2.4 Proposed Approach for Nonlinear PLSPM
2.4.2 The PLSPM Algorithm using Piecewise Inner Weights
Estimation
As discussed in the previous chapter, PLS path modelling aims to esti- mate the relationships among J (j = 1, . . . , J) blocks of manifest vari- ables, which are expression of unobservable variables. The algorithm is characterised by a system of interdependent equations based on simple and multiple linear regressions. The algorithm estimates the dependence relationships among LVs as well as the relationships between MVs and their own LVs.
This section will present the proposed algorithm using piecewise inner weights estimation process, showing how this part is embedded in the standard algorithm.
With the aim of ease the readability of the proposed algorithm some of the steps presented in the previous chapter will be repeated.
Starting Weights Definition
The first step of the PLSPM algorithm regards the definition of a set of arbitrary weights wpj to be used as starting point. These weights are
then normalised in order to produce LVs with unitary variance. Common choices for starting weights are presented in subsection 1.2.3.
Measurement Model: Latent Variables Calculation
Once defined the initial weights the algorithm moves to the outer estimate yj of the standardised (with mean = 0 and standard deviation = 1) latent
variables (⇠j mj). The composites are estimated as linear combination
of their centered MVs: yj / ± 2 4 Pj X p=1 wpj(xpj x¯pj) 3 5 (2.12)
where the/ symbol means that the variables on the left is proportional to the operator on the right; the ± operator represents the sign ambiguity. This problem is solved by selecting the sign that makes the variable yj
positively correlated with the majority of manifest variables xpj.
The j-th estimated latent variable is obtained as follows:
yj = Pj X p=1 ˜ wpj(xpj x¯pj) (2.13)
The coefficients wpj and ˜wpj are called outer weights.
ˆ mj = Pj X p=1 ˜ wpjx¯pj (2.14)
and the latent variable ⇠j is estimated by:
ˆ ⇠j = Pj X p=1 ˜ wpjxpj = yj + ˆmj (2.15)
Structural Model: Piecewise Inner Weights Estimation
The structural model aims to give an estimate of the LVs based on the causal relations present in the inner model. Those relationship may present nonlinear patterns. In order to correctly estimate nonlinearity, the inner weights can be estimated by using the piecewise inner weight estimation process presented in the previous sections. This process can be applied transversally to most of the existing techniques referred in the standard algorithm.
Based on these assumptions, the piecewise inner weights eh
jj0 can be
estimated in di↵erent ways presented in the next paragraphs. The piecewise inner weights eh
jj0 can be defined as a vector of dimensions
n⇥ 1. The i-th value of eh
jj0 corresponds to the weight estimated for the
subset (defined in the third column of the matrix Yjj0) to which the i-th
observation belongs.
Piecewise Centroid Scheme The piecewise centroid scheme rep- resents an adaptation of the original technique proposed by H. Wold. The n⇥ 1 vector eh
jj0 can be defined as having the i-th value correspond-
i-th observation belongs. This can be obtained as: ehjj0 = sign h corrh Yjj0 i (2.16) where Yjj0 represents the matrix shown in 2.11 and the function corrh
represents the piecewise correlation function among the two latent vari- ables j and j0. As referred in the previous section, the piecewise function returns a n⇥1 vector where the generic value i belonging to the h-th sub- set, represents the correlation calculated within the subset h. In this case eh
jj0 are the signs of this piecewise correlations (see Figure 2.7) between
yj and the latent variable yj0 connected to yj (see Equation 2.16).
Piecewise Factorial Scheme The Piecewise Factorial Scheme rep- resents the proposed approach to one of the two techniques proposed by Lohm¨oller where inner weights eh
jj0 are calculated as follows:
ehjj0 = corrh Yjj0 (2.17)
where Yjj0 represents the matrix shown in 2.11 and the function corrh
represents the piecewise correlation function among the two latent vari- ables j and j0. As referred in the previous sections, the piecewise function returns a n⇥ 1 vector where the generic value i belonging to the h-th subset, represents the correlation calculated within the subset h.
Some Observation It has been decided that the Path Weighting scheme proposed by Lohm¨oller would not be adapted to be included as a potential scheme for the proposed nonlinear approach.
The biggest argument in favour of Path Weighting scheme is the fact that this takes into account the path direction.
The proposed nonlinear approach presents the possibility of switching the inner weights estimation from symmetrical to non-symmetrical. If a symmetrical approach is chosen, then the weighting system obtained to calculate the composite zj is calculated using a symmetrical square
matrix of dimensions J⇥ J. In this case, all weights present in the upper diagonal matrix are exactly the same as the ones present in the lower diagonal matrix.
In case the researcher wants to use the non-symmetrical approach, the weights present in the upper diagonal matrix are di↵erent from the ones in the lower diagonal matrix. In this situation, when estimating the weights, if j > j0 (lower diagonal matrix), then the weight is calculated using yj as dependent and yj0 as independent. When j < j0 (upper
diagonal matrix), then the weight is calculated using yj0 as dependent
and yj as independent.
Given to its flexibility, the proposed approach can be applied to other ex- isting techniques and allow the development of novel customised weights estimation techniques.
Structural Model: Latent Variables Calculation
In the inner LVs calculation stage, the standardised (⇠j mj) latent
variables inner estimation zj is given by:
zj /
X
j0 : ⇠
j0 adjacent to ⇠j
where denotes the Hadamard product.
Starting from this stage, the algorithm is exactly the same as the stan- dard algorithm presented in section 1.2. The remaining step is the outer weights estimation related to the measurement model.
Iterative Process and Convergence
After the first cycle the algorithm iterates the following steps: 1. Measurement Model: Latent Variables Calculation 2. Structural Model: Piecewise Inner Weights Estimation 3. Structural Model: Latent Variables Calculation
4. Measurement Model: Outer Weights Estimation 5. Outer Weights Convergence Check
The aforementioned algorithm is described in figure 2.8 and its pseudo- code is shown in Algorithm 2.
After reaching the algorithm convergence, the outer weights wpj are used
to obtain the final estimation of ⇠j calculated as ˆ⇠j =Pwpjxpj.
In the last step of the proposed nonlinear approach to PLSPM, path polynomial functions are estimated. The final output is a function for each relationship existing in the path diagram structure. The generic function jj0 representing the relation between yj (i.e., the composite
for the endogenous LV ⇠j) and yj0 (i.e., the composite for the exogenous
Figure 2.8: PLSPM Iterative Estimation Process using Piecewise Inner Weights Estimation
jj0 = 0+ 1yj0 + 2y2j0 + . . . + dyHj0 (2.19)
where the degree H is defined following the same proprieties used in the second step of the proposed algorithm (Evaluate and Select Polynomial Functions). These polynomial functions jj0 are not easy to interpret.
For this reason the algorithm produces a set of scatter plots (one for each inner relation) with the jj0 function. An example can be seen in figure
2.9.
The algorithm convergence is analysed in the next chapter using several simulation scenarios.
Algorithm 2: Nonlinear PLSPM Input : X = [X1, . . . , Xj, . . . , XJ]
Output: wpj, ˆ⇠j, jj0
1 Arbitrary Weights Initialisation: wpj = wpj(0)
2 while Convergence of wpj is not reached (or max number of
iterations) do
3 Latent Variables Proxies Calculation (Measurement
Model):
yj / ±hPpwpj(xpj x¯pj)
i
4 Piecewise Inner Weights Estimation (Structural Model):
for Every structural connection between yj and yj0 do
Fit Polynomial Functions
Evaluate and Select Polynomial Functions Find Stationary Points
Calculate Piecewise Correlations as eh
jj0 = f h⇣y
j, yj0
⌘
5 Latent Variables Proxies Calculation (Structural Model):
zj /Pj0 : ⇠
j0 adjacent to ⇠j
⇣
ejj0 yj0
⌘
6 Outer Weights Estimation (Measurement Model):
wpj = f (X, Z) according to the chosen estimation technique 7 Final Latent Variables Proxies Calculation:
ˆ
⇠j = Pwpjxpj
8 Path Polynomial Functions Estimation:
jj0 = 0+ 1yj0 + 2y2j0 + . . . + dyHj0 according to the chosen
degree