prod\limits_{{k_2}} {{{\left( {\lambda_{{k_1}{k_2}}^G} \right)}^{n_{{k_1}{k_2}}^G}}d\vec{\lambda }_{{k_1} \cdot }^G} } } \right)} \prod\limits_k {\left( {\int {\prod\limits_{j \in {M_G}(k)} {{{\left( {\varphi_{kj}^G} \right)}^{{m_j}}}d\vec{\varphi }_{k \cdot }^G} } } \right)} } \hfill \\ { = \frac{{\prod\limits_{{k_1}} {\left( {\frac{{\prod\limits_{{k_2}} {\Gamma (n_{{k_1}{k_2}}^G + 1)} }}{{\Gamma \left( {\sum\limits_{{k_2}} {\left( {n_{{k_1}{k_2}}^G + 1} \right)} } \right)}}} \right)\prod\limits_k {\left( {\frac{{\prod\limits_{j \in {M_G}(k)} {\Gamma \left( {{m_j} + 1} \right)} }}{{\Gamma \left( {\sum\limits_{j \in {M_G}(k)} {\left( {{m_j} + 1} \right)} } \right)}}} \right)} } }}{{{{\left( {\left( {{K^G} - 1} \right)!} \right)}^{{K^G}}}\prod\limits_k {\left( {\left( {|{M_G}(k)| - 1} \right)!} \right)} }}} \hfill \\ }<!end array> $$

Thus,

$$ \matrix{ {P(G|Y) = \pi (G)P(Y|G)} \hfill \\ { = \pi (G) \cdot \frac{{\prod\limits_{{k_1}} {\left( {\frac{{\prod\limits_{{k_2}} {\Gamma (n_{{k_1}{k_2}}^G + 1)} }}{{\Gamma \left( {\sum\limits_{{k_2}} {\left( {n_{{k_1}{k_2}}^G + 1} \right)} } \right)}}} \right)\prod\limits_k {\left( {\frac{{\prod\limits_{j \in {M_G}(k)} {\Gamma \left( {{m_j} + 1} \right)} }}{{\Gamma \left( {\sum\limits_{j \in {M_G}(k)} {\left( {{m_j} + 1} \right)} } \right)}}} \right)} } }}{{{{\left( {\left( {{K^G} - 1} \right)!} \right)}^{{K^G}}}\prod\limits_k {\left( {\left( {|{M_G}(k)| - 1} \right)!} \right)} }}} \hfill \\ }<!end array> $$

Technical appendix (Online Supplement)

Here we propose a supplemental algorithm to stochastically search for configurations that are “near” the configuration that has the highest posterior model probability (denoted as G*). This involves a slight modification to the simulated annealing algorithm proposed in Appendix A. Specifically, before the end of each iteration, if the current configuration G _n+1 has a posterior model probability that is higher than rP(G*|y), where r is a pre-defined constant that defines what is considered “near” (e.g., r = exp(-2), which equates to -2 units on the log-probability scale), the algorithm terminates and returns G _n+1 as a “near” configuration. Thus, the supplemental algorithm can be described by the following steps:

• 1. Start with an initial configuration G ₀ and initial “temperature” T ₀.

• 2. For each iteration n (n = 1, …, N):

  • a. Propose a candidate move G’ (discussed below)

  • b. Move to G’ (i.e., $\left( {{\text{i}}.{\text{e}}.,{G_{n + 1}} = G\prime } \right)$with probability min (1, r), where $r = \exp \left( {\frac{{\ln (\Pr (G'|y) - \ln (\Pr ({G_n}|y)}}{{{T_n}}}} \right)$. Otherwise, G _n+1 = G _n .

  • c. Update the temperature using a “cooling schedule”, ${T_{n + 1}} = k{T_n}\quad (0 < k < 1)$.

  • d. If $P({G_{n + 1}}|y) \geqslant P(G*|y)$, terminate the algorithm and return G _n+1; if not, continues the algorithm.

We apply the above supplemental algorithm to the sales mapping problem described in Section 3.4 to identify 100 “near” configuration to G*. As an illustration, two examples of “near” configurations are shown in Figs. 17 and 18 below. Other “near” configurations are available from the authors upon request.

Fig. 17

An example of a “near” configuration

Fig. 18

An example of a “near” configuration

Looking across the 100 nearby solutions, we find that, similar to Figs. 17 and 18, all of them appear very similar to G*. This suggests that the posterior model probability has a fairly “sharp” peak at G*, and the likelihood surface is likely to be unimodal.