图 1  世界各经济体在由相对人均GDP的对数g和相对人均专利申请数的对数n_p(两者均以10为底数)所构成的空间中的变化轨迹. 彩色曲线为11个代表性经济体在该空间的轨迹, 灰色曲线为其他经济体. 灰色虚线为拟合直线n_p = 1.12g – 0.82. 灰色点划线大致区分了发达经济体的轨迹所在区域和发展中经济体所在区域, 右上方主要为发达经济体, 左下方主要为发展中经济体. 插图显示了各个数据点相对拟合直线的离差Δn_p的概率分布, 蓝色虚线为其高斯函数拟合

Fig. 1.  The trajectories of economies from 1985 to 2017 in the space of the logarithmic relative GDP per capita (g) and the logarithmic relative number of patent applications per capita (n_p). The colored curves and gray curves represent the trajectory of 11 representative economies and the remain economies, respectively. The gray dashed line is the fitting function n_p = 1.12g – 0.82 of all data points. The gray dot dash line roughly distinguishes between the developed economies and the developing economies. Developed economies are mainly in the upper right area, while developing economies are in the lower left. The inset plots the distribution of the deviation Δn_p of each data point from the fitting line, in which the blue dashed line is its Gaussian fitting.

图 2  各经济体的区域创新指数I随年份的变化, 彩色线为11个代表性经济体, 灰色线为其他经济体

Fig. 2.  The change of the regional innovation index I of each economy with the years. The colored curves and gray curves represent 11 representative economies and the remain economies, respectively.

图 3  各经济体在2016年的区域创新指数I与该年份的全球创新指数(GII)的关系. 数据点的颜色表示该年份各经济体的相对人均GDP的对数值g

Fig. 3.  The regional innovation index I vs. global innovation index (GII) for each economy at 2016. The color of each data point shows the logarithmic relative GDP per capita (g) of each economy.

图 4  20年时间段(1998年至2017年)内各经济体平均区域创新指数$\left\langle {I} \right\rangle $与相对人均GDP的平均增长率$\left\langle {\Delta g} \right\rangle $之间的相关性 (a) $Z=\left\langle {I} \right\rangle^{\beta} $, 其中β = 3.80为相关性最强时所对应β值, 直线为拟合直线; 插图显示为$\left\langle {I} \right\rangle $同$\left\langle {\Delta g} \right\rangle $之间的相关性(即设定β = 1.0时); (b) 通过相对人均GDP进行修正后的相关性, $Z_{\rm f}=\left\langle { I} \right\rangle^{\beta}+(a\left\langle { g} \right\rangle ^2+b\left\langle { g} \right\rangle +c)/k $, 其中$\left\langle { g} \right\rangle $为各经济体的相对人均GDP的对数值g的20年均值, 其中β = 3.80, a = －0.011, b = –0.020, c = 0.0013, 而k = 0.033为图(a)的拟合直线斜率; 插图显示了修正函数$f(\left\langle { g} \right\rangle)=a\left\langle { g} \right\rangle^2+b\left\langle { g} \right\rangle+c$的获得, 即对图(a)的回归残差ε同$\left\langle {g} \right\rangle $的关系进行拟合所得

Fig. 4.  The correlations between the average regional innovation index of each country $\left\langle {I} \right\rangle $ and the average growth rate of relative per capita GDP $ \left\langle {\Delta g} \right\rangle $ in the period from 1998 to 2017: (a) $ Z=\left\langle {I} \right\rangle^{\beta} $, where β = 3.80 corresponding to the strongest correlation between $\left\langle {\Delta g} \right\rangle $ and Z, and the dashed line is the fitting line. The inset of panel (a) shows the correlation between $\left\langle {I} \right\rangle $ and $\left\langle {\Delta g} \right\rangle $ (setting β = 1.0); (b) the correlation between $\left\langle {\Delta g} \right\rangle $ and the corrected prediction value Z_f of each economy, where $Z_{\rm f}=\left\langle { I} \right\rangle^{\beta}+(a\left\langle { g} \right\rangle ^2+b\left\langle { g} \right\rangle +c)/k $, $\left\langle { g} \right\rangle $ is the 20-year average of the logarithmic relative GDP per capita g of each economy, and β = 3.80, a = -0.011, b = –0.020, c = 0.0013, and k = 0.033 is the slope of the fitting line in Fig.(a). The dashed line in the inset of Fig. (b) shows the correction function $f(\left\langle { g} \right\rangle)=a\left\langle { g} \right\rangle^2+b\left\langle { g} \right\rangle+c$, which is obtained by the fitting for the correlation between ε and $\left\langle { g} \right\rangle $, where ε is the regression residuals in the linear regression shown in Fig. (a)

图 5  以1年期、3年期和5年期为滑动窗口长度, 各类指标在滑动窗口期内各经济体的均值同相对人均GDP增长率的平均值$\left\langle {\Delta g} \right\rangle $之间的相关性随年份的变化. 黑色、蓝色和粉色实线及其空心数据点对应指标为创新指数(该相关性表示为r_I); 其中不同灰度的虚线标志出相关性r_I在最低限度情况下(即有效数据点最少的情况, 对应滑动窗口长度为1年时)的不同显著性水平的边界, 浅灰、中灰和深灰虚线分别对应P = 0.05, 0.01, 0.001的r_I值. 深黄、深青色、品红虚线及其实心数据点对应指标为全球创新指数GII(该相关性表示为r_GII). 橄榄绿色虚线及其空心数据点对应指标为相对人均专利申请数的对数值n_p(该相关性表示为r_p, 只显示了滑动窗口为5年期的情况). 插图显示的是, 采用5年期滑动窗口, 高收入经济体的相对人均GDP的平均增长率$\left\langle {\Delta g} \right\rangle _{\rm H} $与所有经济体的相对人均GDP增长率的均值$\left\langle {\Delta g} \right\rangle_{\rm W} $的差值$\left(\left\langle {\Delta g} \right\rangle_{\rm H}- \left\langle {\Delta g} \right\rangle_{\rm W}\right) $, 同相关性r_I(粉色点)和相关性r_p(橄榄绿色点)的相关性; 实线分别为同色数据点的拟合直线

Fig. 5.  Designing the moving window length of 1 year, 3 years and 5 years, for given index, the correlation between the average value of the index of each economy and the average growth rate $\left\langle {\Delta g} \right\rangle $ of the relative GDP per capita within the moving window are shown by curves and data points. The black, blue and pink lines and hollow data points show correlation r_I, corresponding to the index I. The different gray dashed lines show the thresholds of the correlation r_I for different level of significance in the case with the minimum data points (corresponding to the case with 1-year moving window length), and the light gray, medium gray and dark gray dashed lines correspond to the significance P = 0.05, 0.01 and 0.001, respectively. The dark yellow, dark cyan, magenta dashed lines and solid data points show correlation r_GII, corresponding to global innovation index (GII). The olive dashed line and hollow data points show correlation r_p, corresponding to the index of the logarithmic relative number of patent applications per capita (n_p) (5-year-moving-window only). The inset shows the correlations between $\left(\left\langle {\Delta g} \right\rangle_{\rm H}- \left\langle {\Delta g} \right\rangle_{\rm W}\right) $ and r_I, and the correlation beween $\left(\left\langle {\Delta g} \right\rangle_{\rm H}- \left\langle {\Delta g} \right\rangle_{\rm W}\right) $ and r_p, where $\left\langle {\Delta g} \right\rangle_{\rm H} $ and $ \left\langle {\Delta g} \right\rangle_{\rm W} $ is the average growth rate of the relative GDP per capita within the moving window for high-income economies and all economies, respectively, and the solid lines respectively are the fitting curve for the data points with the same color.

图 6  (a)和(b)分别对比了在谷-峰变换和峰-谷变换前后的两个典型年份的5年滑动窗口内各经济体的创新指数$\left\langle {I} \right\rangle $的平均值与相对人均GDP增长率$\left\langle {\Delta g} \right\rangle $的平均值之间的相关性. (a) 青色六角圈对应1994年(r_I值谷值), 桃红色圆点对应2004年(r_I值峰值), 同一经济体由灰色线连接, 绿色虚线和桃红色虚线分别为1994年和2004年数据点的拟合直线, 斜率分别为–0.050和0.073; (b) 桃红色圈和蓝色圆点分别对应2004年(r_I值峰值)和2014年(r_I值谷值)的数据点, 同一经济体由灰色线连接, 桃红色虚线(斜率0.073)和蓝色虚线(斜率–0.033)分别为2004年和2014年数据点的拟合直线; (c) 1994年至2004年的谷-峰变换中各国的$\left\langle {\Delta g} \right\rangle $改变量$\left(\left\langle {\Delta g} \right\rangle_{2004} - \left\langle {\Delta g} \right\rangle_{1994}\right) $同2004年至2014年的峰-谷变换中各经济体的$ \left\langle {\Delta g} \right\rangle $改变量$\left(\left\langle {\Delta g} \right\rangle_{2014} - \left\langle {\Delta g} \right\rangle_{2004}\right) $的关系, 其中各数据点的直径正比于该经济体自1995至2014年间的创新指数I的20年平均值, 颜色对应于该期间各经济体的相对人均GDP增长率Δg的20年平均值, 虚线为拟合直线

Fig. 6.  (a) and (b) respectively compare the correlations between the average value of the index I of each economy $\left\langle {I} \right\rangle $ and the average growth rate $\left\langle {\Delta g} \right\rangle $ of the relative GDP per capita at the 5-year moving windows before and after the transition from bottom on r_I wave to peak and the one from peak to bottom. Fig. (a) shows the comparison between 1994 (the cyan hexagons, at the bottom) and 2004 (the pink dots, at the peak), where the data points of the same economy are connected by gray lines, and the green dashed line and the pink dashed line respectively show the linear fittings of 1994 (with a slope of –0.050) and the one of 2004 (with a slope of 0.073); Fig. (b) shows the comparison between 2004 (the pink circles, at the peak) and 2014 (the blue dots, at the valley), where the data points of the same economy are connected by gray lines, and the pink dashed line (with a slope 0.073) and the blue dashed line (with a slope of –0.033) show the linear fittings of 2004 and 2014, respectively; Panel (c) plots the relationship between the change $\left(\left\langle {\Delta g} \right\rangle_{2004} - \left\langle {\Delta g} \right\rangle_{1994}\right)$ in the valley-peak transition and the change $\left(\left\langle {\Delta g} \right\rangle_{2014}-\left\langle {\Delta g} \right\rangle_{2004}\right) $ in the peak-valley transition, where the diameter of each circle is proportional to the 20 year average $\left\langle {I} \right\rangle $ of the economy’s index, and the color corresponds to the 20-year average growth rate $\left\langle {\Delta g} \right\rangle $ of the economy’s relative GDP per capita.