Optimization Design of College Teaching Reform Paths in the Context of Big Data Mining-Driven High-Quality Development of Commerce and Circulation Based on Big Data Mining

To study an object using factor analysis is to study the underlying relationships between its attributes. The raw data, which are the sample values, are provided with 2 random variables x, y, which represent two variables A and B. Their content values are measured for n specimens: (1)x→=(x1,x2,⋯⋯,xn)$$\overrightarrow x = ({x_1},{x_2}, \cdots \cdots ,{x_n})$$ (2)y→=(y1,y2,⋯⋯,yn)$$\overrightarrow y = ({y_1},{y_2}, \cdots \cdots ,{y_n})$$

The samples were first standardized and the mean and variance were calculated according to the following formula: (3)x¯=1n∑i=1nxi y¯=1n∑i=1nyi σx2=1n∑i=1n(xi−x¯)2 σy2=1n∑i=1n(yi−y¯)2$$\begin{array}{l} \overline x = \frac{1}{n}\sum\limits_{i = 1}^n {{x_i}} \\ \overline y = \frac{1}{n}\sum\limits_{i = 1}^n {{y_i}} \\ \sigma_x^2 = \frac{1}{n}\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} \\ \sigma_y^2 = \frac{1}{n}\sum\limits_{i = 1}^n {{{({y_i} - \bar y)}^2}} \\ \end{array}$$

Re-order: (4)xi′=xi−x¯σx,yi′=yi−y¯σy,i=1,2⋯⋯,n$$x_i^\prime = \frac{{{x_i} - \overline x }}{{{\sigma_x}}},\quad y_i^\prime = \frac{{{y_i} - \overline y }}{{{\sigma_y}}},\quad i = 1,2 \cdots \cdots ,n$$

The sample after standardization meets the following conditions: (5)x′¯=1n∑i=1nxi′=0,y′¯=1n∑i=1nyi′=0$$\overline {x'} = \frac{1}{n}\sum\limits_{i = 1}^n {{x'_i}} = 0,\quad \overline {y'} = \frac{1}{n}\sum\limits_{i = 1}^n {{{y'}_i}} = 0$$ (6)σx′2=1n∑i=1nxi′2=1,σy′2=1n∑i=1nyi′2=1$$\sigma_{x'}^2 = \frac{1}{n}\sum\limits_{i = 1}^n {{x'_i}^2} = 1,\quad \sigma_{y'}^2 = \frac{1}{n}\sum\limits_{i = 1}^n {{{y'}_i}^2} = 1$$

Here, x¯,y¯$$\overline x ,\overline y$$ is still used to represent the samples after standardization, and their variance and correlation coefficients can be calculated according to the following formula: (7){ σx2=1n∑i=1nxi2=1nx′¯x¯=1 σy2=1n∑i=1nyi2=1ny′¯y¯=1 Yxy=1n∑i=1nxiyi=1nx′¯y¯$$\left\{ {\begin{array}{*{20}{l}} {\sigma_x^2 = \frac{1}{n}\sum\limits_{i = 1}^n {x_i^2} = \frac{1}{n}\overline {{x^\prime }} \bar x = 1} \\ {\sigma_y^2 = \frac{1}{n}\sum\limits_{i = 1}^n {y_i^2} = \frac{1}{n}\overline {{y^\prime }} \bar y = 1} \\ {{Y_{xy}} = \frac{1}{n}\sum\limits_{i = 1}^n {{x_i}} {y_i} = \frac{1}{n}\overline {{x^\prime }} \bar y} \end{array}} \right.$$

It can be shown that the random variables x→,y→$$\overrightarrow x,\overrightarrow y$$ are uncorrelated, Y_xy = 0 and algebraically equivalent to their inner product x→′y→=0$${\overrightarrow x'}\overrightarrow y=0$$ and geometrically the two vectors are directly intersecting.

For n samples with m variables each, the original data matrix is as follows: (8)X=[ x11 x12 ⋯ x1m x21 x22 ⋯ x2m ⋯ ⋯ ⋯ ⋯ xn1 xn2 ⋯ xnm]=[ x1→,x2→,⋯,xm→]$$X = \left[ {\begin{array}{*{20}{c}} {{x_{11}}}&{ {x_{12}}}& \cdots &{ {x_{1m}}} \\ {{x_{21}}}&{ {x_{22}}}& \cdots &{ {x_{2m}}} \\ \cdots & \cdots & \cdots & \cdots \\ {{x_{n1}}}&{ {x_{n2}}}& \cdots &{ {x_{nm}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\overrightarrow {{x_1}} ,\overrightarrow {{x_2}} , \cdots ,\overrightarrow {{x_m}} } \end{array}} \right]$$

The column vector at the right end of the equation: (9)x→j=(x1j,x2j,⋯,xnj)′,j=1,2,⋯,m$${\vec x_j} = {({x_{1j}},{x_{2j}}, \cdots ,{x_{nj}})^\prime },j = 1,2, \cdots ,m$$

The observation representing the jst variable on the n sample can be viewed as a point or vector in a n dimensional Euclidean space, here denoted by x→j$${\vec x_j}$$. The relationship between the original variables is studied by examining the positional relationship of these m points or vectors.

If the sample data is normalized, i.e., X a normalized matrix, there is: (10)x→j=1n∑i=1nxij=0$${\vec x_j} = \frac{1}{n}\sum\limits_{i = 1}^n {{x_{ij}}} = 0$$ (11)σj2=1n∑i=1nxij2=1nx→jx→j=1,2,⋯,m$$\sigma_j^2 = \frac{1}{n}\sum\limits_{i = 1}^n {x_{ij}^2} = \frac{1}{n}{\vec x_j}{\vec x_j} = 1,2, \cdots ,m$$

Then, the correlation coefficient between x→j$${\vec x_j}$$ and x→k$${\vec x_k}$$ is, by Eq: (12)Yjk=1n∑i=1nxijxik=1nx→j′x→k,j,k=1,2,⋯,m$${Y_{jk}} = \frac{1}{n}\sum\limits_{i = 1}^n {{x_{ij}}} {x_{ik}} = \frac{1}{n}\vec x_j^\prime{\vec x_k}\:,j,k = 1,2, \cdots ,m$$

The correlation coefficient matrix R consists of the correlation coefficients between the m variables: (13)R=[ r11 r12 ⋯ r1m r21 r22 ⋯ r2m ⋯ ⋯ ⋯ ⋯ rm1 rm2 ⋯ rmm]=1nx′x$$R = \left[ {\begin{array}{*{20}{c}} {{r_{11}}}&{ {r_{12}}}& \cdots &{ {r_{1m}}} \\ {{r_{21}}}&{ {r_{22}}}& \cdots &{ {r_{2m}}} \\ \cdots & \cdots & \cdots & \cdots \\ {{r_{m1}}}&{ {r_{m2}}}& \cdots &{ {r_{mm}}} \end{array}} \right] = \frac{1}{n}x'x$$

The correlation coefficient matrix R is symmetric and at least semi-positive definite, which means that all its eigenvalues are non-negative.

The correlation coefficient matrix is the starting point of the factor analysis method and an important part of factor analysis is to study the structure of the correlation matrix [21]. Also in factor analysis, we are often involved in the correlation coefficient matrix between two sets of variables, assuming that in addition to the previous m random variables, there are another p random variables, the matrix is as follows: (14)y=[ y11 y12 ⋯ y1p y21 y22 ⋯ y2p ⋯ ⋯ ⋯ ⋯ yn1 yn2 ⋯ ynp]=[ y¯1,y¯2,⋯,y¯p]$$y = \left[ {\begin{array}{*{20}{c}} {{y_{11}}}&{ {y_{12}}}& \cdots &{ {y_{1p}}} \\ {{y_{21}}}&{ {y_{22}}}& \cdots &{ {y_{2p}}} \\ \cdots & \cdots & \cdots & \cdots \\ {{y_{n1}}}&{ {y_{n2}}}& \cdots &{ {y_{np}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{{\bar y}_1},{{\bar y}_2}, \cdots ,{{\bar y}_p}} \end{array}} \right]$$

Assuming all standardized data, the correlation coefficient between y→k$${\vec y_k}$$ and x→j$${\vec x_j}$$ is given by Eq: (15)Skj=1ny→kx→j,k=1,2,⋯,p;j=1,2,⋯,m$${S_{kj}} = \frac{1}{n}{\vec y_k}{\vec x_j},k = 1,2, \cdots ,p;j = 1,2, \cdots ,m$$

Written in matrix form as follows: (16)Sp×m=[ S11 S12 ⋯ S1m S21 S22 ⋯ S2m ⋯ ⋯ ⋯ ⋯ Sp1 Sp2 ⋯ Spm]=[ 1−ny1x1 1−ny1x2 ⋯ 1−ny1′xm 1−ny2x1 1−ny2′x2 ⋯ 1−ny2′xm ⋯ ⋯ ⋯ ⋯ 1−nyp′x1 1−nyp′x2 ⋯ 1−nyp′xm]=1n[ y→1′ y→2′ ⋮ y→p′][ x→1,x→2,⋯,x→m x→1,x→2,⋯,x→m]=1nYX$${S_{p \times m}} = \left[ {\begin{array}{*{20}{c}} {{S_{11}}}&{ {S_{12}}}& \cdots &{ {S_{1m}}} \\ {{S_{21}}}&{ {S_{22}}}& \cdots &{ {S_{2m}}} \\ \cdots & \cdots & \cdots & \cdots \\ {{S_{p1}}}&{ {S_{p2}}}& \cdots &{ {S_{pm}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{{1 - }}{n}{y_1}{x_1}}&{ \frac{{1 - }}{n}{y_1}{x_2}}& \cdots &{ \frac{{1 - }}{n}y_1^\prime {x_m}} \\ {\frac{{1 - }}{n}{y_2}{x_1}}&{ \frac{{1 - }}{n}y_2^\prime {x_2}}& \cdots &{ \frac{{1 - }}{n}y_2^\prime {x_m}} \\ \cdots & \cdots & \cdots & \cdots \\ {\frac{{1 - }}{n}y_p^\prime {x_1}}&{ \frac{{1 - }}{n}y_p^\prime {x_2}}& \cdots &{ \frac{{1 - }}{n}y_p^\prime {x_m}} \end{array}} \right] = \frac{1}{n}\left[ {\begin{array}{*{20}{c}} {\vec y_1^\prime } \\ {\vec y_2^\prime } \\ \vdots \\ {\vec y_p^\prime } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\vec x}_1},{{\vec x}_2}, \cdots ,{{\vec x}_m}} \\ {{{\vec x}_1},{{\vec x}_2}, \cdots ,{{\vec x}_m}} \end{array}} \right] = \frac{1}{n}YX$$

The common factor of factor analysis can, in fact, be expressed in the following linear algebraic form: (17){ x1→=a11f¯1+a21f¯2+⋯+ap1f¯p+μ1ε¯1 x2→=a12f¯1+a22f¯2+⋯+ap2f¯p+μ2ε¯2 ⋯ xm→=a1mf¯1+a2mf¯2+⋯+apmf¯p+μmε¯m$$\left\{ {\begin{array}{*{20}{c}} {\overrightarrow {{x_1}} = {a_{11}}{{\bar f}_1} + {a_{21}}{{\bar f}_2} + \cdots + {a_{p1}}{{\bar f}_p} + {\mu_1}{{\bar \varepsilon }_1}} \\ {\overrightarrow {{x_2}} = {a_{12}}{{\bar f}_1} + {a_{22}}{{\bar f}_2} + \cdots + {a_{p2}}{{\bar f}_p} + {\mu_2}{{\bar \varepsilon }_2}} \\ \cdots \\ {\overrightarrow {{x_m}} = {a_{1m}}{{\bar f}_1} + {a_{2m}}{{\bar f}_2} + \cdots + {a_{pm}}{{\bar f}_p} + {\mu_m}{{\bar \varepsilon }_m}} \end{array}} \right.$$

Abbreviated into: (18)x→j=∑k=1pakjf¯k+μjε→j,j=1,2,⋯,m$${\vec x_j} = \sum\limits_{k = 1}^p {{{\text{a}}_{kj}}} {\bar f_k} + {\mu_j}{\vec \varepsilon_j},j = 1,2, \cdots ,m$$

Where f→1,f→2,……,f→p $${\vec f_1},{\vec f_2}, \ldots \ldots ,{\vec f_p}$$ and ε→1,ε→2,……,ε→m$${\vec \varepsilon_1},{\vec \varepsilon_2}, \ldots \ldots ,{\vec \varepsilon_m}$$ are the new variables sought, the former is the common factor can be understood as commonality. The latter is called the single factor, or the individuality factor. Positive integer P represents the number of common factors, which is much smaller than the original number of variables m, the formula means to simplify the original m variables into a small number of factors, the coefficients a_kj and μ_j(j = 1, 2, ⋯⋯, m; k = 1, 2, ⋯⋯, p) are called factor loadings or factor loadings, the former is called the common factor loadings, the latter is called the single factor loadings, since we are concerned only with the common factors, usually referred to as factor loadings refers only to the former.

Notation: (19)A=[ a11 a12 ⋯ a1m a21 a22 ⋯ a2m ⋯ ⋯ ⋯ ⋯ ap1 ap2 ⋯ apm]p×m$$A = {\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{ {a_{12}}}& \cdots &{ {a_{1m}}} \\ {{a_{21}}}&{ {a_{22}}}& \cdots &{ {a_{2m}}} \\ \cdots & \cdots & \cdots & \cdots \\ {{a_{p1}}}&{ {a_{p2}}}& \cdots &{ {a_{pm}}} \end{array}} \right]_{p \times m}}$$

where a_kj is the loading of the jnd variable on the krd factor (k = 1, 2, ……, p; j = 1, 2, ……., m). (20)F=[ f¯1,f¯2,⋯,f¯p]=[ f11 f12 ⋯ f1p f21 f22 ⋯ f2p ⋯ ⋯ ⋯ ⋯ fn1 fn2 ⋯ fnp]n×p$$F = \left[ {\begin{array}{*{20}{c}} {{{\bar f}_1},{{\bar f}_2}, \cdots ,{{\bar f}_p}} \end{array}} \right] = {\left[ {\begin{array}{*{20}{c}} {{f_{11}}}&{ {f_{12}}}& \cdots &{ {f_{1p}}} \\ {{f_{21}}}&{ {f_{22}}}& \cdots &{ {f_{2p}}} \\ \cdots & \cdots & \cdots & \cdots \\ {{f_{n1}}}&{ {f_{n2}}}& \cdots &{ {f_{np}}} \end{array}} \right]_{n \times p}}$$

Where column k is the value of the knd factor on each specimen, this matrix is called the factorial measure. (21)U=[ u1 0 ⋯ 0 0 u2 ⋯ 0 ... ... ... ... 0 0 ⋯ um]m×m$$U = {\left[ {\begin{array}{*{20}{c}} {{u_1}}&0& \cdots &0 \\ 0&{ {u_2}}& \cdots &0 \\ {...}&{ ...}&{ ...}&{ ...} \\ 0&0& \cdots &{ {u_m}} \end{array}} \right]_{m \times m}}$$

This is the mst order diagonal matrix where the jnd diagonal element u_j is the loading (j = 1, 2, ……, m) of variable X_j on a single factor ε_j. (22)E=[ ε→1,ε→2,⋯,ε→m]=[ ε11 ε12 ⋯ ε1m ε21 ε22 ⋯ ε2m ⋯ ⋯ ⋯ ⋯ εn1 εn2 ⋯ εnm]$$E = \left[ {\begin{array}{*{20}{c}} {{{\vec \varepsilon }_1},{{\vec \varepsilon }_2}, \cdots ,{{\vec \varepsilon }_m}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\varepsilon_{11}}}&{ {\varepsilon_{12}}}& \cdots &{ {\varepsilon_{1m}}} \\ {{\varepsilon_{21}}}&{ {\varepsilon_{22}}}& \cdots &{ {\varepsilon_{2m}}} \\ \cdots & \cdots & \cdots & \cdots \\ {{\varepsilon_{n1}}}&{ {\varepsilon_{n2}}}& \cdots &{ {\varepsilon_{nm}}} \end{array}} \right]$$

where column j is the value of ε_j on each specimen. Then Eq. can be rewritten in the following form: (23)[ x˜1,x˜2,⋯,x˜m]=[ f¯1,f¯2,⋯,f¯p][ a11 a12 ⋯ a1m a21 a22 ⋯ a2m ⋯ ⋯ ⋯ ⋯ ap1 ap2 ⋯ apm]+[ ε→1,ε→2,⋯,ε→m][ u1 0 ⋯ 0 0 u2 ⋯ 0 ⋯ ⋯ ⋯ ⋯ 0 0 ⋯ um]$$\left[ {\begin{array}{*{20}{c}} {{{\tilde x}_1},{{\tilde x}_2}, \cdots ,{{\tilde x}_m}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{{\bar f}_1},{{\bar f}_2}, \cdots ,{{\bar f}_p}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{ {a_{12}}}& \cdots &{ {a_{1m}}} \\ {{a_{21}}}&{ {a_{22}}}& \cdots &{ {a_{2m}}} \\ \cdots & \cdots & \cdots & \cdots \\ {{a_{p1}}}&{ {a_{p2}}}& \cdots &{ {a_{pm}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{{\vec \varepsilon }_1},{{\vec \varepsilon }_2}, \cdots ,{{\vec \varepsilon }_m}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{u_1}}&0& \cdots &0 \\ 0&{ {u_2}}& \cdots &0 \\ \cdots & \cdots & \cdots & \cdots \\ 0&0& \cdots &{ {u_m}} \end{array}} \right]$$ (24)X=FA+EU$$X = FA + EU$$

We already know that the original variables xj→$$\overrightarrow{x_j}$$ in Eq. are all standardized variables, now assume again that both the public factor f→k(k=1,2,⋯⋯,p)$${\vec f_k}(k = 1,2, \cdots \cdots ,p)$$ and the single factor ε→j(j=1,2,⋯⋯,m)$${\vec \varepsilon_j}(j = 1,2, \cdots \cdots ,m)$$ to be solved are also standardized variables.

And the correlation coefficients between all the common factors and between the single factors are 0. Then, there is the following relationship according to Eq: (25){ 1n∑i=1nfik=0,k=1,2,⋯,p 1n∑i=1nεij=0,j=1,2,⋯,m 1nf→k′f→l=δkl={ 1, k=l 0, k≠lk,l=1,2,⋯,p 1nε→j′ε→q=δjq={ 1, j=q 0, j≠qj,q=1,2,⋯,m 1nf→k′ε→j=0,k=1,2,⋯,p;j=1,2,⋯,m$$\left\{ {\begin{array}{*{20}{c}} {\frac{1}{n}\sum\limits_{i = 1}^n {{f_{ik}}} = 0,k = 1,2, \cdots ,p} \\ {\frac{1}{n}\sum\limits_{i = 1}^n {{\varepsilon_{ij}}} = 0,j = 1,2, \cdots ,m} \\ {\frac{1}{n}\vec f_k^\prime{{\vec f}_\ell } = {\delta_{k\ell }} = \left\{ {\begin{array}{*{20}{l}} {1,}&{ k = \ell } \\ {0,}&{ k \ne \ell } \end{array}} \right.k,\ell = 1,2, \cdots ,p} \\ {\frac{1}{n}\vec \varepsilon_j^\prime {{\vec \varepsilon }_q} = {\delta_{jq}} = \left\{ {\begin{array}{*{20}{l}} {1,}&{ j = q} \\ {0,}&{ j \ne q} \end{array}} \right.j,q = 1,2, \cdots ,m} \\ {\frac{1}{n}{{\vec f}_k}^\prime {{\vec \varepsilon }_j} = 0,k = 1,2, \cdots ,p;j = 1,2, \cdots ,m} \end{array}} \right.$$

These relational equations are written in matrix form and the correlation matrix between the metrics is obtained from Eq: (26)1nF′F=1n[ f¯1′ f¯2′ ⋮ f¯p′][ f¯1,f¯2,⋯,f¯p]=[ 1nf¯1′f¯1 1nf¯1′f¯2 ⋯ 1nf¯1′f¯p 1nf¯2′f¯1 1nf¯2′f¯2 ⋯ 1nf¯2′f¯p ⋯ ⋯ ⋯ ⋯ 1nf¯p′f¯1 1nf¯p′f¯2 ⋯ 1nf¯p′f¯p]=[ 1 0 ⋯ 0 0 1 ⋯ 0 ⋯ ⋯ ⋯ ⋯ 0 0 ⋯ 1]=Ip$$\frac{1}{n}F'F = \frac{1}{n}\left[ {\begin{array}{*{20}{c}} {\bar f_1^\prime } \\ {\bar f_2^\prime } \\ \vdots \\ {\bar f_p^\prime } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\bar f}_1},{{\bar f}_2}, \cdots ,{{\bar f}_p}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{1}{n}\bar f_1^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_1^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_1^\prime {{\bar f}_p}} \\ {\frac{1}{n}\bar f_2^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_2^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_2^\prime {{\bar f}_p}} \\ \cdots & \cdots & \cdots & \cdots \\ {\frac{1}{n}\bar f_p^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_p^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_p^\prime {{\bar f}_p}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0& \cdots &0 \\ 0&1& \cdots &0 \\ \cdots & \cdots & \cdots & \cdots \\ 0&0& \cdots &1 \end{array}} \right] = {I_p}$$

where I_p is a unit matrix of order p. Similarly, the correlation matrix between the single factors can be obtained as: (27)1nE′E=Im$$\frac{1}{n}{E^\prime }E = {I_m}$$

Then the correlation matrix between the common factor and the single factor is: (28)1nF′E=1n[ f¯1′ f¯2′ ⋮ f¯p′][ ε→1,ε→2,⋯,ε→m]=[ 1nf¯1′f¯1 1nf¯1′f¯2 ⋯ 1nf¯1′f¯p 1nf¯2′f¯1 1nf¯2′f¯2 ⋯ 1nf¯2′f¯p ⋯ ⋯ ⋯ ⋯ 1nf¯p′f¯1 1nf¯p′f¯2 ⋯ 1nf¯p′f¯p]=[ 1 0 ⋯ 0 0 1 ⋯ 0 ⋯ ⋯ ⋯ ⋯ 0 0 ⋯ 1]=H$$\frac{1}{n}{F^\prime }E = \frac{1}{n}\left[ {\begin{array}{*{20}{c}} {\bar f_1^\prime } \\ {\bar f_2^\prime } \\ \vdots \\ {\bar f_p^\prime } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\vec \varepsilon }_1},{{\vec \varepsilon }_2}, \cdots ,{{\vec \varepsilon }_m}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{1}{n}\bar f_1^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_1^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_1^\prime {{\bar f}_p}} \\ {\frac{1}{n}\bar f_2^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_2^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_2^\prime {{\bar f}_p}} \\ \cdots & \cdots & \cdots & \cdots \\ {\frac{1}{n}\bar f_p^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_p^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_p^\prime {{\bar f}_p}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0& \cdots &0 \\ 0&1& \cdots &0 \\ \cdots & \cdots & \cdots & \cdots \\ 0&0& \cdots &1 \end{array}} \right] = H$$

According to Eq. the correlation matrix between the obtained public factors and the original variables can be obtained: (29)[ 1nf¯1′f¯1 1nf¯1′f¯2 ⋯ 1nf¯1′f¯p 1nf¯2′f¯1 1nf¯2′f¯2 ⋯ 1nf¯2′f¯p ⋯ ⋯ ⋯ ⋯ 1nf¯p′f¯1 1nf¯p′f¯2 ⋯ 1nf¯p′f¯p]1n[ f¯1′ f¯2′ ⋮ f¯p′][ x→1,x→2,⋯,x→m]=1nFX=1nFFA+1nFEU=IpA+HU=A$$\left[ {\begin{array}{*{20}{c}} {\frac{1}{n}\bar f_1^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_1^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_1^\prime {{\bar f}_p}} \\ {\frac{1}{n}\bar f_2^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_2^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_2^\prime {{\bar f}_p}} \\ \cdots & \cdots & \cdots & \cdots \\ {\frac{1}{n}\bar f_p^\prime {{\bar f}_1}}&{ \frac{1}{n}\bar f_p^\prime {{\bar f}_2}}& \cdots &{ \frac{1}{n}\bar f_p^\prime {{\bar f}_p}} \end{array}} \right]\frac{1}{n}\left[ {\begin{array}{*{20}{c}} {\bar f_1^\prime } \\ {\bar f_2^\prime } \\ \vdots \\ {\bar f_p^\prime } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\vec x}_1},{{\vec x}_2}, \cdots ,{{\vec x}_m}} \end{array}} \right] = \frac{1}{n}FX = \frac{1}{n}FFA + \frac{1}{n}FEU = {I_p}A + HU = A$$

where H is the zero matrix.

This shows is the correlation coefficient between the common factor and the original variable for the elements in the factor loading A: (30)1nf→k′x→j=akj,k=1,2,⋯,p;j=1,2,⋯,m$$\frac{1}{n}{\vec f_{k'}}{\vec x_j} = {{\text{a}}_{kj}},k = 1,2, \cdots ,p;j = 1,2, \cdots ,m$$

Factor loading a_kj reflects the link between factor f¯k$${\bar f_k}$$ and variable x¯j$${\bar x_j}$$. When a_kj > 0, it indicates a positive correlation between factor f¯k$${\bar f_k}$$ and variable x_j. When a_kj < 0, it indicates an inverse correlation between factor f→k$${\vec f_k}$$ and variable x→j$${\vec x_j}$$. When a_kj ≈ 0, indicates a weak link between factor f→k$${\vec f_k}$$ and variable x→j$${\vec x_j}$$, the role of a_kj can be seen more clearly.

The correlation array R can be expressed as from Eq: (31)R = 1nX′X=1n(FA+EU)′(FA+EU)=1nA′F′FA+1nU′E′FA+1nA′F′EU+1nU′E′EU = A′(1uF′F)A+U′(1uE′F)A+A′(1uF′E)U+U′(1uE′E)U=A′A+U′U = R*+[ u12 0 ⋯ 0 0 u22 ⋯ 0 ⋯ ⋯ ⋯ ⋯ 0 0 ⋯ um2]$$\begin{array}{rcl} R &=& \frac{1}{n}X'X = \frac{1}{n}{(FA + EU)^\prime }(FA + EU) = \frac{1}{n}A'F'FA + \frac{1}{n}U'E'FA + \frac{1}{n}A'F'EU + \frac{1}{n}U'E'EU \\ &=& {A^\prime }(\frac{1}{u}{F^\prime }F)A + {U^\prime }(\frac{1}{u}{E^\prime }F)A + {A^\prime }(\frac{1}{u}{F^\prime }E)U + {U^\prime }(\frac{1}{u}{E^\prime }E)U = {A^\prime }A + {U^\prime }U \\ &=& {R^*} + \left[ {\begin{array}{*{20}{c}} {u_1^2}&0& \cdots &0 \\ 0&{ u_2^2}& \cdots &0 \\ \cdots & \cdots & \cdots & \cdots \\ 0&0& \cdots &{ u_m^2} \end{array}} \right] \\ \end{array}$$

Among them: (32)R*=A′A$${R^*} = {A^\prime}A$$

is called the approximate correlation matrix, the raw variable correlation coefficients, and the non-diagonal elements are the same as in R. (33)rij=∑k=1pakiakj,i,j=1,2,⋯,m$${r_{ij}} = \sum\limits_{k = 1}^p {{a_{ki}}} {a_{kj}},i,j = 1,2, \cdots ,m$$

The diagonal element of R^* is: (34)hj2=∑k=1pakj2=1−uj2,j=1,2,⋯,m$${h_j}^2 = \sum\limits_{k = 1}^p {a_{kj}^2} = 1 - {u_j}^2,j = 1,2, \cdots ,m$$

hj2$$h_j^2$$ is called the metric variance of variable x→j$${\vec x_j}$$, which represents the share of each metric in the variance of the original variable, and is numerically equal to the sum of the squares of the elements in column j of A. The common factor variance represents the extent to which all of the original variables can be explained by these p common factors, and takes values between 0 and 1, which are positively correlated.

The sum of the common factor variances is shown below: (35)∑j=1mhj2=∑j=1m∑k=1pakj2=∑k=1p∑j=1makj2=∑k=1pSk2$$\sum\limits_{j = 1}^m {{h_j}^2} = \sum\limits_{j = 1}^m {\sum\limits_{k = 1}^p {{a_{kj}}^2} } = \sum\limits_{k = 1}^p {\sum\limits_{j = 1}^m {{a_{kj}}^2} } = \sum\limits_{k = 1}^p {{S_k}^2}$$

Among them: (36)Sk2=∑j=1makj2,k=1,2,⋯,p$${S_k}^2 = \sum\limits_{j = 1}^m {a_{kj}^2} ,k = 1,2, \cdots ,p$$

is called the variance contribution of factor f¯k$${\bar f_k}$$ and is numerically equal to the sum of the squares of the elements in row k of A. It indicates the degree of contribution, or significance, that factor f¯k$${\bar f_k}$$ plays in all of the common factors, and is again positively correlated.

All the attributes, variables or indicators, known or calculated, in Eq. can be regarded as vectors in a n-dimensional Euclidean space. Since they are all normalized variables, the squares of their modulus lengths are n, i.e., the lengths of the vectors are all n$$\sqrt n$$. From Eq. the vectors corresponding to all the factors are orthogonal to each other two by two, and they form the base of a p + m subspace, Eq. is the expansion of the variables x→j$${\vec x_j}$$ within this set of bases, and the factor loadings are the coordinates of x→j$${\vec x_j}$$ the variables within this set of bases. The variables and the projections of the variables into the common factor space are shown in Figure 1. The common factor space refers to the p dimensional subspace generated by the p common factors, and the projection of variable x→j$${\vec x_j}$$ in the common factor space is: (37)x→j*=a1jf→1+a2jf→2+⋯+apjf→p=∑k=1pakjf→k,j=1,2,⋯,m$$\vec x_j^* = {a_{1j}}{\vec f_1} + {a_{2j}}{\vec f_2} + \cdots + {a_{pj}}{\vec f_p} = \sum\limits_{k = 1}^p {{a_{kj}}} {\vec f_k},j = 1,2, \cdots ,m$$

Figure 1.

Variables and their projections in the common factor space

This x→j*$$\vec x_j^*$$ is called the projective variable of x→j$${\vec x_j}$$, which is the square of its length: (38)||x→j*||2=x→{_j^*′ }x→j*=(∑k=1pakjf¯k)′(∑l=1paljf¯l)=∑k=1p∑l=1pakjaijf¯k′f¯l=∑k=1pakj2f¯k′f¯k=∑k=1pakj2=nhj2$$||\vec x_j^*|{|^2} = \vec x_j^{*\prime} \vec x_j^* = {(\sum\limits_{k = 1}^p {{a_{kj}}} {\bar f_k})^\prime }(\sum\limits_{\ell = 1}^p {{a_{\ell j}}} {\bar f_\ell }) = \sum\limits_{k = 1}^p {\sum\limits_{\ell = 1}^p {{a_{kj}}} } {a_{ij}}\bar f_k^\prime {\bar f_\ell } = \sum\limits_{k = 1}^p {a_{kj}^2} \bar f_k^\prime {\bar f_k} = \sum\limits_{k = 1}^p {a_{kj}^2} = nh_j^2$$

Also from ||x→j||2=n$$||{\vec x_j}|{|^2} = n$$ you can get the square of the cosine of the angle between vector x→j$${\vec x_j}$$ and the projected vector x→j*$$\vec x_j^*$$ as: (39)cos2θ=(||x→j||||x→j||)2=nhj2n=hj2$${\cos^2}\theta = {(\frac{{||{{\vec x}_j}||}}{{||{{\vec x}_j}||}})^2} = \frac{{n{h_j}^2}}{n} = {h_j}^2$$

A comparison of the top 30 students’ composite score rankings with the mean score rankings is shown in Figure 3. From the figure, it can be seen that the results of the factor analysis composite score and the traditional mean score ranking are still different. For example, the student whose number is #1 has the highest composite score and ranked #1 in the factor analysis, but the mean score ranking is #30.

Figure 3.

The overall ranking was compared with the average ranking

To summarize, in the context of the high-quality development of commerce and distribution, the performance of different students in their ability to deal with common diseases in various clinical disciplines, clinical diagnosis and identification of traditional Chinese medicines, the ability to diagnose and examine the condition, take medical history, and the ability to recognize and treat the illnesses in Chinese medicine is also very different. Teachers can tailor their teaching to the needs of their students. Teaching managers can also use the structure of professional knowledge and ability obtained from factor analysis as an objective reference in improving or refining the setting of professional courses and the formulation of professional training objectives.

The new swarm intelligence FWA originates from the process of sparking when fireworks are ignited, which is regarded as the process of searching for the location of fireworks ignition in the local space of the neighborhood of a specific point by igniting the sparks generated by the ignition, and then continuously igniting fireworks in the search space until the sparks reach the optimal location.

Sparks based on fitness are categorized into 2 types, i.e., good sparks and bad sparks, where sparks with smaller fitness have a stronger search capability, have a smaller explosion radius in a smaller search space, and produce a higher number of fireworks. On the contrary, sparks with greater adaptation have stronger digging ability, explode in larger radius in larger search space and produce less number of fireworks [23]. In FWA, there are 2 parameters that play a decisive role, one is the explosion radius R_i of firework i, and the other is the number of sparks S_i produced by the explosion of firework i, which are calculated as follows (40)Ri=R^fi−ymin+ε∑i=1N(fi−ymin)+ε$${R_i} = \hat R\frac{{{f_i} - {y_{\min }} + \varepsilon }}{{\sum\limits_{i = 1}^N {({f_i} - {y_{\min }})} + \varepsilon }}$$ (41)Si=Mymax−fi+ε∑i=1N(ymax−fi)+ε$${S_i} = M\frac{{{y_{\max }} - {f_i} + \varepsilon }}{{\sum\limits_{i = 1}^N {({y_{\max }} - {f_i})} + \varepsilon }}$$

Where: R^$$\hat R$$ is the average explosion radius of the firework. f_i is the fitness function of fireworks i. y_min = minf_i, y_max = maxf_i are the minimum and maximum values of the fitness function in the fireworks population, respectively. ε is the machine minimum. M is the constant. N is the firework size.

The number of sparklers S_i is also subject to certain constraints in Eq: (42)Si={ round(aW),si<aW round(bW),si<bW round(si),other$${S_i} = \left\{ {\begin{array}{*{20}{l}} {{\text{round}}(aW),{s_i} < aW} \\ {{\text{round}}(bW),{s_i} < bW} \\ {{\text{round}}({s_i}),{\text{other}}} \end{array}} \right.$$

Where: a, b are constants, set by default a = 0.1, b = 0.5. round(·) is a rounding function. s_i is the initial number of fireworks. W is the weight parameter.

Set the dimension of the fireworks for k, in-law i position for x_i = (x_i, x_i2, ⋯, x_ik), according to Eq. respectively calculated fireworks i explosion radius R_i, the number of sparks S_i, randomly selected u(1 ≤ u ≤ w) fireworks position component, and according to Eq. position update to generate the explosion of sparks, that is: (43)xik′=xik+riU(−1,1)$$x_{ik}^\prime = {x_{ik}} + {r_i}U( - 1,1)$$

Where: x_ik is the position element of firework i in dimension k. x_i is the explosion updated x_ik. U(−1, 1) is a random number on the interval [-1, 1]. r_i is the scaling parameter for fireworks i.

Exploding fireworks may be out of range of the feasible domain boundary, sparks within the range are not processed, but for fireworks outside the range, they need to be assigned to a new search space by the mapping rule. The mapping formula is: (44)||x^||=xib,k+|x^ik||xub,k−xib,k|$$||\hat x|| = {x_{ib,k}} + |{\hat x_{ik}}||{x_{ub,k}} - {x_{ib,k}}|$$

Where: x^$$\hat x$$ is the position of the next generation firework. x^ik$${\hat x_{ik}}$$ is the position element of the next generation firework i in k dimensions. x_ub,k, x_lb,k are the upper and lower bounds of the solution space in k dimensions, respectively. ||·|| is the mode operation.

The variational operator is introduced in FWA in order to generate Gaussian sparks, which allows the diversity of the population to increase. The process of increasing starts with randomly selecting x_i, then selecting a specific dimension for Gaussian variation, and finally performing Gaussian variation calculations through the dimension k of x_i, i.e: (45)x^=xike$$\hat x = {x_{ik}}e$$

where e is equivalent to N(1, 1) and N(1, 1) is a Gaussian random number with variance and mean of 1.

The computational flow of the fireworks algorithm is shown in Fig. 6.

Figure 6.

Calculation flow of fireworks algorithm

k The principle of the mean clustering algorithm is to make the objects of different classes of clusters as different as possible and the objects of the same class of clusters as identical as possible according to the corresponding similarity rules. The algorithm uses distance as a criterion for classifying the cluster classes, and the Euclidean distance formula is usually used to calculate the distance between samples of data objects, i.e.: (46)d=∑i=1N(xi−yi)2$$d = \sqrt {\sum\limits_{i = 1}^N ( {x_i} - {y_i}{{\text{)}}^2}}$$

Where: d is the Euclidean distance. x_i, y_i are the data object samples, X_i = {x₁, x₂, ⋯, x_N}, Y = {y₁, y₂, ⋯, y_N}, i ∈ [1, N] respectively.

In the k mean clustering algorithm clustering process, each iteration needs to recalculate the average of all samples in the cluster that is the cluster centroid c_i, update c_i is calculated as: (47)Ci=1|ci|∑xi∈cixi$${C_i} = \frac{1}{{|{c_i}|}}\sum\limits_{{x_i} \in {c_i}} {{x_i}}$$

Where C_i is the cluster.

k The mean clustering algorithm needs to iteratively update the partitioned categories and cluster centroids c_i until the termination conditions are met. The default termination condition is that the number of iterations has reached the maximum value or the objective function of the algorithm is less than a threshold value.

From the above description, it can be seen that the core of the k-mean clustering algorithm is based on the minimum error sum of squares criterion, and the basic idea is to divide the given data objects into the same class clusters through a certain number of iterations, and then recalculate the clustering centers, and carry out cyclic iterations according to a certain number of times, and output the results when the criterion function converges [24]. k The criterion function E is defined as: (48)E=∑i=1k∑xi∈Ci(xi−x¯i)2$$E = \sum\limits_{i = 1}^k {\sum\limits_{{x_i} \in {C_i}} ( } {x_i} - {\bar x_i}{)^2}$$

where x¯i$${\bar x_i}$$ is the mean value of x_i.

It can be seen from Eq. The larger E is, the lower the similarity within class clusters. On the contrary, the smaller E is, the higher the similarity within class clusters.

The k mean clustering algorithm is more reliant on initializing the cluster centroids and easily falls into the local optimum problem, while FWA has the ability to balance the global search and local search, therefore, FWA is used to optimize the k mean clustering algorithm. Firstly, FWA is used to find k clustering centers as the initial cluster centroids of the k-mean clustering algorithm, and then the k-mean clustering algorithm is used for clustering to get the optimal values. The selection strategy is set in FWA, and the candidate set Y_H (H is the total number of elements) is set to contain the original fireworks, exploding sparks and Gaussian sparks. The set Y_n contains the optimal elements, in addition to the optimal elements, select Q − 1 elements in the set to form a new population of Q elements, where the probability of each element selection is: (49)pi=∑j=1Hdij∑i=1H∑j=1Hdij$${p_i} = \frac{{\sum\limits_{j = 1}^H {{d_{ij}}} }}{{\sum\limits_{i = 1}^H {\sum\limits_{j = 1}^H {{d_{ij}}} } }}$$ (50)dij=|fi−fj|$${d_{ij}} = \left| {{f_i} - {f_j}} \right|$$

where d_ij is the Euclidean distance between firework i and firework j.

In this section of the experiment, the grades of three courses related to commerce and distribution in a university were selected for statistical analysis, and Q-Q plots were selected to observe the changes of students at the high grades level, with an effective sample of 1,366. The normal Q-Q plots for the year of enrollment are shown in Figure 7. There is no significant change, mainly due to the limitation of the laboratory equipment with the limitation of the size of the innovation laboratory to accommodate a limited number of students. The trend normal Q-Q plot for year of enrollment is shown in Figure 8. The current main results are achieved by the students of class 2021 and 2022, with class 2023 as a strong reserve. The trend normal Q-Q graph of examination results in the main courses is shown in Fig. 9 (Fig. a shows the trend normal Q-Q graph of Data Structures and Fig. b shows the trend normal Q-Q graph of Database Principles). From the data of the above table and figure, it can be seen that through participation in the innovation laboratory and enterprise practical training, after a period of time, the students’ motivation to learn has improved, which is reflected in the improvement of the average grade, especially the number of excellent grades is growing. At the same time, polarization is also increasing, after all, the innovation laboratory does not cover all students.

Figure 7.

The normal state of the year of the admission Q-Q

Figure 8.

The trend of the year of admission Q-Q

Figure 9.

The main course test results are normal

The database principles were selected as the comparison object, the ANOVA model was constructed, the year of enrollment was selected as the dependent variable, and the fixed factor was the database principles, and the ANOVA was utilized to view the test of the between-subjects effect, and the test of the between-subjects effect of the database principles course is shown in Table 8. Where df represents the degree of freedom, F is the group variance value, Sig is the test value of the difference is significant, the value is generally compared with 0.05 or 0.01, if it is greater than 0.05, it means that the difference is significant. As can be seen from the table, Sig>0.05 indicates that there is a significant change in the achievement with the innovative activities in each grade. Next, a random factor analysis was conducted to derive the test of between-subjects effect by comparing the achievement of data structure and database principles based on the year of enrollment, and the test of between-subjects effect of comparing data structure and database principles courses is shown in Table 9. Comparing on the year of enrollment, the Sig<0.05 indicates that there is no significant difference in the degree of students’ choosing to participate in innovative activities in different years of enrollment, but with the deepening of the innovative activities, the before-and-after comparison of the two courses of Data Structure and Database Principles, there is a significant difference between the students’ learning outcomes and the rest of the students with the Sig>0.05.

In order to compare the mined data more intuitively, the year of enrollment, the main course scores and the trend of scores were compared by introducing RFM model to measure the level of students’ learning motivation and innovation ability, and the students’ year of enrollment and the main course scores were compared as shown in Figure 10. The distribution of the trend of training students’ test scores is shown in Figure 11. It shows that the impact of innovation education on students’ course learning is very obvious, and achieves the goal of optimizing the path of education reform in colleges and universities.

Figure 10.

The student enrollment was compared with the main course score

Figure 11.

Training student test performance trend distribution