I. INTRODUCTION

In modern society, psychological crisis has become a social problem, and the pressure people bear increases unceasingly. The assessment of psychological health state and the crisis intervention have been hot topics for psychological research^[1]. The achievement related to the research can help to build up reasonable evaluation, and reasonable rule base built by analysis of psychological health status for a large number of individuals can provide guidance for people to take moderate and appropriate psychological intervention, and improve the efficiency of disposing psychological crisis.

In 1964, Caplan provided principles of preventive psychiatry^[2], since then the study of psychological health status has been noticed. Till now, many findings of theories and applications for the evaluation of psychological health state and crisis intervention have appeared. Caplan gave the representations for psychological crisis, and presented its source, classification, evolution process and intervention^[2,3,4], and these works lay the foundation for the theory study of psychological crisis intervention. James and Gilliland used three basic modes of crisis intervention to help individual to prevent or alleviate the psychological trauma^[5]. Roberts took a synthesis ACT crisis intervention mode to deal with sudden crisis and traumatic crisis, and achieved good result to intervene the high risk group of the local events honor 9/11 by the mode^[6]. In China, Zhang and Chen discussed the intervention for college student's interpersonal communication^[7,8]. Zhang et al. studied relative questions of psychological crisis and crisis intervention^[9]. However, these intervention modes are too many to unify for guiding intervention process, and the empirical research is very little. Thus, it is necessary to build a set of effective mechanism for the evaluation and intervention of psychological health state .

At the same time, the knowledge on evaluation for psychological health state and crisis intervention are mainly perception information. In daily life, the description and analysis of these questions are mainly in natural language, so how to use natural language to deal with the evaluation for psychological health state and crisis intervention is the key to solve the problem. In the meantime, the factors affecting people$'$s psychological health state include many reasons, and the impacts of every factor maybe not be the same. Then it is necessary for us to analyze the psychological health state and provide crisis intervention from multi-factors and dynamic methods to describe the process objectively. Type-2 fuzzy sets (T2 FSs) can help us solve the problems of language ambiguity and data noise better than the corresponding type-1. As an effective method for the solution to perceptional information, T2 FSs have been used to deal with some practical problems^{[10,11,12,13]}. The questions of multi-factors can be managed by the method of fuzzy comprehensive evaluation. The two methods can let us to describe uncertainty of language, but their description and analysis are static.

In linguistic dynamic systems (LDS), computing with words is used to substitute conventional numerical symbolic computation, and the modeling, analysis, evaluation and decision-making can be at the level of language^{[14,15,16,17]}. In the past years, LDS were mainly on 1-D universe. However, there are many factors to be evaluated, so it is necessary to consider, LDS on multi-dimension universe. Wang's fuzzy comprehension evaluation can help people to provide the assessment results for many factors^[18], but it is static. Meanwhile, how to give the evaluation, for different experts, there are different means. It is necessary to use the theories of type-2 fuzzy sets and linguistic dynamic systems to describe and analyze the evolutionary process of the evaluation for psychological health state.

In the paper, interval type-2 fuzzy sets (IT2 FSs), linguistic dynamic systems (LDS) and fuzzy comprehension evaluation are used to build up the models for psychological health state and its intervention, and provide the linguistic dynamic description for the evolutionary of psychological health state. The paper is arranged as follows: Section II is the introduction of terminologies; In Section III, the evaluation of psychological health state is discussed; Section IV presents the linguistic dynamic description of psychological health state; linguistic dynamic modeling of psychological crisis intervention is described in Section V; Section VI provides the conclusion. II. PRELIMINARY

In 1995, Wang provided the theory of linguistic dynamic systems^[19]. In LDS, the state equation, output equation and the feedback equation are converted to the corresponding linguistic forms^[20,21,22]:

The state equation of $LDS$:

$$\begin{align} X(k+1)=F(X(k), U(k), k), F:I^{N_{1}}\times I^{N_{2}}\to I^{N_{1}}. \end{align}$$

(1)

The output equation of $LDS$:

$$\begin{align} Y(k)=H(X(k), k), H:I^{N_{1}}\times Z\to I^{N_{3}}. \end{align}$$

(2)

The feedback equation of $LDS$:

$$\begin{align} U(k)=R(Y(k), V(k), k), R:I^{N_{3}}\times I^{N_{4}}\times Z\to I^{N_{2}}, \end{align}$$

(3)

where $Z=\{0, 1, \ldots, K\}$, $N_{1}, N_{2}, N_{3}, N_{4}$ are all positive numbers, $X(k)\in I^{N_{1}}$ is the state word of the system, $Y(k)\in I^{N_{3}}$ the output word, $V(k)\in I^{N_{4}}$ the input word, $U(k)\in I^{N_{2}}$ the control word, $k$ the discrete time series, and $F$, $H$, $R$ are all fuzzy logic operators defining the system, output and control mapping of $LDS$ respectively Let $C(2^{I})$ is a set of all the nonempty close subsets of $I$. A T2 FS $\omega$ on $\Omega$ is defined as^[22,23]

$$\begin{align} &\omega=\{(x, u, z)|\forall x\in{\Omega}, \forall u\in L_x\subseteq{C(2^{I})}, \notag\\ &z=\mu^{2}_{\omega}(x, u)\in{C(2^{I}}\}, \end{align}$$

(4)

where $x$ (or $u$, $z$) is the primary (or secondary, third) variable, $L_x\subseteq{[0, 1]}$ is the primary membership grade, defined by the following multi-value mapping: $$\mu^{1}_{\omega}:\Omega\rightarrow{C(2^{I})},$$ i. e. , for every $x\in{\Omega}$, there is $L_{x}\in{C(2^{I})}$, such that $$\mu^{1}_{\omega}(x)=L_{x},$$ and $0\leq\mu^{2}_{\omega}(x, u)\leq1$, $\mu^{2}_{\omega}$ is the second membership function, defined as $$\mu^{2}_{\omega}:\bigcup_{x\in {\Omega}}x\times{L_{x}}\rightarrow{I}.$$ If $\Omega$ and $L_{x}$ are connected, and $\mu^{2}_{\omega}$ is continuous on $\bigcup_{x\in{\Omega}}x\times{L_{x}}$, then $\omega$ is a continuous T2 FS on $\Omega$, and can be written as

$$\begin{align} \omega=\int_{x\in {\Omega}}\int_{u\in L_x}\frac{\mu^{2}_{\omega}(x, u)}{(x, u)}, L_x\subseteq[0, 1]. \end{align}$$

(5)

If $\Omega$ is discrete, $L_{x}$ is connected, and $\mu^{2}_{\omega}$ is continuous on every $x\times{L_{x}}$, then $\omega$ is a partially discrete T2 FS on $\Omega$, and can be written as

$$\begin{align} \omega=\sum_{x\in{\Omega}}\int_{u\in L_x}\frac{\mu^{2}_{\omega}(x, u)}{(x, u)}. \end{align}$$

(6)

If $\Omega$ and $L_{x}$ are all discrete, then $\omega$ is a discrete T2 FS on $\Omega$, and can be written as

$$\begin{align} \omega=\sum_{x\in{\Omega}}\sum_{u\in L_x}\frac{\mu^{2}_{\omega}(x, u)}{(x, u)}. \end{align}$$

(7)

If for every $x\in{\Omega}$ and $u\in{L_{x}}$, there is $\mu^{2}_{\omega}(x, u)=1$, then $\omega$ is an interval type-2 fuzzy sets (IT2 FS). Especially, a partially discrete IT2 FS can be written as $$\omega=\sum_{x\in{\Omega}}\frac{\frac{1}{[a_x, b_x]}}{x},$$ where $L_x=[a_x, b_x]$. III. DESCRIPTION AND EVALUATION OF PSYCHOLOGICAL HEALTH STATE ON IT2 FSS

Psychological health state is divided into the following four scenarios: health, sub-health, ill health and disease^[23], represented by $q_3$, $q_2$, $q_1$, $q_0$ respectively. The observation and evaluation for the psychological health state and intervention form the close-loop-intervention model of psychological state as Fig. 1.

The psychological pressure is an inevitable factor of psychological disease^[24]. So in the paper, the study of psychological health state is converted to the evaluation of psychological pressure and its intervention. At the same time, psychological pressure has many typical characteristics, for example, diversity, time-variance, etc.

Let $\Omega=\{o_1, o_2, \ldots, o_N\}$ be the set being observed. For every $o_n\, (n=1, 2, \ldots, N)$, $P_{n}, Q_{n}$ are the psychological pressure and psychological endurance of $o_n$ respectively , and $\Delta P_{n}$ is the error between $P_{n}$ and $Q_{n}$. i. e. ,

$$\Delta P_{n}=P_{n}-Q_{n}.$$ There are four IT2 FSs $\{\omega_1, \omega_2, \omega_3, \omega_4\}$ = {positive high, positive medium, positive small, non-positive} on the universe of $\Delta P_{n}$.

According to the above assumption, the corresponding fuzzy rules are as follows:

R₁: if △P is ω₁, then X is q₀;

R₂: if △P is ω₂, then X is q₁;

R₃: if △P is ω₃, then X is q₂;

R₄: if △P is ω₄, then X is q₃.

where $X$ is the variable of pressure sources, also the universe of pressure source. By the following four steps, type-2 fuzzy comprehensive evaluation can be used to evaluate the psychological health state of every person.

Step 1. Determination of the factor set

The factor set $X$ contains $K$ factors, written as

$$\begin{align} X=\{X_1, X_2, \ldots, X_K\}, \end{align}$$

(8)

where $X_k, k=1, 2, \ldots, K$ is one of the pressure sources, and $K$ is a nature number.

Step 2. Determination of the evaluation set

For every pressure source $X_k, k=1, 2, \ldots, K$, the assess value is expressed by four basis words {very high, high, medium, small}, i. e. ,

$$\begin{align} V=\{v_{1}, v_{2}, \ldots, v_{4}\}=\{VH, H, M, S\}, \end{align}$$

(9)

where $\{VH, H, M, S\}$ is the abbreviation of {very high, high, medium, small} respectively, and used in the following context.

Step 3. Determination of the weight of every factor

Different element of the factor set may play different pole in the comprehensive evaluation, then the weight vector is represented as

$$\begin{align} B=\{b_{1}, b_{2}, \ldots, b_{K}\} \end{align}$$

(10)

where $b_{j}$ is the factor's weight for $X_{k}\, (k=1, 2, \ldots, K)$, and $\sum_{j=1}^K b_{j}=1$.

Step 4. Determination of the fuzzy comprehension evaluation matrix

For the pressure source $X_{1}, \ldots, X_{k}, \ldots, X_{K}$ and the corresponding assess basis words $v_{1}, \ldots, v_{4}$, let $Z=\{z_{1}, \ldots, z_{M}\}$ be the set of evaluation experts, and the assessed value by $z_{m}, m=1, 2, \ldots, M$ for all the observed $\{o_1, o_2, \dots, o_N\}$ is in Table I.

In Table I, $o_i|z_{m}$ means $o_{i}$ is assessed by $z_{m}$. The evaluation matrix $R_n^{m}$ by $z_{m}$ for $o_{n}$ is as follows: $$R_n^{m}=\left[\begin{array}{ccccccccc} r_{11}^ {mn}& r_{12}^ {mn} & r_{13}^ {mn} & r_{14}^ {mn}\\ r_{21}^ {mn}& r_{22}^ {mn} & r_{23}^ {mn} & r_{24}^ {mn}\\ \vdots & \vdots & \vdots & \vdots \\ r_{K1}^ {mn} & r_{K2}^ {mn} & r_{K3}^ {mn} &r_{K4}^ {mn} \end{array} \right],$$ where, $r_{ki}^{mn}$ is the assessment value for $o_{n}$ to $v_{i}$ of pressure source $X_{k}$ assessed by $z_{m}$, and $m=1, 2, \ldots, M; n=1, 2, \ldots, N; k=1, 2, \ldots, K; i=1, 2, 3, 4$.

Step 5. Comprehension evaluation

Psychological endurance is the regular and overall performance of the personal to the pressure. Let the psychological endurance of $o_{n}$ be $Q_{n}$, then the error between the evaluation result by $z_{m}$ and the psychological endurance is

$$\begin{align} \Delta P^{m}_{n}=P_{n}^{m}-Q_{n}. \end{align}$$

(11)

In the following context, $\Delta P^{m}_{n}$ is written as $P^{m}_{n}$ for abbreviation.

Let ``$\bullet$'' be the fuzzy comprehension operation, and $P_{n}^{m}$ is the fuzzy comprehension evaluation of $o_{m}$ evaluated by $z_{m}$. For the weight vector $B$ and the evaluation matrix $R$, the operation is defined as follows:

$$\begin{align} P_{n}^{m}=B\bullet R_n^{m}=(p_{n1}^{m}, p_{n2}^{m}, p_{n3}^{m}, p_{n4}^{m}), \end{align}$$

(12)

where

$$\begin{align} p_{ni}^{m}=\sum_{j=1}^K{b_{j}\cdot r_{ji}^{mn}}, \end{align}$$

(13)

and $p_{ni}^{m}$ is the comprehensive evaluated grade of the $o_{m}$ to $\omega_{i}$ by $z_{m}$.

Thus, $\omega_{i}, i=1, \ldots, 4$ can be represented by a half discrete IT2 FS, and its primary membership grade is defined as follows:

$$\begin{align} L_{ni}=[\underline{P_{ni}}, \overline{P_{ni}}], \end{align}$$

(14)

where $$\underline{P_{ni}}=\min\ \{P_{ni}^{1}, \ldots, P_{ni}^{M}\},$$ $$\overline{P_{ni}}=\max\{P_{ni}^{1}, \ldots, P_{ni}^{M}\}.$$ Then $\omega_i, i=1, \ldots, 4$ can be written as

$$\begin{align} \omega_i=\sum_{n=1}^N\cfrac{\cfrac{1}{[\underline{P_{ni}}, \overline{P_{ni}}]}}{o_n}. \end{align}$$

(15)

By the above fuzzy rules and the match degree method^[22], it is not difficult to obtain its match degree with basis words of every rule, and its psychological state of $o_{n}$, together with the linguistic dynamic modeling of the psychological health state for $o_{n}$.

Example 1. The psychological state of three employees $\{o_{1}, o_{2}, o_{3}\}$ is observed by four experts $\{z_{1}, z_{2}, z_{3}, z_{4}\}$. The pressure source consists of four factors $\{X_{1}, X_{2}, X_{3}, X_{4}\}$ and the evaluation of every factor is described by four basis words

$$\{v_{1}, v_{2}, v_{3}, v_{4}\}=\{VH, H, M, S\},$$ and the weight vector is

$$\begin{align} B=\{0. 4, 0. 2, 0. 3, 0. 1\}. \end{align}$$

(16)

The evaluation is as Table II, where $N=3, M=K=4$.

From the above discussion, the fuzzy comprehension evaluation results are in Table III.

In Table III, when $4n-3\leq{i}\leq{4n}, n=1, 2, 3$, if $1\leq{j}\leq{4}$, $r_{ij}$ is the membership grade to $\omega_{j}$ for $o_{n}$ evaluated by $z_{i-4(n-1)}$. And the corresponding the primary membership grade are as follows:

$$L_{11}=[0. 27, 0. 33], L_{12}=[0. 30, 0. 36],$$ $$L_{13}=[0. 28, 0. 34], L_{14}=[0. 03, 0. 07],$$ $$L_{21}=[0. 21, 0. 31], L_{22}=[0. 40, 0. 58],$$ $$L_{23}=[0. 18, 0. 32], L_{24}=[0. 00, 0. 03],$$ $$L_{31}=[0. 07, 0. 09], L_{32}=[0. 15, 0. 24],$$ $$L_{33}=[0. 32, 0. 43], L_{34}=[0. 29, 0. 41].$$ Then the corresponding half discrete IT2 FSs can be written as $$\omega_{1}=\frac{\frac{1}{[0. 27, 0. 33]}}{o_{1}}+\frac{\frac{1}{[0. 21, 0. 31]}}{o_{2}}+\frac{\frac{1}{[0. 07, 0. 09]}}{o_{3}},$$ $$\omega_{2}=\frac{\frac{1}{[0. 30, 0. 36]}}{o_{1}}+\frac{\frac{1}{[0. 40, 0. 58]}}{o_{2}}+\frac{\frac{1}{[0. 15, 0. 24]}}{o_{3}},$$ $$\omega_{3}=\frac{\frac{1}{[0. 28, 0. 34]}}{o_{1}}+\frac{\frac{1}{[0. 18, 0. 32]}}{o_{2}}+\frac{\frac{1}{[0. 32, 0. 43]}}{o_{3}},$$ $$\omega_{4}=\frac{\frac{1}{[0. 03, 0. 07]}}{o_{1}}+\frac{\frac{1}{[0. 00, 0. 03]}}{o_{2}}+\frac{\frac{1}{[0. 29, 0. 41]}}{o_{3}}.$$ From the representation, it is not difficult to see that the psychological pressure of $o_{1}$ is higher than $o_{2}$ and $o_{3}$, and that of $o_{3}$ is much less. IV. LINGUISTIC DYNAMIC DESCRIPTION OF PSYCHOLOGICAL HEALTH STATE

By the above analysis, it is easy to see that for the given datum of some object, type-2 fuzzy sets can be used to describe the states. The work of the description for psychological state is static, but the evolution of psychological healthy state is dynamic, and linguistic dynamic systems can be used to describe the evolution procedure.

Different element of the factor set may play different role in the comprehensive evaluation, and even for the same person, the factor's weight is different at different time, then Steps 3-5 should be modified as follows:

Step 3$^{*}$. Determine the weight of every factor

$$B(t)=\{b_{1}(t), b_{2}(t), \ldots, b_{K}(t)\}$$ where $b_{j}(t)$ is the factor's weight of $X_{k}\, (k=1, 2, \ldots, K)$ at $t$, and $\sum_{j=1}^K b_{j}(t)=1$.

Step 4$^{*}$. Determine the fuzzy comprehension evaluation matrix $R_n^{m}(t)$ can be represented as follows:

$$R_n^{m}(t)=\left[\begin{array}{ccccccccc} r_{11}^ {mn}(t)& r_{12}^ {mn}(t) & r_{13}^ {mn}(t) & r_{14}^ {mn}(t)\\ r_{21}^ {mn}(t)& r_{22}^ {mn}(t) & r_{23}^ {mn}(t) & r_{24}^ {mn}(t)\\ \vdots & \vdots & \vdots & \vdots \\ r_{K1}^ {mn}(t)& r_{K2}^ {mn}(t) & r_{K3}^ {mn}(t) &r_{K4}^ {mn}(t), \end{array} \right],$$ where $r_{ki}^{mn}(t)$ is the evaluation for $o_{n}$ to $v_{i}$ of pressure source $X_{k}$ by $z_{m}$ at $t$, and $m=1, 2, \ldots, M; n=1, 2, \ldots, N; k=1, 2, \ldots, K; i=1, 2, 3, 4$.

Step 5$^{*}$. Comprehension evaluation

For the weight vector $B(t)$ and the valuation matrix $R(t)$, $P_{n}^{m}(t)$ is the fuzzy comprehension evaluation of $o_{m}$ evaluated by $z_{m}$ at $t$, and

$$\begin{align} P_{n}^{m}(t)=B(t)\bullet R_n^{m}(t)=(p_{n1}^{m}(t), p_{n2}^{m}(t), p_{n3}^{m}(t), p_{n4}^{m}(t)), \end{align}$$

(17)

where

$$\begin{align} p_{ni}^{m}(t)=\sum_{j=1}^K{b_{j}(t)\cdot r_{ji}^{mn}(t)} \end{align}$$

(18)

and $p_{ni}^{m}(t)$ is the comprehension evaluated grade of the $o_{n}$ to $\omega_{i}$ provided by $z_{m}$ at $t$.

Primary membership grade $L_{o_{n}}^{i}$ of $\omega_{i}(t)$ is defined as:

$$\begin{align} L_{o_{n}}^{i}=[\underline{P_{ni}(t)}, \overline{P_{ni}(t)}], \end{align}$$

(19)

where $i=1, \ldots, 4$. $$\underline{P_{ni}(t)}=\min\ \{P_{ni}^{1}(t), \ldots, P_{ni}^{M}(t)\},$$ $$\overline{P_{ni}(t)}=\max\{P_{ni}^{1}(t), \ldots, P_{ni}^{M}(t)\},$$ $\omega_i(t)$ can be written as

$$\begin{align} \omega_i(t)=\sum_{n=1}^N\cfrac{\cfrac{1}{[\underline{P_{ni}(t)}, \overline{P_{ni}(t)}]}}{o_n}, \end{align}$$

(20)

when $t=kT, k=0, 1, 2, \ldots$, then the evaluation result $\omega_i(t)$ forms linguistic dynamic orbit as follows: $$\omega_i(0), \omega_i(T), \omega_i(2T), \ldots, \omega_i(kT), \ldots.$$ For simplicity, it is written as $$\omega_i, \omega_i(1), \omega_i(2), \ldots, \omega_i(k), \ldots.$$ Example 2. In Example 1, the psychological state of $\{o_{1}, o_{2}, o_{3}\}$ is observed by $\{z_{1}, z_{2}, z_{3}, z_{4}\}$ after every period $T$. The pressure source and the weight vector are the same as Example 1, and for the evaluation $r_{ki}^{mn}$ is the assess value for $o_{n}$ to $v_{i}$ of pressure source $X_{k}$ assessed by $z_{m}$, and $m=1, 2, \ldots, 4; n=1, 2, 3; k=1, 2, \ldots, 4; i=1, 2, 3, 4$ at $t=mT$.

When $m=0$, $r_{ki}^{mn}(mT)=r_{ki}^{mn}$ as in Table II, and $\omega_{i}(m)=\omega_{i}, i=1, 2, 3, 4$; when $m=1, 2$, the corresponding evaluations of $r_{ki}^{mn}(mT)$ are shown in Tables IV and V.

From Tables IV and V, by the method of fuzzy comprehension evaluation, the corresponding assessments can be represented as

$$\omega_{1}(1)=\frac{\frac{1}{[0. 40, 0. 46]}}{o_{1}}+\frac{\frac{1}{[0. 35, 0. 40]}}{o_{2}}+\frac{\frac{1}{[0. 06, 0. 08]}}{o_{3}},$$ $$\omega_{2}(1)=\frac{\frac{1}{[0. 24, 0. 30]}}{o_{1}}+\frac{\frac{1}{[0. 44, 0. 55]}}{o_{2}}+\frac{\frac{1}{[0. 06, 0. 13]}}{o_{3}},$$ $$\omega_{3}(1)=\frac{\frac{1}{[0. 24, 0. 29]}}{o_{1}}+\frac{\frac{1}{[0. 09, 0. 15]}}{o_{2}}+\frac{\frac{1}{[0. 39, 0. 46]}}{o_{3}},$$ $$\omega_{4}(1)=\frac{\frac{1}{[0. 03, 0. 06]}}{o_{1}}+\frac{\frac{1}{[0. 01, 0. 01]}}{o_{2}}+\frac{\frac{1}{[0. 41, 0. 42]}}{o_{3}},$$ $$\omega_{1}(2)=\frac{\frac{1}{[0. 47, 0. 56]}}{o_{1}}+\frac{\frac{1}{[0. 41, 0. 51]}}{o_{2}}+\frac{\frac{1}{[0. 07, 0. 09]}}{o_{3}},$$ $$\omega_{2}(2)=\frac{\frac{1}{[0. 19, 0. 29]}}{o_{1}}+\frac{\frac{1}{[0. 42, 0. 52]}}{o_{2}}+\frac{\frac{1}{[0. 06, 0. 14]}}{o_{3}},$$ $$\omega_{3}(2)=\frac{\frac{1}{[0. 15, 0. 25]}}{o_{1}}+\frac{\frac{1}{[0. 04, 0. 07]}}{o_{2}}+\frac{\frac{1}{[0. 37, 0. 45]}}{o_{3}},$$ $$\omega_{4}(2)=\frac{\frac{1}{[0. 0, 0. 03]}}{o_{1}}+\frac{\frac{1}{[0. 0, 0. 01]}}{o_{2}}+\frac{\frac{1}{[0. 41, 0. 42]}}{o_{3}}.$$ By the assessment results, it is easy to see that with time changing, the psychology states of $o_{1}, o_{2}$ become higher and higher, at the same time, that of $o_{3}$ is much less than before.

When $t=3T, 4T, \ldots$, by the evaluation process, for $o_{1}, o_{2}, o_{3}$, the assessment results are $\omega_{i}(3), \omega_{i}(4), \ldots$, then the linguistic dynamic orbit of psychological pressure for $o_{1}, o_{2}, o_{3}$ is as follows:

$$\omega_{i}, \omega_{i}(1), \omega_{i}(2), \ldots, \omega_{i}(k), \ldots$$ V. LINGUISTIC DYNAMIC MODELING FOR PSYCHOLOGICAL CRISIS INTERVENTION

If the psychological pressure is high or very high, then it is not the expected state, and necessary to be intervened, otherwise it may become worse than before, as $o_{1}, o_{2}$ in Example 2. Psychological intervention is the process to make the psychological health state be converted to the expected target according to the instruction of psychological theory^[5,25,26]. The target of all intervention models is to alleviate the symptoms of acute emergencies, restore the initiative, alleviate the psychological trauma and posttraumatic stress disorder. Health mental state can be reached by being provided proper interventions.

In the paper, IT2 FS is used to describe the psychological intervention strategy. If the factor affecting psychological health state is different, intervention measures are also different. Let the set of factors be $\{X_{1}, X_{2}, \ldots, X_{K}\}$, and for every factor $X_{k}$, the variable on $X_{k}$ is also represented as $X_{k}$ and the corresponding judgement set is $\{\alpha_{k1}, \ldots, \alpha_{kl_{k}}\}$. If there are $m_{k}$ kinds of intervention measures $\{\nu_{k1}, \ldots, \nu_{km_{k}}\}$, then by the expert's experiment knowledge, the fuzzy rules are represented as follows:

$R_{1}$: if $X_{1}(n)$ is $\alpha_{1l_{j_{1}}}$, and $U_{1}$ is $\nu_{1m_{r_{1}}}$, then $X_{1}(n+1)$ is $\alpha_{1l_{s_{1}}}$;

$R_{2}$: if $X_{2}(n)$ is $\alpha_{2l_{j_{2}}}$, and $U_{2}$ is $\nu_{2m_{r_{2}}}$, then $X_{2}(n+1)$ is $\alpha_{2l_{s_{2}}}$;

$$\vdots$$ $R_{K}$: if $X_{K}(n)$ is $\alpha_{Kl_{j_{K}}}$, and $U_{K}$ is $\nu_{Km_{r_{K}}}$, then $X_{K}(n+1)$ is $\alpha_{Kl_{s_{K}}}$;

where $l_{j_{k}}, l_{s_{k}}\in\{1, 2, \ldots, l_{k}\}, m_{r_{k}}\in\{1, 2, \ldots, m_{k}\}, R_{k}$ is the $k$-th fuzzy rule, $k=1, 2, \ldots, K$, and every $R_{k}$ contains $l_{k}\cdot{m_{k}}$ fuzzy subrules as follows:

$R_{k}(1, 1)$: if $X_{k}(n)$ is $\alpha_{k1}$, and $U_{k}$ is $\nu_{k1}$, then $X_{k}(n+1)$ is $\alpha_{kl_{t_{1}}}$;

$$\vdots$$ $R_{k}(1, m_{k})$: if $X_{k}(n)$ is $\alpha_{k1}$, and $U_{k}$ is $\nu_{km_{k}}$, then $X_{k}(n+1)$ is $\alpha_{kl_{t_{k}}}$;

$R_{k}(2, 1)$: if $X_{k}(n)$ is $\alpha_{k2}$, and $U_{k}$ is $\nu_{k1}$, then $X_{k}(n+1)$ is $\alpha_{kl_{w_{1}}}$;

$$\vdots$$ $R_{k}(2, m_{k})$: if $X_{k}(n)$ is $\alpha_{k2}$, and $U_{1}$ is $\nu_{km_{k}}$, then $X_{k}(n+1)$ is $\alpha_{kl_{w_{k}}}$; $$\vdots$$ $R_{k}(l_{k}, 1)$: if $X_{k}(n)$ is $\alpha_{kl_{k}}$, and $U_{k}$ is $\nu_{k1}$, then $X_{k}(n+1)$ is $\alpha_{kl_{d_{k}}}$; $$\vdots$$ $R_{k}(l_{k}, m_{k})$: if $X_{k}(n)$ is $\alpha_{kl_{k}}$, and $U_{k}$ is $\nu_{km_{k}}$, then $X_{k}(n+1)$ is $\alpha_{kl_{d_{k}}}$;

where $l_{t_{1}}, \ldots, l_{t_{k}}, l_{m_{1}}, \ldots, l_{m_{k}}, l_{d_{1}}, \ldots, l_{d_{k}}\in\{1, 2, \ldots, l_{k}\}$. Let $R=\cup_{k=1}^{K}R_{k}$, where

$$R_{k}=\bigcup_{l_{j_{k}}=1}^{l_{k}}\bigcup_{m_{r_{k}}=1}^{m_{k}} R_{k}(l_{j_{k}}, m_{r_{k}}),$$ then $$R=\cup_{k=1}^{K}\bigcup_{l_{j_{k}}=1}^{l_{k}}\bigcup_{m_{r_{k}}=1}^{m_{k}} R_{k}(l_{j_{k}}, m_{r_{k}}).$$ For the $k$-th factor $X_{k}$, if the initial words on $X_{k}$ is $\alpha_{k}(0)$, there is $$R(\alpha_{k}(0))=R_{k}(\alpha_{k}(0)),$$ i. e. , only fuzzy rule $R_{k}$ is activated. and $$R_{k}(\alpha_{k}(0))=\bigcup_{l_{j_{k}}=1}^{l_{k}}\bigcup_{m_{r_{k}}=1}^{m_{k}} R_{k} (l_{j_{k}}, m_{r_{k}})(\alpha_{k}(0)).$$ Let $\lambda(l_{j_{k}}, 0)=\alpha_{k}(0)\wedge{\alpha_{kl_{j_{k}}}}$ be the matching degree of $\alpha_{k}(0)$ and $\alpha_{kl_{j_{k}}}$, and for every $\alpha_{kl_{j_{k}}}$, there are $m_{r_{k}}$ interventions, and for every intervention, if $\lambda(l_{j_{k}}, 0)\neq{0}$, then there is an image word $\lambda(l_{j_{k}}, 0)\wedge{\alpha_{kl_{s_{k}}}}$. At the same time, if the intervention target $\alpha^{*}_{k}$ is given, and $$\lambda^{*}(l_{j_{k}}, 0)=\alpha^{*}_{k}\wedge{\alpha_{kl_{s_{k}}}}.$$ Might as well let $$\lambda^{*}(l_{j_{k}}, 0)=max\{\lambda^{*}(1, 0), \ldots, \lambda^{*}(l_{k}, 0)\},$$ then the corresponding intervention strategy $\nu_{km_{r_{k}}}$ is the best one. That is to say, once the intervention target is given, the best intervention strategy can be determines by the fuzzy rules. VI. CONCLUSION

In the paper, fuzzy comprehension evaluation and interval type-2 fuzzy sets are used to assess the psychological health state, and the corresponding linguistic dynamic modeling is built to analyze the evolution process, and the best intervention strategy is provided when the initial state and intervention target are given.

Big data plays a very important role in modern life. The future work will use big data to provide T2 FSs and linguistic dynamic orbit.