A study on the reform of college English education inspired by the cultural differences between China and the United States

Set up X_t(t = 1,2, ⋯,T) represents the time series of matrix values, where X_t is representative of the p₁ × p₂ dimension matrix [19], and the formula is as follows: (1) Xt=[xt,11⋯xt,1p2⋮⋱⋮xt,p11⋯xt,p1P2] {X_t} = \left[{\matrix {{{x_{t,11}}} & \cdots & {{x_{t,1{p_2}}}} \cr \vdots & \ddots & \vdots \cr {{x_{t,{p_1}1}}} & \cdots & {{x_{t,{p_1}{P_2}}}} \cr}} \right] If X_t represents the covariance matrix of a positive definite symmetry, so that p₁ = p₂ = n and implement decomposition, and the formula is as follows: (2) Xt=LtLt {X_t} = {L_t}{L_t} where L_t is the triangular matrix [20] after decomposition is described, which is modelled by the using matrix-valued factor method. The formula is as follows: (3) Lt=RFtC'+Et {L_t} = R{F_t}{C^{'}} + {E_t} Among them, F_t expresses k₁ × k₂ (k₁,k₂ ≪ n) a dimensional potential factor matrix, R expresses n × k₁ a pre-dimensional load matrix, C expresses n × k₂ a post-dimensional load matrix and E_t represents a difference matrix. Set up k₁ = n and R is n dimensional unit matrix, and the model becomes L_t = F_tC^′ + E_t. On this basis, in the matrix L_t, every column is F_t. The linear combination of the overall column vectors can be known as the matrix C. As column load matrix [21]. Similarly, the matrix R is a row load matrix.

Keep the row and column load matrix unchanged, and the factor matrix after dimension reduction F_t. Use the VAR mode to implement modelling and prediction, so that vec(·) vectorisation operator lag p order, and the formula is as follows: (4) vec(Ft+1)=∑p=1pΦF,t−p+1vec(Ft−p+1)+et+1 vec\left({{F_{t + 1}}} \right) = \sum\limits_{p = 1}^p {\Phi _{F,t - p + 1}}vec\left({{F_{t - p + 1}}} \right) + {e_{t + 1}} Φ_F is the k₁k₂ × k₁k₂ dimensional coefficient matrix and e is the cultural difference vector [22]. It can be concluded that the prediction matrix-valued factor model is as follows: (5) X^t+1=L^t+1L^t+1'=RF^t+1C'CF^t+1'R' {\hat X_{t + 1}} = {\hat L_{t + 1}}\hat L_{t + 1}^{'} = R{\hat F_{t + 1}}{C^{'}}C\hat F_{t + 1}^{'}{R^{'}} According to the VAR-LASSO model, the realised covariance matrix can be modelled: (6) vec(Xt)=∑p=1pΦV,t−pvec(Xt−p)+et vec\left({{X_t}} \right) = \sum\limits_{p = 1}^p {\Phi _{V,t - p}}vec\left({{X_{t - p}}} \right) + {e_t} It is assumed that the decomposition of the predictable matrix-valued factor model and the lag order of VAR-LASSO are 1, and the number of different elements in the covariance matrix is n(n + 1)/2; therefore, Φ_V is the representative [n(n+1)2×n(n+1)2] \left[{{{n(n + 1)} \over 2} \times {{n(n + 1)} \over 2}} \right] dimensional coefficient matrix and e_t is the representative error vector [23]. The VAR model without considering the shrinkage algorithm Φ_V. The number of parameters to be estimated is [n(n + 1)/2]². However, in the matrix-valued factor model, the high-dimensional covariance matrix to be estimated can be described by the low-dimensional factor matrix, and C Matrix, R Matrix, Φ_F. The total number of matrix parameters to be estimated is Φ_F. Therefore, the matrix-valued coefficient model can effectively reduce the matrix dimension and greatly reduce the number of prediction parameters [24].

Lack of interest is one of the main reasons why students’ English performance is not ideal. In order to enable students to learn English well, students should first have a strong interest in this kind of English [25]. In the teaching strategy, we should implement the teaching concept of ‘teachers leading students’ and adopt a variety of practice strategies to stimulate students’ interest in learning [26]. Teachers should give flexible guidance, choose the best teaching methods, encourage students to actively participate in the classroom and improve the teaching effect.

Using the construction method of the realised covariance matrix [27], for the reform of college English education, it is assumed that the number of courses in the whole day is m, students interested in English courses p, English classroom performance i and English classroom performance j between test scores t. The realised covariance of days is given by the following formula: (7) RCOVijt(m)=∑k=1m(pitk−pitk−1)(pjtk−pjtk−1) RCOV_{ijt}^{(m)} = \sum\limits_{k = 1}^m \left({{p_{itk}} - {p_{itk - 1}}} \right)\left({{p_{jtk}} - {p_{jtk - 1}}} \right) After obtaining the realised covariance, you can build n realised covariance matrix of dimension. According to the structure of the existing covariance matrix, the realised variances are sorted in order, and all the realised covariance matrices can be obtained [28]. The expression is as follows: (8) RCOVt=[RCOV11t⋯RCOV1nt⋮⋱⋮RCOVn1t⋯RCOVnnt] RCOV_{t} = \left[{\matrix {{RCOV_{{11t}}} & \cdots & {RCOV_{{1nt}}} \cr \vdots & \ddots & \vdots \cr {RCOV_{{n1t}}} & \cdots & {RCOV_{{nnt}}} \cr}} \right] According to the realised covariance, the structural model that can accurately reflect the essential characteristics of the entity is obtained for the formal description and reasoning of the entity.

The cultivation of English intercultural communicative competence should be emphasised in teachers’ teaching process. Therefore, a high level of cross-cultural English and correct values are necessary for every teacher [30]. On this basis, teachers should constantly strengthen their knowledge of anthropology and geography. It can make teachers more quickly and effectively understand the process of the formation and development of Chinese and American culture and the differences between Chinese and American cultures and expand teachers’ cross-cultural vision [31]. Second, teachers should strengthen the cultivation and establishment of cultural awareness, improve their mastery of professional skills and constantly strengthen and learn cross-cultural education and psychology.

In classroom teaching, teachers should actively guide students’ learning, stimulate students’ diversified appreciation ability, establish correct values for them, make them face the differences and obstacles between Chinese and Western English cultures and help students form an objective and fair cultural awareness system [32]. Teachers should combine language and culture closely with English teaching, guide students to experience the differences between Chinese and Western cultures and ensure the realisation of the purpose of cross-cultural communication.

As one of the subjects of teaching, teachers should constantly enrich their English language knowledge, teaching theory and thought; timely grasp the development trend of educational theory of the times; choose reasonable teaching methods [33]; learn from the successful teaching experience in history, actively adapt to the corresponding teaching reform work; push through the old and adopt a people-oriented way to find teaching methods suitable for students to learn English cross-cultural knowledge, so as to improve the teaching effect.

In teaching, teachers should correctly treat the cultural differences between China and the United States and guide students to analyse them from an objective and fair perspective [34]. First, colleges and universities should increase the investment in hardware facilities and build a good multimedia equipment environment. In classroom teaching, teachers can use the Internet to ‘learn and check’, use the Internet to search the relevant teaching cultural background in a short time, give students a comprehensive explanation, construct a sound Chinese and American cultural database and update it in time and provide the latest journals, magazines and documents, enabling students to learn the most cutting-edge knowledge of American culture [35]. Second, the teaching curriculum should be rich and diversified. Rich and diverse English courses can make students feel the culture of different levels and backgrounds, and the range of culture students will receive will be broader. Teachers should encourage students to develop the concepts of equality and language equality. In the research, teachers should overcome prejudice and eliminate the restrictions starting from their own values [36]. In education, teachers should provide students with more opportunities for learning strategies and training strategies.

In the development of college English education and teaching, we should pay attention to the infiltration of cross-cultural education, and it is very important to show it in the cultural connotation [37]. Teachers should actively adopt new teaching methods; infiltrate cross-cultural education into the teaching of reading, grammar and oral English; and establish new English learning ideas for students by expanding the scope of English learning. At the same time, teachers can transform the existing teaching mode, make students play a role in English cross-culture and stimulate students’ interest in English learning. In the setting of English teaching content, we need to get rid of the influence of stereotyped thinking. We should introduce the rich content of cross-cultural awareness and look at the problem from a global perspective. The world is an organic whole. American English culture is only a part of it. Teachers should help and guide students to understand the formation of English culture in various countries and the West, establish an open learning attitude for them and encourage students to take English as a tool to understand the world, so as to establish a standard and correct English cross-cultural learning attitude for students.

In view of the problems in the process of college English teaching, how to find and eliminate the possible adverse effects as soon as possible is the focus of this article. Using the powerful data analysis and processing ability of discriminant model, we can provide scientific means and basis for the prediction of actual teaching effect.

Set up n represents the covariance matrix dimension so that any U₁ and U₂ all n × n dimensional invertible matrix, decomposed L_t. The matrix can be represented by the matrix-valued factor model as follows: (9) Lt=RU1U1−1FtU2U2−1C'+Et {L_t} = R{U_1}U_1^ {- 1}{F_t}{U_2}U_2^ {- 1}{C^{'}} + {E_t} It is not difficult to see that the of the model may encounter identification problems. Although the C matrix and R matrix are difficult to identify, the R matrix and C column space of the matrix M(R) and M(C) are uniquely certain [38]. Adopt QR decomposition treatment R matrix and C matrix, and the formula is as follows: (10) R=Q1W1 R = {Q_1}{W_1} (11) C=Q2W2 C = {Q_2}{W_2} Q_m is a representative p_m × k_m dimensional column orthogonal matrix, W_m is the representative k_m × k_m dimensional upper triangular matrix m = 1,2 and Q₁ and Q₂ are frontal space. M (Q₁), M (Q₂), M (R) and M (C) are equivalent, and the estimated R matrix and C column space of the matrix are transformed into an estimate Q₁ and Q₂ column space, and the formula is as follows: (12) Zt=W1FtW2' {Z_t} = {W_1}{F_t}W_2^{'} Then the matrix-valued factor model is transformed into (13) Lt=Q1ZtQ2'+Et {L_t} = {Q_1}{Z_t}Q_2^{'} + {E_t} Take l_t,j, R_j, C_j, Q_m,j, ɛ_{m, j} separately as L_t, R, C, Q_m, E_t the first j column will Rk' R_k^{'} , Ck' C_k^{'} , Qm,k' Q_{m,k}^{'} , separately as R, C, Q_m on line k of, the expression is: (14) lt,j=RFtCj+εt,j=Q1ZtQ2,j+εt,j {l_{t,j}} = R{F_t}{C_j} + {\varepsilon _{t,j}} = {Q_1}{Z_t}{Q_{2,j}} + {\varepsilon _{t,j}} Set h to represent a positive integer step size. For i j = 1,2,3, ⋯,n, it is defined as Ω. (15) Ωzq,ij(h)=1T−h∑t=1T−hcov(ZtQ2,i,Zt+hQ2,j) {\Omega _{zq,ij}}(h) = {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} \mathop {\rm cov}\left({{Z_t}{Q_{2,i}},{Z_{t + h}}{Q_{2,j}}} \right) (16) Ωl,ij(h)=1T−h∑t=1T−hcov(lt,i,lt+h,j) {\Omega _{l,ij}}(h) = {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} \mathop {\rm cov}\left({{l_{t,i}},{l_{t + h,j}}} \right) (17) Ωl,ij(h)=Q1Ωzq,ij(h)Q1' {\Omega _{l,ij}}(h) = {Q_1}{\Omega _{zq,ij}}(h)Q_1^{'} Set h = h₀, defined as: (18) M=∑h=1h0∑i=1n∑j=1nΩl,ij(h)Ωl,ij'(h) M = \sum\limits_{h = 1}^{{h_0}} \sum\limits_{i = 1}^n \sum\limits_{j = 1}^n {\Omega _{l,ij}}(h)\Omega _{l,ij}^{'}(h) (19) M=Q1(∑h=1h0∑i=1n∑j=1nΩzq,ij(h)Ωzq,ij'(h))Q1' M = {Q_1}\left({\sum\limits_{h = 1}^{{h_0}} \sum\limits_{i = 1}^n \sum\limits_{j = 1}^n {\Omega _{zq,ij}}(h)\Omega _{zq,ij}^{'}(h)} \right)Q_1^{'} It can be seen that each column M in Q₁ is a linear combination of the whole column. Therefore, M(M) = M(Q₁) is estimated according to the eigenvector of the observed value of rotating M samples. The expression of M(Q₁) is as follows: (20) Ω^l,ij(h)=1T−h∑t=1T−hlt,ilt+h,j' {\hat \Omega _{l,ij}}(h) = {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} {l_{t,i}}l_{t + h,j}^{'} (21) M^=∑h=1h0∑i=1n∑j=1nΩ^l,ij(h)Ω^l,ij'(h) \hat M = \sum\limits_{h = 1}^{{h_0}} \sum\limits_{i = 1}^n \sum\limits_{j = 1}^n {\hat \Omega _{l,ij}}(h)\hat \Omega _{l,ij}^{'}(h) Implement cultural difference feature decomposition for M^ \hat M and take the former k₁ eigenvectors corresponding to the maximum eigenvalues q^1,⋯,q^k1 {\hat q_1}, \cdots,{\hat q_{{k_1}}} get Q^1={q^1,⋯,q^k1} {\hat Q_1} = \left\{{{{\hat q}_1}, \cdots , {{\hat q}_{{k_1}}}} \right\} hold L_t transpose processing and get Q^2 {\hat Q_2} .

According to k₁, the determination method recorded λ^1λ^2⋯λ^n {\hat \lambda _1}{\hat \lambda _2} \cdots {\hat \lambda _n} is M^ \hat M . Then the formula is as follows: (22) k1=argmin1in/2λ^i+1λ^i {k_1} = \arg \mathop {\min}\limits_{1in/2} {{{{\hat \lambda}_{i + 1}}} \over {{{\hat \lambda}_i}}} (23) Z^t=Q^1'LtQ^2 {\hat Z_t} = \hat Q_1^{'}{L_t}{\hat Q_2}

The vectorisation operation of Z^t {\hat Z_t} is implemented, and the VAR identification model is constructed to obtain Z^t {\hat Z_t} . (24) L^t+1=Q^1Z^t+1Q^2' {\hat L_{t + 1}} = {\hat Q_1}{\hat Z_{t + 1}}\hat Q_2^{'} where L^t+1 {\hat L_{t + 1}} represents the decomposed triangular matrix. The non-zero elements of the upper triangle are errors, so L^t+1 {\hat L_{t + 1}} elements in the l^t+1,ij {\hat l_{t + 1,ij}} threshold function can be obtained L^t+1*={l^t+1,ij*}n×n \hat L_{t + 1}^* = {\left\{{\hat l_{t + 1,ij}^*} \right\}_{n \times n}} : (25) l^t+1,ij*={l^t+1,ijij0i<j \hat l_{t + 1,ij}^* = \left\{{\matrix {{{{\hat l}_{t + 1,ij}}} & {ij} \cr 0 & {i < j} \cr}} \right. To sum up, the predicted realised covariance matrix is as follows: (26) X^t+1=L^t+1*L^t+1* {\hat X_{t + 1}} = \hat L_{t + 1}^*\hat L_{t + 1}^* The following formula is obtained through processing: (27) Ω^s,ij(h)=1T−h∑t=1T−hRFtCiCj'Ft+h'R' {\hat \Omega _{s,ij}}(h) = {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} R{F_t}{C_i}C_j^{'}F_{t + h}^{'}{R^{'}} (28) Ω^sε,ij(h)=1T−h∑t=1T−hRFtCiεt+h,j' {\hat \Omega _{s\varepsilon,ij}}(h) = {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} R{F_t}{C_i}\varepsilon _{t + h,j}^{'} (29) Ω^εs,ij(h)=1T−h∑t=1T−hεt,iCj'Ft+h'R' {\hat \Omega _{\varepsilon s,ij}}(h) = {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} {\varepsilon _{t,i}}C_j^{'}F_{t + h}^{'}{R^{'}} (30) Ω^ε,ij(h)=1T−h∑t=1T−hεt,iεt+h,j' {\hat \Omega _{\varepsilon,ij}}(h) = {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} {\varepsilon _{t,i}}\varepsilon _{t + h,j}^{'} Available: (31) Ω^l,ij(h)=1T−h∑t=1T−hlt,ilt+h,j'=1T−h∑t=1T−h(RFtCj+εt,j)(RFt+hCj+εt+h,j)=Ω^s,ij(h)+Ω^sε,ij(h)+Ω^εs,ij(h)+Ω^ε,ij(h) \matrix {{{{\hat \Omega}_{l,ij}}(h)} \hfill & {= {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} {l_{t,i}}l_{t + h,j}^{'}} \hfill \cr {} \hfill & {= {1 \over {T - h}}\sum\limits_{t = 1}^{T - h} \left({R{F_t}{C_j} + {\varepsilon _{t,j}}} \right)\left({R{F_{t + h}}{C_j} + {\varepsilon _{t + h,j}}} \right)} \hfill \cr {} \hfill & {= {{\hat \Omega}_{s,ij}}(h) + {{\hat \Omega}_{s\varepsilon,ij}}(h) + {{\hat \Omega}_{\varepsilon s,ij}}(h) + {{\hat \Omega}_{\varepsilon,ij}}(h)} \hfill \cr} About the n decomposition matrix of dimensional realised covariance matrix L. Using the Davydov inequality, the sample covariance space Ω^l,ij(h) {\hat \Omega _{l,ij}}(h) . The convergence rate of estimation error is as follows: (32) ∑i=1n∑j=1n∥Ω^l,ij(h)−Ωl,ij(h)∥224∑i=1n∑j=1n(∥Ω^s,ij(h)−Ωs,ij(h)∥22+∥Ω^sε,ij(h)∥22+∥Ω^εs,ij(h)∥22+∥Ω^ε,ij(h)∥22)=Op(n4T−1) \matrix {{\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \parallel {{\hat \Omega}_{l,ij}}(h) - {\Omega _{l,ij}}(h)\parallel _2^24\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \left({\parallel {{\hat \Omega}_{s,ij}}(h) - {\Omega _{s,ij}}(h)\parallel _2^2 + \parallel {{\hat \Omega}_{s\varepsilon,ij}}(h)\parallel _2^2} \right.} \hfill \cr {\left. {+ \parallel {{\hat \Omega}_{\varepsilon s,ij}}(h)\parallel _2^2 + \parallel {{\hat \Omega}_{\varepsilon,ij}}(h)\parallel _2^2} \right) = {O_p}\left({{n^4}{T^ {- 1}}} \right)} \hfill \cr}