From 569050d07bd16d16ee239c47b7d4b5d99f1d7173 Mon Sep 17 00:00:00 2001 From: Rob Hyndman Date: Fri, 6 May 2022 11:25:41 +1000 Subject: [PATCH 01/12] Updated regression slides --- 7-regression.Rmd | 144 ++++++++++++++--------------------------------- 7-regression.pdf | Bin 1575982 -> 1545459 bytes 2 files changed, 42 insertions(+), 102 deletions(-) diff --git a/7-regression.Rmd b/7-regression.Rmd index 31d34e9..9e77390 100644 --- a/7-regression.Rmd +++ b/7-regression.Rmd @@ -10,7 +10,7 @@ output: binb::monash: colortheme: monashwhite fig_width: 7.5 - fig_height: 3.5 + fig_height: 2.7 keep_tex: no includes: in_header: header.tex @@ -24,8 +24,6 @@ source("setup.R") ## Multiple regression and forecasting -\fontsize{13}{15}\sf - \begin{block}{}\vspace*{-0.3cm} \[ y_t = \beta_0 + \beta_1 x_{1,t} + \beta_2 x_{2,t} + \cdots + \beta_kx_{k,t} + \varepsilon_t. @@ -43,41 +41,9 @@ That is, the coefficients measure the **marginal effects**. * $\varepsilon_t$ is a white noise error term - -## Example: US consumption expenditure -\fontsize{10}{11}\sf - -```{r ConsInc, echo=TRUE, fig.height=3.5} -us_change %>% - pivot_longer(c(Consumption, Income), names_to="Series") %>% - autoplot(value) + - labs(y="% change") -``` - -## Example: US consumption expenditure -\fontsize{9}{10}\sf - -```{r ConsInc2, echo=TRUE} -us_change %>% - ggplot(aes(x = Income, y = Consumption)) + - labs(y = "Consumption (quarterly % change)", - x = "Income (quarterly % change)") + - geom_point() + geom_smooth(method = "lm", se = FALSE) -``` - -## Example: US consumption expenditure - -\fontsize{9}{9}\sf - -```{r, echo=TRUE} -fit_cons <- us_change %>% - model(lm = TSLM(Consumption ~ Income)) -report(fit_cons) -``` - ## Example: US consumption expenditure -```{r MultiPredictors, echo=FALSE, fig.height=5} +```{r MultiPredictors, echo=FALSE, fig.height=4} us_change %>% gather("Measure", "Change", Consumption, Income, Production, Savings, Unemployment) %>% ggplot(aes(x = Quarter, y = Change, colour = Measure)) + @@ -89,7 +55,7 @@ us_change %>% ## Example: US consumption expenditure -```{r ScatterMatrix, echo=FALSE, fig.height=5, fig.width=8} +```{r ScatterMatrix, echo=FALSE, fig.height=4, fig.width=8} us_change %>% as_tibble() %>% select(-Quarter) %>% @@ -98,7 +64,7 @@ us_change %>% ## Example: US consumption expenditure -\fontsize{7.4}{7.4}\sf +\fontsize{7}{7}\sf ```{r usestim, echo=TRUE} fit_consMR <- us_change %>% @@ -113,9 +79,7 @@ augment(fit_consMR) %>% ggplot(aes(x = Quarter)) + geom_line(aes(y = Consumption, colour = "Data")) + geom_line(aes(y = .fitted, colour = "Fitted")) + - labs(y = NULL, - title = "Percent change in US consumption expenditure" - ) + + labs(y = NULL, title = "Percent change in US consumption expenditure") + scale_colour_manual(values=c(Data="black",Fitted="#D55E00")) + guides(colour = guide_legend(title = NULL)) ``` @@ -150,6 +114,26 @@ fit_consMR %>% gg_tsresiduals() * $t=1,2,\dots,T$ * Strong assumption that trend will continue. +## Nonlinear trend + +**Piecewise linear trend with bend at $\tau$** +\begin{align*} +x_{1,t} &= t \\ +x_{2,t} &= \left\{ \begin{array}{ll} + 0 & t <\tau\\ + (t-\tau) & t \ge \tau +\end{array}\right. +\end{align*} +\pause\vspace*{1cm} + +**Quadratic or higher order trend** +\[ + x_{1,t} =t,\quad x_{2,t}=t^2,\quad \dots +\] +\pause\vspace*{-0.5cm} + +\centerline{\textcolor{orange}{\textbf{NOT RECOMMENDED!}}} + ## Dummy variables \begin{textblock}{6}(0.4,1.5) @@ -158,18 +142,18 @@ or `No'), then an equivalent numerical variable can be constructed taking value 1 if yes and 0 if no. This is called a \textbf{dummy variable}. \end{textblock} -\placefig{7.7}{1.}{width=5cm}{dummy2} +\placefig{7.7}{1.5}{width=5cm}{dummy2} ## Dummy variables -\begin{textblock}{5}(0.4,1.5) +\begin{textblock}{6}(0.4,1.5) If there are more than two categories, then the variable can be coded using several dummy variables (one fewer than the total number of categories). \end{textblock} -\placefig{5.5}{1.5}{width=7.3cm}{dummy3} +\placefig{7.7}{1.5}{width=7.3cm}{dummy3} ## Beware of the dummy variable trap! * Using one dummy for each category gives too many dummy variables! @@ -181,7 +165,6 @@ number of categories). * The coefficients of the dummies are relative to the omitted category. ## Uses of dummy variables - \fontsize{13}{15}\sf **Seasonal dummies** @@ -216,7 +199,7 @@ recent_production %>% autoplot(Beer) + \begin{block}{Regression model} \centering $y_t = \beta_0 + \beta_1 t + \beta_2d_{2,t} + \beta_3 d_{3,t} + \beta_4 d_{4,t} + \varepsilon_t$ -\end{block} +\end{block}\vspace*{-0.3cm} * $d_{i,t} = 1$ if $t$ is quarter $i$ and 0 otherwise. @@ -229,9 +212,8 @@ report(fit_beer) ``` ## Beer production revisited -\fontsize{8.5}{9}\sf -```{r, fig.height=3.3} +```{r, fig.height=2.4} augment(fit_beer) %>% ggplot(aes(x = Quarter)) + geom_line(aes(y = Beer, colour = "Data")) + @@ -242,25 +224,22 @@ augment(fit_beer) %>% ## Beer production revisited -```{r, echo=FALSE} +```{r, fig.height=2.4} augment(fit_beer) %>% ggplot(aes(x=Beer, y=.fitted, colour=factor(quarter(Quarter)))) + geom_point() + - labs(y="Fitted", x="Actual values", - title = "Quarterly beer production") + + labs(y="Fitted", x="Actual values", title = "Quarterly beer production") + scale_colour_brewer(palette="Dark2", name="Quarter") + geom_abline(intercept=0, slope=1) ``` ## Beer production revisited -\fontsize{9}{9}\sf ```{r, echo=TRUE} fit_beer %>% gg_tsresiduals() ``` ## Beer production revisited -\fontsize{9}{9}\sf ```{r, echo=TRUE} fit_beer %>% forecast %>% autoplot(recent_production) @@ -294,9 +273,7 @@ report(fourier_beer) ## Harmonic regression: eating-out expenditure -\fontsize{9}{10}\sf - -```{r cafe, echo=TRUE, fig.height=3.8, fig.width=8} +```{r cafe, echo=TRUE} aus_cafe <- aus_retail %>% filter( Industry == "Cafes, restaurants and takeaway food services", year(Month) %in% 2004:2018 @@ -306,10 +283,10 @@ aus_cafe %>% autoplot(Turnover) ## Harmonic regression: eating-out expenditure -\fontsize{9}{10}\sf +\fontsize{8}{9}\sf ```{r cafefit, dependson='cafe', fig.height=5, echo=TRUE} -fit <- aus_cafe %>% +fit <- aus_cafe %>% model(K1 = TSLM(log(Turnover) ~ trend() + fourier(K = 1)), K2 = TSLM(log(Turnover) ~ trend() + fourier(K = 2)), K3 = TSLM(log(Turnover) ~ trend() + fourier(K = 3)), @@ -394,16 +371,6 @@ cafe_plot(K = 6) * Easter: use a dummy variable $v_t=1$ if any part of Easter is in that month, $v_t=0$ otherwise. * Ramadan and Chinese new year similar. -## Trading days - -With monthly data, if the observations vary depending on how many different types of days in the month, then trading day predictors can be useful. - -\begin{align*} -z_1 &= \text{\# Mondays in month;} \\ -z_2 &= \text{\# Tuesdays in month;} \\ -&\vdots \\ -z_7 &= \text{\# Sundays in month.} -\end{align*} ## Distributed lags @@ -418,37 +385,14 @@ Example: $x$ is advertising which has a delayed effect x_{m} &= \text{advertising for $m$ months previously.} \end{align*} -## Nonlinear trend - -**Piecewise linear trend with bend at $\tau$** -\begin{align*} -x_{1,t} &= t \\ -x_{2,t} &= \left\{ \begin{array}{ll} - 0 & t <\tau\\ - (t-\tau) & t \ge \tau -\end{array}\right. -\end{align*} -\pause\vspace*{1cm} - -**Quadratic or higher order trend** -\[ - x_{1,t} =t,\quad x_{2,t}=t^2,\quad \dots -\] -\pause\vspace*{-0.5cm} - -\centerline{\textcolor{orange}{\textbf{NOT RECOMMENDED!}}} - ## Example: Boston marathon winning times -\fontsize{9}{9}\sf - -```{r, fig.height=3.5, echo=TRUE} +```{r, fig.height=2.6, echo=TRUE} marathon <- boston_marathon %>% filter(Event == "Men's open division") %>% select(-Event) %>% mutate(Minutes = as.numeric(Time)/60) -marathon %>% autoplot(Minutes) + - labs(y="Winning times in minutes") +marathon %>% autoplot(Minutes) + labs(y="Winning times in minutes") ``` ## Example: Boston marathon winning times @@ -473,13 +417,11 @@ fit_trends ## Example: Boston marathon winning times -\fontsize{10}{10}\sf - ```{r, echo=TRUE, eval=FALSE} fit_trends %>% forecast(h=10) %>% autoplot(marathon) ``` -```{r, echo=FALSE, message=TRUE, warning=FALSE} +```{r, echo=FALSE, message=TRUE, warning=FALSE, fig.height=3.3} fc_trends <- fit_trends %>% forecast(h = 10) marathon %>% autoplot(Minutes) + @@ -491,10 +433,10 @@ marathon %>% ``` ## Example: Boston marathon winning times -\fontsize{9}{9}\sf ```{r residPiecewise, message=FALSE, warning=FALSE} -fit_trends %>% select(piecewise) %>% +fit_trends %>% + select(piecewise) %>% gg_tsresiduals() ``` @@ -546,7 +488,7 @@ $$ * It is the proportion of variance accounted for (explained) by the predictors. ## Comparing regression models -\fontsize{13}{15}\sf +\fontsize{13}{14}\sf However \dots @@ -779,7 +721,6 @@ $$y_{t}=\beta_0+\beta_1x_{1,t-h}+\dots+\beta_kx_{k,t-h}+\varepsilon_{t}$$ * A different model for each forecast horizon $h$. ## US Consumption -\fontsize{10}{11}\sf ```{r usconsumptionf, echo=TRUE} fit_consBest <- us_change %>% @@ -798,9 +739,8 @@ fc <- forecast(fit_consBest, new_data = future_scenarios) ``` ## US Consumption -\fontsize{10}{10}\sf -```{r usconsumptionf2, echo=TRUE, fig.height=3.5} +```{r usconsumptionf2, echo=TRUE} us_change %>% autoplot(Consumption) + labs(y="% change in US consumption") + autolayer(fc) + diff --git a/7-regression.pdf b/7-regression.pdf index fb1d0e1804659b9c13e99a7991ed325ef565b116..9a68e0fd70d81a6da0b6eff41307676bfd98975c 100644 GIT binary patch delta 373490 zcmYg%bzD@@6R&iGlr#uPDBVjf(jXupQqmABG^CYixC^g|b32|za6*_6EWD3cf)L|=hYTzZN-7Byb@`3+j#nG#` z3OII*)ow&lNhuJ&P26Y=ElAo9Fa%rJMno&wsM zo?Sh=^cO0MHo$Iz@p$(&TopfqcW216W3a`MWb#wOW0*>5UMYOE>gq;{968KcEfKc_ 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zYKPy7qmSV$_-d+s+EYAuZlj9jp^XnR97cQ&GlFBILwjV=MPgC9dq0P2KzFoic=O$0 z#H-s8=f;X*evsT&v?hGdQ5%_JN_Mov2cfRZ&a@GnCj~ zb|_13_`g@iDc)*l1F>b_T%m!MdqT(~R<2!aL)Px^1Hn2WHOE+jtljJb!BU|KI}T*F zd{X6n!e9B6tlh{s_uuGA;2=W>f($|&w$Hj+$2jVZt*C?tT+z#ykCkN#iz9Hngl From 40a03a66dff7d32891107ef5f07315312dee8e5d Mon Sep 17 00:00:00 2001 From: Rob Hyndman Date: Wed, 11 May 2022 10:46:55 +1000 Subject: [PATCH 02/12] Updated ch7 --- 7-regression.R | 84 +++++++++++++++++++---------------------------- 7-regression.Rmd | 12 ------- 7-regression.pdf | Bin 1545459 -> 1544275 bytes 3 files changed, 33 insertions(+), 63 deletions(-) diff --git a/7-regression.R b/7-regression.R index c326dac..88a9b62 100644 --- a/7-regression.R +++ b/7-regression.R @@ -2,24 +2,6 @@ library(fpp3) # US consumption quarterly changes -us_change %>% - pivot_longer(c(Consumption, Income), names_to = "Series") %>% - autoplot(value) + - labs(y = "% change") - -us_change %>% - ggplot(aes(x = Income, y = Consumption)) + - labs( - y = "Consumption (quarterly % change)", - x = "Income (quarterly % change)" - ) + - geom_point() + - geom_smooth(method = "lm", se = FALSE) - -fit_cons <- us_change %>% - model(lm = TSLM(Consumption ~ Income)) -report(fit_cons) - us_change %>% pivot_longer(-Quarter, names_to = "Measure", values_to = "Change") %>% ggplot(aes(x = Quarter, y = Change, colour = Measure)) + @@ -104,6 +86,39 @@ recent_production %>% ) %>% glance() + +# Fourier terms for cafe data + +aus_cafe <- aus_retail %>% + filter( + Industry == "Cafes, restaurants and takeaway food services", + year(Month) %in% 2004:2018 + ) %>% + summarise(Turnover = sum(Turnover)) +aus_cafe %>% + autoplot(Turnover) +aus_cafe %>% + autoplot(log(Turnover)) + +fit <- aus_cafe %>% + model( + K1 = TSLM(log(Turnover) ~ trend() + fourier(K = 1)), + K2 = TSLM(log(Turnover) ~ trend() + fourier(K = 2)), + K3 = TSLM(log(Turnover) ~ trend() + fourier(K = 3)), + K4 = TSLM(log(Turnover) ~ trend() + fourier(K = 4)), + K5 = TSLM(log(Turnover) ~ trend() + fourier(K = 5)), + K6 = TSLM(log(Turnover) ~ trend() + fourier(K = 6)) + ) + +augment(fit) %>% + filter(.model %in% c("K1", "K2", "K3")) %>% + ggplot(aes(x = Month, y = Turnover)) + + geom_line() + + geom_line(aes(y = .fitted, col = .model)) + + facet_grid(.model ~ .) + +glance(fit) %>% + select(.model, sigma2, log_lik, AIC, AICc, BIC) ## Boston Marathon marathon <- boston_marathon %>% @@ -147,39 +162,6 @@ fit_trends %>% glance(fit_trends) %>% select(.model, r_squared, adj_r_squared, AICc, CV) -# Fourier terms for cafe data - -aus_cafe <- aus_retail %>% - filter( - Industry == "Cafes, restaurants and takeaway food services", - year(Month) %in% 2004:2018 - ) %>% - summarise(Turnover = sum(Turnover)) -aus_cafe %>% - autoplot(Turnover) -aus_cafe %>% - autoplot(log(Turnover)) - -fit <- aus_cafe %>% - model( - K1 = TSLM(log(Turnover) ~ trend() + fourier(K = 1)), - K2 = TSLM(log(Turnover) ~ trend() + fourier(K = 2)), - K3 = TSLM(log(Turnover) ~ trend() + fourier(K = 3)), - K4 = TSLM(log(Turnover) ~ trend() + fourier(K = 4)), - K5 = TSLM(log(Turnover) ~ trend() + fourier(K = 5)), - K6 = TSLM(log(Turnover) ~ trend() + fourier(K = 6)) - ) - -augment(fit) %>% - filter(.model %in% c("K1", "K2", "K3")) %>% - ggplot(aes(x = Month, y = Turnover)) + - geom_line() + - geom_line(aes(y = .fitted, col = .model)) + - facet_grid(.model ~ .) - -glance(fit) %>% - select(.model, sigma2, log_lik, AIC, AICc, BIC) - # US consumption quarterly changes fit_all <- us_change %>% diff --git a/7-regression.Rmd b/7-regression.Rmd index 9e77390..a01ca3b 100644 --- a/7-regression.Rmd +++ b/7-regression.Rmd @@ -654,18 +654,6 @@ loocv_plot() + \only<2>{\begin{textblock}{4}(6,6)\begin{block}{}\fontsize{13}{15}\sf CV = MSE on \textcolor[HTML]{D55E00}{test sets}\end{block}\end{textblock}} -## Comparing regression models -\fontsize{11}{14}\sf - -```{r, echo=TRUE} -glance(fit_trends) %>% - select(.model, r_squared, adj_r_squared, AICc, CV) -``` - -\fontsize{13}{14}\sf - - * Be careful making comparisons when transformations are used. - ## Choosing regression variables **Best subsets regression** diff --git a/7-regression.pdf b/7-regression.pdf index 9a68e0fd70d81a6da0b6eff41307676bfd98975c..15251ccc30ca8ab1d73394ceffa73cde35a264e6 100644 GIT binary patch delta 95568 zcmY(KW0WOP)1}L1ciFbPY}>ZkWxH94R=11hO$d$Qro%8IB z*eCAG^cQmV{ewY+!GghqA%Yxqx{;QKgFfbri%iPmVRdj z-ufv}wzGAwgbJ{Qb0Tc63s#F7n5I$K2GAK%o}x+cZW8h{1nY=d(P>86#~i z7>+`O+KC1T{^8zA0XL%Z`%7avm_mg$XjsrVIb1=8%oN*T%59L$?2!C?BJRPWT5wir zWnPwo0++Hf?~-0pV8)(9py9+qcr621mxwjSB?tZNmB6VuaYvw$zEcZi9AnX$v=@%N z1(}G=)C@I1BFI(N$Uk6!S_{U^r^N*Y7uu8)h340hJbiTiT~1v zZ5;i&IQ42Cx2MsKP5;?Wnu+J`!5g4qmQLIE>Z$}Gw*2A?*Bg6dUr%GL8fOonwC-x@ z+jfGh0_~V2egLE&et0Y0e`p6aqQ>M+Gu^OCN33Xk+<_>wVgE*mInP;ZmRmZ1IBu{$<7pey zM-s!<3!n0|nJbtTCh!FW1w?F?p5(+y_({ShCZm1V=?=yvF!(?dE;cPWO*v>8vHg!O zzTIby_#{jNoQwH9)|Gfwu(&leY0m>QrI^+bM6G3UIAI$ajpP`kua7q!nm7&Qk;MSrNCD|TRzS& z8p!l5A{r(m9Act{#NL6bgp;u;Ll-8))=X7HrI)1A{4}C?Run382Lv~0r9r;A 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2001 From: Rob Hyndman Date: Wed, 11 May 2022 12:25:23 +1000 Subject: [PATCH 03/12] Corrections from today's lecture --- 7-regression.R | 2 +- 7-regression.Rmd | 2 +- 7-regression.pdf | Bin 1544275 -> 1544325 bytes 9-arima_c.R | 6 ++++++ 4 files changed, 8 insertions(+), 2 deletions(-) diff --git a/7-regression.R b/7-regression.R index 88a9b62..b261044 100644 --- a/7-regression.R +++ b/7-regression.R @@ -29,7 +29,7 @@ augment(fit_consMR) %>% guides(colour = guide_legend(title = NULL)) augment(fit_consMR) %>% - ggplot(aes(x = .fitted, y = Consumption)) + + ggplot(aes(y = .fitted, x = Consumption)) + geom_point() + labs( y = "Fitted (predicted values)", diff --git a/7-regression.Rmd b/7-regression.Rmd index a01ca3b..6f8a105 100644 --- a/7-regression.Rmd +++ b/7-regression.Rmd @@ -88,7 +88,7 @@ augment(fit_consMR) %>% ```{r usfitted2, echo=FALSE, message=FALSE, warning=FALSE} augment(fit_consMR) %>% - ggplot(aes(x=.fitted, y=Consumption)) + + ggplot(aes(y=.fitted, x=Consumption)) + 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a/9-arima_c.R b/9-arima_c.R index 1ddcf57..ed6f89e 100644 --- a/9-arima_c.R +++ b/9-arima_c.R @@ -186,6 +186,12 @@ aus_economy <- global_economy %>% filter(Code == "AUS") %>% mutate(Population = Population / 1e6) aus_economy %>% autoplot(Population) +aus_economy %>% + model( + ets = ETS(Population), + arima = ARIMA(Population) + ) %>% + glance() aus_economy %>% slice(-n()) %>% stretch_tsibble(.init = 10) %>% From 11542e4cc76ce33e7a7a0701d159329b5f51b070 Mon Sep 17 00:00:00 2001 From: Rob Hyndman Date: Thu, 12 May 2022 12:54:38 +1000 Subject: [PATCH 04/12] Updated code --- 7-regression.R | 15 ++++++++++++++- 1 file changed, 14 insertions(+), 1 deletion(-) diff --git a/7-regression.R b/7-regression.R index b261044..6c8f596 100644 --- a/7-regression.R +++ b/7-regression.R @@ -119,7 +119,9 @@ augment(fit) %>% glance(fit) %>% select(.model, sigma2, log_lik, AIC, AICc, BIC) -## Boston Marathon + + +# Boston Marathon marathon <- boston_marathon %>% filter(Event == "Men's open division") %>% @@ -141,6 +143,11 @@ fit_trends <- marathon %>% fit_trends + +fit_trends %>% + select(piecewise) %>% + report() + fc_trends <- fit_trends %>% forecast(h = 10) marathon %>% @@ -162,6 +169,7 @@ fit_trends %>% glance(fit_trends) %>% select(.model, r_squared, adj_r_squared, AICc, CV) + # US consumption quarterly changes fit_all <- us_change %>% @@ -184,6 +192,11 @@ fit_all <- us_change %>% TSLM(Consumption ~ 1), ) +glance(fit_all) %>% + select(.model, AICc, CV) %>% + arrange(AICc) + + us_change %>% model( TSLM(Consumption ~ Income * Savings + Production + Unemployment), From 0cd4fbb01003d6a57367cb5cf95ce2c98e7eaa93 Mon Sep 17 00:00:00 2001 From: Rob Hyndman Date: Thu, 12 May 2022 14:20:04 +1000 Subject: [PATCH 05/12] Updated ch10 slides --- 10-dynamic-regression.Rmd | 167 +++++++++++++++----------------------- 10-dynamic-regression.pdf | Bin 3136086 -> 3119592 bytes 2 files changed, 64 insertions(+), 103 deletions(-) diff --git a/10-dynamic-regression.Rmd b/10-dynamic-regression.Rmd index fb0dc5a..dfb811b 100644 --- a/10-dynamic-regression.Rmd +++ b/10-dynamic-regression.Rmd @@ -10,7 +10,7 @@ output: binb::monash: colortheme: monashwhite fig_width: 7.5 - fig_height: 3.5 + fig_height: 2.7 keep_tex: no includes: in_header: header.tex @@ -38,17 +38,16 @@ vic_elec_daily <- vic_elec %>% # Regression with ARIMA errors ## Regression with ARIMA errors -\fontsize{13}{15}\sf \begin{block}{Regression models}\vspace*{-0.2cm} \[ y_t = \beta_0 + \beta_1 x_{1,t} + \dots + \beta_k x_{k,t} + \varepsilon_t, \] -\end{block} +\end{block}\vspace*{-0.3cm} * $y_t$ modeled as function of $k$ explanatory variables $x_{1,t},\dots,x_{k,t}$. - * In regression, we assume that $\varepsilon_t$ was WN. + * In regression, we assume that $\varepsilon_t$ is WN. * Now we want to allow $\varepsilon_t$ to be autocorrelated. \vspace*{0.3cm} \pause @@ -73,7 +72,6 @@ $x_{1,t},\dots,x_{k,t}$. * In ordinary regression, $\eta_t$ is assumed to be white noise and so $\eta_t = \varepsilon_t$. ## Estimation -\fontsize{14}{14}\sf If we minimize $\sum \eta_t^2$ (by using ordinary regression): @@ -105,11 +103,9 @@ If we minimize $\sum \eta_t^2$ (by using ordinary regression): where $y'_t=y_t-y_{t-1}$, $x'_{t,i}=x_{t,i}-x_{t-1,i}$ and $\eta'_t=\eta_t-\eta_{t-1}$. ## Regression with ARIMA errors -\fontsize{13}{15}\sf +\fontsize{14}{15}\sf -Any regression with an ARIMA -error can be rewritten as a regression with an ARMA error by differencing all -variables with the same differencing operator as in the ARIMA model.\pause +Any regression with an ARIMA error can be rewritten as a regression with an ARMA error by differencing all variables with the same differencing operator as in the ARIMA model.\pause \begin{block}{Original data}\vspace*{-0.2cm} \begin{align*} @@ -122,22 +118,21 @@ variables with the same differencing operator as in the ARIMA model.\pause $$ y'_t = \beta_1 x'_{1,t} + \dots + \beta_k x'_{k,t} + \eta'_t. $$ -where $\phi(B)\eta'_t = \theta(B)\varepsilon_t$, +where $\phi(B)\eta'_t = \theta(B)\varepsilon_t$,\vspace*{0.1cm} -$y_t' = (1-B)^dy_t$, $x_{i,t}' = (1-B)^dx_{i,t}$, and $\eta_t' = (1-B)^d \eta_t$ +$y_t' = (1-B)^dy_t$,\quad $x_{i,t}' = (1-B)^dx_{i,t}$,\quad and $\eta_t' = (1-B)^d \eta_t$ \end{block} ## Regression with ARIMA errors -\fontsize{13}{15}\sf - * In R, we can specify an ARIMA($p,d,q$) for the errors, and $d$ levels of differencing will be applied to all variables ($y, x_{1,t},\dots,x_{k,t}$). + * In R, we can specify an ARIMA($p,d,q$) for the errors, and $d$ levels of differencing will be applied to all variables ($y, x_{1,t},\dots,x_{k,t}$) during estimation. * Check that $\varepsilon_t$ series looks like white noise. * AICc can be calculated for final model. * Repeat procedure for all subsets of predictors to be considered, and select model with lowest AICc value. ## US personal consumption and income -```{r usconsump, fig.height=6, fig.width=8.5, echo=FALSE} +```{r usconsump, fig.height=6, fig.width=9, echo=FALSE, out.height="92%"} us_change %>% gather(key='variable', value='value') %>% ggplot(aes(y=value, x=Quarter, group=variable, colour=variable)) + @@ -159,36 +154,32 @@ Write down the equations for the fitted model. \end{alertblock} ## US personal consumption and income -\fontsize{11}{12}\sf -```{r , echo=TRUE, fig.height=3.7} +```{r , echo=TRUE, fig.height=3} residuals(fit, type='regression') %>% gg_tsdisplay(.resid, plot_type = 'partial') + labs(title = "Regression errors") ``` ## US personal consumption and income -\fontsize{11}{12}\sf -```{r , echo=TRUE, fig.height=3.7} +```{r , echo=TRUE, fig.height=3} residuals(fit, type='innovation') %>% gg_tsdisplay(.resid, plot_type = 'partial') + labs(title = "ARIMA errors") ``` ## US personal consumption and income - -\fontsize{9}{12}\sf +\fontsize{11}{12}\sf ```{r , echo=TRUE, fig.height=3.7} augment(fit) %>% - features(.resid, ljung_box, dof = 5, lag = 12) + features(.innov, ljung_box, dof = 5, lag = 12) ``` ## US personal consumption and income -\fontsize{9}{12}\sf -```{r usconsump3, echo=TRUE, fig.height=3} +```{r usconsump3, echo=TRUE, fig.height=2.4} us_change_future <- new_data(us_change, 8) %>% mutate(Income = mean(us_change$Income)) forecast(fit, new_data = us_change_future) %>% @@ -207,13 +198,8 @@ results. * Forecast intervals ignore the uncertainty in forecasting the predictors. ## Daily electricity demand -\fontsize{12}{13}\sf -Model daily electricity demand as a function of temperature using quadratic regression with ARMA errors. - -\fontsize{9}{10}\sf - -```{r, echo=TRUE, fig.height=3} +```{r, echo=TRUE} vic_elec_daily %>% ggplot(aes(x = Temperature, y = Demand, colour = Day_Type)) + geom_point() + @@ -221,9 +207,8 @@ vic_elec_daily %>% ``` ## Daily electricity demand -\fontsize{11}{12}\sf -```{r, echo=TRUE, fig.height=3.6} +```{r, echo=TRUE} vic_elec_daily %>% pivot_longer(c(Demand, Temperature)) %>% ggplot(aes(x = Date, y = value)) + geom_line() + @@ -231,7 +216,7 @@ vic_elec_daily %>% ``` ## Daily electricity demand -\fontsize{9}{10}\sf +\fontsize{9}{9.5}\sf ```{r, echo=TRUE} fit <- vic_elec_daily %>% @@ -241,14 +226,13 @@ report(fit) ``` ## Daily electricity demand -\fontsize{9}{10}\sf ```{r, echo=TRUE} gg_tsresiduals(fit) ``` ## Daily electricity demand -\fontsize{9}{10}\sf +\fontsize{10}{11}\sf ```{r, echo=TRUE} augment(fit) %>% @@ -266,7 +250,7 @@ forecast(fit, vic_next_day) ``` ## Daily electricity demand -\fontsize{10}{13}\sf +\fontsize{10}{11}\sf ```{r, echo=TRUE} vic_elec_future <- new_data(vic_elec_daily, 14) %>% mutate( @@ -281,7 +265,7 @@ vic_elec_future <- new_data(vic_elec_daily, 14) %>% ``` ## Daily electricity demand -\fontsize{10}{13}\sf + ```{r, echo = TRUE} forecast(fit, new_data = vic_elec_future) %>% autoplot(vic_elec_daily) + labs(y="GW") @@ -306,7 +290,6 @@ where $\eta'_t$ is ARMA process. \end{block} ## Air transport passengers Australia -\fontsize{10}{10}\sf ```{r} aus_airpassengers %>% @@ -316,18 +299,16 @@ aus_airpassengers %>% ``` ## Air transport passengers Australia -\fontsize{10}{10}\sf **Deterministic trend** -\vspace*{-0.1cm} +\fontsize{10}{10}\sf ```{r, echo=TRUE} fit_deterministic <- aus_airpassengers %>% model(ARIMA(Passengers ~ 1 + trend() + pdq(d = 0))) report(fit_deterministic) ``` -\pause\vspace*{-0.3cm} ```{r austaparams, echo=FALSE, dependson='deterministictrend'} coef <- rlang::set_names(tidy(fit_deterministic)$estimate, tidy(fit_deterministic)$term) @@ -337,27 +318,29 @@ slope <- coef["trend()"] sigma2 <- glance(fit_deterministic)$sigma2 ``` -### -\vspace*{-.2cm} +\only<2>{\begin{textblock}{5}(9,4.5) +\begin{block}{}\vspace*{-0.2cm}\fontsize{12}{13}\sf \begin{align*} y_t &= `r sprintf("%.3f", intercept)` + `r sprintf("%.3f", slope)` t + \eta_t \\ \eta_t &= `r sprintf("%.3f", phi1)` \eta_{t-1} + \varepsilon_t\\ \varepsilon_t &\sim \text{NID}(0,`r sprintf("%.3f", sigma2)`). -\end{align*}\vspace*{10cm} +\end{align*} +\end{block} +\end{textblock}} + +\vspace*{10cm} ## Air transport passengers Australia -\fontsize{10}{10}\sf **Stochastic trend** -\vspace*{-0.1cm} +\fontsize{10}{10}\sf ```{r, echo=TRUE} fit_stochastic <- aus_airpassengers %>% model(ARIMA(Passengers ~ pdq(d = 1))) report(fit_stochastic) ``` -\pause\vspace*{-0.3cm} ```{r austaparams2, echo=FALSE, dependson='stochastictrend'} coef <- rlang::set_names(tidy(fit_stochastic)$estimate, tidy(fit_stochastic)$term) @@ -365,19 +348,23 @@ drift <- coef["constant"] sigma2 <- glance(fit_stochastic)$sigma2 ``` -### -\vspace*{-.2cm} + +\only<2>{\begin{textblock}{5}(9,4.5) +\begin{block}{}\vspace*{-0.2cm}\fontsize{12}{13}\sf \begin{align*} y_t-y_{t-1} &= `r sprintf("%.3f", drift)` + \varepsilon_t,\\ y_t &= y_0 + `r sprintf("%.3f", drift)` t + \eta_t \\ \eta_t &= \eta_{t-1} + \varepsilon_{t}\\ \varepsilon_t &\sim \text{NID}(0,`r sprintf("%.3f", sigma2)`). \end{align*} +\end{block} +\end{textblock}} + +\vspace*{10cm} ## Air transport passengers Australia -\fontsize{9}{9}\sf -```{r, fig.height=3.} +```{r, fig.height=2.2} aus_airpassengers %>% autoplot(Passengers) + autolayer(fit_stochastic %>% forecast(h = 20), @@ -397,10 +384,9 @@ aus_airpassengers %>% # Dynamic harmonic regression ## Dynamic harmonic regression +\fontsize{14}{15}\sf - **Combine Fourier terms with ARIMA errors** - -\fontsize{13}{14}\sf +\alert{Combine Fourier terms with ARIMA errors} ### Advantages * it allows any length seasonality; @@ -413,9 +399,7 @@ aus_airpassengers %>% ## Eating-out expenditure -\fontsize{10}{11}\sf - -```{r cafe, echo=TRUE, fig.height=3.8, fig.width=8} +```{r cafe, echo=TRUE, fig.height=2.6} aus_cafe <- aus_retail %>% filter( Industry == "Cafes, restaurants and takeaway food services", year(Month) %in% 2004:2018 @@ -425,9 +409,9 @@ aus_cafe %>% autoplot(Turnover) ## Eating-out expenditure -\fontsize{9}{11}\sf +\fontsize{10}{10.5}\sf -```{r cafefit, dependson='cafe', fig.height=5, echo=TRUE, results='hide'} +```{r cafefit, dependson='cafe', echo=TRUE, results='hide'} fit <- aus_cafe %>% model( `K = 1` = ARIMA(log(Turnover) ~ fourier(K = 1) + PDQ(0,0,0)), `K = 2` = ARIMA(log(Turnover) ~ fourier(K = 2) + PDQ(0,0,0)), @@ -460,37 +444,37 @@ cafe_plot <- function(...){ } ``` -```{r cafe1, dependson='cafe', fig.height=5, echo=FALSE} +```{r cafe1, dependson='cafe', fig.height=4, echo=FALSE} cafe_plot(K = 1) ``` ## Eating-out expenditure -```{r cafe2, dependson='cafe', fig.height=5, echo=FALSE} +```{r cafe2, dependson='cafe', fig.height=4, echo=FALSE} cafe_plot(K = 2) ``` ## Eating-out expenditure -```{r cafe3, dependson='cafe', fig.height=5, echo=FALSE} +```{r cafe3, dependson='cafe', fig.height=4, echo=FALSE} cafe_plot(K = 3) ``` ## Eating-out expenditure -```{r cafe4, dependson='cafe', fig.height=5, echo=FALSE} +```{r cafe4, dependson='cafe', fig.height=4, echo=FALSE} cafe_plot(K = 4) ``` ## Eating-out expenditure -```{r cafe5, dependson='cafe', fig.height=5, echo=FALSE} +```{r cafe5, dependson='cafe', fig.height=4, echo=FALSE} cafe_plot(K = 5) ``` ## Eating-out expenditure -```{r cafe6, dependson='cafe', fig.height=5, echo=FALSE} +```{r cafe6, dependson='cafe', fig.height=4, echo=FALSE} cafe_plot(K = 6) ``` @@ -507,16 +491,14 @@ report(fit) ``` ## Example: weekly gasoline products -\fontsize{11}{12}\sf -```{r gasf, echo=TRUE, fig.height=3.5} +```{r gasf, echo=TRUE, fig.height=3.2} forecast(fit, h = "3 years") %>% autoplot(us_gasoline) ``` ## 5-minute call centre volume - -\fontsize{9}{10}\sf +\fontsize{10}{10}\sf ```{r calls, echo=TRUE, fig.height=4} (calls <- readr::read_tsv("http://robjhyndman.com/data/callcenter.txt") %>% @@ -537,26 +519,14 @@ calls %>% fill_gaps() %>% autoplot(volume) ## 5-minute call centre volume -\fontsize{12}{13}\sf - -```{r calls-season, echo=TRUE, fig.height=4} +```{r calls-season, echo=TRUE, fig.height=3} calls %>% fill_gaps() %>% gg_season(volume, period = "day", alpha = 0.1) + guides(colour = FALSE) ``` ## 5-minute call centre volume - -\fontsize{10}{11}\sf -```{r calls-cal, echo=TRUE, fig.height=4} -library(sugrrants) -calls %>% filter(month(date, label = TRUE) == "Apr") %>% - ggplot(aes(x = time, y = volume)) + - geom_line() + facet_calendar(date) -``` - -## 5-minute call centre volume -\fontsize{8}{8}\sf +\fontsize{10}{10}\sf ```{r callsmodel, echo=TRUE} calls_mdl <- calls %>% @@ -568,24 +538,21 @@ report(fit) ``` ## 5-minute call centre volume -\fontsize{10}{13}\sf -```{r callsres, echo=TRUE} +```{r callsres, echo=TRUE, fig.height=4} gg_tsresiduals(fit, lag = 338) ``` ## 5-minute call centre volume -\fontsize{11}{12}\sf -```{r callsf, echo=TRUE, fig.height=4} +```{r callsf, echo=TRUE, fig.height=3.2} fit %>% forecast(h = 1690) %>% autoplot(calls_mdl) ``` ## 5-minute call centre volume -\fontsize{11}{12}\sf -```{r callsf2, echo=TRUE, fig.height=4} +```{r callsf2, echo=TRUE, fig.height=3.2} fit %>% forecast(h = 1690) %>% autoplot(filter(calls_mdl, idx > 25000)) ``` @@ -610,7 +577,7 @@ Sometimes a change in $x_t$ does not affect $y_t$ instantaneously\pause ## Lagged predictors -The model include present and past values of predictor: $x_t,x_{t-1},x_{t-2},\dots.$ +The model include present and past values of predictor: \begin{block}{} \centerline{$ y_t = a + \gamma_0x_t + \gamma_1x_{t-1} + \dots + \gamma_kx_{t-k} + \eta_t$} @@ -621,18 +588,15 @@ where $\eta_t$ is an ARIMA process.\pause \begin{align*} y_{t} & = a+ (\gamma_{0} + \gamma_{1} B + \gamma_{2} B^{2} + \dots + \gamma_{k} B^{k}) x_{t} +\eta_t \\ & = a+ \gamma(B) x_{t} +\eta_t. -\end{align*}\pause\vspace*{-0.6cm} - -\fontsize{13}{14}\sf +\end{align*}\pause\vspace*{-0.9cm} * $\gamma(B)$ is called a \textit{transfer function} since it describes how change in $x_t$ is transferred to $y_t$. * $x$ can influence $y$, but $y$ is not allowed to influence $x$. ## Example: Insurance quotes and TV adverts -\fontsize{10}{10}\sf -```{r tvadvert} +```{r tvadvert, fig.height=2.5} insurance %>% pivot_longer(Quotes:TVadverts) %>% ggplot(aes(x = Month, y = value)) + geom_line() + @@ -658,7 +622,6 @@ insurance %>% ``` ## Example: Insurance quotes and TV adverts -\fontsize{10}{10}\sf ```{r, echo=TRUE} fit <- insurance %>% @@ -689,7 +652,7 @@ glance(fit) %>% ## Example: Insurance quotes and TV adverts -\fontsize{8}{9}\sf +\fontsize{10}{10}\sf ```{r tvadvertagain, echo=TRUE} fit_best <- insurance %>% @@ -711,9 +674,10 @@ intercept <- coef["intercept"] gamma0 <- coef["TVadverts"] gamma1 <- coef["lag(TVadverts)"] ``` +\vspace*{0.2cm} ### -\begin{align*} +\vspace*{-0.2cm}\begin{align*} y_t &= `r sprintf("%.3f", intercept)` + `r sprintf("%.3f", gamma0)` x_t + `r sprintf("%.3f", gamma1)` x_{t-1} + \eta_t,\\ @@ -724,27 +688,24 @@ gamma1 <- coef["lag(TVadverts)"] \end{align*} ## Example: Insurance quotes and TV adverts -\fontsize{11}{13}\sf -```{r, echo=TRUE, fig.height=3.3} +```{r, echo=TRUE, fig.height=3} advert_a <- new_data(insurance, 20) %>% mutate(TVadverts = 10) forecast(fit_best, advert_a) %>% autoplot(insurance) ``` ## Example: Insurance quotes and TV adverts -\fontsize{11}{13}\sf -```{r, echo=TRUE, fig.height=3.3} +```{r, echo=TRUE, fig.height=3} advert_b <- new_data(insurance, 20) %>% mutate(TVadverts = 8) forecast(fit_best, advert_b) %>% autoplot(insurance) ``` ## Example: Insurance quotes and TV adverts -\fontsize{11}{13}\sf -```{r, echo=TRUE, fig.height=3.3} +```{r, echo=TRUE, fig.height=3} advert_c <- new_data(insurance, 20) %>% mutate(TVadverts = 6) forecast(fit_best, advert_c) %>% autoplot(insurance) diff --git a/10-dynamic-regression.pdf b/10-dynamic-regression.pdf index 07dda2a2abdd4b5be8cedab0e5784b63f2c61d90..7af24fada882bc7bab8a947befa3af45c5207463 100644 GIT binary patch delta 2351076 zcmeF&Ra9T;o8WugEw~1E*WeH|KyY_=cW2`ScL);P-Q8V+ySoH}yH8T}ukM=e>X|vc z)|_=N&K=jRy}zIRKJW9Juj>RotnC2x4FnAY3j_~@2!yPIA~ObMORheHrB0SHhJu6n zImKVKmf}sAtY-{G2|(bnc2}TP(>A;6WA`~wOhr07Z;J%2*l>Gj zb;!fRFKydQ!^~I&X;_mMcFO!-I-fh(a*OKgjzYyKWEAbub)uSbseXqH2iOVQCqw2{ z6Xr=f<5<7!5dbZ&1Y0g~3sum7O_U};nJtg1Tc!Vp$>y{d?}T8%gMs|_k+J}@=>r3< z(A?6~yG&Jmvl%ZLAC7Oz8=~LO`-QikzZdJL(h%95e6cleo_ABi(8<=2XTregay59g 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