A Graphical Point Process Framework for Understanding Removal Effects in Multi-Touch Attribution

The attribution problem has been studied broadly by researchers from both academia and industry for over two decades. In this subsection, we provide a brief overview of existing approaches, and more details can be found in the survey papers (Kannan et al., 2016; Gaur & Bharti, 2020). Existing attribution models can be classified into two major categories: rule-based heuristics and data-driven approaches.

To the best of our knowledge, none of the existing models can study the full relational structure of numerous types of touches across multi-channels, which is essential to understanding both the direct and indirect conversion effects of each type of touch and the corresponding channels. Given the potentially large number of various types of touchpoints under study, a model that can simultaneously study the interactive and conversion effects of these many types of touchpoints is in need. To fill these research gaps, we make the following efforts in this work:

Firstly, we develop a novel graphical point process model for attribution to describe customer behavior in the multi-channel setting using customer-level path-to-purchase data. The graphical model not only learns the direct conversion effects of numerous types of touchpoints but also estimates exciting effects among different types of touches simultaneously.

Secondly, we propose graphical attribution methods to allocate proper conversion credit, called the attribution score, to individual touchpoints and the corresponding channels for each customer’s path to purchase. Our proposed methods consider the attribution score as the removal effect. We derive two versions of the removal effect using the temporal point process of conversion and the graphical structure.

Thirdly, we propose a regularization method for simultaneous edge selection and parameter estimation. We design a customized alternating direction method of multipliers (ADMM) to solve this optimization problem in an efficient and scalable way. In addition, we provide a theoretical guarantee by establishing the asymptotic theory for parameter estimates.

In what follows, we briefly introduce the idea of our proposed graphical point process model. As the first step, we model path-to-purchase data as multivariate temporal point processes. More specifically, the proposed method considers individual paths of touchpoints as independent event streams. Each stream consists of various types of touchpoints (events) occurring irregularly and asynchronously on a common timeline. To capture the dynamic inter-dependencies (e.g., exciting patterns) among touches from a large number of independent event streams, the proposed method models event streams as multivariate temporal point processes. The multivariate temporal point processes are commonly characterized using conditional intensity functions (Gunawardana et al., 2011), which describe the instantaneous rates of occurrences of future touchpoints given the history of prior touchpoints. The proposed model can consider the temporal distances and the clumpy nature of touches. Xu et al. (2014) is the first paper to tackle attribution modeling using multivariate temporal point processes. They consider advertisement clicks and purchases as dependent random events in continuous time and cast the model in the Bayesian hierarchical framework.

The proposed model further introduces a Granger causality graph that is a directed graph to represent the dependencies among various event types. The nodes in the graph represent event types, and directed edges depict the historical influence of one type of event on the others, which are called the Granger causality relations (Granger, 1969). The proposed graphical model allows for a large number of online marketing channels and types of touchpoints. By fully capturing the Granger causality relations among various event types, the proposed model simultaneously measures how the numerous types of touches affect conversion, as well as the exciting effects of different types of touches within and across channels.

Based on the learned graph, we propose graphical attribution methods to assign proper credit for conversions to each type of touchpoint or a corresponding channel. The conversion credit, also called attribution scores, are calculated at the path level from a granular point of view. They can be aggregated to the channel level when necessary. The first attribution method measures the direct effect of the event(s) of interest on the conversion. This is the relative change in conversion intensity when we remove only the event(s) of interest from the path and assume other touchpoints on the path remain unaffected. The second attribution method fully uses the graphical causality structure and measures the total removal effect of the event(s) of interest. The corresponding attribution score is the marginal lift of the expected intensity of conversion by considering not only the removal of the events of interest but also the potential loss of subsequent customer-initiated events.

We examine the performance of our proposed methods in simulation studies and compare their performance with commonly used attribution models using two sets of simulated data. One data set is simulated from the multivariate Hawkes process (Hawkes, 1971). The other data set is simulated from a modified version of the Digital Advertising System Simulation (DASS) developed by Google Inc. The simulated data includes online customer browsing behavior and injected advertising events that impact customer behavior. Our proposed methods outperform the benchmark models in measuring channels’ contribution to conversions. Moreover, we demonstrate the performance of the proposed methods in a real-world attribution application.

We provide practitioners with a new attribution modeling tool to understand how different marketing efforts contribute to conversions at a finer granularity in online multi-channel settings, where there exist a potentially large number of different types of touchpoints nowadays. Our tool distributes proper credit to individual touchpoints and the corresponding channels by conducting attribution modeling at the individual customer path level. Our tool helps firms’ granular marketing operations, budget allocation, profit maximization, etc.

In addition to the substantive contribution, we have the following methodological contributions to the literature.

Firstly, we contribute to the attribution modeling literature by proposing a graphical point process model to describe customer behavior using individual customer-level path-to-purchase data. We rigorously model the exciting effects of numerous types of touches in this framework and develop an efficient penalized algorithm for model estimation. We apply Granger causality in a marketing context to study the temporal relations among marketing activities. We also establish the asymptotic theory for parameter estimates.

Secondly, our proposed graphical attribution methods contribute to the literature on the incremental value (or removal effect) approaches. We provide a rigorous probabilistic definition of attribution scores and derive two types of removal effects, namely, the direct and total removal effects. We develop a new efficient thinning-based simulation method and a backpropagation algorithm for the calculation of two types of removal effects, respectively.

We organize the rest of the paper as follows. We introduce the proposed graphical point process model in Section 2 and the proposed graphical attribution methods in Section 3. We present the model estimation, computational details, and asymptotic properties in Section 4. We then demonstrate the performance of the proposed method and algorithm with the simulated data in Section 5 and provide an empirical application in Section 6. We conclude this work in Section 7.

For an event type labeled as $e$, we can describe its occurrence on the timeline as a temporal point process $N_{e}(t)$. The function $N_{e}(t)$ is the number of type-$e$ events that happened until time $t$, which is a right-continuous and non-decreasing piece-wise function $\mathbb{R}_{\geq 0}\rightarrow\mathbb{N}$:

We assume $N_{e}(t)$ has almost surely step size $1$ and does not jump simultaneously. If $t^{\prime}<t$, then $N_{e}(t)-N_{e}(t^{\prime})$ is the number of type-$e$ events that occurred during the interval $(t^{\prime},t]$, which can also be denoted by $N_{e}((t^{\prime},t])$. Putting $p$ event types together, we let $\mathbf{N}(t)$ denote the vector of counting functions $(N_{1}(t),\dots,N_{p}(t))^{\top}$. The coordinate $N_{e}(t)$, $e=1,\dots,p$, is characterized by its conditional intensity

which describes the instantaneous rates of occurrence of future type-$e$ events. The filtration $\mathcal{H}_{t}:=\sigma\{\mathbf{N}(u):u<t\}$ is the $\sigma$-algebra of all events up to but excluding $t$, referring to the historical information before time $t$.

An example is the multivariate Hawkes process (Hawkes, 1971) with the conditional intensity function

Here $\mu_{e}>0$ is the baseline intensity serving as a background rate of occurrence regardless of historical impact, and $h_{e^{\prime}e}(\cdot)\geq 0$ is called an impact function. Intuitively, each of the past $e^{\prime}$-events with $h_{e^{\prime}e}(\cdot)>0$ has a positive contribution to the occurrence of the current type-$e$ event by increasing the conditional intensity of event type $e$, and this influence may decrease through time. Such positive influence is called an exciting effect from event type $e^{\prime}$ to event type $e$.

Let $\mathcal{E}=\{1,2,\dots,p\}$ be the set of various types of labeled events (i.e., touchpoints and conversions), whose historical influences on each other are of great interest. We use the Granger causality relations (Granger, 1969) to describe the temporal dependencies among the studied types of touchpoints. If the history of event type $e^{\prime}$ helps to predict event type $e$ above and beyond the history of event type $e$ alone, event type $e^{\prime}$ is said to “Granger-cause” event type $e$. The Granger causality was introduced and discussed in the original paper Granger (1969) and follow-up papers (Granger, 1980, 1988), while Sims (1972) gave an alternative definition of Granger causality.

For any event label subset $W\subseteq\mathcal{E}$, let $\mathbf{N}_{W}(t)$ be the subprocess $(N_{e}(t))_{e\in W}$. For example, $\mathbf{N}_{\mathcal{E}\setminus\{e,e^{\prime}\}}(t)$ means the subprocess of all the event types other than $e$ and $e^{\prime}$. For a temporal point process, the Granger causality is defined below.

The above definition was introduced by Didelez (2008) for marked point processes in an interval $[0,T]$. Eichler (2012) studied the stationary multivariate point processes like multivariate Hawkes processes in $\mathbb{R}$ and used the notion “Granger non-causality”. The above definition is equivalent to saying the temporal point process $N_{e^{\prime}}(t)$ does not Granger-cause $N_{e}(t)$ with respect to historical information $\mathcal{H}_{t}$. Otherwise, $N_{e^{\prime}}(t)$ Granger-causes $N_{e}(t)$ with respect to $\mathcal{H}_{t}$. For ease of interpretation, we also say that event type $e^{\prime}$ Granger-causes event type $e$ in an unambiguous manner.

We use a directed graph $\mathcal{G}=(\mathcal{E},\mathcal{A})$ like Figure 2 to represent the temporal dependencies among various types of touchpoints in the conversion paths. The node set of the graph represents the set of event types $\mathcal{E}$. The edge set $\mathcal{A}\subseteq\mathcal{E}\times\mathcal{E}$ represents the Granger causality relations between the various event types. If $N_{e^{\prime}}(t)$ Granger-causes $N_{e}(t)$ with respect to $\mathcal{H}_{t}$, then the directed edge from node $e^{\prime}$ to node $e$, denoted by $(e^{\prime}\rightarrow e)$, is said to belong to $\mathcal{A}$. Such a graph $\mathcal{G}=(\mathcal{E},\mathcal{A})$ is called a Granger causality graph. For a multivariate temporal point process, the Granger causality graph is defined below.

We treat each path to purchase as a $p$-dimensional point process $\mathbf{N}(t)$. In the attribution problem, there are two categories of event types, firm-initiated event types and customer-initiated event types. The firm-initiated event types (Wiesel et al., 2011) are initiated by firms such as email sent, social ad impression, and display impression. The customer-initiated event types (Bowman & Narayandas, 2001; Wiesel et al., 2011) are initiated by customers or prospective customers, including conversion, search ad impression and click, email click, and so forth. Let $\mathcal{E}_{\mathrm{f}}$ and $\mathcal{E}_{\mathrm{c}}$ denote the set of firm-initiated event types and the set of customer-initiated event types and thus $\mathcal{E}=\mathcal{E}_{\mathrm{f}}\cup\mathcal{E}_{\mathrm{c}}$ and $\mathcal{E}_{\mathrm{f}}\cap\mathcal{E}_{\mathrm{c}}=\emptyset$. Suppose there are $q$ customer-initiated event types for some $1\leq q<p$. Without loss of generality, let $\mathcal{E}_{\mathrm{c}}=\{1,\dots,q\}$ with conversion labeled as $e=1$ and $\mathcal{E}_{\mathrm{f}}=\{q+1,\dots,p\}$. We assume that the point process of the firm-initiated event types $\mathbf{N}_{\mathcal{E}_{\mathrm{f}}}(t)$ is controlled by the firms or ad advertisers strategically. In other words, $\mathbf{N}_{\mathcal{E}_{\mathrm{f}}}(t)$ only serves as an observed input whose conditional intensity function does not require learning. For the point process of customer-initiated event types $\mathbf{N}_{\mathcal{E}_{\mathrm{c}}}(t)$, we model it through the following conditional intensity:

• •

  $\mu_{e}\geq 0$ is the baseline intensity, which corresponds to the sources of the occurrence of type-$e$ events other than the occurrence history of itself and other types of touchpoints, namely the intrinsic tendency of the occurrence.

• •

  $\psi_{e^{\prime}e}(\cdot)\geq 0,\ e^{\prime}=1,\dots,p$, is a bounded, left-continuous kernel function, $\psi_{e^{\prime}e}(t)=0$ when $t\leq 0$ and $\int_{0}^{\infty}\psi_{e^{\prime}e}(t)dt=1.$ The kernel function $\psi_{e^{\prime}e}(\cdot)$ describes the shapes of the touchpoints’ impact. For example, $\psi_{e^{\prime}e}(t)=\frac{1}{T_{0}}\cdot\mathbbm{1}_{\{0<t\leq T_{0}\}}$ accounts for a constant impact of a previous type-$e^{\prime}$ event within $T_{0}$ of its occurrence; $\psi_{e^{\prime}e}(t)=\frac{1}{T_{0}}\exp(-\frac{t}{T_{0}})\cdot\mathbbm{1}_{\{t>0\}}$ works for an exponential decaying impact; $\psi_{e^{\prime}e}(t)=\sqrt{\frac{2}{\pi T_{0}^{2}}}\exp(-\frac{t^{2}}{2T_{0}^{2}})\cdot\mathbbm{1}_{\{t>0\}}$ can be used when the decaying impact is even faster.

• •

  $\alpha_{e^{\prime}e}\geq 0,\ e^{\prime}=1,\dots,p$, is the Granger causality coefficient. The value of $\alpha_{e^{\prime}e}$ describes the scale of the temporal dependence.

To interpret the coefficient $\alpha_{e^{\prime}e}$, we introduce the following theorem.

This result is an adaptation from the case of the multivariate Hawkes process (Eichler, 2012; Xu et al., 2016). Analogous to the multivariate Hawkes process, the meaning of $\alpha_{e^{\prime}e}$ can be described in two cases: $\alpha_{e^{\prime}e}>0$ implies an exciting effect from event type $e^{\prime}$ to event type $e$ by increasing its conditional intensity $\lambda_{e}(t\mid\mathcal{H}_{t})$; the case $\alpha_{e^{\prime}e}=0$ implies no Granger causality from event type $e^{\prime}$ to event type $e$ since $\lambda_{e}(t\mid\mathcal{H}_{t})$ is not affected by the occurrence of any type-$e^{\prime}$ event. By such construction, we can use the matrix $A=\left(\alpha_{e^{\prime}e}\right)_{e^{\prime}\in\mathcal{E},e\in\mathcal{E}_{c}}\in\mathbb{R}_{\geq 0}^{p\times q}$ to stand for the graphical Granger causality structure for customer-initiated event types.

As the firm-initiated event types are fully controlled by marketers or ad advertisers, in our proposed model, we assume that they are not dependent on other types of events. That is to say, the ground truth of $\mathcal{A}$ for $e\in\mathcal{E}_{\mathrm{f}}$ is known with $N_{e^{\prime}}\nrightarrow N_{e}\mid\mathbf{N}_{\mathcal{E}\setminus\{e,e^{\prime}\}}$ for any $e^{\prime}\in\mathcal{E}$.

In this subsection, we calculate attribution scores using the direct removal effect from the point of view of the graphical point process. We define the direct removal effects (DRE) as the relative change in conversion intensity when we remove only the event(s) of interest from the path and assume other touchpoints on the path remain unaffected. That is, the direct removal effect considers merely the influence of the studied event(s) on an occurrence of conversion and ignores their influence on other events. Graphically speaking, the direct removal effect focuses on the direct Granger causality parent nodes of conversion and does not depend on the hierarchy beyond them.

For a positive path $D$, suppose there is a conversion at $t=t_{i^{\star}}$ for some $1<i^{\star}\leq m$, that is, $e_{i^{\star}}=1$. Let $F^{W}_{t}(D):=\{(t_{i},e_{i})\in D:t_{i}<t,\ e_{i}\in W\}$ be the set of occurred events before $t$ whose event labels belong to $W$, where $W\subseteq\mathcal{E}$ is an arbitrary event label subset. Especially, let $F_{t}(D)$ stand for $F^{\mathcal{E}}_{t}(D)$, which is the truncated path before $t$. We are interested in the influence of a subset of touchpoints $R\subseteq F_{t^{\star}}(D)$ on this conversion and call $R$ a removal set. Let $\mathbf{N}^{D}(t)=(N_{1}^{D}(t),\dots,N_{p}^{D}(t))^{\top}$ denote the point process with respect to path $D$ with $N_{e}^{D}(t)=\sum_{(t_{i},e_{i})\in D:e_{i}=e}\mathbbm{1}_{\{t_{i}\leq t\}}$ for $e=1,\dots,p$. Let $\mathcal{H}_{t}^{D}$ be the corresponding filtration $\sigma\{\mathbf{N}^{D}(u):u<t\}$. We can calculate the attribution score as the direct removal effect of $R$ with respect to path $D$:

The above definition is a general statement that does not depend on any assumptions of point process modeling.

Suppose that the conversion event satisfies model (1), which means the conditional intensity function of this conversion takes the form

We can derive more specific attribution scores based on Equation (2) and (3). For example, the direct removal effect of touchpoint $(t_{i},e_{i})$ can be calculated by

This expression shows the relationship between the Granger causality graph and the attribution score. A touchpoint $(t_{i},e_{i})$ can be attributed with a score only if its touchpoint type is a parent node of the conversion node on the graph or $(e_{i}\rightarrow 1)\in\mathcal{A}$. Generally, for a subset $R$ of $F_{t^{\star}}(D)$, its direct removal effect is

Besides, under model (1), the baseline effect can be defined to be $\frac{\mu_{1}}{\lambda_{1}(t_{i^{\star}}\mid\mathcal{H}_{t_{i^{\star}}}^{D})}$. For example, Figure 3 shows the direct removal effects of touchpoints with respect to a path consisting of three touchpoints and a conversion.

In the following context, we will explain the attribution score’s general form (2) in probability language. Consider an ideal experiment of two customers, A and B, with a toy example shown in Figure 4. Suppose A and B react to touchpoints in the same way. Their conditional intensity functions for customer-initiated event types follow the same model. We observe $D_{\mathrm{A}}$, the path of A, with a conversion at $t=t_{i^{\star}}>0$. Let $D_{\mathrm{B}}$ denote the path of B. Suppose $F^{\mathcal{E}\setminus\{1\}}_{t_{i^{\star}}}(D_{\mathrm{B}})\subsetneqq F^{\mathcal{E}\setminus\{1\}}_{t_{i^{\star}}}(D_{\mathrm{A}})$, the touchpoints on $D_{\mathrm{B}}$ are a subset of those on $D_{\mathrm{A}}$. Assume that there is no information about $N_{1}^{D_{\mathrm{B}}}(t)$. Namely, we do not observe whether B converts or not. We can sample the process $N_{1}^{D_{\mathrm{B}}}(t)$ to obtain a complete path $D_{\mathrm{B}}$. We exploit the thinning operation for a temporal point process, which uses some definite rule to delete points of a basic point process, yielding a new point process.

We refer to the point process $N(t)$ as a background process for the desired point process $N^{\prime}(t)$. In general, Ogata (1981) suggested using a Poisson process as the background process for point process simulation. To sample the $N_{1}^{D_{\mathrm{B}}}(t)$, we can look for a constant $\overline{\lambda}$ satisfying $\overline{\lambda}\geq\lambda_{1}(t\mid\mathcal{H}_{t}^{D_{\mathrm{B}}})$ for $t\in[0,T]$ and use the Poisson process with rate $\overline{\lambda}$ as the background process.

With a closer look at the problem, we can find that sampling a Poisson process as the background process is not necessary. The point process of conversion for A, $N_{1}^{D_{\mathrm{A}}}(t)$, can also serve as a background process. Based on (3), we have $\lambda_{1}(t\mid\mathcal{H}_{t}^{D_{\mathrm{B}}})\leq\lambda_{1}(t\mid\mathcal{H}_{t}^{D_{\mathrm{A}}})$ before any occurrence of conversion for B. Theorem 2 tells us that if we delete the conversion at $t=t_{i^{\star}}$ in $N_{1}^{D_{\mathrm{A}}}(t)$ with probability $1-\lambda_{1}(t_{i^{\star}}\mid\mathcal{H}_{t_{i^{\star}}}^{D_{\mathrm{B}}})/\lambda_{1}(t_{i^{\star}}\mid\mathcal{H}_{t_{i^{\star}}}^{D_{\mathrm{A}}})$, then we obtain a process following the distribution described in (3) for B. Compared with (2), this probability is nothing but the direct removal effect of $D_{\mathrm{A}}\setminus D_{\mathrm{B}}$ with respect to path $D_{\mathrm{A}}$.

The direct removal effect of a given subset of events describes the expected loss of conversion by comparing the path without it to the original path. In this subsection, we study another attribution score that measures the overall influence resulting from removing the subset of events along the path. The corresponding attribution score is the marginal lift of the expected intensity of conversion by considering not only the removal of the events of interest but also the potential loss of other related customer-initiated events.

For example, the direct removal effect of the first touchpoint $(t_{1},e_{1})\in D$ can be studied through $D\setminus\{(t_{1},e_{1})\}$. But removing such a touchpoint in the early stage of the path will result in more than the removal of itself. Some later touchpoints may get affected and tend not to occur. So $D\setminus\{(t_{1},e_{1})\}$ may not well represent a possible remaining path described by model (1) because $\mathbf{N}_{\mathcal{E}_{\mathrm{c}}}^{D\setminus\{(t_{1},e_{1})\}}(t)$ is not guaranteed to follow the same model as $\mathbf{N}_{\mathcal{E}_{\mathrm{c}}}^{D}(t)$. In other words, the direct removal effect may not well reflect the overall influence of the removal of the touchpoint.

By model (1), for any $e\in\mathcal{E}_{\mathrm{c}}$ (including conversion),

The removal of $R$ from $D$ may result in the missing of certain subsequent touchpoints as well as paid conversion. The actual remaining path may contain even fewer event occurrences, which yields a subset of $D\setminus R$. This inspires us to perform the thinning operation to $\mathbf{N}_{\mathcal{E}_{\mathrm{c}}}(t)$ according to Theorem 2 instead of thinning the univariate process $N_{1}(t)$ only.

Let $i_{\min}(R)=\min\{1\leq i\leq m:(t_{i},e_{i})\in R\}$ be the event index of the first event in $R$. Recall the ideal experiment of two customers, A and B. Now we assume that $D_{\mathrm{A}}=D$ for A and the process of the customer-initiated event types $\mathbf{N}_{\mathcal{E}_{\mathrm{c}}}^{D_{\mathrm{B}}}(t)$ is unknown for B. Let $D_{\mathrm{B}}=D\setminus R$ first. By Theorem 2, if we delete the touchpoint $(t_{i},e_{i})$ in $F^{\mathcal{E}_{\mathrm{c}}}_{t_{i^{\star}}}(D_{\mathrm{B}})$ from $D_{\mathrm{B}}$ with probability $1-\lambda_{e_{i}}(t_{i}\mid\mathcal{H}_{t_{i}}^{D_{\mathrm{B}}})/\lambda_{e_{i}}(t_{i}\mid\mathcal{H}_{t_{i}}^{D_{\mathrm{A}}})$ sequentially for $i>i_{\min}(R)$, then the obtained process $\mathbf{N}_{\mathcal{E}_{\mathrm{c}}}^{D_{\mathrm{B}}}(t)$ follows model (1). Suppose $D\setminus R^{\diamond}$ is the thinned version of $D_{\mathrm{B}}$ with $R^{\diamond}\supseteq R$ being the actual removal set. Figure 5 shows a toy example of this idea. As a result, given a subset $R$ of a path $D$, its total removal effect (TRE) is defined by

where the conditional expectation is with respect to the actual removal set $R^{\diamond}$ from the random thinning operation. The algorithm with the thinning operation is summarized in Algorithm 1.

The largest value of $R^{\diamond}$ is $R\cup\{(t_{i},e_{i})\in F^{\mathcal{E}_{\mathrm{c}}}_{t_{i^{\star}}}(D\setminus R):i>i_{\min}(R)\}$, denoted by $\Omega$ for ease of notation. Using the thinning algorithm, we can derive an explicit form of the total removal effect:

During the thinning operation, the value of $R^{\diamond}$ depends on sampling a sequence of Bernoulli random variables. Let $U_{i}\sim\mathrm{Bernoulli}(1-\frac{\lambda_{e_{i}}(t_{i}\mid\mathcal{H}_{t_{i}}^{D\setminus R^{\prime}})}{\lambda_{e_{i}}(t_{i}\mid\mathcal{H}_{t_{i}}^{D})})$ be the Bernoulli random variable for thinning event $(t_{i},e_{i})\in\Omega$. Then the conditional probability mass function of $R^{\diamond}$ is

The above probability mass function could be difficult to implement in practice due to the combinatorial subset calculation. Under model (1), we can change the order of summation to derive the total removal effect according to the linear decomposition of the direct removal effect in (4).

This result implies another iterative algorithm, which is simulation-free and more efficient. The basic idea is to redistribute the obtained scores from the direct removal effect. It adopts an intuitive backpropagation way of scoring as summarized in Algorithm 2, and the backpropagation, introduced by Rumelhart et al. (1986), has been gaining popularity in artificial neural networks.

As shown by the example path in Figure 6, instead of going forward along the path, the score flows backward from the last touchpoint to the previous touchpoints. This flow follows the reverse graph of the Granger causality graph, whose arrows point from an event type to its Granger causality parents.

In this subsection, we discuss the difference between the proposed two attribution scoring methods and their implementation in an extreme case.

The direct removal effect regards the contributions from each touchpoint as individual components. While the total removal effect of a specified subset of touchpoints is the cumulative influence resulting from its removal along the path. The direct removal effect regards the conversion event as the only response, while the whole vector/subprocess of customer-initiated event types is the response for the total removal effect. Intuitively, on the score-flow graph in Figure 6, the direct removal effect lets the score directly come out from the conversion node, and the total removal effect is based on the continual flow of the score until it reaches a specified node or a node with no parents.

These two scores differ in the context of additivity. Suppose $R_{1}$ and $R_{2}$ are two mutually exclusive subsets of the truncated path $F_{t^{\star}}(D)$. The direct removal effect under model (1) has the additive property implied by the definition that

However, for the total removal effect, we let the score flow between the two sets during backpropagation wherever there is Granger causality between their event types. Hence the total removal effect is subadditive and thus not incremental

As a result, to obtain the total removal effect of a channel, we should select the removal set as the set of all the corresponding individual touchpoints rather than take the sum of touchpoint-wise scores.

We consider an extreme case illustrated by the Granger causality graph in Figure 7. In practice, we believe that touchpoints in each channel Granger-cause conversion separately, and hence the conversion node has multiple parents on the Granger causality graph like Figure 2. But here we use this example to justify the proposed two attributions scores. On the graph, conversion is the event type labeled as $1$, and all the other event types $2,\dots,p$ are touchpoints, where the touchpoint type labeled as $p$ is the only firm-initiated event type. Suppose there is no baseline intensity for any customer-initiated event type, namely $\mu_{e}=0$ for $e=1,\dots,p-1$. Then a type-$(e+1)$ event will be the only reason for the occurrence of a type-$e$ event. Suppose we observe a path $D$ with the same pattern as Figure 7. That is,

If we look at the direct removal effect of each touchpoint, we get

where $e_{i}=p-i+1$ for $i=1,\dots,p-1$. The direct removal effect ignores the importance of touchpoints in the early stage since they cannot instantly trigger a conversion. On the other hand, the total removal effect gives

It seems that the total removal effect can over-allocate since there is only $1$ conversion on path $D$. This issue was also pointed out by Singal et al. (2022) as a drawback of attribution methods based on the removal effect. However, it does not mean the scores are wrong. This extreme over-allocation problem implies that it might be inappropriate to calculate such an attribution score for each touchpoint. In Figure 7, it is likely that event types $2$ to $p$ all belong to the same channel. A detailed example can be the case where email sent, email open, and email click are the touchpoints, and conversion can be only triggered by email click (invite-only purchase through email link). In this case, we can regard event types $2$ to $p$ as an entirety using the removal set $R=\{(t_{1},p),\dots,(t_{p-2},3),(t_{p-1},2)\}$. Then we have

If event types $2$ to $p$ are actually from different channels, it means the graph is not likely true, or there is no need to distinguish these channels from each other since they are highly dependent.

To summarize, the direct removal effect serves as an explanatory score by allocating the credit of a conversion event to each touchpoint according to its incremental influence. Both proposed methods can give channel-level scores, but the total removal effect provides a marginal point of view for conversion. In other words, it better applies to cases where the leave-one-channel-out loss of conversion is the desired quantity.

Suppose there are $n$ paths in total, $D_{1},\dots,D_{n}$, where $D_{j}$ is the $j$-th path on time interval $[0,T_{j}]$. Suppose the kernel functions $\psi_{e^{\prime}e}(\cdot),\ e^{\prime}=1,\dots,p,\ e=1,\dots,q,$ are known. Let $\bm{\mu}=(\mu_{1},\dots,\mu_{q})^{\top}\in\mathbb{R}_{\geq 0}^{q}$ be the vector of baseline intensities of $\mathbf{N}_{\mathcal{E}_{\mathrm{c}}}(t)$ and $\bm{\alpha}_{e}=(\alpha_{1e},\dots,\alpha_{pe})^{\top}\in\mathbb{R}_{\geq 0}^{p}$ be the $e$-th column of $A$ for $e=1,\dots,q$. Learning the model turns into an optimization problem:

where $\Phi(D_{j};\bm{\mu},\bm{\alpha}_{1},\dots,\bm{\alpha}_{q})$ is the loss function given the $j$-th path $D_{j}$. Consider a least-squares functional:

Compared with the negative log-likelihood, the least-squares functional enjoys better computational efficiency. Its equivalent form was proposed for estimating the additive risk model by Lin & Ying (1994). It was also adopted by Hansen et al. (2015) and Bacry et al. (2020) for point process models.

Assume that the edge set is sparse, we add the sparsity constraints on the coefficients as follows:

where $\gamma_{1},\dots,\gamma_{q}\geq 0$ are the regularization parameters to control the individual sparsity of coefficient vectors $\bm{\alpha}_{1},\dots,\bm{\alpha}_{q}$. It is worth pointing out that $\{\mu_{e},\bm{\alpha}_{e}\}_{e=1}^{q}$ are separable in the objective function (5). Thus, the optimization problem can be decomposed into node-wise model learning. For each node $e\in\mathcal{E}_{\mathrm{c}}$, learning its parent nodes yields

where

In the following context, we write $\bm{\theta}=(\theta_{0},\theta_{1},\dots,\theta_{p})^{\top}=(\mu_{e},\bm{\alpha}_{e}^{\top})^{\top}\in\mathbb{R}_{\geq 0}^{p+1}$. Let $\bm{X}_{j}(t)=(X_{j,0}(t),X_{j,1}(t),\dots,X_{j,p}(t))^{\top}$ with

Now the conditional intensity can be written as $\lambda_{e}(t\mid\mathcal{H}_{t}^{D_{j}})=\bm{\theta}^{\top}\bm{X}_{j}(t)$. Let $V=(V_{kk^{\prime}})_{k,k^{\prime}=0}^{p}\in\mathbb{R}^{(p+1)\times(p+1)}$ and $\bm{b}=(b_{0},\cdots,b_{p})^{\top}\in\mathbb{R}^{p+1}$, where for $k=0,\dots,p$,