The taxonomy of research collaboration in science and technology: evidence from mechanical research through probabilistic clustering analysis

Abstract

This paper suggests an empirical framework to classify research collaboration activities with developed indicators that carry on a previous theoretical framework (Wagner [Science and Technology Policy for Development, Dialogues at the Interface, 2006]; Wagner et al. [Linking effectively: Learning lessons from successful collaboration in science and technology. DB-345-OSTP, 2002]) by employing the Gaussian mixture model, an advanced probabilistic clustering analysis. By further exploring the method upon a profound evidence-based reflection of actual phenomena, this paper also proposes an exploratory analysis to manage and evaluate research projects upon their differentiated classification in a preceding perspective of research collaboration and R&D management. In addition, the results show that international collaboration tends to be associated with more evenly committed collaboration, and that collaboration featuring a higher degree of funding or dispersed commitments generally results in larger outcomes than research clustered on the opposite side of the framework.

Appendix

Finding the maximum likelihood mixture density parameters via the EM algorithm can be performed as follows. AIC and MDL differ in their penalty terms. The MDL criterion reflects the total number of NM ¹⁶ data values, unlike the AIC criterion, in which data tends to result in an over-fitting of the model. Thus, when the number of observations heads toward infinity, the estimated model order C does not converge to the true value (Kashyap 1980). The MDL criterion, on the contrary, attempts to find the model order C by minimizing the MDL value, the code of which reflects both the number of data samples and the parameter vector. Consequently, unlike AIC, the MDL criterion does not have the limitations of over-fitting the model by generally guaranteeing a consistent estimator (Bouman et al. 2005). This study adopts the MDL criterion proposed by Rissanen (1983) for order identification.

Direct minimization of the MDL criterion is difficult, but if only the clusters of each observation, $x_{n} ,$ are known, then the estimation of parameters $\theta = \left( {\pi ,\mu ,W} \right)$ will be quite simple (Bouman et al. 2005).

The EM algorithm is a method that is generally used to find the MLE of the parameters, especially when the data is incomplete or has missing values (Dempster et al. 1977). Although the EM algorithm can be applied in cases where the data actually has missing values, it is generally applied in cases where optimizing the likelihood function is analytically intractable or the likelihood function can be simplified by assuming the existence of values for additional but missing or hidden parameters.

The EM algorithm first finds the expected value of the complete-data log-likelihood $Q(\theta ;\theta^{(i)} )$ with respect to the unknown data, $x_{n} ,$ given $Y_{n}$ and $\theta^{(i)} ,$ which are the observations and current parameter estimates of the clusters, respectively. Membership is represented by a probability function. This study adopts the merging approach taken by Rissanen (1983), which constrains the parameters of two clusters to be equal to a decrease in the number of clusters from C to C − 1. In other words, if two clusters, a and b, are merged into a single cluster, their mean and covariance parameters are constrained to be equal, as denoted in Eqs. A.1 and A.2.

$$ \mu_{a} = \mu_{b} = \mu_{(a,b)} , $$

(A.1)

$$ W_{a} = W_{b} = W_{(a,b)} , $$

(A.2)

$$ d(a,b) = Q(\theta^{*} ;\theta^{(i)} ) - Q(\theta^{*}_{(a,b)} ;\theta^{(i)} ),\,{\text{and}} $$

(A.3)

$$ \left( {a^{*} ,b^{*} } \right) = \mathop {\arg \min }\limits_{(a,b)} d(a,b), $$

(A.4)

where $\mu_{(a,b)}$ and $W_{(a,b)}$ denote the mean and covariance of the new cluster and $d(a,b)$ denotes the distant function, In addition, $\theta^{*}$ and $\theta^{*}_{(a,b)}$ are the unconstrained and constrained optima. In particular, if the EM algorithm has been conducted to converge for a fixed order $C,\,\theta^{*}$ equals $\theta^{(i)} ,$ which satisfies $Q(\theta^{(i)} ;\theta^{(i)} ) - Q(\theta^{*} ;\theta^{(i)} ) = 0$. The value of $\theta^{*}_{(a,b)}$ is obtained by maximizing $Q(\theta_{(a,b)} ;\theta^{(i)} )$ as a function of $\theta_{(a,b)} ,$ subject to the constraints.

Having found the cluster pair $(a^{*} ,b^{*} )$ that minimizes the distant function, and an upper bound on the change in the MDL criterion among all pairs, $(a^{*} ,b^{*} )$ are merged. The parameters of this merged cluster are then calculated and used as an initial value for another EM optimization process with C − 1 clusters (Bouman et al. 2005).