比较多个时间序列

Question

我有多个时间序列数据。

c1= c(0.558642328, 
0.567173803, 
0.572518969, 
0.579917556, 
0.592155421, 
0.600239837, 
0.598955071, 
0.608857572, 
0.615442061, 
0.613502347, 
0.618076897, 
0.626769781, 
0.633930194, 
0.645518577, 
0.66773088, 
0.68128165, 
0.695552504, 
0.6992836, 
0.702771866, 
0.700840271, 
0.684032428, 
0.665082645, 
0.646948862, 
0.621813893, 
0.597888613, 
0.577744126, 
0.555984044, 
0.533597678, 
0.523645413, 
0.522041142, 
0.525437844, 
0.53053292, 
0.543152606, 
0.549038792, 
0.555300856, 
0.563411331, 
0.572663951, 
0.584438777, 
0.589476192, 
0.604197562, 
0.61670388, 
0.624161184, 
0.624345171, 
0.629342985, 
0.630379665, 
0.620067096, 
0.597480375, 
0.576228619, 
0.561285031, 
0.543921304, 
0.530826211, 
0.519563568, 
0.514228535, 
0.515202665, 
0.516663855, 
0.525673366, 
0.543545395, 
0.551681638, 
0.558951402, 
0.566816133, 
0.573842585, 
0.578611696, 
0.589180577, 
0.603297615, 
0.624550509, 
0.641310155, 
0.655093217, 
0.668385196, 
0.671600127, 
0.658876967, 
0.641041982, 
0.605081463, 
0.585503519, 
0.556173635, 
0.527428073, 
0.502755737, 
0.482510734, 
0.453295642, 
0.439938772, 
0.428757811, 
0.422361642, 
0.40945864, 
0.399504355, 
0.412688798, 
0.42684828, 
0.456935656, 
0.48355422, 
0.513727218, 
0.541630101, 
0.559122121, 
0.561763656, 
0.572532833, 
0.576761365, 
0.576146233, 
0.580199403, 
0.584954906)


c2 = c(0.548312998, 
0.557590675, 
0.569792616, 
0.57648288, 
0.582519495, 
0.584094632, 
0.583095658, 
0.58734405, 
0.590847301, 
0.588833773, 
0.591280174, 
0.60634923, 
0.614732516, 
0.628865051, 
0.64757216, 
0.666375887, 
0.677957845, 
0.681262505, 
0.684586871, 
0.682824385, 
0.667175573, 
0.648371458, 
0.633537549, 
0.608043456, 
0.581588411, 
0.559924698, 
0.540028617, 
0.517327446, 
0.507012364, 
0.501844284, 
0.503439355, 
0.509529454, 
0.521872091, 
0.526565039, 
0.534752142, 
0.54074347, 
0.548374414, 
0.563411844, 
0.568870294, 
0.58314842, 
0.59602412, 
0.611573255, 
0.617278945, 
0.622240722, 
0.619769347, 
0.610022175, 
0.593835229, 
0.568100679, 
0.551525283, 
0.529572153, 
0.51428445, 
0.499277723, 
0.494636112, 
0.497823489,
0.498831296,
0.504156655,
0.518578172,
0.526088589,
0.537710232,
0.542733413,
0.555014384,
0.561118293,
0.56687597,
0.581570888,
0.604392135,
0.618327677,
0.635660493,
0.651578796,
0.644108343,
0.628193212,
0.608698857,
0.577001792,
0.559129715,
0.535084575,
0.511081594,
0.490507799,
0.470004833,
0.438273591,
0.425731319,
0.414378071,
0.411413676,
0.400676805,
0.396925634,
0.399999064,
0.412862909,
0.431658995,
0.458585095,
0.488017178,
0.509648013,
0.529197955,
0.535413015,
0.543576932,
0.544105291,
0.547104096,
0.54662621,
0.543324244)


c3 = c(0.232481461,
0.234331533,
0.236983097,
0.239418665,
0.251142623,
0.258787839,
0.27416779,
0.288656804,
0.311543879,
0.318174845,
0.322510272,
0.322567761,
0.327798945,
0.328320512,
0.334473217,
0.348781505,
0.361979774,
0.370497525,
0.371140084,
0.364378414,
0.351432962,
0.334426059,
0.311300224,
0.28731017,
0.253390689,
0.225132306,
0.199986064,
0.174223486,
0.164470221,
0.161742944,
0.165116308,
0.171975482,
0.192469646,
0.199604255,
0.217081873,
0.227614346,
0.234985653,
0.249016567,
0.258492273,
0.270134681,
0.284708416,
0.300111979,
0.301602611,
0.299326849,
0.301946108,
0.28775356,
0.265379742,
0.233876735,
0.214674118,
0.186715491,
0.166108309,
0.147216564,
0.145897944,
0.147968549,
0.146944046,
0.162383635,
0.184625961,
0.200818598,
0.215416914,
0.226366764,
0.233739947,
0.237156208,
0.241002475,
0.250965495,
0.268156333,
0.278750855,
0.301519382,
0.319674549,
0.307104857,
0.28811741,
0.264678046,
0.223922733,
0.199140929,
0.177024715,
0.152031524,
0.127592109,
0.111354212,
0.079259231,
0.067813135,
0.06893201,
0.072917068,
0.075522878,
0.070564032,
0.05824402,
0.049086138,
0.050384771,
0.050128536,
0.061492301,
0.086675285,
0.10030736,
0.107977158,
0.119556361,
0.120863232,
0.116997129,
0.119532892,
0.117785394) 

数据的日期顺序是：

dates = seq(as.Date("2016-09-01"), as.Date("2020-07-30"), by=15) #96 times observation  

我的问题是可以应用哪些统计方法来评估时间序列之间的相似性或差异性？我们能否评估时间序列的趋势是否在统计上彼此不同，以及在什么水平上？这个问题可以很宽泛和笼统。感谢有关统计测试的建议，以及可能用于评估这些问题的代码行。

另外，数据具有很强的季节性成分。

我发现了一些讨论 https://stats.stackexchange.com/questions/19103/how-to-statistically-compare-two-time-series, https://stats.stackexchange.com/questions/372086/statistically-evaluate-difference-between-time-series?noredirect=1&lq=1 但无法对我想做的事情推断出有意义的想法。

谢谢！

Answer 1

Chow检验用于检验两个时间序列数据是否来自同一个回归。

Chow 检验用于检验两个不同回归模型在不同数据集上的系数是否相等。

该测试通常用于计量经济学领域的时间序列数据，以确定数据在某个点是否存在结构性断裂。

显然，从情节来看，时间序列似乎非常相似，系列 3 低于其他两个。有视觉证据表明所有三个系列的滞后 1 差异可能相同。

执行测试的方法是将滞后 1 差值放在单个向量中并指定分离点。这是通过比较 c1 与 c2、c2 与 c3 以及 c1 与 c3 来完成的。

c1= c(0.558642328, 
      0.567173803, 
      0.572518969, 
      0.579917556, 
      0.592155421, 
      0.600239837, 
      0.598955071, 
      0.608857572, 
      0.615442061, 
      0.613502347, 
      0.618076897, 
      0.626769781, 
      0.633930194, 
      0.645518577, 
      0.66773088, 
      0.68128165, 
      0.695552504, 
      0.6992836, 
      0.702771866, 
      0.700840271, 
      0.684032428, 
      0.665082645, 
      0.646948862, 
      0.621813893, 
      0.597888613, 
      0.577744126, 
      0.555984044, 
      0.533597678, 
      0.523645413, 
      0.522041142, 
      0.525437844, 
      0.53053292, 
      0.543152606, 
      0.549038792, 
      0.555300856, 
      0.563411331, 
      0.572663951, 
      0.584438777, 
      0.589476192, 
      0.604197562, 
      0.61670388, 
      0.624161184, 
      0.624345171, 
      0.629342985, 
      0.630379665, 
      0.620067096, 
      0.597480375, 
      0.576228619, 
      0.561285031, 
      0.543921304, 
      0.530826211, 
      0.519563568, 
      0.514228535, 
      0.515202665, 
      0.516663855, 
      0.525673366, 
      0.543545395, 
      0.551681638, 
      0.558951402, 
      0.566816133, 
      0.573842585, 
      0.578611696, 
      0.589180577, 
      0.603297615, 
      0.624550509, 
      0.641310155, 
      0.655093217, 
      0.668385196, 
      0.671600127, 
      0.658876967, 
      0.641041982, 
      0.605081463, 
      0.585503519, 
      0.556173635, 
      0.527428073, 
      0.502755737, 
      0.482510734, 
      0.453295642, 
      0.439938772, 
      0.428757811, 
      0.422361642, 
      0.40945864, 
      0.399504355, 
      0.412688798, 
      0.42684828, 
      0.456935656, 
      0.48355422, 
      0.513727218, 
      0.541630101, 
      0.559122121, 
      0.561763656, 
      0.572532833, 
      0.576761365, 
      0.576146233, 
      0.580199403, 
      0.584954906)


c2 = c(0.548312998, 
       0.557590675, 
       0.569792616, 
       0.57648288, 
       0.582519495, 
       0.584094632, 
       0.583095658, 
       0.58734405, 
       0.590847301, 
       0.588833773, 
       0.591280174, 
       0.60634923, 
       0.614732516, 
       0.628865051, 
       0.64757216, 
       0.666375887, 
       0.677957845, 
       0.681262505, 
       0.684586871, 
       0.682824385, 
       0.667175573, 
       0.648371458, 
       0.633537549, 
       0.608043456, 
       0.581588411, 
       0.559924698, 
       0.540028617, 
       0.517327446, 
       0.507012364, 
       0.501844284, 
       0.503439355, 
       0.509529454, 
       0.521872091, 
       0.526565039, 
       0.534752142, 
       0.54074347, 
       0.548374414, 
       0.563411844, 
       0.568870294, 
       0.58314842, 
       0.59602412, 
       0.611573255, 
       0.617278945, 
       0.622240722, 
       0.619769347, 
       0.610022175, 
       0.593835229, 
       0.568100679, 
       0.551525283, 
       0.529572153, 
       0.51428445, 
       0.499277723, 
       0.494636112, 
       0.497823489,
       0.498831296,
       0.504156655,
       0.518578172,
       0.526088589,
       0.537710232,
       0.542733413,
       0.555014384,
       0.561118293,
       0.56687597,
       0.581570888,
       0.604392135,
       0.618327677,
       0.635660493,
       0.651578796,
       0.644108343,
       0.628193212,
       0.608698857,
       0.577001792,
       0.559129715,
       0.535084575,
       0.511081594,
       0.490507799,
       0.470004833,
       0.438273591,
       0.425731319,
       0.414378071,
       0.411413676,
       0.400676805,
       0.396925634,
       0.399999064,
       0.412862909,
       0.431658995,
       0.458585095,
       0.488017178,
       0.509648013,
       0.529197955,
       0.535413015,
       0.543576932,
       0.544105291,
       0.547104096,
       0.54662621,
       0.543324244)


c3 = c(0.232481461,
       0.234331533,
       0.236983097,
       0.239418665,
       0.251142623,
       0.258787839,
       0.27416779,
       0.288656804,
       0.311543879,
       0.318174845,
       0.322510272,
       0.322567761,
       0.327798945,
       0.328320512,
       0.334473217,
       0.348781505,
       0.361979774,
       0.370497525,
       0.371140084,
       0.364378414,
       0.351432962,
       0.334426059,
       0.311300224,
       0.28731017,
       0.253390689,
       0.225132306,
       0.199986064,
       0.174223486,
       0.164470221,
       0.161742944,
       0.165116308,
       0.171975482,
       0.192469646,
       0.199604255,
       0.217081873,
       0.227614346,
       0.234985653,
       0.249016567,
       0.258492273,
       0.270134681,
       0.284708416,
       0.300111979,
       0.301602611,
       0.299326849,
       0.301946108,
       0.28775356,
       0.265379742,
       0.233876735,
       0.214674118,
       0.186715491,
       0.166108309,
       0.147216564,
       0.145897944,
       0.147968549,
       0.146944046,
       0.162383635,
       0.184625961,
       0.200818598,
       0.215416914,
       0.226366764,
       0.233739947,
       0.237156208,
       0.241002475,
       0.250965495,
       0.268156333,
       0.278750855,
       0.301519382,
       0.319674549,
       0.307104857,
       0.28811741,
       0.264678046,
       0.223922733,
       0.199140929,
       0.177024715,
       0.152031524,
       0.127592109,
       0.111354212,
       0.079259231,
       0.067813135,
       0.06893201,
       0.072917068,
       0.075522878,
       0.070564032,
       0.05824402,
       0.049086138,
       0.050384771,
       0.050128536,
       0.061492301,
       0.086675285,
       0.10030736,
       0.107977158,
       0.119556361,
       0.120863232,
       0.116997129,
       0.119532892,
       0.117785394) 

plot(ylim=range(c(c1,c2,c3)), c1, type="l", xlab="Time", ylab="")
lines(c2, type="l", col="red")
lines(c3, type="l", col="blue")

#install.packages("strucchange")
library(strucchange)


dat = data.frame(x = 1:190, c12 = c(diff(c1), diff(c2)), 
                 c13=c(diff(c1),diff(c3)),
                 c23=c(diff(c2),diff(c3)))

sctest(dat$c12 ~ dat$x, type="Chow", point=95)
sctest(dat$c13 ~ dat$x, type="Chow", point=95)
sctest(dat$c23 ~ dat$x, type="Chow", point=95)

根据结果​​，这三个都可能来自同一个回归。尽管如此，与 c3 相比，c1 看起来与 c2 更相似。

输出

> sctest(dat$c12 ~ dat$x, type="Chow", point=95)

    Chow test

data:  dat$c12 ~ dat$x
F = 0.010695, p-value = 0.9894

> sctest(dat$c13 ~ dat$x, type="Chow", point=95)

    Chow test

data:  dat$c13 ~ dat$x
F = 0.25134, p-value = 0.778

> sctest(dat$c23 ~ dat$x, type="Chow", point=95)

    Chow test

data:  dat$c23 ~ dat$x
F = 0.2534, p-value = 0.7764