Methodological Approach

Following the Agovino et al. approach, we proceed in three steps. First, we analyze the transition probabilities of Italian provinces among three different e-waste collection states. Second, we investigate the state of the e-waste collection points in the Italian provinces. To perform these analyses, the probability transition matrix methodology was adopted. Thirdly, we have analysed the presence of a correlation between the WEEE collection rate and the dynamics of the CCs in different areas and provinces in order to better comprehend the role that can play both the investments in the CC system and other soft measures (e.g.,: communication, information and education campaigns)in achieving the WEEE collection targets.

In particular, the transition matrix is a methodology to measure the probability of moving an element from initial state (at time t) to a new state at time t + 1. Ina transition probability matrix (P), the generic term P_{ij} is defined as P_{Xi−Xj}=\dfrac{a_{ij}}{∑_ia_i(t)}, where as a_{ij} represents the number of elements moving from an initial i state to a final j state (Table 2). The rows and columns of the matrix describe, respectively, the initial state and the final state, while the terms on the main diagonal represent the steady state, namely the probability of an element to remain in the same condition during a given unit of time. Furthermore, it is possible to recognize an "absorbing state" which occurs, in general, whenever the probability that an element exits at t + 1 from that state is zero. In other words, when one of the diagonal transition probabilities of the matrix is unity.

Table 2. Transition probability matrix.

t+1
t
 X_1 ... X_i ... X_n
 total (a)
X_1 P_{X_1-X_1} ... P_{X_1}−X_i ... P_{X_1}−X_n ∑_{k=1}^{n}P_{X_1−X_k}
... ... ... ... ... ... ...
X_i P_{X_i-X_1} ... P_{X_i}−X_i ... P_{X_i}−X_n ∑_{k=1}^{n}P_{X_i−X_k}
... ... ... ... ... ... ...
X_n P_{X_n-X_1} ... P_{X_n}−X_i ... P_{X_n}−X_n ∑_{k=1}^{n}P_{X_n−X_k}
total (b)
 ∑_{z=1}^{n}P_{X_z−X_1} ... ∑_{z=1}^{n}P_{X_z−X_i} ... ∑_{z=1}^{n}P_{X_z−X_n} ∑_{z=1}^{n} ∑_{k=1}^{n}P_{X_z−X_k}


The elements of matrix P are probability that they are subject to the following axioms of probability:

P_{ij} ≥0 ∀i, j,

∑^{N}_{j=1}P_{ij}=1 ∀i.

The data about WEEE's management system have been extracted from the records of the Italian Coordination center which is the institution set by WEEE Legislative Decree 151/05 (art. 13) for "the optimization of the activities of competence of the collective systems, to guarantee common, homogeneous and uniform operating conditions".

In particular, we used data about kg of WEEE collected and about the CCs related to the 110 Italian provinces, corresponding to the European level NUTS-3 over the period 2009–2017. Level NUTS-3 represents a more widespread WEEE collection data, having substantial internal homogeneity and the possibility to interpret the data deviations caused by limiting the influences of the territorial and demographic dimension.

Data about the provinces population in the 2008–2017 period were extracted by ISTAT (Italian National Institute of Statistics) records.