R
如何檢測和量化時間序列中的結構性中斷 (R)
背景
因此，首先需要一些背景來衡量我可能擁有的理解水平。目前正在完成碩士論文，儘管我確實有基本的了解，但統計數據一直是其中可以忽略不計的一部分。我目前的問題讓我懷疑在實踐中我能/應該做什麼，越來越多的在線閱讀和文學似乎適得其反。
我想達到什麼目的？
因此，對於我的論文，我加入了一家公司，我試圖回答的一般問題基本上是預測過程如何受到某些系統的實施的影響（這會影響用於預測過程的數據）。
期望的結果是理解：
- 有明顯的變化嗎？（例如統計證明）
- 變化有多大？（均值和方差）
- 在這個預測過程中哪些因素很重要（以及因素的影響如何從休息前 > 休息後變化）
為了回答 1 和 2，我獲得了時間序列對象形式的歷史數據（在這個階段更多但無關緊要）。我使用的軟件是R。
數據
數據包含每天（2.5 年）的加權分數，表明預測過程執行得有多糟糕（與實際事件的偏差）。這個時間序列對象包含這 2.5 年中從一小時前到事件實際發生（1 小時間隔）發生的預測的加權分數（因此，該間隔的每一天都有一個加權分數）。同樣，為其他時間間隔（例如 1-2、2-3 小時等）構建了多個時間序列。
myts1 <- structure(c(412.028462047, 468.938224875, 372.353242472, 662.26844965, 526.872020535, 396.434818388, 515.597528222, 536.940884418, 642.878650146, 458.935314286, 544.096691918, 544.378838523, 486.854043968, 478.952935122, 533.171083451, 507.543369365, 475.992539251, 411.626822157, 574.256785085, 489.424743512, 558.03917366, 488.892234577, 1081.570101272, 488.410996801, 420.058151274, 548.43547725, 759.563191992, 699.857042552, 505.546581256, 2399.735167563, 959.058553387, 565.776425823, 794.327364085, 1060.096712241, 636.011672603, 592.842508666, 643.576323635, 639.649884944, 420.788373053, 506.948276856, 503.484363746, 466.642585817, 554.521681602, 578.44355769, 589.29487224, 636.837396631, 647.548662447, 740.222655163, 391.545826142, 537.551842222, 908.940523615, 590.446686171, 543.002925217, 1406.486794264, 1007.596435757, 617.098818856, 633.848676718, 576.040175894, 881.49475483, 687.276105325, 628.977801859, 1398.136047241, 749.644445942, 639.958039461, 649.265606673, 645.57852203, 577.862446744, 663.218073256, 593.034544803, 672.096591437, 544.776355324, 720.242877214, 824.963939263, 596.581822515, 885.215989867, 693.456405627, 552.170633931, 618.855329732, 1030.291011295, 615.889921256, 799.498196448, 570.398558528, 680.670975027, 563.404802085, 494.790365745, 756.684436338, 523.051238729, 535.502475619, 520.8344231, 623.971011973, 928.274580287, 639.702434094, 583.234364572, 623.144865566, 673.342687695, 567.501447619, 602.473664361, 655.181508321, 593.662768316, 617.830786992, 652.461315007, 496.505155747, 550.24687917, 588.952116381, 456.603281447, 425.963966309, 454.729462342, 487.22846023, 613.269432488, 474.916140657, 505.93051487, 536.401546008, 555.824475073, 509.429036303, 632.232746263, 677.102831732, 506.605957979, 701.99882145, 499.770942819, 555.599224002, 557.634152694, 448.693828549, 661.921921922, 447.00540349, 561.194112634, 590.797954608, 590.739061378, 445.949400588, 725.589882976, 480.650749378, 587.03144903, 483.054524693, 428.813155209, 540.609606719, 495.756149832, 409.713220791, 492.43287131, 618.492643291, 723.203623076, 461.433833742, 420.414959481, 480.501175081, 564.955582744, 453.0704893, 506.711353939, 521.12661934, 487.509966405, 483.442305774, 506.932771141, 442.871555249, 873.285819221, 1201.628963682, 1392.479592817, 693.292446258, 629.477998542, 660.777526646, 414.376675251, 475.517946081, 501.626384564, 470.216781646, 444.195433559, 697.258566625, 546.966755779, 428.945521943, 388.203080434, 579.759476551, 548.433130604, 453.950530959, 460.613845164, 534.329569431, 560.663080722, 660.799405665, 432.3134958, 569.59842379, 518.195281689, 650.007266105, 521.642137647, 442.763872575, 687.470213886, 951.651918891, 589.611971045, 493.203713291, 431.966577408, 616.912296912, 685.80916291, 502.518373775, 595.630289879, 563.104035749, 523.383707347, 532.042896625, 470.949823756, 603.408124923, 615.301428799, 708.26541245, 725.853182875, 705.777543119, 530.351781147, 698.828825921, 462.173187592, 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145.238645275, 175.211717467, 165.116950464, 187.006564993, 160.959119092, 198.552333833, 170.406040157, 162.10770001, 146.356131036, 170.336552623, 163.095730104, 155.192077125, 217.182324478, 231.270198833, 256.905250226, 184.59676245, 175.557146633, 164.246298131, 221.695058452, 197.911691457, 188.427830442, 259.361745153, 164.243672823, 190.67188784, 182.331604811, 190.352555581, 248.738493256, 196.854564795, 164.974185334, 332.650385373, 169.081552611, 193.578840033, 192.166911863, 214.174943222, 271.287900593, 224.675083031, 171.950208574, 173.867031268, 139.260432794, 177.012491325, 171.268066406, 132.714578168, 197.224558817, 152.561299656, 143.415562042), .Tsp = c(2016.3306010929, 2018.99909424358, 365), class = "ts")處理到現在
現在我已經明白，對於問題 1，我可以對結構性中斷進行測試，以確定中斷是否以及何時發生（具有已知的中斷日期）。為此，我在 R 中使用 strucchange 包並利用斷點函數。
但是，我的主管也推薦了**CUSUM （用於未知休息日期）測試。**不確定這裡最好的是什麼？
編輯：
我看到 Andrew 的 supF 測試針對所有可能的中斷進行 Chow 測試。如果 F（或 Chow）統計量的最大值變得太大，則拒絕。（發現 -對時間序列進行 chow 測試）
使用 struccchange 獲取休息日期的代碼
library(strucchange) test2 <- Fstats(myts1~1) #Gets a sequence of fstatistics for all possible # break points within the middle 70% of myts1 myts1.fs <- test2$Fstats #These are the fstats bp.myts1 <- breakpoints(myts1~1) #Gets the breakpoint based on the F-stats plot(myts1) #plots the series myts1 lines(bp.myts1) #plots the break date implied by the sup F test bd.myts1 <- breakdates(bp.myts1) #Obtains the implied break data (2018.35, # referring to day 128 (0.35*365 = day number)) sctest(test2) #Obtains a p-value for the implied breakpoint ci.myts1 <- confint(bp.myts1) #95% CI for the location break date plot(myts1) lines(ci.myts1) #This shows the interval around the estimated break date使用它我可以獲得中斷日期和95% CI，這告訴我發生了中斷。但是，由於公式為 myts1~1，因此此中斷是均值，反映了對常數的回歸。如果我理解正確，殘差是myts1的貶值值，因此我正在研究平均值的變化。該圖使用中斷日期和置信區間可視化數據。
問題
Q0：在開始這個分析之前，我想知道我是否應該關注這些預測誤差是如何分佈的以及如何針對某些特徵進行校正？除了發生的中斷和一些異常值之外，這似乎是一個相當穩定的過程。
Q1：如何計算方差的變化？我可以想像方差的變化也可能發生在與平均值不同的時間點？說方差的中斷也是均值的中斷是否正確，但是平方貶值序列的均值中斷是正確的嗎？沒有太多可以找到的。
Q2：鑑於我現在已經獲得了均值和方差中斷的充分證據，我該如何量化這種變化？例如，在休息日之後，方差已從 X 轉移到 Y？是否像沿休息日期拆分時間序列並彙總有關這兩個部分的統計數據一樣簡單？
Q3：如果我對其他時間間隔重新運行中斷分析，我如何比較不同預測範圍內均值和方差的變化如何演變。這又是對統計數據的簡單總結，還是有一個測試來評估錯誤的不同程度？
補充 Q3:##
在創建這些時間序列時，會考慮到預測事件發生前 10 小時的預測誤差。
以一天為例：將預測分成 1 小時的 bin（創建 10 個 bin），然後在每個 bin 內，將所有預測匯總為加權平均值（根據不同的變量加權）。這意味著對於每一天，每個 bin 有一個加權分數（總共 10 個）。
將其轉換為我在這篇文章中提供的時間序列對象（myts1，涵蓋最後一小時）會產生以下結果：一個時間序列，其中每個點對應於給定時間間隔內當天的加權平均值。基本上每個 bin 包含 975 天，每個天都有平均加權值（純歷史）。
我對這部分的想法：我添加了一張圖片，其中包含 10 個垃圾箱中的 9 個垃圾箱，這清楚地表明，隨著時間的推移，中斷變得不那麼明顯了。鑑於這 10 個時間序列，我為每個時間序列重新運行“Score-CUSUM”（均值/方差）檢驗。從那裡可以確定該系統的效果在哪個時間變得“顯著”（如均值/方差的絕對變化）並且從操作的角度來看是可用的。
Q3.1這樣分析時間序列有意義嗎？我認為我重新運行 SCORE-CUSUM 測試 10 次並不重要？
Q3.1分割休息時如何處理跨越 6 個月的 95% CI？（在 4 小時後的垃圾箱中找到）
Q3.2在比較這 10 個時間間隔的不同模型（錯誤）時，我是否應該關注？
我希望我的解釋足夠，如有必要可以提供更多信息。
編輯：我添加了一個列格式的 csv 文件（由 ; 分隔），這還包括每天發生的事件數，但是，繪製時似乎沒有相關性。鏈接：https ://www.dropbox.com/s/5pilmn43bps9ss4/Data.csv?dl=0
EDIT2：應該補充一點，實際實施發生在時間序列中的時間點 2018 年第 136 天左右。
EDIT3：將第 1 小時到第 2 小時的第二個預測間隔添加為 pastebin 中 R 中的 TS 對象： https ://pastebin.com/50sb4RtP （主要帖子字符的限制）
問題
Q0：時間序列看起來相當右傾，並且水平偏移伴隨著尺度偏移。因此，我將分析日誌而不是級別的時間序列，即使用乘法而不是加法錯誤。在日誌中，似乎 AR(1) 模型在每個段中都運行良好。參見例如
acf()休息pacf()前後。pacf(log(window(myts1, end = c(2018, 136)))) pacf(log(window(myts1, start = c(2018, 137))))Q1：對於沒有均值中斷的時間序列，您可以簡單地使用平方（或絕對）殘差並再次運行水平偏移測試。或者，您可以基於最大似然模型運行測試和斷點估計，其中誤差方差是除回歸係數之外的另一個模型參數。這是 Zeileis等人。（2010，doi：10.1016/j.csda.2009.12.005）。相應的基於分數的 CUSUM 測試也可用，
strucchange但斷點估計在fxregime. 最後，在僅尋找均值和方差變化時沒有回歸變量的情況下，changepointR 包還提供了專用函數。話雖如此，對於您發布的時間序列，最小二乘法（將方差視為令人討厭的參數）似乎就足夠了。見下文。
Q2：是的。我會簡單地將單獨的模型擬合到每個部分，並分析這些“像往常一樣”的 Bai & Perron (2003, Journal of Applied Econometrics ) 也認為這是漸近地證明的，因為斷點估計的收斂速度更快（率 $ n $ 而不是 $ \sqrt{n} $ ）。
Q3：我不完全確定您在這裡尋找什麼。如果您想按順序運行測試以監控傳入的數據，那麼您應該採用正式的監控方法。Zeileis等人也對此進行了討論。（2010）。
分析代碼片段：
將日誌序列與其滯後相結合以進行後續回歸。
d <- ts.intersect(y = log(myts1), y1 = lag(log(myts1), -1))使用 supF 和基於分數的 CUSUM 測試進行測試：
fs <- Fstats(y ~ y1, data = d) plot(fs) lines(breakpoints(fs))
sc <- efp(y ~ y1, data = d, type = "Score-CUSUM") plot(sc, functional = NULL)
這突出表明，在原始時間序列中可見的時間點，截距和自相關係數都發生了顯著變化。方差也有一些波動，但在 5% 的水平上並不顯著。
基於 BIC 的約會也清楚地找到了這個斷點：
bp <- breakpoints(y ~ y1, data = d) coef(bp) ## (Intercept) y1 ## 2016(123) - 2018(136) 3.926381 0.3858473 ## 2018(137) - 2019(1) 3.778685 0.2845176顯然，均值下降，但自相關也略有下降。日誌中的擬合模型是：
plot(log(myts1), col = "lightgray", lwd = 2) lines(fitted(bp)) lines(confint(bp))
然後可以通過以下方式將模型重新擬合到每個段：
summary(lm(y ~ y1, data = window(d, end = c(2018, 136)))) ## Call: ## lm(formula = y ~ y1, data = window(d, end = c(2018, 136))) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.73569 -0.18457 -0.04354 0.12042 1.89052 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.92638 0.21656 18.13 <2e-16 *** ## y1 0.38585 0.03383 11.40 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.2999 on 742 degrees of freedom ## Multiple R-squared: 0.1491, Adjusted R-squared: 0.148 ## F-statistic: 130.1 on 1 and 742 DF, p-value: < 2.2e-16
summary(lm(y ~ y1, data = window(d, start = c(2018, 137)))) ## Call: ## lm(formula = y ~ y1, data = window(d, start = c(2018, 137))) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.43663 -0.13953 -0.03408 0.09028 0.99777 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.61558 0.33468 10.80 < 2e-16 *** ## y1 0.31567 0.06327 4.99 1.2e-06 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.2195 on 227 degrees of freedom ## Multiple R-squared: 0.09883, Adjusted R-squared: 0.09486 ## F-statistic: 24.9 on 1 and 227 DF, p-value: 1.204e-06
