Research on Incentive Mechanism for New Villager Participation in Rural Industry in the Age of Artificial Intelligence

This block structure has the following predefined specifications: (3) E(x1h|ξ1)=π1h0+π1hξ1\[E\left( {{x}_{1h}}|{{\xi }_{1}} \right)={{\pi }_{1h0}}+{{\pi }_{1h}}{{\xi }_{1}}\] (4) E(x2k|ξ2)=π2k0+π2kξ2\[E\left( {{x}_{2k}}|{{\xi }_{2}} \right)={{\pi }_{2k0}}+{{\pi }_{2k}}{{\xi }_{2}}\]

In the mathematical equations above, x₁,x₂ is the observed variable. ξ₁,ξ₂ is the latent variable; π_1ho,π_2k0 is the locus parameter in the respective equation; π_1h,π_2k is the loading of the observed variable x₁,x₂ respectively; and v_1h,v_2k represents the corresponding residual term.

In structural equation modeling (as should be the case in other models as well), it is necessary to standardize all data in order to overcome the differences in the corresponding coefficients due to the different units of the variables. The need for data standardization is especially true in structural equation models where the latent variable ξ₁,ξ₂ is unknown. Standardizing means making the mean of the variable 0 and the variance 1. The mathematical representation is as follows: (5) E(ξ1)=0\[E\left( {{\xi }_{1}} \right)=0\] (6) E(ξ2)=0\[E\left( {{\xi }_{2}} \right)=0\] (7) Var(ξ1)=1\[Var\left( {{\xi }_{1}} \right)=1\] (8) Var(ξ2)=1\[Var\left( {{\xi }_{2}} \right)=1\]

There is also the following canonical setting: the residuals v_1h,v_2k in each block-structured equation are independent of their corresponding latent variables ξ₁,ξ₂, respectively. The mathematical expression is as follows: (9) r(v1h,ξ1)=0\[r\left( {{v}_{1h}},{{\xi }_{1}} \right)=0\] (10) r(v2k,ξ2)=0\[r\left( {{v}_{2k}},{{\xi }_{2}} \right)=0\]

Similarly, as determined by the specification of the soft model of the structural equations: the residuals, the latent variables in each block of structural equations are also independent of each other. Expressed mathematically as follows: (11) r(v1h,ξ2)=0\[r\left( {{v}_{1h}},{{\xi }_{2}} \right)=0\] (12) r(v2k,ξ1)=0\[r\left( {{v}_{2k}},{{\xi }_{1}} \right)=0\] (13) r(v1h,v2k)=0\[r\left( {{v}_{1h}},{{v}_{2k}} \right)=0\]

Further, it is also necessary to assume that the residuals in each block structure equation are also independent of each other. The mathematical expression is as follows: (14) r(υ1h,υ1h′)=0\[r\left( {{\upsilon }_{1h}},{{\upsilon }_{1h\prime }} \right)=0\] (15) r(υ2k,υ2k′)=0\[r\left( {{\upsilon }_{2k}},{{\upsilon }_{2k\prime }} \right)=0\]

The expected value of ξ₂ is: (17) E(ξ2|ξ1)=β20+β21ξ1\[E\left( {{\xi }_{2}}|{{\xi }_{1}} \right)={{\beta }_{20}}+{{\beta }_{21}}{{\xi }_{1}}\]

Made in an inference that there is: (18) r(ε2,ξ1)=0\[r\left( {{\varepsilon }_{2}},{{\xi }_{1}} \right)=0\]

For, in the internal relational equation (16) containing ξ₁,ξ₂, ε₂ is linear, showing that: (19) r(υ1h,ε2)=0\[r\left( {{\upsilon }_{1h}},{{\varepsilon }_{2}} \right)=0\] (20) r(υ2k,ε2)=0\[r\left( {{\upsilon }_{2k}},{{\varepsilon }_{2}} \right)=0\]

Mode C₂₁ partial least squares method is to take a mixture of formula (26) of mode A and formula (27) of mode B as its iterative calculation formula respectively. Therefore, the specification of its mathematical calculation formula is as follows: (29) ξ2=∑h(ω1hx1h)+δ1\[{{\xi }_{2}}=\sum\limits_{h}{\left( {{\omega }_{1h}}{{x}_{1h}} \right)}+{{\delta }_{1}}\] (30) x2k=ω2kξ1+δ2k\[{{x}_{2k}}={{\omega }_{2k}}{{\xi }_{1}}+{{\delta }_{2k}}\]

Mode C₁₂ partial least squares method is to take a mixture of Eq. (27) of mode A and Eq. (30) of mode B as its iterative calculation formula, respectively. Therefore, the specification of its mathematical calculation formula is as follows: (31) x1h=ω1hξ2+δ1h\[{{x}_{1h}}={{\omega }_{1h}}{{\xi }_{2}}+{{\delta }_{1h}}\] (32) ξ1=∑k(ω2kx2k)+δ2\[{{\xi }_{1}}=\sum\limits_{k}{\left( {{\omega }_{2k}}{{x}_{2k}} \right)}+{{\delta }_{2}}\]

The specific model is as follows.

Block structure: (33) xh=πh0+πhξ1+υh h=1,2,3,4\[{{x}_{h}}={{\pi }_{{{h}_{0}}}}+{{\pi }_{h}}{{\xi }_{1}}+{{\upsilon }_{h}}\text{ }h=1,2,3,4\] (34) yk=πk0+πkξ2+υk k=1,2,3\[{{y}_{k}}={{\pi }_{{{k}_{0}}}}+{{\pi }_{k}}{{\xi }_{2}}+{{\upsilon }_{k}}\text{ }k=1,2,3\]

Coefficient π_h in Eq. (33) is referred to as the load of indicator x_h, and accordingly, coefficient π_k in Eq. (34) is the load of indicator y_k. υ_h,υ_k is the residual, and π_h0,π_k0 is the intercept value.

The above structure is assumed to satisfy the following relationship in the PLS-SEM algorithm:

Desired Relationship: (35) E(xh|ξ1)=πh0+πhξ1\[E\left( {{x}_{h}}|{{\xi }_{1}} \right)={{\pi }_{{{h}_{0}}}}+{{\pi }_{h}}{{\xi }_{1}}\] (36) E(yk|ξ2)=πk0+πkξ2\[E\left( {{y}_{k}}|{{\xi }_{2}} \right)={{\pi }_{{{k}_{0}}}}+{{\pi }_{k}}{{\xi }_{2}}\]

Unitization of latent variable variance: (37) var(ξ1)=1\[\operatorname{var}\left( {{\xi }_{1}} \right)=1\] (38) var(ξ2)=1\[\operatorname{var}\left( {{\xi }_{2}} \right)=1\]

Non-relevance: (39) r(υh,ξ1)=r(υh,ξ2)=r(υk,ξ1)=r(υk,ξ2)=r(υh,υk)=0\[r\left( {{\upsilon }_{h}},{{\xi }_{1}} \right)=r\left( {{\upsilon }_{h}},{{\xi }_{2}} \right)=r\left( {{\upsilon }_{k}},{{\xi }_{1}} \right)=r\left( {{\upsilon }_{k}},{{\xi }_{2}} \right)=r\left( {{\upsilon }_{h}},{{\upsilon }_{k}} \right)=0\]

Internal relations: (40) ξ2=β0+β1ξ1+ε\[{{\xi }_{2}}={{\beta }_{0}}+{{\beta }_{1}}{{\xi }_{1}}+\varepsilon \]

It is also assumed that the following relationship is satisfied (41) E(ξ2|ξ1)=β0+β1ξ1\[E\left( {{\xi }_{2}}|{{\xi }_{1}} \right)={{\beta }_{0}}+{{\beta }_{1}}{{\xi }_{1}}\] (42) r(ε,ξ1)=0\[r\left( \varepsilon ,{{\xi }_{1}} \right)=0\]

It should be pointed out that, since the modeling method based on PLS structural equation modeling is essentially a cyclic iterative approach to gradually approximate the real parameter values, regardless of whether the model is identifiable or not, the method can always be used to derive the corresponding parameter estimates, only that the estimation bias may be larger. Therefore, the modeling method based on PLS structural equation modeling does not require model identification before parameter estimation.

After the model is set up, the PLS algorithm can be used to estimate each parameter in the model and then solve the whole structural equation model.

Here the sample size is taken as N and the sample observations for indicator x_h,y_k are denoted as x_hn,y_kn respectively, where n = 1⋯N and it is assumed that all the data have been normalized.

The PLS algorithm is divided into three main steps

Step 1: The latent variable estimates are obtained through repeated iterations as follows:

Let: (43) LXn=f1∑h(ωhxhn)\[{{L}_{Xn}}={{f}_{1}}\sum\limits_{h}{\left( {{\omega }_{h}}{{x}_{hn}} \right)}\] (44) LYn=f2∑k(ωkyhn)\[{{L}_{Yn}}={{f}_{2}}\sum\limits_{k}{\left( {{\omega }_{k}}{{y}_{hn}} \right)}\]

Here f₁,f₂ is the normalization operator, so there: (45) f1=±{ 1N∑n[ ∑h(ωhxhn) ]2 }−12\[{{f}_{1}}=\pm {{\left\{ \frac{1}{N}\sum\limits_{n}{{{\left[ \sum\limits_{h}{\left( {{\omega }_{h}}{{x}_{hn}} \right)} \right]}^{2}}} \right\}}^{-\frac{1}{2}}}\] f₂ The same reasoning can be obtained.

Depending on the weight relationship chosen, there is again: (46) LYn=∑h(ωhxhn)+dn\[{{L}_{Yn}}=\sum\limits_{h}{\left( {{\omega }_{h}}{{x}_{hn}} \right)}+{{d}_{n}}\] (47) ykn=ωkLXn+dkn k=1,2,3\[{{y}_{kn}}={{\omega }_{k}}{{L}_{Xn}}+{{d}_{kn}}\text{ }k=1,2,3\]

Synthesizing the relationships expressed in equations (43), (44), (46), and (47) above, iteration begins next in order to obtain the latent variable estimate L_Xn,L_Yn.

Take the initial weights: (48) ωk(1)=1When k=k0ωk(1)=0When k≠k0here 1≤k0≤3\[\begin{align} & \begin{matrix} \omega _{k}^{(1)}=1 & \text{When }k={{k}_{0}} \\ \end{matrix} \\ & \begin{matrix} \omega _{k}^{(1)}=0 & \text{When }k\ne {{k}_{0}} & \text{here }1\le {{k}_{0}}\le 3 \\ \end{matrix} \\ \end{align}\]

The loop termination condition is usually set as follows: (49) | ω(n)−ω(n+1) |<10−5\[\left| {{\omega }^{(n)}}-{{\omega }^{(n+1)}} \right|<{{10}^{-5}}\]

Or: (50) | (ω(n)−ω(n+1))/ω(n) |<10−5\[\left| \left( {{\omega }^{(n)}}-{{\omega }^{(n+1)}} \right)/{{\omega }^{(n)}} \right|<{{10}^{-5}}\]

Step 2: The latent variable estimates L_Xn,L_Yn derived from the first step are regressed against the corresponding indicator observations, respectively, to obtain: (51) xhn=phLXn+μhn\[{{x}_{hn}}={{p}_{h}}{{L}_{Xn}}+{{\mu }_{hn}}\] (52) ykn=pkLYn+μkn\[{{y}_{kn}}={{p}_{k}}{{L}_{Yn}}+{{\mu }_{kn}}\]

The regression equation between L_Xn and L_Yn is as follows: (53) LYn=b1LXn+e\[{{L}_{Yn}}={{b}_{1}}{{L}_{Xn}}+e\]

Where e is the residual and b₁ is the regression coefficient.

Step 3: Find the mean value to give the initial relational equation.

Since: (54) L¯Xn=f1∑h(ωhxh¯)L¯Yn=f2∑k(ωkyk¯)\[\begin{matrix} {{{\bar{L}}}_{Xn}}={{f}_{1}}\sum\limits_{h}{\left( {{\omega }_{h}}\overline{{{x}_{h}}} \right)} \\ {{{\bar{L}}}_{Yn}}={{f}_{2}}\sum\limits_{k}{\left( {{\omega }_{k}}\overline{{{y}_{k}}} \right)} \\ \end{matrix}\]

So the intercept terms in Eqs. (50), (51), and (51) are, respectively: (55) ph0=x¯h−phL¯Xnpk0=y¯k−pkL¯Ynb0=L¯Yn−b1L¯Xn\[\begin{matrix} {{p}_{{{h}_{0}}}}={{{\bar{x}}}_{h}}-{{p}_{h}}{{{\bar{L}}}_{Xn}} \\ {{p}_{{{k}_{0}}}}={{{\bar{y}}}_{k}}-{{p}_{k}}{{{\bar{L}}}_{Yn}} \\ {{b}_{0}}={{{\bar{L}}}_{Yn}}-{{b}_{1}}{{{\bar{L}}}_{Xn}} \\ \end{matrix}\]

At this point, the whole model solving is completed.

Since the above introduction is a general modeling technique, which does not involve specific problems and sample data, the model evaluation process will not be repeated here.

It should be noted that the block structure setting principle of the structural equation model with multiple latent variables is the same as that of the aforementioned structural equation model with two latent variables, which assumes that each indicator in the block has a linear relationship with the corresponding latent variable respectively.

The specific settings are as follows: (56) xh=πh0+πhξ1+υh h=1,2,3,4\[{{x}_{h}}={{\pi }_{{{h}_{0}}}}+{{\pi }_{h}}{{\xi }_{1}}+{{\upsilon }_{h}}\text{ }h=1,2,3,4\] (57) ek=πk0+πkξ2+υk k=1,2,3\[{{e}_{k}}={{\pi }_{{{k}_{0}}}}+{{\pi }_{k}}{{\xi }_{2}}+{{\upsilon }_{k}}\text{ }k=1,2,3\] (58) zl=πl0+πlξ3+υl l=1,2,3\[{{z}_{l}}={{\pi }_{{{l}_{0}}}}+{{\pi }_{l}}{{\xi }_{3}}+{{\upsilon }_{l}}\text{ }l=1,2,3\] (59) ym=πm0+πmξ4+υm m=1,2,3\[{{y}_{m}}={{\pi }_{{{m}_{0}}}}+{{\pi }_{m}}{{\xi }_{4}}+{{\upsilon }_{m}}\text{ }m=1,2,3\]

Coefficient π_h in Eq. (55) is called the load of indicator x_h, and accordingly, coefficient π_k,π_l,π_m in Eqs. (57), (58), and (59) is the load of indicator e_k,z_l,y_m. v_h,v_k,v_l, v_m are the residuals, and π_h0,π_k0,π_l0,π_m0 is the intercept value.

The above structure is assumed to satisfy the following relationship:

Desired Relationship: (60) E(xh|ξ1)=πh0+πhξ1E(ek|ξ2)=πk0+πkξ2E(zl|ξ3)=πl0+πlξ3E(ym|ξ4)=πm0+πmξ4\[\begin{align} & E\left( {{x}_{h}}|{{\xi }_{1}} \right)={{\pi }_{{{h}_{0}}}}+{{\pi }_{h}}{{\xi }_{1}} \\ & E\left( {{e}_{k}}|{{\xi }_{2}} \right)={{\pi }_{{{k}_{0}}}}+{{\pi }_{k}}{{\xi }_{2}} \\ & E\left( {{z}_{l}}|{{\xi }_{3}} \right)={{\pi }_{{{l}_{0}}}}+{{\pi }_{l}}{{\xi }_{3}} \\ & E\left( {{y}_{m}}|{{\xi }_{4}} \right)={{\pi }_{{{m}_{0}}}}+{{\pi }_{m}}{{\xi }_{4}} \end{align}\]

Unitization of latent variable variance: (61) var(ξ1)=1\[\operatorname{var}\left( {{\xi }_{1}} \right)=1\] (62) var(ξ2)=1\[\operatorname{var}\left( {{\xi }_{2}} \right)=1\] (63) var(ξ3)=1\[\operatorname{var}\left( {{\xi }_{3}} \right)=1\] (64) var(ξ4)=1\[\operatorname{var}\left( {{\xi }_{4}} \right)=1\]

Non-relevance: (65) r(vh,ξ1)=r(vh,ξ2)=r(vh,ξ3)=r(vh,ξ4)=0\[r\left( {{v}_{h}},{{\xi }_{1}} \right)=r\left( {{v}_{h}},{{\xi }_{2}} \right)=r\left( {{v}_{h}},{{\xi }_{3}} \right)=r\left( {{v}_{h}},{{\xi }_{4}} \right)=0\] (66) r(vk,ξ1)=r(vk,ξ2)=r(vk,ξ3)=r(vk,ξ4)=0\[r\left( {{v}_{k}},{{\xi }_{1}} \right)=r\left( {{v}_{k}},{{\xi }_{2}} \right)=r\left( {{v}_{k}},{{\xi }_{3}} \right)=r\left( {{v}_{k}},{{\xi }_{4}} \right)=0\] (67) r(vl,ξ1)=r(vl,ξ2)=r(vl,ξ3)=r(vl,ξ4)=0\[r\left( {{v}_{l}},{{\xi }_{1}} \right)=r\left( {{v}_{l}},{{\xi }_{2}} \right)=r\left( {{v}_{l}},{{\xi }_{3}} \right)=r\left( {{v}_{l}},{{\xi }_{4}} \right)=0\] (68) r(vm,ξ1)=r(vm,ξ2)=r(vm,ξ3)=r(vm,ξ4)=0\[r\left( {{v}_{m}},{{\xi }_{1}} \right)=r\left( {{v}_{m}},{{\xi }_{2}} \right)=r\left( {{v}_{m}},{{\xi }_{3}} \right)=r\left( {{v}_{m}},{{\xi }_{4}} \right)=0\] (69) r(vh,vk)=r(vh,vl)=r(vh,vm)=r(vk,vl)=r(vk,vm)=r(vl,vm)=0\[r\left( {{v}_{h}},{{v}_{k}} \right)=r\left( {{v}_{h}},{{v}_{l}} \right)=r\left( {{v}_{h}},{{v}_{m}} \right)=r\left( {{v}_{k}},{{v}_{l}} \right)=r\left( {{v}_{k}},{{v}_{m}} \right)=r\left( {{v}_{l}},{{v}_{m}} \right)=0\]

The specific settings are as follows: (70) ξ4=β40+β41ξ1+β42ξ2+β43ξ3+ε4\[{{\xi }_{4}}={{\beta }_{40}}+{{\beta }_{41}}{{\xi }_{1}}+{{\beta }_{42}}{{\xi }_{2}}+{{\beta }_{43}}{{\xi }_{3}}+{{\varepsilon }_{4}}\] (71) ξ3=β30+β31ξ1+ε3\[{{\xi }_{3}}={{\beta }_{30}}+{{\beta }_{31}}{{\xi }_{1}}+{{\varepsilon }_{3}}\]

The following relationship is assumed to be satisfied: (72) E(ξ4|ξ1,ξ2,ξ3)=β40+β41ξ1+β42ξ2+β43ξ3\[E\left( {{\xi }_{4}}|{{\xi }_{1}},{{\xi }_{2}},{{\xi }_{3}} \right)={{\beta }_{40}}+{{\beta }_{41}}{{\xi }_{1}}+{{\beta }_{42}}{{\xi }_{2}}+{{\beta }_{43}}{{\xi }_{3}}\] (73) E(ξ3|ξ1)=β30+β31ξ1\[E\left( {{\xi }_{3}}|{{\xi }_{1}} \right)={{\beta }_{30}}+{{\beta }_{31}}{{\xi }_{1}}\] (74) r(ε4,ξ1)=r(ε4,ξ2)=r(ε4,ξ3)=0\[r\left( {{\varepsilon }_{4}},{{\xi }_{1}} \right)=r\left( {{\varepsilon }_{4}},{{\xi }_{2}} \right)=r\left( {{\varepsilon }_{4}},{{\xi }_{3}} \right)=0\] (75) r(ε3,ξ1)=r(ε3,ξ2)=0\[r\left( {{\varepsilon }_{3}},{{\xi }_{1}} \right)=r\left( {{\varepsilon }_{3}},{{\xi }_{2}} \right)=0\]

The model estimation principle of the structural equation model with multiple latent variables is basically the same as that of the previous structural equation model with two latent variables, which is still to firstly use the cyclic iteration to derive the estimated values of the latent variables, and then regress the estimated values of the latent variables on the corresponding indicators and between the latent variables, so as to derive the parameters of each model.

The symbolic weight sum is defined as follows:

In structural equation modeling, the sum of the sign weights of the jst latent variable ξ_j is the sign-weighted sum of the latent variable estimates of all other latent variables adjacent to ξ_j. It is common to notate the sum of the sign weights of latent variable ξ_j as A_j. The mathematical expression is as follows: (76) Aj=∑a(sjaLa)\[{{A}_{j}}=\sum\limits_{a}{\left( {{s}_{ja}}{{L}_{a}} \right)}\]

where L_a represents the latent variable estimate of the latent variable adjacent to ξ_j, and s_ja is a sign function with: (77) sja=signr(Lj,La)=−1 When r(Lj,La)<0=1 When r(Lj,La)>0\[\begin{align} & {{s}_{ja}}=\operatorname{sign}r\left( {{L}_{j}},{{L}_{a}} \right) \\ & =-1\text{ When }r\left( {{L}_{j}},{{L}_{a}} \right)<0 \\ & =1\text{ When }r\left( {{L}_{j}},{{L}_{a}} \right)>0 \end{align}\]

Note that when the latent variable ξ_j has only one other latent variable in close proximity, then the value of the sign function s_ja is taken to be 1.

Based on the above definition, the specific procedure for model estimation of structural equation models containing multiple latent variables is given below:

First let: (78) LXn=f1∑h(ωhxhn)\[{{L}_{Xn}}={{f}_{1}}\sum\limits_{h}{\left( {{\omega }_{h}}{{x}_{hn}} \right)}\] (79) LEn=f2∑k(ωkehn)\[{{L}_{En}}={{f}_{2}}\sum\limits_{k}{\left( {{\omega }_{k}}{{e}_{hn}} \right)}\] (80) LZn=f3∑l(ωlzln)\[{{L}_{Zn}}={{f}_{3}}\sum\limits_{l}{\left( {{\omega }_{l}}{{z}_{ln}} \right)}\] (81) LYn=f4∑m(ωmymn)\[{{L}_{Yn}}={{f}_{4}}\sum\limits_{m}{\left( {{\omega }_{m}}{{y}_{mn}} \right)}\]

Depending on the selected weighting relations (Mode B for the latent independent variable and Mode A for the latent dependent variable), there are again: (82) A1=∑h(ωhxhn)+dhn\[{{A}_{1}}=\sum\limits_{h}{\left( {{\omega }_{h}}{{x}_{hn}} \right)}+{{d}_{hn}}\] (83) A2=∑k(ωkeln)+dkn\[{{A}_{2}}=\sum\limits_{k}{\left( {{\omega }_{k}}{{e}_{ln}} \right)}+{{d}_{kn}}\] (84) zln=ωlA3+dln\[{{z}_{ln}}={{\omega }_{l}}{{A}_{3}}+{{d}_{ln}}\] (85) ymn=ωmA4+dmn\[{{y}_{mn}}={{\omega }_{m}}{{A}_{4}}+{{d}_{mn}}\] where the sign weights of ξ₁ and: (86) A1=s13LZn+s14LYn\[{{A}_{1}}={{s}_{13}}{{L}_{Zn}}+{{s}_{14}}{{L}_{Yn}}\]

The symbolic weights of ξ₂ and: (87) A2=LYn\[{{A}_{2}}={{L}_{Yn}}\]

The symbolic weights of ξ₃ and: (88) A3=S31LXn+S34LYn\[{{A}_{3}}={{S}_{31}}{{L}_{Xn}}+{{S}_{34}}{{L}_{Yn}}\]

The symbolic weights of ξ₄ and: (89) A4=s41LXn+s42LEn+s43LZn\[{{A}_{4}}={{s}_{41}}{{L}_{Xn}}+{{s}_{42}}{{L}_{En}}+{{s}_{43}}{{L}_{Zn}}\]