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Time Series in R
Jared P. Lander

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Time Series and R
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Agenda
1. Our Data
2. Time Series Objects
3. Stationarity
4. Autocorrelation
5. ARIMA
6. Prophet
7. Neural Networks
8. XGBoost
9. Accuracy
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+
-
Leaflet

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Riders
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ts
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zoo
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xts
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Convert to xts
citi <- readr::read_csv(file.path(dataDir, 'citibike_201307_201707.csv'))
citiTS <- xts(citi %>% dplyr::select(Trips), order.by=citi$Date)
citiTS['2017-07-23/']
Trips
2017-07-23 47779
2017-07-24 44702
2017-07-25 66620
2017-07-26 71672
2017-07-27 67066
2017-07-28 65089
2017-07-29 53897
2017-07-30 60402
2017-07-31 65523
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Stationarity
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Stationarity
1.
2.
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Autocorrelation Function
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ACF
acf(citiTS)
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Partial Autocorrelation Function
where is the last element of
and
and
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PACF
pacf(citiTS)
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Dickey Fuller Test
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ADF Test
adf.test(citiTS)
Augmented Dickey-Fuller Test
data: citiTS
Dickey-Fuller = -2.9189, Lag order = 11, p-value = 0.1893
alternative hypothesis: stationary
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KPSS Test
Test that
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KPSS Test
kpss.test(citiTS)
KPSS Test for Level Stationarity
data: citiTS
KPSS Level = 5.9245, Truncation lag parameter = 8, p-value = 0.01
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Difference the Data
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Diff
citiDiff <- diff(citiTS)
dygraph(citiDiff, elementId='CitiDiff') %>% dyRangeSelector() %>% dyUnzoom()
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ACF and PACF
acf(citiDiff[-1])
pacf(citiDiff[-1])
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ADF on Diff
adf.test(citiDiff[-1])
Augmented Dickey-Fuller Test
data: citiDiff[-1]
Dickey-Fuller = -17.493, Lag order = 11, p-value = 0.01
alternative hypothesis: stationary
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KPSS on Diff
kpss.test(citiDiff[-1])
KPSS Test for Level Stationarity
data: citiDiff[-1]
KPSS Level = 0.015114, Truncation lag parameter = 8, p-value = 0.1
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Box-Cox Transformation
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Box-Cox
citiBC <- BoxCox(citiTS, lambda=0)
dygraph(citiBC, elementId='CitiBC') %>% dyRangeSelector() %>% dyUnzoom()
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ACF and PACF
acf(citiBC)
pacf(citiBC)
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Starter Models
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Autoregressive
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AR(p)
ar14 <- ar(citiTS, order.max=14)
ar14$ar
[1] 0.48285396 0.03748923 0.01301544 0.01364710 0.05740610
[6] 0.10362113 0.13929120 -0.01445352 0.02920200 -0.04967610
[11] -0.05221794 0.05979242 0.05699536 0.09126062
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Moving Average
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MA(q)
ma14 <- ma(citiTS, 14)
summary(ma14)
Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
5337 21823 30914 30836 39158 60702 14
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Autoregressive Moving Average
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ARMA(p, q)
arma1_1 <- arma(citiTS, order=c(1, 1))
summary(arma1_1)
Call:
arma(x = citiTS, order = c(1, 1))
Model:
ARMA(1,1)
Residuals:
Min 1Q Median 3Q Max
-47976 -4214 1077 5069 20960
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
ar1 0.990907 0.004651 213.067 <2e-16 ***
ma1 -0.749162 0.033873 -22.117 <2e-16 ***
intercept 309.551314 152.815760 2.026 0.0428 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Fit:
sigma^2 estimated as 63353316, Conditional Sum-of-Squares = 93889663352, AIC = 30876.34
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Autoregressive Integrated Moving Average
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ARIMA(p, d, q)
arima1_1_1 <- arima(citiTS, order=c(1, 1, 1))
summary(arima1_1_1)
Call:
arima(x = citiTS, order = c(1, 1, 1))
Coefficients:
ar1 ma1
0.3834 -0.9016
s.e. 0.0278 0.0108
sigma^2 estimated as 56481996: log likelihood = -15340.13, aic = 30686.27
Training set error measures:
ME RMSE MAE MPE MAPE MASE
Training set 169.4511 7512.918 5554.759 -14.66341 31.23256 0.9318706
ACF1
Training set 0.03266496
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Seasonal ARIMA
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ARIMA(p, d ,q)(P, D, Q)
arima_season <- arima(citiTS, order=c(1, 1, 1), seasonal=c(12, 2, 12))
summary(arima_season)
Call:
arima(x = citiTS, order = c(1, 1, 1), seasonal = c(12, 2, 12))
Coefficients:
ar1 ma1 sar1 sar2 sar3 sar4 sar5 sar6
-0.4859 -0.9998 -0.2511 -0.0837 -0.0011 0.1572 0.0111 0.3046
s.e. 0.2157 0.0035 0.2579 0.0321 0.0160 0.0234 0.0259 0.0089
sar7 sar8 sar9 sar10 sar11 sar12 sma1 sma2
0.3297 0.5942 0.2559 -0.5208 0.0967 -0.1624 -0.7959 -0.5989
s.e. 0.1686 0.1876 0.2031 0.1055 0.0883 0.1959 0.0716 0.0857
sma3 sma4 sma5 sma6 sma7 sma8 sma9 sma10
0.0765 -0.0006 0.2784 -0.1645 -0.0077 -0.2307 0.3967 0.9158
s.e. 0.0286 0.0494 0.0534 0.0080 0.1215 0.0658 0.0129 0.0936
sma11 sma12
-0.8379 -0.0310
s.e. 0.1090 0.0778
sigma^2 estimated as 48239970: log likelihood = -15222.75, aic = 30499.5
Training set error measures:
ME RMSE MAE MPE MAPE MASE
Training set 244.1609 6938.476 4892.245 -11.09788 28.59577 0.8207268
ACF1
Training set -0.01560002
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Automatic Fitting
citi_arima <- auto.arima(citiTS,
max.p=10, max.d=2, max.q=10,
max.P=30, max.D=2, max.Q=30,
seasonal=TRUE)
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ARIMA Forecast
arimaForecast <- forecast(citi_arima, h=30)
autoplot(arimaForecast)
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Prophet
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Prophet
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Trend
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Seasonality
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Holidays
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Data For Prophet
citiDF <- tibble::tibble(ds=index(citiTS), y=as.numeric(citiTS))
citiDF
# A tibble: 1,484 x 2
ds y
<date> <dbl>
1 2013-07-01 16650
2 2013-07-02 22745
3 2013-07-03 21864
4 2013-07-04 22326
5 2013-07-05 21842
6 2013-07-06 20467
7 2013-07-07 20477
8 2013-07-08 21615
9 2013-07-09 26641
10 2013-07-10 25732
# ... with 1,474 more rows
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Prophet with Defaults
prophet1 <- prophet(citiDF)
Initial log joint probability = -48.3415
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
futureDF1 <- make_future_dataframe(prophet1, periods=30)
preds1 <- predict(prophet1, df=futureDF1)
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Prophet Results
plot.prophet(prophet1, preds1, elementID='preds1') %>%
dyRangeSelector(dateWindow=as.Date(c('2017-01-01', '2017-10-30')))
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Model Components
prophet_plot_components(prophet1, fcst=preds1)
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Specify Seasonality
prophet2 <- prophet(citiDF,
yearly.seasonality=TRUE,
weekly.seasonality=TRUE
)
Initial log joint probability = -48.3415
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
futureDF2 <- make_future_dataframe(prophet2, periods=30)
preds2 <- predict(prophet2, df=futureDF2)
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Seasonal Components
prophet_plot_components(prophet2, fcst=preds2)
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Holiday Data
holidays <- readr::read_csv(file.path(dataDir, 'holidays_2013_2017.csv'))
tail(holidays)
# A tibble: 6 x 2
ds holiday
<date> <chr>
1 2017-07-04 Independence Day
2 2017-09-04 Labor Day
3 2017-10-09 Columbus Day
4 2017-11-10 Veterans Day
5 2017-11-23 Thanksgiving Day
6 2017-12-25 Christmas Day
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Prophet with Holidays
prophet3 <- prophet(citiDF,
yearly.seasonality=TRUE,
weekly.seasonality=TRUE,
holidays=holidays)
Initial log joint probability = -48.3415
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
futureDF3 <- make_future_dataframe(prophet3, periods=30)
preds3 <- predict(prophet3, df=futureDF3)
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Seasonal Components
prophet_plot_components(prophet3, fcst=preds3)
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Neural Network
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Simple Neural Network
net1 <- nnetar(citiTS)
netPreds1 <- forecast(net1, h=30)
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Neural Net Forecast
autoplot(netPreds1, include=180)
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Boosted Trees
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XGBoost
where
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XGBoost
citiXG1 <- xgbar(as.ts(citiTS), maxlag=14,
trend_method="differencing",
seas_method="fourier")
citiXGPreds1 <- forecast(citiXG1, h=30)
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XGBoost Predictions
autoplot(citiXGPreds1, include=180)
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Accuracy
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Accuracy
forecastAccuracy <- function(y_hat, y)
{
mean(abs(as.vector(y_hat) - as.vector(y)))
}
results <- tibble::tibble(
Method=c('ARIMA', 'Prophet', 'Neural Net', 'XGBoost'),
MAE=c(
forecastAccuracy(arimaForecast$mean, citiTest_ts),
forecastAccuracy(preds3 %>%
filter(ds >= '2017-07-31') %>% pull(yhat), citiTest_ts),
forecastAccuracy(netPreds1$mean, citiTest_ts),
forecastAccuracy(citiXGPreds1$mean, citiTest_ts)
)
)
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Accuracy
ggplot(results, aes(x=Method, y=MAE)) + geom_bar(stat='identity', width=.01)
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What Have We Seen?
Different time series types in R
Interactive Plots
Stationarity
ACF/PACF
Dickey Fuller and KPSS Tests
Differencing
Box-Cox
ARIMA
Prophet
Neural Nets
XGBoost
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Going Foward
Use the forecast package
Fit CNNs and RNNs
timetk
tibbletime
Multivariate Models
GARCH Models
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Thank You
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Jared P. Lander
Chief Data Scientist of Lander Analytics
Author of
Adjunct Professor at Columbia University
Organizer of New York Open Statistical Programming (The R) Meetup
Organizer of New York R Conference
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Packages
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