CS evaluation model
CS is the cosine of the angle between two vectors in the vector space as a measure of the difference between two subjects. It can be used in multi-attribute decision analysis to judge the order of pros and cons of each scheme by comparing the cosine of the angle between each scheme and the ideal scheme. Here, a scheme (Ô) to be evaluated is assumed. If the ideal scheme is given, the distance between the scheme to be evaluated and the ideal scheme can be determined by the cosine of the angle.
When an evaluation grade standard is given, the index data of the object can be randomly generated coming from the same grade interval based on Monte Carlo simulation (MCS). MCS is an advanced simulation technology, which is used for numerical estimation under the guidance of probability theory and mathematical statistics. This method can scientifically and reasonably solve complex problems with multiple factors and uncertainties. In this way several objects with known evaluation results can be generated by MCS. Then, according to the objects with known evaluation results, the evaluation results of the investigated schemes can be easily calculated by the cosine similarity measures. Herein there is an evaluation grade standard with five evaluation grade and j indexes, as shown in Table 1.
Five evaluation objects (O1, O2, O3, O4, and O5) can be dynamically generated based on stochastic simulation, following the uniform distribution (the number of randomly generated evaluation objects is equal to the number of evaluation grade)24. It is noticed that each index value of the five evaluation objects (O1, O2, O3, O4, and O5) comes from the same grade interval respectively. For example, each index value of O1 randomly comes from the first-grade interval, obeying the uniform distribution (χj0, χj1], each index value of O2 randomly comes from the second-grade interval, obeying the uniform distribution (χj1, χj2], and so on, as shown in Table 2. Therefore, the evaluation results of O1, O2, O3, O4, and O5 are determinable, which corresponds to very good (I), good (II), normal (III), poor (IV), very poor (V), respectively.
Cosine distance can be applied to calculate the similarity of the investigated object (Ô) with respect to the five evaluation objects (O1, O2, O3, O4, and O5) respectively. Sqrt-cosine similarity measurement can be considered an effective distance measurement in machine learning for high-dimensional applications based on Hellinger distance25. Thus, sim(Oi, Ô), i = 1, 2, 3, 4, 5, can be described as:
$$sim\left( {{\varvec{O}}^{i} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right) = \frac{{\sum\limits_{j = 1}^{5} {\sqrt {\mathop O\limits^{ \wedge }_{j} \times O_{j}^{i} } } }}{{\sqrt {\sum\limits_{j = 1}^{5} {\left( {\mathop O\limits^{ \wedge }_{j} } \right)} } \times \sqrt {\sum\limits_{j = 1}^{5} {\left( {O_{j}^{i} } \right)} } }},\quad i = 1,2,3,4,5$$
(1)
where sim(Oi, Ô) is the cosine similarity between two objects Oi and Ô. Ôj is the jth element of the investigated object (Ô).
According to the critical similarity value sim(Oi, Ô)*, If sim(Oi, Ô) ≥ sim(Oi, Ô)*, then Oi and Ô are the same evaluation result. According to the formula of sqrt-cosine similarity measurement, a proposition can be easily obtained.
Proposition
There are three classifications of evaluation objects: Oi, Oj and Ok. If the classification results of Oi and Oj are same, the classification results of Oi and Ok are different, then min{sim(Oi, Oj)} > max{sim(Oi, Ok)}.
Proof
Assume that min{sim(Oi, Oj)} ≤ max{sim(Oi, Ok)}.
then, sim(Oi, Oj) ≥ sim(Oi, Oj)*, sim(Oi, Oj)* ≤ min{sim(Oi, Oj)} ≤ max{sim(Oi, Ok)}.
Thus, Oi and Ok are the same classification, which contradicts the proposition. Therefore, the proposition mentioned above is true.
According to the Proposition, the evaluation result of Ô can be obtained by calculating max{sim(Oi, Ô)}, i = 1, 2, 3, 4, 5, as follow:
$$\left\{ \begin{gathered} Grade\left( {\text{I}} \right),\quad if\;\max \left\{ {sim\left( {{\varvec{O}}^{i} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right)} \right\} = sim\left( {{\varvec{O}}^{1} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right) \hfill \\ Grade\left( {{\text{II}}} \right),\quad if\;\max \left\{ {sim\left( {{\varvec{O}}^{i} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right)} \right\} = sim\left( {{\varvec{O}}^{2} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right) \hfill \\ Grade\left( {{\text{III}}} \right),\quad if\;\max \left\{ {sim\left( {{\varvec{O}}^{i} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right)} \right\} = sim\left( {{\varvec{O}}^{3} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right) \hfill \\ Grade\left( {{\text{IV}}} \right),\quad if\;\max \left\{ {sim\left( {{\varvec{O}}^{i} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right)} \right\} = sim\left( {{\varvec{O}}^{4} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right) \hfill \\ Grade\left( {\text{V}} \right),\quad if\;\max \left\{ {sim\left( {{\varvec{O}}^{i} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right)} \right\} = sim\left( {{\varvec{O}}^{5} ,\mathop {\varvec{O}}\limits^{ \wedge } } \right) \hfill \\ \end{gathered} \right.$$
(2)
The uncertainty of five evaluation objects will influence the similarity results, because the indexes values of five evaluation objects are generated using stochastic simulation strategy.
Suppose Numgrade(k) (k = I, II, III, IV, V) is the random trial result of Ô. For a sequence of N tests, Numgrade(k) notes the occurrences numbers of grade (k) (k = I, II, III, IV, V). Then the probability of grade (k) of Ô in the tests can be calculated as:
$$P\left( k \right) = \frac{{Num_{grade\left( k \right)} }}{N},\quad k = {\text{I}},{\text{II}},{\text{III}},{\text{IV}},{\text{V}}$$
(3)
Thus, the grade corresponding to the maximum probability is the evaluation result of Ô.
Dynamic evaluation model of MC
Markov process is a special random motion process. A process of change in X of a moving system is called a Markov process if the state of Xr+1is only related to the state of Xr and not to the previous state of Xr.
$$X_{r + 1} = X_{r} \times P$$
(4)
where P is the transition probability matrix of the process.
$$P = \left[ {\begin{array}{*{20}c} {P_{11} } & \cdots & {P_{1n} } \\ \vdots & \ddots & \vdots \\ {P_{n1} } & \cdots & {P_{nn} } \\ \end{array} } \right]$$
(5)
Markov process has no aftereffect and stability, and its key lies in the determination of transition probability matrix P. Puv is the transition probability of type u to type v. Puv should meet two basic conditions: ① Puv ∈ [0,1]; ② ∑Puv = 1.
Since the state of the evaluation result is divided into five levels in part 2.1, the state space is composed of five states in MC model. The transition probability matrix P can be expressed as follows:
$$P = \left[ {\begin{array}{*{20}c} {P_{11} } & \cdots & {P_{15} } \\ \vdots & \ddots & \vdots \\ {P_{51} } & \cdots & {P_{55} } \\ \end{array} } \right]$$
(6)
The evaluation result of Ô (P(I), P(II), P(III), P(IV), P(V)) can be used as the initial state in MC model. At present, many researchers determine the transformation probability matrix P of MC model based on the multi-year state transformation data. For the absence of multi-year historical data of slope ecological restoration, this paper establishes a constraint model for solving the transformation probability matrix P of MC model.
There are two assumptions, as follows:
-
①
The probabilities of transitions between states are not equal;
-
②
The state X1(0, 0, 0, 0, 1) or X1(1, 0, 0, 0, 0) will not change after several steps. That is to say, when the evaluation result probability of V or I is 100%, its state will not change with time.
Based on these two assumptions, we establish the objective function and the constraint function.
$$\min Q = \sum\limits_{r = 1}^{m} {\sum\limits_{k = 1}^{5} {\left| {\frac{{S_{k} \left( r \right) – \mathop S\limits^{ \wedge }_{k} \left( r \right)}}{{S_{k} \left( r \right)}}} \right|} }$$
(7)
$$\left\{ {\begin{array}{*{20}c} {\sum\limits_{v = 1}^{n} {P_{uv} = 1,\quad u = 1,{2,} \ldots {,}n} } \\ {P_{uv} \ge 0,\quad u,v = 1,{2,} \ldots {,}n \, } \\ \end{array} } \right.$$
(8)
where Sk(r) is the actual value at state k after r-step transfer, Ŝk(r) is the estimated value at state k after r-step transfer.
By solving the above model, the transformation probability matrix P of MC model can be obtained. Based on this, it is possible to construct a dynamic evaluation model for MC.
Calculation procedure
The dynamic evaluation process of slope ecological restoration effect includes two parts, the first part is the initial evaluation using CS, and the second part is the dynamic evaluation using MC. In order to obtain the dynamic evaluation results, the initial evaluation of slope ecological restoration is carried out based on CS firstly. Then, the transition probability matrix was determined based on Eqs. (7) and (8). Finally, according to the initial evaluation results of slope ecological restoration and the transition probability matrix P, the dynamic evaluation results were solved by Eq. (4), and the change of slope ecological restoration effect was predicted to realize dynamic evaluation. Based on the above calculation process, we integrated CS and MC to construct a method system for dynamic evaluation of slope ecological restoration. The most critical part of the first part is to randomly generate schemes known evaluation results. Crystal Ball is an easy-to-use simulation software that can run Monte Carlo simulation (MCS) for stochastic simulations, and is associated with Microsoft Excel® spreadsheet program26. Therefore, the proposed evaluation method for effect of the slope ecological restoration can easily be carried out by Crystal Ball in an Excel® worksheet. The detailed calculation process is shown in Fig. 1.
