From e6adb1bb67578ec99cdec0da727c486272ee16e3 Mon Sep 17 00:00:00 2001 From: Jongjin Kim Date: Sat, 5 Feb 2022 21:29:04 -0500 Subject: [PATCH] Chapter2 Jongjin --- 02-timeseries-decomposition.Rmd | 214 +++++++++++++- .../figure-html/unnamed-chunk-12-1.png | Bin 96839 -> 97959 bytes .../figure-html/unnamed-chunk-13-1.png | Bin 30471 -> 31083 bytes .../figure-html/unnamed-chunk-14-1.png | Bin 27641 -> 28466 bytes .../figure-html/unnamed-chunk-24-1.png | Bin 15508 -> 17383 bytes .../figure-html/unnamed-chunk-9-1.png | Bin 15890 -> 22556 bytes .../figure-html/unnamed-chunk-3-1.png | Bin 17160 -> 16484 bytes .../figure-html/unnamed-chunk-4-1.png | Bin 19143 -> 17952 bytes .../figure-html/unnamed-chunk-4-2.png | Bin 208191 -> 187792 bytes .../figure-html/unnamed-chunk-5-1.png | Bin 18664 -> 17305 bytes .../figure-html/unnamed-chunk-5-2.png | Bin 104024 -> 91534 bytes .../figure-html/unnamed-chunk-6-1.png | Bin 12813 -> 16399 bytes .../figure-html/unnamed-chunk-7-1.png | Bin 208191 -> 187792 bytes .../figure-html/unnamed-chunk-8-1.png | Bin 104024 -> 91534 bytes bookdown.rds | Bin 807 -> 825 bytes docs/01-intro-to-tsibble.md | 18 +- docs/02-timeseries-decomposition.md | 263 +++++++++++++++++- docs/index.html | 35 ++- docs/index.md | 2 +- docs/reference-keys.txt | 17 ++ docs/search_index.json | 2 +- packages.bib | 15 +- 22 files changed, 535 insertions(+), 31 deletions(-) diff --git a/02-timeseries-decomposition.Rmd b/02-timeseries-decomposition.Rmd index 0d38e7f..8e1bf0f 100644 --- a/02-timeseries-decomposition.Rmd +++ b/02-timeseries-decomposition.Rmd @@ -1,6 +1,214 @@ -# 시계열 분해의 이해 {#chap2} +# 시계열 모델의 기초 {#ch2} -## `tsibble` 그리고 `fable` 패키지 +## 시계열 예측 단계 (fpp3 1.6) -시계열과 관련 된 R 패키지는 정말 많이 있다. 그 중에서 가장 유명한 패키지는 `forecast` 패키지이다. `fpp` 2판에서 메인 패키지로 활용되었던 `forecast` 패키지는 예측성능이 좋기로 유명하다. 하지만 `forecast`는 base 패키지를 기반으로 짜여져있는 패키지라서 `fpp` 3판에서 사용하게 될 패키지는 `tsibble`과 `fable` 패키지를 주로 사용하게 될 예정이다. 두 패키지 모두 `tidyverse`스러운 코딩과 잘 맞는 함수들을 지원하는 것이 특징이다. +시계열 예측은 아래의 5개의 단계를 포함한다. +### 1단계. 문제 정의 + +예측 분석에서 가장 어려운 부분은 바로 문제를 정의하는 것이다. 예측 문제를 정의할 때, 1. 이 예측 모델이 어떻게 사용될지, 2. 예측 모델을 누가 요구하는지, 3. 예측 결과가 어떻게 사용자의 요구와 부합하는지에 대한 이해를 필요로 한다. 예측 분석자들은 자료를 수집하고, 데이터베이스를 관리하며, 미래 계획을 수립하는 사람들과 많은 이야기를 나누어야 한다. + +### 2단계. 정보 수집 + +수집하는 정보는 통계 데이터 뿐만 아니라 데이터를 모으고 예측을 하려는 분야의 축적된 전문성을 포함한다. 종종, 좋은 통계 모델을 수립하기 위한 충분한 양의 과거의 데이터를 모으기는 어렵다. 이럴 경우에 판단예측법 (Judgmental forecasting)이 사용된다. (fpp3 Chapter 6). 오래된 데이터는 예측하려는 시스템의 구조가 변할 경우에 덜 유용한 경우가 많다. 이럴 때에는 가장 최근의 데이터만을 사용하기도 한다. 하지만, 좋은 통계 모델은 시스템 구조의 변화를 반영할 수 있어야 한다. 불필요하게 데이터를 버리지 마십시오. + +### 3단계. 기초 분석 + +항상 데이터의 그래프를 그리는 것으로 시작하라. 일관된 패턴이 있는가? 유의한 트렌드가 있는가? 계절성이 중요한가? 주기성이 존재한다는 증거가 있는가? 전문적 지식으로 설명해야 하는 아웃라이어가 있는가? 변수 사이의 강한 상관관계가 있는가? + +### 4단계. 모델 선택 및 피팅 + +최고의 모델은 과거 자료의 사용유무, 예측 변수들 사이의 관계의 정도, 예측이 사용되는 방식에 따라 정해진다. 2 ~ 3 개의 모델을 비교하는 것이 일반적이다. 각각의 모델은 여러 가정 위에 올려진 인위적인 산물로써, 모델 내의 파라미터들은 데이터 샘플을 통해서 추정된다. `fpp3` 책에서는 다루는 모델은 아래와 같다. + +1. Regression Model +2. Exponential Smoothing Methods +3. Box-Jenkins ARIMA models +4. Dynamic regression model +5. Hierarchical forecasting +6. Neural networks and vector autoregression + +### 5단계. 모델 사용과 평가 + +일단 모델을 선택하고 파라미터를 추정한 뒤에, 모델은 예측을 수행할 수 있다. 모델의 성능은 예측 기간의 데이터가 가능해진 뒤에 평가가 가능하다. 예측의 정확도를 평가하기 위한 많은 방법이 개발되어 왔다. 예측 모델을 실용적으로 사용하는 데 중요한 이슈들로써, Missing Value 및 Outlier 처리, 혹은 짧은 시계열을 어떻게 다루는가 등이 있다. + +## 시계열 데이터 분포 + +먼저 `fpp3` 패키지에 있는 예제를 살펴보기로 하자. `fpp3` 패키지를 불러오면 `tsibble`, `feasts`, 와 `fable` 패키지 내의 함수를 또한 사용할 수 있다. + +```{r} +library(fpp3) +library(dplyr) +library(ggplot2) +``` + +호주 빅토리아 주의 전력 수요량을 30분 단위로 기록한 `vic_elec` 데이터의 2012년도 일일별 전력량을 표현한 그래프는 아래와 같다. + + +```{r} +vic_elec %>% + janitor::clean_names() %>% + filter(year(time) == 2012) %>% + gg_season(demand, period = "day") + + theme(legend.position = "none") + + labs(y="MWh", title="Electricity demand: Victoria") +``` + +위의 그래프를 볼 때에 우리는 0시에서부터 23시 30분까지 총 48개씩의 데이터 포인트를 갖는 365개의 데이터 샘플로 볼 수 있다. + +## 시계열 모델 분석을 위한 기초 + +### 확률과정 (Stochastic Process) + +$\{Y_t: t = 0, \pm 1 , \pm 2, \pm 3, \dots \}$ + +확률 과정은 랜덤 변수의 시퀀스로써, 관측된 시계열 데이터 분석을 위한 모델로써 사용된다. +위 확률 과정의 분석은 평균, 분산, 공분산에서 얻은 정보를 바탕으로 이루어진다. + +### 평균함수 (Mean Function) + +평균함수는 아래와 같이 정의된다. + +$\mu_t = \rm{E} [\it Y_t \rm ] \quad for \quad t = 0, \pm1, \pm2, \dots$ + +여기에서 $\mu_t$는 $t$의 함수이며, 일반적으로 time point $t$에 따라서 변할 수 있다. + +### 자기공분산 함수 (Autocovariance function) + +자기공분산 함수는 아래와 같이 정의된다. + +$\gamma_{t, s} = \rm{Cov} (\it Y_t, Y_s \rm ) \quad for \quad t, s = 0, \pm1, \pm2, \dots$ + +$\rm{Cov} (\it Y_t, Y_s \rm ) = \rm{E} [(Y_t - \mu_t)(Y_t - \mu_s) \rm ]$ + + +### 자기상관함수 (Autocorrelation function) + +자기상관함수는 아래와 같이 정의된다. + +$\rho_{t, s} = \rm{Corr} (\it Y_t, Y_s \rm ) \quad for \quad t, s = 0, \pm1, \pm2, \dots$ + +$\rm{Corr} (\it Y_t, Y_s \rm ) = \cfrac{Cov(\it Y_t, Y_s \rm )}{\sqrt{Var(Y_t)}\sqrt{Var(Y_s)}}$ + +자기상관함수의 성질 + +## 예제 1. 백색소음 모형 (White Noise) + +백색소음 모형이란? + +$\{e_t = e_1, e_2, \dots \}$ 는 서로 독립적이고 동일한 분포를 갖는 랜덤 변수들로써, 평균은 0이고, 분산은 $\sigma_e^2$으로 가정한다. 우리는 이를 백색소음 (White Noise)라고 부른다. + +$\{Y_t = e_t: t = 1 , 2, 3, \dots \} \\ \mu_t = E[Y_t] = E[e_t] = 0 \\ Var(Y_t) = Var(e_t) = \sigma_e^2 \\ Cov(Y_t, Y_s) = Cov(e_t, e_s) = 0 \quad \rm{for} \quad t \neq s \\ Corr(Y_t, Y_s) = Corr(e_t, e_s) = 0 \quad \rm{for} \quad t \neq s$ + +서로 독립적인 랜덤 변수들이기 때문에, 공분산은 0이 된다. + +백색소음 모형 데이터의 분포 + +```{r} +white.noise <- tsibble(t = seq_len(24), y = rnorm(24), index = t) +white.noise %>% autoplot() +``` + +```{r} +N <- 100 +df.wn <- data.frame(ind = integer(), t = integer(), y = double()) +for (i in seq(N)){ + white.noise <- tsibble(t = seq_len(24), y = rnorm(24), index = t) + white.noise <- white.noise %>% + mutate(ind = i) %>% + select(ind, t, y) %>% + as.data.frame() + df.wn <- df.wn %>% bind_rows(white.noise) +} + +df.wn$ind <- df.wn$ind %>% as.factor() + +# One sample +ggplot(df.wn %>% filter(ind == "1"), aes(x = t, y = y, col = ind)) + geom_line() + +# Distribution +ggplot(df.wn, aes(x = t, y = y, col = ind)) + + geom_line() + + theme(legend.position="none") +``` + +## 예제 2. 랜덤워크 모형 (Random Walk) + +랜덤워크 모형은 아래와 같이 정의된다. $t$번째 값은 $t-1$번째 값에서 백색소음 값을 더한 것으로 정의된다. + +$\{Y_1 = e_1, \quad Y_t = Y_{t-1} + e_t: t = 2, 3, \dots \} \\ \mu_t = E[Y_t] = E[Y_{t-1}] + E[e_t] = \mu_{t-1} + 0 = \mu_{t-1} = \cdots = \mu_1 = 0 \\ Var(Y_t) = Var(Y_{t-1}) + Var(e_t) = Var(Y_{t-1}) + \sigma_e^2 = \cdots = t \sigma_e^2 \\ Cov(Y_t, Y_s) = Cov(Y_s + e_{s+1} + \cdots + e_t, Y_s) = Cov(Y_s, Y_s) = Var(Y_s) = s \sigma_e^2 \quad \rm{for} \quad t > s \\ Corr( Y_t, Y_s) = \cfrac{Cov(\it Y_t, Y_s \rm )}{\sqrt{Var(Y_t)}\sqrt{Var(Y_s)}} = \cfrac{s\sigma_e^2}{\sqrt{t \sigma_e^2}\sqrt{s \sigma_e^2}} = \sqrt{\cfrac{s}{t}} \quad \rm{for} \quad t > s$ + +위에서 우리는 랜덤 워크 모형의 평균은 0이고, 분산은 $t$에 따라 증가하는 패턴을 보임을 알 수 있다. + +```{r} +N <- 100 +df.rw <- data.frame(ind = integer(), t = integer(), y = double()) +for (i in seq(N)){ + random.walk <- tsibble(t = seq_len(24), y = cumsum(rnorm(24)), index = t) + random.walk <- random.walk %>% + mutate(ind = i) %>% + select(ind, t, y) %>% + as.data.frame() + df.rw <- df.rw %>% bind_rows(random.walk) +} + +df.rw$ind <- df.rw$ind %>% as.factor() + +# One sample +ggplot(df.rw %>% filter(ind == "1"), aes(x = t, y = y, col = ind)) + geom_line() + +# Distribution +ggplot(df.rw, aes(x = t, y = y, col = ind)) + + geom_line() + + theme(legend.position="none") +``` + +## 예제 3. 이동평균 모형 (Moving Average) + +이동평균 모형은 아래와 같이 정의된다. + +$\{Y_t = \frac{1}{2} (e_t + e_{t-1}): t = 2, 3, \dots \} \\ \mu_t = E[Y_t] = \frac{1}{2} (E[e_t] + E[e_{t-1}] = 0 \\ Var(Y_t) = \frac{1}{4} Var(e_t) + \frac{1}{4} Var(e_{t-1}) = \frac{1}{4} \sigma_e^2 + \frac{1}{4} \sigma_e^2 = \frac{1}{2} \sigma_e^2 \\ Cov(Y_t, Y_{t-1}) = Cov(\frac{1}{2} (e_t + e_{t-1}), \frac{1}{2} (e_{t-1} + e_{t-2})) = \cfrac{1}{4}\sigma_e^2 \\ Cov(Y_t, Y_s) = Cov(\frac{1}{2} (e_t + e_{t-1}), \frac{1}{2} (e_{s} + e_{s-1})) = 0 \quad \rm for \quad |t-s| > 1 \\ Corr(Y_t, Y_t) = 1 \\ Corr(Y_t, Y_{t-1}) = 0.5 \\ Corr(Y_t, Y_s) = 0 \quad for \quad |t-s|>1$ + +```{r} +library(zoo) +white.noise <- tsibble(t = seq_len(100), y = rnorm(100), index = t) + +# moving average with rollmean +white.noise.r <- white.noise %>% + mutate(yavg = rollmean(y, k = 5, align = "right", fill = NA )) + +white.noise.r + +# Mean +white.noise <- white.noise %>% + mutate(y1 = lag(y, n = 1)) %>% + rowwise() %>% + mutate(avg = mean(c_across(y:y1), na.rm=TRUE)) + +white.noise.r %>% + ACF(yavg, lag_max = 17) %>% + autoplot() +``` + + +## 정상성 (Stationarity) + +시계열 모델이 아래와 같은 조건을 만족할 때 이 모델은 정상성을 띤다고 한다. + ++ $\mu_t$ 가 상수이며, 시점 $t$와 무관하다. ++ 자기공분산함수 $\gamma_{s,t}$는 $|s-t|$에만 의존한다. 즉, 시점 $t$와 무관하다. + +위의 예제에서 백색소음 모형은 정상성을 띤다고 할 수 있다. + +```{r} +# Distribution +ggplot(df.wn, aes(x = t, y = y, col = ind)) + geom_line() + + theme(legend.position="none") +``` + +위의 예제에서 랜덤워크 모형은 정상성을 띤다고 할 수 없다. 그 이유는 분산이 시간 $t$가 증가함에 따라 증가하는 패턴을 보이기 때문이다. + +```{r} +# Distribution +ggplot(df.rw, aes(x = t, y = y, col = ind)) + geom_line() + + theme(legend.position="none") +``` diff --git a/_bookdown_files/01-intro-to-tsibble_files/figure-html/unnamed-chunk-12-1.png 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zr#>RsSEE=6?2SSasnNz1dt*FQE}J+AR>Et00p?~I_C_KM`aYQwm==*YPjXy|TQi`; zk&q={2RZjHkm_Ozn8 zvF=9Xc&$gkCY%c#j18XX^3~Ie?SkmJR-gf#F=;e3OAJfXaREyCsl7Ib+6(996w1S5 zJ$_i8T>ZEvY;H2250fcX_xSPO&q@9a|RvDC1w3>>(g9WfaRxXW+(BWz! zB9AU5Cygy2XEWPLOnw5sh5X#`oN)lKE_SXQoJ!<{L@F;+yu;;!f1K3K{e6CxUAgqB zZ=|<(9mxcJ#~WQoW9LhG_kO+`HpR92VR`sh{aID2W&z)T+!{d{|bih1jl^hwOy_egky`5TSY$001mRhcy5I diff --git a/docs/01-intro-to-tsibble.md b/docs/01-intro-to-tsibble.md index a09197c..0053cbd 100644 --- a/docs/01-intro-to-tsibble.md +++ b/docs/01-intro-to-tsibble.md @@ -164,7 +164,8 @@ prison 분기별 데이터를 `yearquarter` 사용해서 설정해줬던 것 처럼 다음 표과 같이 여러 시간 간격을 설정할 수 있는 함수를 `tsibble` 패키지에서는 제공한다. 자신의 데이터에 맞는 시간 간격을 설정해서 사용하자. - + @@ -258,8 +259,8 @@ vic_elec %>% ```
-하루 기준 전력 수요량의 변화 패턴 -

(\#fig:unnamed-chunk-11)하루 기준 전력 수요량의 변화 패턴

+<U+D558><U+B8E8> <U+AE30><U+C900> <U+C804><U+B825> <U+C218><U+C694><U+B7C9><U+C758> <U+BCC0><U+D654> <U+D328><U+D134> +

(\#fig:unnamed-chunk-11)

한 주를 기준으로 그려보면 새로운 패턴을 발견할 수 있다. @@ -275,8 +276,8 @@ vic_elec %>% ```
-한주 기준 전력 수요량의 변화 패턴 -

(\#fig:unnamed-chunk-12)한주 기준 전력 수요량의 변화 패턴

+<U+D55C><U+C8FC> <U+AE30><U+C900> <U+C804><U+B825> <U+C218><U+C694><U+B7C9><U+C758> <U+BCC0><U+D654> <U+D328><U+D134> +

(\#fig:unnamed-chunk-12)

@@ -364,7 +365,8 @@ lag_x_2 <- lag(x, n = 2) ```
(\#tab:unnamed-chunk-8)tsibble 객체에 부여할 수 있는 시간 간격 종류들(\#tab:unnamed-chunk-8)tsibble +
Frequency
- + @@ -430,8 +432,8 @@ Lag 플랏은 시계열 벡터와 각 lag 값에 대응하는 벡터를 사용 ```{=html} -
- +
+ ``` lag plot은 다음과 같이 맥주 생산량 데이터의 계절성을 체크하는데 유용하게 사용할 수 있다. lag 4와 lag 8을 살펴보면 대각선을 따라서 점들이 분포하는 것을 알 수 있다. 반대로 lag 2와 lag 4의 경우에는 좌에서 우로 뻗어 내리는 대각선 형태로 점들이 분포한 것을 알 수 있다. diff --git a/docs/02-timeseries-decomposition.md b/docs/02-timeseries-decomposition.md index 0d38e7f..aa753f4 100644 --- a/docs/02-timeseries-decomposition.md +++ b/docs/02-timeseries-decomposition.md @@ -1,6 +1,263 @@ -# 시계열 분해의 이해 {#chap2} +# 시계열 모델의 기초 {#ch2} -## `tsibble` 그리고 `fable` 패키지 +## 시계열 예측 단계 (fpp3 1.6) -시계열과 관련 된 R 패키지는 정말 많이 있다. 그 중에서 가장 유명한 패키지는 `forecast` 패키지이다. `fpp` 2판에서 메인 패키지로 활용되었던 `forecast` 패키지는 예측성능이 좋기로 유명하다. 하지만 `forecast`는 base 패키지를 기반으로 짜여져있는 패키지라서 `fpp` 3판에서 사용하게 될 패키지는 `tsibble`과 `fable` 패키지를 주로 사용하게 될 예정이다. 두 패키지 모두 `tidyverse`스러운 코딩과 잘 맞는 함수들을 지원하는 것이 특징이다. +시계열 예측은 아래의 5개의 단계를 포함한다. +### 1단계. 문제 정의 + +예측 분석에서 가장 어려운 부분은 바로 문제를 정의하는 것이다. 예측 문제를 정의할 때, 1. 이 예측 모델이 어떻게 사용될지, 2. 예측 모델을 누가 요구하는지, 3. 예측 결과가 어떻게 사용자의 요구와 부합하는지에 대한 이해를 필요로 한다. 예측 분석자들은 자료를 수집하고, 데이터베이스를 관리하며, 미래 계획을 수립하는 사람들과 많은 이야기를 나누어야 한다. + +### 2단계. 정보 수집 + +수집하는 정보는 통계 데이터 뿐만 아니라 데이터를 모으고 예측을 하려는 분야의 축적된 전문성을 포함한다. 종종, 좋은 통계 모델을 수립하기 위한 충분한 양의 과거의 데이터를 모으기는 어렵다. 이럴 경우에 판단예측법 (Judgmental forecasting)이 사용된다. (fpp3 Chapter 6). 오래된 데이터는 예측하려는 시스템의 구조가 변할 경우에 덜 유용한 경우가 많다. 이럴 때에는 가장 최근의 데이터만을 사용하기도 한다. 하지만, 좋은 통계 모델은 시스템 구조의 변화를 반영할 수 있어야 한다. 불필요하게 데이터를 버리지 마십시오. + +### 3단계. 기초 분석 + +항상 데이터의 그래프를 그리는 것으로 시작하라. 일관된 패턴이 있는가? 유의한 트렌드가 있는가? 계절성이 중요한가? 주기성이 존재한다는 증거가 있는가? 전문적 지식으로 설명해야 하는 아웃라이어가 있는가? 변수 사이의 강한 상관관계가 있는가? + +### 4단계. 모델 선택 및 피팅 + +최고의 모델은 과거 자료의 사용유무, 예측 변수들 사이의 관계의 정도, 예측이 사용되는 방식에 따라 정해진다. 2 ~ 3 개의 모델을 비교하는 것이 일반적이다. 각각의 모델은 여러 가정 위에 올려진 인위적인 산물로써, 모델 내의 파라미터들은 데이터 샘플을 통해서 추정된다. `fpp3` 책에서는 다루는 모델은 아래와 같다. + +1. Regression Model +2. Exponential Smoothing Methods +3. Box-Jenkins ARIMA models +4. Dynamic regression model +5. Hierarchical forecasting +6. Neural networks and vector autoregression + +### 5단계. 모델 사용과 평가 + +일단 모델을 선택하고 파라미터를 추정한 뒤에, 모델은 예측을 수행할 수 있다. 모델의 성능은 예측 기간의 데이터가 가능해진 뒤에 평가가 가능하다. 예측의 정확도를 평가하기 위한 많은 방법이 개발되어 왔다. 예측 모델을 실용적으로 사용하는 데 중요한 이슈들로써, Missing Value 및 Outlier 처리, 혹은 짧은 시계열을 어떻게 다루는가 등이 있다. + +## 시계열 데이터 분포 + +먼저 `fpp3` 패키지에 있는 예제를 살펴보기로 하자. `fpp3` 패키지를 불러오면 `tsibble`, `feasts`, 와 `fable` 패키지 내의 함수를 또한 사용할 수 있다. + + +```r +library(fpp3) +library(dplyr) +library(ggplot2) +``` + +호주 빅토리아 주의 전력 수요량을 30분 단위로 기록한 `vic_elec` 데이터의 2012년도 일일별 전력량을 표현한 그래프는 아래와 같다. + + + +```r +vic_elec %>% + janitor::clean_names() %>% + filter(year(time) == 2012) %>% + gg_season(demand, period = "day") + + theme(legend.position = "none") + + labs(y="MWh", title="Electricity demand: Victoria") +``` + + + +위의 그래프를 볼 때에 우리는 0시에서부터 23시 30분까지 총 48개씩의 데이터 포인트를 갖는 365개의 데이터 샘플로 볼 수 있다. + +## 시계열 모델 분석을 위한 기초 + +### 확률과정 (Stochastic Process) + +$\{Y_t: t = 0, \pm 1 , \pm 2, \pm 3, \dots \}$ + +확률 과정은 랜덤 변수의 시퀀스로써, 관측된 시계열 데이터 분석을 위한 모델로써 사용된다. +위 확률 과정의 분석은 평균, 분산, 공분산에서 얻은 정보를 바탕으로 이루어진다. + +### 평균함수 (Mean Function) + +평균함수는 아래와 같이 정의된다. + +$\mu_t = \rm{E} [\it Y_t \rm ] \quad for \quad t = 0, \pm1, \pm2, \dots$ + +여기에서 $\mu_t$는 $t$의 함수이며, 일반적으로 time point $t$에 따라서 변할 수 있다. + +### 자기공분산 함수 (Autocovariance function) + +자기공분산 함수는 아래와 같이 정의된다. + +$\gamma_{t, s} = \rm{Cov} (\it Y_t, Y_s \rm ) \quad for \quad t, s = 0, \pm1, \pm2, \dots$ + +$\rm{Cov} (\it Y_t, Y_s \rm ) = \rm{E} [(Y_t - \mu_t)(Y_t - \mu_s) \rm ]$ + + +### 자기상관함수 (Autocorrelation function) + +자기상관함수는 아래와 같이 정의된다. + +$\rho_{t, s} = \rm{Corr} (\it Y_t, Y_s \rm ) \quad for \quad t, s = 0, \pm1, \pm2, \dots$ + +$\rm{Corr} (\it Y_t, Y_s \rm ) = \cfrac{Cov(\it Y_t, Y_s \rm )}{\sqrt{Var(Y_t)}\sqrt{Var(Y_s)}}$ + +자기상관함수의 성질 + +## 예제 1. 백색소음 모형 (White Noise) + +백색소음 모형이란? + +$\{e_t = e_1, e_2, \dots \}$ 는 서로 독립적이고 동일한 분포를 갖는 랜덤 변수들로써, 평균은 0이고, 분산은 $\sigma_e^2$으로 가정한다. 우리는 이를 백색소음 (White Noise)라고 부른다. + +$\{Y_t = e_t: t = 1 , 2, 3, \dots \} \\ \mu_t = E[Y_t] = E[e_t] = 0 \\ Var(Y_t) = Var(e_t) = \sigma_e^2 \\ Cov(Y_t, Y_s) = Cov(e_t, e_s) = 0 \quad \rm{for} \quad t \neq s \\ Corr(Y_t, Y_s) = Corr(e_t, e_s) = 0 \quad \rm{for} \quad t \neq s$ + +서로 독립적인 랜덤 변수들이기 때문에, 공분산은 0이 된다. + +백색소음 모형 데이터의 분포 + + +```r +white.noise <- tsibble(t = seq_len(24), y = rnorm(24), index = t) +white.noise %>% autoplot() +``` + + + + +```r +N <- 100 +df.wn <- data.frame(ind = integer(), t = integer(), y = double()) +for (i in seq(N)){ + white.noise <- tsibble(t = seq_len(24), y = rnorm(24), index = t) + white.noise <- white.noise %>% + mutate(ind = i) %>% + select(ind, t, y) %>% + as.data.frame() + df.wn <- df.wn %>% bind_rows(white.noise) +} + +df.wn$ind <- df.wn$ind %>% as.factor() + +# One sample +ggplot(df.wn %>% filter(ind == "1"), aes(x = t, y = y, col = ind)) + geom_line() +``` + + + +```r +# Distribution +ggplot(df.wn, aes(x = t, y = y, col = ind)) + + geom_line() + + theme(legend.position="none") +``` + + + +## 예제 2. 랜덤워크 모형 (Random Walk) + +랜덤워크 모형은 아래와 같이 정의된다. $t$번째 값은 $t-1$번째 값에서 백색소음 값을 더한 것으로 정의된다. + +$\{Y_1 = e_1, \quad Y_t = Y_{t-1} + e_t: t = 2, 3, \dots \} \\ \mu_t = E[Y_t] = E[Y_{t-1}] + E[e_t] = \mu_{t-1} + 0 = \mu_{t-1} = \cdots = \mu_1 = 0 \\ Var(Y_t) = Var(Y_{t-1}) + Var(e_t) = Var(Y_{t-1}) + \sigma_e^2 = \cdots = t \sigma_e^2 \\ Cov(Y_t, Y_s) = Cov(Y_s + e_{s+1} + \cdots + e_t, Y_s) = Cov(Y_s, Y_s) = Var(Y_s) = s \sigma_e^2 \quad \rm{for} \quad t > s \\ Corr( Y_t, Y_s) = \cfrac{Cov(\it Y_t, Y_s \rm )}{\sqrt{Var(Y_t)}\sqrt{Var(Y_s)}} = \cfrac{s\sigma_e^2}{\sqrt{t \sigma_e^2}\sqrt{s \sigma_e^2}} = \sqrt{\cfrac{s}{t}} \quad \rm{for} \quad t > s$ + +위에서 우리는 랜덤 워크 모형의 평균은 0이고, 분산은 $t$에 따라 증가하는 패턴을 보임을 알 수 있다. + + +```r +N <- 100 +df.rw <- data.frame(ind = integer(), t = integer(), y = double()) +for (i in seq(N)){ + random.walk <- tsibble(t = seq_len(24), y = cumsum(rnorm(24)), index = t) + random.walk <- random.walk %>% + mutate(ind = i) %>% + select(ind, t, y) %>% + as.data.frame() + df.rw <- df.rw %>% bind_rows(random.walk) +} + +df.rw$ind <- df.rw$ind %>% as.factor() + +# One sample +ggplot(df.rw %>% filter(ind == "1"), aes(x = t, y = y, col = ind)) + geom_line() +``` + + + +```r +# Distribution +ggplot(df.rw, aes(x = t, y = y, col = ind)) + + geom_line() + + theme(legend.position="none") +``` + + + +## 예제 3. 이동평균 모형 (Moving Average) + +이동평균 모형은 아래와 같이 정의된다. + +$\{Y_t = \frac{1}{2} (e_t + e_{t-1}): t = 2, 3, \dots \} \\ \mu_t = E[Y_t] = \frac{1}{2} (E[e_t] + E[e_{t-1}] = 0 \\ Var(Y_t) = \frac{1}{4} Var(e_t) + \frac{1}{4} Var(e_{t-1}) = \frac{1}{4} \sigma_e^2 + \frac{1}{4} \sigma_e^2 = \frac{1}{2} \sigma_e^2 \\ Cov(Y_t, Y_{t-1}) = Cov(\frac{1}{2} (e_t + e_{t-1}), \frac{1}{2} (e_{t-1} + e_{t-2})) = \cfrac{1}{4}\sigma_e^2 \\ Cov(Y_t, Y_s) = Cov(\frac{1}{2} (e_t + e_{t-1}), \frac{1}{2} (e_{s} + e_{s-1})) = 0 \quad \rm for \quad |t-s| > 1 \\ Corr(Y_t, Y_t) = 1 \\ Corr(Y_t, Y_{t-1}) = 0.5 \\ Corr(Y_t, Y_s) = 0 \quad for \quad |t-s|>1$ + + +```r +library(zoo) +white.noise <- tsibble(t = seq_len(100), y = rnorm(100), index = t) + +# moving average with rollmean +white.noise.r <- white.noise %>% + mutate(yavg = rollmean(y, k = 5, align = "right", fill = NA )) + +white.noise.r +``` + +``` +#> # A tsibble: 100 x 3 [1] +#> t y yavg +#> +#> 1 1 1.28 NA +#> 2 2 1.56 NA +#> 3 3 0.547 NA +#> 4 4 1.97 NA +#> 5 5 1.49 1.37 +#> 6 6 -0.681 0.979 +#> 7 7 -0.318 0.603 +#> 8 8 -0.502 0.393 +#> 9 9 -1.16 -0.235 +#> 10 10 0.556 -0.422 +#> # ... with 90 more rows +``` + +```r +# Mean +white.noise <- white.noise %>% + mutate(y1 = lag(y, n = 1)) %>% + rowwise() %>% + mutate(avg = mean(c_across(y:y1), na.rm=TRUE)) + +white.noise.r %>% + ACF(yavg, lag_max = 17) %>% + autoplot() +``` + + + + +## 정상성 (Stationarity) + +시계열 모델이 아래와 같은 조건을 만족할 때 이 모델은 정상성을 띤다고 한다. + ++ $\mu_t$ 가 상수이며, 시점 $t$와 무관하다. ++ 자기공분산함수 $\gamma_{s,t}$는 $|s-t|$에만 의존한다. 즉, 시점 $t$와 무관하다. + +위의 예제에서 백색소음 모형은 정상성을 띤다고 할 수 있다. + + +```r +# Distribution +ggplot(df.wn, aes(x = t, y = y, col = ind)) + geom_line() + + theme(legend.position="none") +``` + + + +위의 예제에서 랜덤워크 모형은 정상성을 띤다고 할 수 없다. 그 이유는 분산이 시간 $t$가 증가함에 따라 증가하는 패턴을 보이기 때문이다. + + +```r +# Distribution +ggplot(df.rw, aes(x = t, y = y, col = ind)) + geom_line() + + theme(legend.position="none") +``` + + diff --git a/docs/index.html b/docs/index.html index 3313e0f..3ceadda 100644 --- a/docs/index.html +++ b/docs/index.html @@ -24,7 +24,7 @@ - + @@ -57,10 +57,10 @@ - - - - + + + + @@ -203,9 +203,28 @@
  • 1.9 Lag plots
  • 1.10 Autocorrelation
  • -
  • 2 시계열 분해의 이해 +
  • 2 시계열 모델의 기초
  • References
  • @@ -231,7 +250,7 @@

    들어가며

    diff --git a/docs/index.md b/docs/index.md index fd7b5d9..a1919c3 100644 --- a/docs/index.md +++ b/docs/index.md @@ -1,7 +1,7 @@ --- title: "시계열 공략집 with R" author: "[슬기로운통계생활](https://www.youtube.com/c/statisticsplaybook)" -date: "2022-01-22" +date: "2022-02-05" site: bookdown::bookdown_site output: bookdown::gitbook documentclass: book diff --git a/docs/reference-keys.txt b/docs/reference-keys.txt index ce837a4..fb16bed 100644 --- a/docs/reference-keys.txt +++ b/docs/reference-keys.txt @@ -19,3 +19,20 @@ tab:unnamed-chunk-19 tsibble-feasts-그리고-fable-패키지 tab:unnamed-chunk-20 autocorrelation +ch2 +시계열-예측-단계-fpp3-1.6 +단계.-문제-정의 +단계.-정보-수집 +단계.-기초-분석 +단계.-모델-선택-및-피팅 +단계.-모델-사용과-평가 +시계열-데이터-분포 +시계열-모델-분석을-위한-기초 +확률과정-stochastic-process +평균함수-mean-function +자기공분산-함수-autocovariance-function +자기상관함수-autocorrelation-function +예제-1.-백색소음-모형-white-noise +예제-2.-랜덤워크-모형-random-walk +예제-3.-이동평균-모형-moving-average +정상성-stationarity diff --git a/docs/search_index.json b/docs/search_index.json index d9a17c6..b02be65 100644 --- a/docs/search_index.json +++ b/docs/search_index.json @@ -1 +1 @@ -[["index.html", "시계열 공략집 with R 들어가며 공략집 사용법 설치하기 기본 패키지", " 시계열 공략집 with R 슬기로운통계생활 2022-01-22 들어가며 슬기로운 통계생활에서 2022년을 맞이하여 새롭게 시작한 시계열 스터디 내욜을 정리해놓은 레포입니다! 본 교재의 많은 내용은 다음 두 권의 R 시계열 교과서를 기반으로 하고 있습니다. 주교재 - Forecasting: Principles and Practice (이하 fpp3, Hyndman and Athanasopoulos (2018)) 실습 위주의 R 코드가 많은 책입니다. R 코드를 따라 치면서 시계열 입문하기 쉽게 되어있는 책이라서 선정하였습니다. 아마존 구매 온라인 페이지 부교재 - Time Series Analysis With Applications in R (Cryer and Chan 2008) 주교재에서 부족한 이론이 잘 채워져있는 책이고, 제가 수리 통계를 배웠던 챈 교수님이 쓰신 책이라서 선정하였습니다. 아마존 구매 또한 전반적인 내용이 위의 두 교재를 기반으로 작성이 되었지만, 슬기로운 통계생활의 입맛에 맞게 내용이 추가 혹은 변경되어 있음을 알려드립니다. 공략집 사용법 중요표시, 주석달기: 읽다가 중요하거나 나중에 따로 보고싶은 기능의 경우, 드래그를 하면 다음과 같은 선택 버튼이 나온다. 형광펜 긋고 싶은 경우 Highlight 선택, 주석을 달고 싶은 경우 Annotate을 선택하자. 하이라이팅 on/off: 가끔은 하이라이팅 해 놓은 것들을 없애고 글 만 읽고싶은 경우가 있을텐데 그 때엔 오른쪽 상단 동그라미에 점이 찍혀있는 버튼를 클릭하면 하이라이팅 된 것이 사라진다. 주석 공개/비공개: 주석은 필자와 다른 독자가 볼 수 있도록 public하게 남길 수도 있고, 자신만 볼 수 있게끔 private으로 설정 할 수 있다. 주석 기능을 이용하여 필자에게 피드백을 줄 수 있다. 오타나 오류 발견시 주석을 달아주시면 필자가 주기적으로 체크해서 고쳐나가도록 하겠다. 알아두기 하이라이팅과 주석들을 나중에 따로 볼 수 있다. 자주 방문하시는 분들은 가입하시고 사용하시면 여러모로 편할 것이다. 설치하기 설치 역시 간단한다. 여느 R패키지와 같이 install.packages() 함수를 사용하면 된다. 서브 라이브러리인 torchaudio와 torchvision이 있으나, 책의 뒷부분에서 다루기로 한다. install.packages("fpp3") 기본 패키지 앞으로의 내용에 있어서 다음의 두 패키지는 기본으로 불러와서 사용하는 것을 약속으로 한다. library(tidyverse) library(fpp3) References "]] +[["index.html", " with R ", " with R 2022-02-05 2022 ! R . - Forecasting: Principles and Practice ( fpp3, Hyndman and Athanasopoulos (2018)) R . R . - Time Series Analysis With Applications in R (Cryer and Chan 2008) , . , . , : , . Highlight , Annotate . on/off: . /: public , private . . . . . . R install.packages() . torchaudio torchvision , . install.packages("fpp3") . library(tidyverse) library(fpp3) References "]] diff --git a/packages.bib b/packages.bib index 6144028..4c51dde 100644 --- a/packages.bib +++ b/packages.bib @@ -17,7 +17,7 @@ @Manual{R-bookdown @Manual{R-dplyr, title = {dplyr: A Grammar of Data Manipulation}, - author = {Hadley Wickham and Romain Francois and Lionel Henry and Kirill Muller}, + author = {Hadley Wickham and Romain François and Lionel Henry and Kirill Müller}, year = {2021}, note = {R package version 1.0.7}, url = {https://CRAN.R-project.org/package=dplyr}, @@ -83,7 +83,7 @@ @Manual{R-knitr title = {knitr: A General-Purpose Package for Dynamic Report Generation in R}, author = {Yihui Xie}, year = {2021}, - note = {R package version 1.37}, + note = {R package version 1.36}, url = {https://yihui.org/knitr/}, } @@ -107,7 +107,7 @@ @Manual{R-readr title = {readr: Read Rectangular Text Data}, author = {Hadley Wickham and Jim Hester}, year = {2021}, - note = {R package version 2.0.2}, + note = {R package version 2.1.0}, url = {https://CRAN.R-project.org/package=readr}, } @@ -137,9 +137,10 @@ @Manual{R-tibble @Manual{R-tidyr, title = {tidyr: Tidy Messy Data}, - author = {Hadley Wickham and Maximilian Girlich}, - year = {2022}, - note = {https://tidyr.tidyverse.org}, + author = {Hadley Wickham}, + year = {2021}, + note = {R package version 1.1.4}, + url = {https://CRAN.R-project.org/package=tidyr}, } @Manual{R-tidyverse, @@ -240,7 +241,7 @@ @Book{rmarkdown2020 @Article{tidyverse2019, title = {Welcome to the {tidyverse}}, - author = {Hadley Wickham and Mara Averick and Jennifer Bryan and Winston Chang and Lucy D'Agostino McGowan and Romain Francois and Garrett Grolemund and Alex Hayes and Lionel Henry and Jim Hester and Max Kuhn and Thomas Lin Pedersen and Evan Miller and Stephan Milton Bache and Kirill Muller and Jeroen Ooms and David Robinson and Dana Paige Seidel and Vitalie Spinu and Kohske Takahashi and Davis Vaughan and Claus Wilke and Kara Woo and Hiroaki Yutani}, + author = {Hadley Wickham and Mara Averick and Jennifer Bryan and Winston Chang and Lucy D'Agostino McGowan and Romain François and Garrett Grolemund and Alex Hayes and Lionel Henry and Jim Hester and Max Kuhn and Thomas Lin Pedersen and Evan Miller and Stephan Milton Bache and Kirill Müller and Jeroen Ooms and David Robinson and Dana Paige Seidel and Vitalie Spinu and Kohske Takahashi and Davis Vaughan and Claus Wilke and Kara Woo and Hiroaki Yutani}, year = {2019}, journal = {Journal of Open Source Software}, volume = {4},

    (\#tab:unnamed-chunk-19)x에 대한 lag 1, lag 2 벡터(\#tab:unnamed-chunk-19)x lag 1, lag 2 +
    x

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