Assessment of Science and Technology Innovation Capability of Digital Creative Industries Based on Empirical Analysis of Inputs and Outputs

In fact the study of an object using factor analysis is the study of potential relationships between its attributes. In geology, the study of the tectonic processes of a particular ore body can be viewed as a study of the relationship between the contents of its m chemical elements such as Cu, Pb, Zn, Ag ----- and so on, and the values of these contents can all be viewed as random variables,. We study them, through sample values.

The original data is the sample value, 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) \[\vec{x}=(\;{{x}_{1}},{{x}_{2}},\cdots \cdots ,{{x}_{n}})\] (2) y→=( y1,y2,⋯⋯,yn) \[\vec{y}=(\;{{y}_{1}},{{y}_{2}},\cdots \cdots ,{{y}_{n}})\]

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 \[\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}}\;,\bar{y}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{y}_{i}}}\] (4) σx2=1n∑i=1n(xi−x¯)2 \[\sigma _{x}^{2}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{({{x}_{i}}-\bar{x})}^{2}}}\] (5) σy2=1n∑i=1n(yi−y¯)2 \[\sigma _{y}^{2}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{({{y}_{i}}-\bar{y})}^{2}}}\]

Re-order: (6) xi′=xi−xσx,yi′=yi−y¯σy,i=1,2⋯⋯,n \[x_{i}^{\prime }=\frac{{{x}_{i}}-x}{{{\sigma }_{x}}},y_{i}^{\prime }=\frac{{{y}_{i}}-\bar{y}}{{{\sigma }_{y}}},i=1,2\cdots \cdots ,n\]

The sample after standardization meets the following conditions: (7) x¯'=1n∑i=1nxi'=0,y¯'=1n∑i=1nyi'=0 \[{{\bar{x}}^{'}}=\frac{1}{n}\sum\limits_{i=1}^{n}{x_{i}^{'}}=0,{{\bar{y}}^{'}}=\frac{1}{n}\sum\limits_{i=1}^{n}{y_{i}^{'}}=0\] (8) σx′2=1n∑i=1nx′i2=1,σy′2=1n∑i=1ny′i2=1

Here, x→,y→\[\vec{x},\vec{y}\] is still used to represent the samples after standardization, and their variance and correlation coefficients can be calculated according to the following formula: (9) { σx2=1n∑i=1nxi2=1nx→'x→=1σy2=1n∑i=1nyi2=1ny→'y→=1Yxy=1n∑i=1nxiyi=1nx→'y→ \[\left\{ \begin{matrix} \sigma _{x}^{2}=\frac{1}{n}\sum\limits_{i=1}^{n}{x_{i}^{2}}=\frac{1}{n}{{{\vec{x}}}^{'}}\vec{x}=1 \\ \sigma _{y}^{2}=\frac{1}{n}\sum\limits_{i=1}^{n}{y_{i}^{2}}=\frac{1}{n}{{{\vec{y}}}^{'}}\vec{y}=1 \\ {{Y}_{xy}}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}}{{y}_{i}}=\frac{1}{n}{{{\vec{x}}}^{'}}\vec{y} \\ \end{matrix} \right.\]

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

For n samples with m variables each, the original data matrix is as follows: (10) X=[ x11x12⋯x1mx21x22⋯x2m⋯⋯⋯⋯xn1xn2⋯xnm ]=[ x→1,x→2,⋯,x→m ] \[X=\left[ \begin{matrix} {{x}_{11}} & {{x}_{12}} & \cdots & {{x}_{1m}} \\ {{x}_{21}} & {{x}_{22}} & \cdots & {{x}_{2m}} \\ \cdots & \cdots & \cdots & \cdots \\ {{x}_{n1}} & {{x}_{n2}} & \cdots & {{x}_{nm}} \\ \end{matrix} \right]=\left[ \begin{matrix} {{{\vec{x}}}_{1}},{{{\vec{x}}}_{2}},\cdots ,{{{\vec{x}}}_{m}} \\ \end{matrix} \right]\]

The column vector at the right end of the equation: (11) x¯j=(x1j,x2j,⋯,xnj)′,j=1,2,⋯,m \[{{\bar{x}}_{j}}={{({{x}_{1j}},{{x}_{2j}},\cdots ,{{x}_{nj}})}^{\prime }},j=1,2,\cdots ,m\]

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: (12) x→j=1n∑i=1nxij=0 \[{{\vec{x}}_{j}}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{ij}}}=0\] (13) σj2=1n∑i=1nxij2=1nx→j′x→j=1,2,⋯,m \[\sigma _{j}^{2}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{ij}}^{2}}=\frac{1}{n}\vec{x}_{j}^{\prime }{{\vec{x}}_{j}}=1,2,\cdots ,m\]

Then, the correlation coefficient between x→j${{\vec{x}}_{j}}$ and x→k${{\vec{x}}_{k}}$ is: (14) Yjk=1n∑i=1nxijxik=1nx→j'x→k,j,k=1,2,⋯,m \[{{Y}_{jk}}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{ij}}}{{x}_{ik}}=\frac{1}{n}\vec{x}_{j}^{'}{{\vec{x}}_{k}},j,k=1,2,\cdots ,m\]

The correlation coefficient matrix R consists of the correlation coefficients between m variables two by two [23]: (15) R=[ r11r12⋯r1mr21r22⋯r2m⋯⋯⋯⋯rm1rm2⋯rmm ]=1nx′x \[R=\left[ \begin{matrix} {{r}_{11}} & {{r}_{12}} & \cdots & {{r}_{1m}} \\ {{r}_{21}} & {{r}_{22}} & \cdots & {{r}_{2m}} \\ \cdots & \cdots & \cdots & \cdots \\ {{r}_{m1}} & {{r}_{m2}} & \cdots & {{r}_{mm}} \\ \end{matrix} \right]=\frac{1}{n}{{x}^{\prime }}x\]

The correlation coefficient matrix R is symmetric and at least semi-positive definite, which means that all of 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. In addition, in factor analysis, we also often involve the correlation coefficient matrix between two groups of variables, assuming that in addition to the previous m random variables, there are another p random variables, the matrix is as follows: (16) y=[ y11y12⋯y1py21y22⋯y2p⋯⋯⋯⋯yn1yn2⋯ynp ]=[ y¯1,y¯2,⋯,y¯p ] \[y=\left[ \begin{matrix} {{y}_{11}} & {{y}_{12}} & \cdots & {{y}_{1p}} \\ {{y}_{21}} & {{y}_{22}} & \cdots & {{y}_{2p}} \\ \cdots & \cdots & \cdots & \cdots \\ {{y}_{n1}} & {{y}_{n2}} & \cdots & {{y}_{np}} \\ \end{matrix} \right]=\left[ \begin{matrix} {{{\bar{y}}}_{1}},{{{\bar{y}}}_{2}},\cdots ,{{{\bar{y}}}_{p}} \\ \end{matrix} \right]\]

Assuming all standardized data, the correlation coefficient between y→k${{\vec{y}}_{k}}$ and x→j${{\vec{x}}_{j}}$ from equation (16) is (17) Skj=1ny→k'x→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: (18) Sp×m=[ S11S12⋯S1mS21S22⋯S2m⋯⋯⋯⋯Sp1Sp2⋯Spm ]=[ 1ny1x11ny1′x2⋯1ny1′xm1ny2′x11ny2′x2⋯1ny2′xm⋯⋯⋯⋯1nyp′x11nyp′x2⋯1nyp′xm ]=1n[ y¯1′y¯2′⋮y¯p′ ][ x→1,x→2,⋯,x→m⋮y¯p′ ]=1nYX \[\begin{align} & {{S}_{p\times m}}=\left[ \begin{matrix} {{S}_{11}} & {{S}_{12}} & \cdots & {{S}_{1m}} \\ {{S}_{21}} & {{S}_{22}} & \cdots & {{S}_{2m}} \\ \cdots & \cdots & \cdots & \cdots \\ {{S}_{p1}} & {{S}_{p2}} & \cdots & {{S}_{pm}} \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} \frac{1}{n}{{y}_{1}}{{x}_{1}} & \frac{1}{n}y_{1}^{\prime }{{x}_{2}} & \cdots & \frac{1}{n}y_{1}^{\prime }{{x}_{m}} \\ \frac{1}{n}y_{2}^{\prime }{{x}_{1}} & \frac{1}{n}y_{2}^{\prime }{{x}_{2}} & \cdots & \frac{1}{n}y_{2}^{\prime }{{x}_{m}} \\ \cdots & \cdots & \cdots & \cdots \\ \frac{1}{n}y_{p}^{\prime }{{x}_{1}} & \frac{1}{n}y_{p}^{\prime }{{x}_{2}} & \cdots & \frac{1}{n}y_{p}^{\prime }{{x}_{m}} \\ \end{matrix} \right] \\ & =\frac{1}{n}\left[ \begin{matrix} \bar{y}_{1}^{\prime } \\ \bar{y}_{2}^{\prime } \\ \vdots \\ \bar{y}_{p}^{\prime } \\ \end{matrix} \right]\left[ \begin{matrix} {{{\vec{x}}}_{1}},{{{\vec{x}}}_{2}},\cdots ,{{{\vec{x}}}_{m}} \\ \vdots \\ \bar{y}_{p}^{\prime } \\ \end{matrix} \right] \\ & =\frac{1}{n}YX \end{align}\]

Mathematical Model of Factor Analysis The common factor of factor analysis can, in fact, be expressed in the following linear algebraic form [24]: (19) { x→1=a11f→1+a21f→2+⋯+ap1f→p+μ1ε→1x→2=a12f→1+a22f→2+⋯+ap2f→p+μ2ε→2⋯x→m=a1mf→1+a2mf→2+⋯+apmf→p+μmε→m \[\left\{ \begin{matrix} {{{\vec{x}}}_{1}}={{a}_{11}}{{{\vec{f}}}_{1}}+{{a}_{21}}{{{\vec{f}}}_{2}}+\cdots +{{a}_{p1}}{{{\vec{f}}}_{p}}+{{\mu }_{1}}{{{\vec{\varepsilon }}}_{1}} \\ {{{\vec{x}}}_{2}}={{a}_{12}}{{{\vec{f}}}_{1}}+{{a}_{22}}{{{\vec{f}}}_{2}}+\cdots +{{a}_{p2}}{{{\vec{f}}}_{p}}+{{\mu }_{2}}{{{\vec{\varepsilon }}}_{2}} \\ \cdots \\ {{{\vec{x}}}_{m}}={{a}_{1m}}{{{\vec{f}}}_{1}}+{{a}_{2m}}{{{\vec{f}}}_{2}}+\cdots +{{a}_{pm}}{{{\vec{f}}}_{p}}+{{\mu }_{m}}{{{\vec{\varepsilon }}}_{m}} \\ \end{matrix} \right.\]

Abbreviated into: (20) x→j=∑k=1pakjf→k+μjε→j,j=1,2,⋯,m \[{{\vec{x}}_{j}}=\sum\limits_{k=1}^{p}{{{a}_{kj}}}{{\vec{f}}_{k}}+{{\mu }_{j}}{{\vec{\varepsilon }}_{j}},j=1,2,\cdots ,m\] Where f→1,f→2,…,f→p\[{{\vec{f}}_{1}},{{\vec{f}}_{2}},\ldots ,{{\vec{f}}_{p}}\] and ε→1,ε→2,…,ε→m${{\vec{\varepsilon }}_{1}},{{\vec{\varepsilon }}_{2}},\ldots ,{{\vec{\varepsilon }}_{m}}$ are the new variables sought, the former is the common factor, which can be understood as the commonality; the latter is called the single factor, which is the individuality factor. The positive integer P represents the number of common factors, which is much smaller than the original number of variables m, reducing the original m variables to a small number of factors, and the coefficients a_ij 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, and the latter is called the single factor loadings, and since we are concerned with the common factors only, the factor loadings usually referred to the former only.

Notation: (21) A=[ a11a12⋯a1ma21a22⋯a2m⋯⋯⋯⋯ap1ap2⋯apm ]p×m \[A={{\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & \cdots & {{a}_{1m}} \\ {{a}_{21}} & {{a}_{22}} & \cdots & {{a}_{2m}} \\ \cdots & \cdots & \cdots & \cdots \\ {{a}_{p1}} & {{a}_{p2}} & \cdots & {{a}_{pm}} \\ \end{matrix} \right]}_{p\times m}}\] where a_kj is the loading of the jnd variable on the krd factor (k = 1,2,……,p; j = 1,2,……,m); (22) F=[ f¯1,f¯2,⋯,f¯p ]=[ f11f12⋯f1pf21f22⋯f2p⋯⋯⋯⋯fn1fn2⋯fnp ]n×p \[F=\left[ \begin{matrix} {{{\bar{f}}}_{1}},{{{\bar{f}}}_{2}},\cdots ,{{{\bar{f}}}_{p}} \\ \end{matrix} \right]={{\left[ \begin{matrix} {{f}_{11}} & {{f}_{12}} & \cdots & {{f}_{1p}} \\ {{f}_{21}} & {{f}_{22}} & \cdots & {{f}_{2p}} \\ \cdots & \cdots & \cdots & \cdots \\ {{f}_{n1}} & {{f}_{n2}} & \cdots & {{f}_{np}} \\ \end{matrix} \right]}_{n\times p}}\] Where column k is the value of the knd factor on each specimen, this matrix is called the factorial measure; (23) U=[ u10⋯00u2⋯0⋯⋯⋯⋯00⋯um ]m×m \[U={{\left[ \begin{matrix} {{u}_{1}} & 0 & \cdots & 0 \\ 0 & {{u}_{2}} & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & {{u}_{m}} \\ \end{matrix} \right]}_{m\times m}}\]

This is the m st order diagonal matrix where the j nd diagonal element u_j is the loading (j = 1,2,⋯⋯,m) of variable X_j on a single factor ε_j : (24) E=[ ε→1,ε→2,⋯,ε→m ]=[ ε11ε12⋯ε1mε21ε22⋯ε2m⋯⋯⋯⋯εn1εn2⋯εnm ] \[E=\left[ \begin{matrix} {{{\vec{\varepsilon }}}_{1}},{{{\vec{\varepsilon }}}_{2}},\cdots ,{{{\vec{\varepsilon }}}_{m}} \\ \end{matrix} \right]=\left[ \begin{matrix} {{\varepsilon }_{11}} & {{\varepsilon }_{12}} & \cdots & {{\varepsilon }_{1m}} \\ {{\varepsilon }_{21}} & {{\varepsilon }_{22}} & \cdots & {{\varepsilon }_{2m}} \\ \cdots & \cdots & \cdots & \cdots \\ {{\varepsilon }_{n1}} & {{\varepsilon }_{n2}} & \cdots & {{\varepsilon }_{nm}} \\ \end{matrix} \right]\] where column j is the value of ε_j on each specimen. Then equation (24) can be rewritten in the following form: (25) [ x→1,x→2,⋯,x→m ]=[ f→1,f→2,⋯,f→p ][ a11a12⋯a1ma21a22⋯a2m⋯⋯⋯⋯ap1ap2⋯apm ]+[ ε→1,ε→2,⋯,ε→m ][ u10⋯00u2⋯0⋯⋯⋯⋯00⋯um ] \[\begin{align} & \left[ \begin{matrix} {{{\vec{x}}}_{1}},{{{\vec{x}}}_{2}},\cdots ,{{{\vec{x}}}_{m}} \\ \end{matrix} \right]=\left[ \begin{matrix} {{{\vec{f}}}_{1}},{{{\vec{f}}}_{2}},\cdots ,{{{\vec{f}}}_{p}} \\ \end{matrix} \right]\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & \cdots & {{a}_{1m}} \\ {{a}_{21}} & {{a}_{22}} & \cdots & {{a}_{2m}} \\ \cdots & \cdots & \cdots & \cdots \\ {{a}_{p1}} & {{a}_{p2}} & \cdots & {{a}_{pm}} \\ \end{matrix} \right] \\ & +\left[ \begin{matrix} {{{\vec{\varepsilon }}}_{1}},{{{\vec{\varepsilon }}}_{2}},\cdots ,{{{\vec{\varepsilon }}}_{m}} \\ \end{matrix} \right]\left[ \begin{matrix} {{u}_{1}} & 0 & \cdots & 0 \\ 0 & {{u}_{2}} & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & {{u}_{m}} \\ \end{matrix} \right] \end{align}\] (26) X=FA+EU \[X=FA+EU\]

We already know that the original variables x→j${{\vec{x}}_{j}}$ in Eq. (27) are all standardized variables, now assume again that the common factor f¯K(k=1,2,.......,p)${{\bar{f}}_{K}}(k=1,2,.......,p)$ and the single factor ε→j(j=1,2,.......,m)${{\vec{\varepsilon }}_{j}}(j=1,2,.......,m)$ to be solved are also all standardized variables and that the correlation coefficients between all the common factors, and between the single factors, are 0. There is the following relationship: (27) { 1n∑i=1nfik=0,k=1,2,⋯,p1n∑i=1nεij=0,j=1,2,⋯,m1nf→k'f→ℓ=δkℓ={ 1,k=ℓ0,k≠ℓ k,ℓ=1,2,⋯,p1nε→j'ε→q=δjq={ 1,j=q0,j≠q j,q=1,2,⋯,m1nf→k'ε→j=0,k=1,2,⋯,p;j=1,2,⋯,m \[\left\{ \begin{matrix} \frac{1}{n}\sum\limits_{i=1}^{n}{{{f}_{ik}}}=0,k=1,2,\cdots ,p \\ \frac{1}{n}\sum\limits_{i=1}^{n}{{{\varepsilon }_{ij}}}=0,j=1,2,\cdots ,m \\ \frac{1}{n}\vec{f}_{k}^{'}{{{\vec{f}}}_{\ell }}={{\delta }_{k\ell }}=\left\{ \begin{matrix} 1,k=\ell \\ 0,k\ne \ell \\ \end{matrix} \right.k,\ell =1,2,\cdots ,p \\ \frac{1}{n}\vec{\varepsilon }_{j}^{'}{{{\vec{\varepsilon }}}_{q}}={{\delta }_{jq}}=\left\{ \begin{matrix} 1,j=q \\ 0,j\ne q \\ \end{matrix} \right.j,q=1,2,\cdots ,m \\ \frac{1}{n}\vec{f}_{k}^{'}{{{\vec{\varepsilon }}}_{j}}=0,k=1,2,\cdots ,p;j=1,2,\cdots ,m \\ \end{matrix} \right.\]

These relational equations are written in matrix form to obtain the correlation matrix between the metrics: (28) 1nF′F=1n[ f¯1′f¯2′⋮f¯p′ ][ f→1,f→2,⋯,f→p ]=[ 1nf→1′f→11nf→1′f→2⋯1nf→1′f→p1nf→2′f→11nf→2′f→2⋯1nf→2′f→p⋯⋯⋯⋯1nf→p′f→11nf→p′f→2⋯1nf→p′f→p ]=[ 10⋯001⋯0⋯⋯⋯⋯00⋯1 ]=Ip \[\begin{align} & \frac{1}{n}{{F}^{\prime }}F=\frac{1}{n}\left[ \begin{matrix} \bar{f}_{1}^{\prime } \\ \bar{f}_{2}^{\prime } \\ \vdots \\ \bar{f}_{p}^{\prime } \\ \end{matrix} \right]\left[ \begin{matrix} {{{\vec{f}}}_{1}},{{{\vec{f}}}_{2}},\cdots ,{{{\vec{f}}}_{p}} \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{f}}}_{1}} & \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{f}}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{f}}}_{p}} \\ \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{f}}}_{1}} & \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{f}}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{f}}}_{p}} \\ \cdots & \cdots & \cdots & \cdots \\ \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{f}}}_{1}} & \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{f}}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{f}}}_{p}} \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & 1 \\ \end{matrix} \right]={{I}_{p}} \end{align}\] where I_p is a unit matrix of order p. Similarly, the correlation matrix between the single factors can be obtained as: (29) 1nE′E=Im \[\frac{1}{n}{{E}^{\prime }}E={{I}_{m}}\]

Then the correlation matrix between the common factor and the single factor is: (30) 1nF′E=1n[ f→1′f→2′⋮f→p′ ][ ε→1,ε→2,⋯,ε→m ]=[ 1nf→1′ε→11nf→1′ε→2⋯1nf→1′ε→m1nf→2′ε→11nf→2′ε→2⋯1nf→2′ε→m⋯⋯⋯⋯1nf→p′ε→11nf→p′ε→2⋯1nf→p′ε→m ]=[ 0⋯0⋯⋯⋯0⋯0 ]=H \[\begin{align} & \frac{1}{n}{{F}^{\prime }}E=\frac{1}{n}\left[ \begin{matrix} \vec{f}_{1}^{\prime } \\ \vec{f}_{2}^{\prime } \\ \vdots \\ \vec{f}_{p}^{\prime } \\ \end{matrix} \right]\left[ \begin{matrix} {{{\vec{\varepsilon }}}_{1}},{{{\vec{\varepsilon }}}_{2}},\cdots ,{{{\vec{\varepsilon }}}_{m}} \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{\varepsilon }}}_{1}} & \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{\varepsilon }}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{\varepsilon }}}_{m}} \\ \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{\varepsilon }}}_{1}} & \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{\varepsilon }}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{\varepsilon }}}_{m}} \\ \cdots & \cdots & \cdots & \cdots \\ \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{\varepsilon }}}_{1}} & \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{\varepsilon }}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{\varepsilon }}}_{m}} \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 0 & \cdots & 0 \\ \cdots & \cdots & \cdots \\ 0 & \cdots & 0 \\ \end{matrix} \right]=H \end{align}\] where H is the zero matrix.

According to equation (30), the correlation matrix between the obtained public factors and the original variables can be obtained: (31) [ 1nf→1′x→11nf→1′x→2⋯1nf→1′x→m1nf→2′x→11nf→2′x→2⋯1nf→2′x→m⋯⋯⋯⋯1nf→p′x→11nf→p′x→2⋯1nf→p′x→m ]=1n[ f→1′f→2′⋮f→p′ ][ x→1,x→2,⋯,x→mf→p′ ]=1nF′X=1nF′FA+1nF′EU=IpA+HU=A $\begin{align} & \left[ \begin{matrix} \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{x}}}_{1}} & \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{x}}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{1}^{^{\prime }}{{{\vec{x}}}_{m}} \\ \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{x}}}_{1}} & \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{x}}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{2}^{^{\prime }}{{{\vec{x}}}_{m}} \\ \cdots & \cdots & \cdots & \cdots \\ \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{x}}}_{1}} & \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{x}}}_{2}} & \cdots & \frac{1}{n}\vec{f}_{p}^{^{\prime }}{{{\vec{x}}}_{m}} \\ \end{matrix} \right]=\frac{1}{n}\left[ \begin{matrix} \vec{f}_{1}^{\prime } \\ \vec{f}_{2}^{\prime } \\ \vdots \\ \vec{f}_{p}^{\prime } \\ \end{matrix} \right]\left[ \begin{matrix} {{{\vec{x}}}_{1}},{{{\vec{x}}}_{2}},\cdots ,{{{\vec{x}}}_{m}} \\ \vec{f}_{p}^{\prime } \\ \end{matrix} \right] \\ & =\frac{1}{n}{{F}^{\prime }}X=\frac{1}{n}{{F}^{\prime }}FA+\frac{1}{n}{{F}^{\prime }}EU={{I}_{p}}A+HU=A \\ \end{align}$

This illustrates that it is the correlation coefficient between the common factor and the original variable that is the element in the factor loading A: (32) 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${{\vec{f}}_{k}}$ and variable x→j${{\vec{x}}_{j}}$. 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${{\vec{x}}_{j}}$; and when a_kj ≈ 0, it indicates a weak link between factor f→k${{\vec{f}}_{k}}$ and variable x→j${{\vec{x}}_{j}}$.

The correlation array R can be expressed as: (33) R=1nX′X=1n(FA+EU)′(FA+EU)=1nA′F′FA+1nU′E′FA+1nA′F′EU+1nU′E′EU=A′(1nF′F)A+U′(1nE′F)A+A′(1nF′E)U+U′(1nE′E)U=A′A+U′U=R*+[ u120⋯00u22⋯0⋯⋯⋯⋯00⋯um2 ] \[\begin{align} & R=\frac{1}{n}{{X}^{\prime }}X=\frac{1}{n}{{(FA+EU)}^{\prime }}(FA+EU) \\ & =\frac{1}{n}{{A}^{\prime }}{{F}^{\prime }}FA+\frac{1}{n}{{U}^{\prime }}{{E}^{\prime }}FA+\frac{1}{n}{{A}^{\prime }}{{F}^{\prime }}EU+\frac{1}{n}{{U}^{\prime }}{{E}^{\prime }}EU \\ & ={{A}^{\prime }}(\frac{1}{n}{{F}^{\prime }}F)A+{{U}^{\prime }}(\frac{1}{n}{{E}^{\prime }}F)A+{{A}^{\prime }}(\frac{1}{n}{{F}^{\prime }}E)U+{{U}^{\prime }}(\frac{1}{n}{{E}^{\prime }}E)U \\ & ={{A}^{\prime }}A+{{U}^{\prime }}U={{R}^{*}}+\left[ \begin{matrix} {{u}_{1}}^{2} & 0 & \cdots & 0 \\ 0 & {{u}_{2}}^{2} & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & {{u}_{m}}^{2} \\ \end{matrix} \right] \end{align}\]

Among them: (34) R*=A′A \[{{R}^{*}}={{A}^{\prime }}A\] is called the approximate correlation matrix, the raw variable correlation coefficients, and the off-diagonal elements are the same as R: (35) 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: (36) 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${{\vec{x}}_{j}}$, and it represents the share of each metric in the variance of the original variable, and is numerically equal to the square of the element in column j of A. The common factor variance represents the extent to which all of the original variables can be explained by the p common factors, and takes values between 0 and 1, which are positively correlated.

The sum of the common factor variances is shown below: (37) ∑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: (38) 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.

Data Envelopment Analysis (DEA model for short) can be categorized into three kinds from the perspective of orientation, i.e., input-oriented, output-oriented, and non-oriented [25]. In the efficiency study of multi-input-output systems, we should choose the appropriate orientation according to the purpose and direction of the study. In this paper, we adopt the input-oriented model. According to the theory, for any decision unit, the mathematical theory of BCC model in dyadic form under input orientation is as follows: (39) ιθ−ε(e^TS−+eTS+)subject to∑j=lnXjλj+S−=θX0∑j=lnYjλj−S+=Y0λj≥0,S−,S+≥0 \[\begin{array}{*{35}{l}} \iota \theta -\varepsilon ({{{\hat{e}}}^{T}}{{S}^{-}}+{{e}^{T}}{{S}^{+}}) \\ subject~to\underset{j=l}{\overset{n}{\mathop \sum }}\,{{X}_{j}}{{\lambda }_{j}}+{{S}^{-}}=\theta {{X}_{0}} \\ \underset{j=l}{\overset{n}{\mathop \sum }}\,{{Y}_{j}}{{\lambda }_{j}}-{{S}^{+}}={{Y}_{0}} \\ {{\lambda }_{j}}\ge 0,{{S}^{-}},{{S}^{+}}\ge 0 \\ \end{array}\]] where j = l,2,⋯,n denotes the decision unit and X,Y is the input indicator variable and output indicator variable, respectively. S⁻,S⁺ is the input and output slack variables. θ is the efficiency value of the BCC model. In this model, there are the following definitions and concepts that have universal applicability:

If θ = I,S⁺ = S⁻ = 0, the decision unit DEA is valid;

Based on different economic perspectives, DEA is categorized into two types of radial and non-radial methods, radial methods such as DEA-CCR, DEA-BCC, etc. are based on Debreu-Farrel’s economic theory; and non-radial methods such as DEA-SBM, etc. are based on Pareto-Koopmans’ economic theory. The advantage of the DEA-SBM model is that it can be adjusted for different inputs or outputs in non-equal proportions, which is an improvement over the BCC model, and the theoretical formulation of the input-oriented SBM model is given below in the following pairwise transformed form: (40) minρ−1m∑i=1msi−xosubjectto∑j=1nxijλj+si−=ρxio,i=1,…,m∑j=1nyrjλj≥yro,r=1,…,s,λjsi−,sr+≥0, ∀i,j,r \[\begin{align} & \text{min}\rho -\frac{1}{m}\sum\limits_{i=1}^{m}{\frac{s_{i}^{-}}{{{x}_{o}}}} \\ & subjectto\sum\limits_{j=1}^{n}{{{x}_{ij}}}{{\lambda }_{j}}+s_{i}^{-}=\rho {{x}_{io}},i=1,\ldots ,m \\ & \sum\limits_{j=1}^{n}{{{y}_{rj}}}{{\lambda }_{j}}\ge {{y}_{ro}},r=1,\ldots ,s,{{\lambda }_{j}} \\ & s_{i}^{-},s_{r}^{+}\ge 0,\;\forall i,j,r \\ \end{align}\]

This paper adopts the EBM model based on constant scale reward and input orientation, and uses this model to measure the efficiency of digital creative industries in each region.The EBM model effectively improves the advantages of the SBM and CCR models, and at the same time makes up for their shortcomings. The study shows that the same proportion reduction of input factors in the traditional DEA(CCR) model is contrary to the actual situation, and the SBM model has the problem of losing the original proportion information of factors in the efficiency frontier. The EBM model can be used to evaluate the efficiency of each region more effectively. This paper adopts the EBM model based on non-expected output, variable returns to scale, and input orientation, and uses this model to measure the efficiency of digital creative industry in each region. The theoretical formula is as follows: (41) Min θ0−ε1∑i=1mwi−∑i=1mwi−si−subject to∑j=1nxijλj+si−=θoxio,i=1,…,m∑j=1nyrjλj≥yro,r=1,…,sλj,si−,sr+≥0,∀i,j,r \[\begin{align} & Min\text{ }{{\theta }_{0}}-\varepsilon \frac{1}{\sum\limits_{i=1}^{m}{w_{i}^{-}}}\sum\limits_{i=1}^{m}{w_{i}^{-}}s_{i}^{-} \\ & subject\text{ }to\sum\limits_{j=1}^{n}{{{x}_{ij}}}{{\lambda }_{j}}+s_{i}^{-}={{\theta }_{o}}{{x}_{io}},i=1,\ldots ,m \\ & \sum\limits_{j=1}^{n}{{{y}_{rj}}}{{\lambda }_{j}}\ge {{y}_{ro}},r=1,\ldots ,s \\ & {{\lambda }_{j}},s_{i}^{-},s_{r}^{+}\ge 0,\forall i,j,r \\ \end{align}\]

In this model, θ₀ is the efficiency of digital creative industry in each region of the country, i,r represents the data of input and output indicators respectively. r represents the number of panel data of each city-region, and Min θ_o represents the optimal solution of the evaluated DMU. wi−$w_{i}^{-}$ represents the relative importance of input indicators. λ_j is the weight of each DMU. ε is a key parameter, which takes the value of [0, 1]. si−,sr+$s_{i}^{-},s_{r}^{+}$ is a slack variable for the input and output indicators. The objective function θ₀ takes the value [0, 1]. When θ₀ = 1, it means that the decision unit is located on the efficiency frontier, i.e., it represents that the DEA is relatively efficient; when θ₀ < 1, it means that there is an efficiency loss in the decision unit.

Since the third stage uses the modified variables for the analysis, the theoretical description of the relaxation improvement is not provided here, and this paper will illustrate the specific model of the relaxation variables in the third stage of the EBM model.

In this paper, we begin with two determinations when regressing the slack variables in the first stage using SFA regressions. In the first determination, we need to consider whether to use the simultaneous adjustment of inputs and outputs or only the adjustment of inputs or outputs. The determination is made according to the type of orientation of our first-stage EBM model, and since the orientation perspective of the first-stage EBM model is chosen to be input-oriented, this paper only performs SFA regression decomposition on the input slack variables of digital creative industries and adjusts the input variables. For the second determination, this paper stacks all the slack variables thus estimating only a single SFA regression. Adopting this approach ensures a higher degree of freedom in model treatment.

In summary, in this paper, the following SFA regression function can be constructed in order to correct the slack variables of the first stage EBM model: (42) Sni=f(Zi;βn)+vni+μni;i=1,2,⋯,I;n=1,2,⋯,N \[{{S}_{ni}}=f({{Z}_{i}};{{\beta }_{n}})+{{v}_{ni}}+{{\mu }_{ni}};i=1,2,\cdots ,I;n=1,2,\cdots ,N\] Where s_mi is the slack value of the n rd input in the i nd province and city; z_i is the environmental variable in each province and city, β_n is the coefficient of the environmental variable in each province and city; v_m + μ_m is the mixed error term in the model, v_mi represents the random disturbances in the model, and μ_ni represents the managerial inefficiency in the model. Where v~N(0,σv2)$v\tilde{\ }N(0,\sigma _{v}^{2})$ is the random error term in the model, which represents the effect of random disturbances on the input slack variable; μ is the management inefficiency, which represents the effect of management factors on the input slack variable, and is assumed to obey a normal distribution truncated at the null point in this paper, i.e. μ~N+(0,σμ2)$\mu \tilde{\ }{{N}^{+}}(0,\sigma _{\mu }^{2})$.

In order to separate the management inefficiency term, the separation equation used in this paper takes the following form: (43) E(μ|ε)=σ*[ ϕ(λεσ)Φ(λεσ)+λεσ ] \[E(\mu |\varepsilon )={{\sigma }_{*}}\left[ \frac{\phi (\lambda \frac{\varepsilon }{\sigma })}{\text{ }\!\!\Phi\!\!\text{ }(\frac{\lambda \varepsilon }{\sigma })}+\frac{\lambda \varepsilon }{\sigma } \right]\] where σ*=σμσvσ,σ=σμ2+σv2,λ=σμ/σv${{\sigma }_{*}}=\frac{{{\sigma }_{\mu }}{{\sigma }_{v}}}{\sigma },\sigma =\sqrt{\sigma _{\mu }^{2}+\sigma _{v}^{2}},\lambda ={{\sigma }_{\mu }}/{{\sigma }_{v}}$.

To calculate the random error term μ in the model, the following formula is used in this paper: (44) E[vni|=sni−f(zi;βn)−E[uni|vni+μni] \[E[{{v}_{ni}}|={{s}_{ni}}-f({{z}_{i}};{{\beta }_{n}})-E[{{u}_{ni}}|{{v}_{ni}}+{{\mu }_{ni}}]\]

The purpose of the SFA regression is to eliminate the effects of environmental and random factors on efficiency measures in order to adjust all decision units to the same external environment. The SFA adjustment formula in this paper is as follows: (45) XniA=Xni+[max(f(Zi,β^n))−f(Zi,β^n)]+[max(vni)−vni] i=1,2,⋯,I;n=1,2,⋯,N \[\begin{align} & X_{ni}^{A}={{X}_{ni}}+[\text{max}(f({{Z}_{i}},{{{\hat{\beta }}}_{n}}))-f({{Z}_{i}},{{{\hat{\beta }}}_{n}})]+[\text{max}({{v}_{ni}})-{{v}_{ni}}]\; \\ & i=1,2,\cdots ,I;n=1,2,\cdots ,N \\ \end{align}\] where xm4$x_{m}^{4}$ is the adjusted inputs; x_mi is the pre-adjusted inputs; [max(f(Zi;β^n))−f(Zi;β^n)]$[\text{max}(f({{Z}_{i}};{{\hat{\beta }}_{n}}))-f({{Z}_{i}};{{\hat{\beta }}_{n}})]$ is the adjustment for external environmental factors; and [max(v_mi) – v_mi] is the placement of all decision-making units in the same external environment.

Using the adjusted input-output variables to measure the efficiency of each decision-making unit again, this time the efficiency has eliminated the influence of environmental factors and random factors, is relatively real and accurate. In order to study the improvement of each index, this paper adopts the relaxation variable analysis. So that the target value and the improvement value can be achieved.

For input and output slack variables, this variable is the improvement value in the projection analysis. Here the target value is the sum of the improved value and the original value. All three values contain input and output indicators. The following formula is used: (46) si−+sio=sipsr++sro=srp \[\begin{align} & s_{i}^{-}+{{s}_{io}}={{s}_{ip}} \\ & s_{r}^{+}+{{s}_{ro}}={{s}_{rp}} \\ \end{align}\] Where si−\[s_{i}^{-}\] is the input redundancy value of digital creative industry efficiency, i.e. input improvement value. s_io is the input original value of digital creative industry efficiency, s_ip is the input target value of digital creative industry efficiency. si+\[s_{i}^{+}\] is the output insufficiency value of digital creative industry efficiency, i.e. output improvement value. s_m is the original value of output of digital creative industry efficiency, s_m is the target value of output of digital creative industry efficiency. Projection analysis is mainly to find the input redundancy value and output deficiency value. Then it guides the region to make corresponding improvements to achieve the DEA efficiency of digital creative industries.

Since the efficiency of some digital creative industry provinces may reach the condition of efficiency 1. Under this condition, it is difficult for these provinces to conduct specific efficiency ranking and comparative analysis. This paper adopts the super-efficiency model to expand the ordinary EBM model, and the specific expansion model is as follows: (47) Min θ0−ε1∑i=1mwi−∑i=1mwi−si−xosubject to∑j=1nxijλj+si−=θoxio,i=1,…,m∑j=1nyrjλj≥yro,r=1,…,sλj,si−,sr+≥0,∀i,j,r.k \[\begin{align} & Min\text{ }{{\theta }_{0}}-\varepsilon \frac{1}{\sum\limits_{i=1}^{m}{w_{i}^{-}}}\sum\limits_{i=1}^{m}{\frac{w_{i}^{-}s_{i}^{-}}{{{x}_{o}}}} \\ & subject\text{ }to\sum\limits_{j=1}^{n}{{{x}_{ij}}}{{\lambda }_{j}}+s_{i}^{-}={{\theta }_{o}}{{x}_{io}},i=1,\ldots ,m \\ & \sum\limits_{j=1}^{n}{{{y}_{rj}}}{{\lambda }_{j}}\ge {{y}_{ro}},r=1,\ldots ,s \\ & {{\lambda }_{j}},s_{i}^{-},s_{r}^{+}\ge 0,\forall i,j,r.k \\ \end{align}\]

In this super-efficient EBM model, θ₀ is the efficiency of the digital creative industry in each region of the country, and i,r represents the data of input and output indicators respectively. r represents the number of panel data in each region, and since the number of years is 1. r therefore takes the value of 1. Min θ_o represents the optimal solution of the evaluated DMU. wi−$w_{i}^{-}$ represents the relative importance of the input indicators. Objective function θ₀ > 0. The model uses desired output, input orientation, and variable returns to scale. In this super-efficient EBM model, inputs and outputs are treated as one system, and multiple input and multiple output variables are allowed in this system.

From 2016 to 2029, the returns to scale are decreasing, and from 2020 to 2022, the returns to scale are increasing, which also indicates that, after 2020, the technical efficiency of the digital culture and creative equipment industry is increasing along with the increase in the input of innovative materials.

The effective year of DEA is 2018, and the mean value of pure technical efficiency is higher than the mean value of scale efficiency, which indicates that the main reason for the ineffectiveness of DEA is the scale factor.

In addition to the year when DEA is effective, the changes in returns to scale in other years are all increasing returns to scale, which also indicates that the reason for the lower comprehensive efficiency of the design service industry is the scale factor, i.e., the scale of the enterprise’s technological innovation resource input has not reached the optimum.

From an overall perspective, the average value of scale efficiency from 2018-2022 also decreases from 0.893 to 0.755, which indicates that the cause of ineffective technical efficiency is the scale factor, i.e., the scale of enterprises should be adjusted.

Overall, the mean value of pure technical efficiency has increased, the mean value of scale efficiency has decreased, and the value of comprehensive efficiency has also become smaller. This indicates that the reason for the lower overall efficiency value of the digital content processing industry is due to scale, and there is still a certain distance from the optimal state of scale.

To summarize, the main factor restricting the development of digital creative science and technology innovation ability is the scale factor, the insufficiency of the input scale and the poor scale status adversely affect the technological development and comprehensive development of the digital creative industry, and the input scale of the digital creative industry should be expanded at the right time.