Time_Series_Analysis_And_Forecasting_Practice

In a time series analysis

#install.packages("forecast") library(forecast) AirPassengers Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1949 112 118 132 129 121 135 148 148 136 119 104 118 1950 115 126 141 135 125 149 170 170 158 133 114 140 1951 145 150 178 163 172 178 199 199 184 162 146 166 1952 171 180 193 181 183 218 230 242 209 191 172 194 1953 196 196 236 235 229 243 264 272 237 211 180 201 1954 204 188 235 227 234 264 302 293 259 229 203 229 1955 242 233 267 269 270 315 364 347 312 274 237 278 1956 284 277 317 313 318 374 413 405 355 306 271 306 1957 315 301 356 348 355 422 465 467 404 347 305 336 1958 340 318 362 348 363 435 491 505 404 359 310 337 1959 360 342 406 396 420 472 548 559 463 407 362 405 1960 417 391 419 461 472 535 622 606 508 461 390 432 #Time Series Decomposition

the Ts() function will convert a numeric vector into an R time series object. The format is ts(vector, start =, end =, frequency = ) - where the start and end are the times of the first and last observation and frequency is the number of observations per unit time (1= annual, 4 = quarterly, 12 = monthly, etc)

#create a time series object Air_passengers_TS <- ts(AirPassengers, frequency = 12)

#decompose the time series Air_passengers_decomposition <- decompose(Air_passengers_TS)

#plot the decomposition plot(AirPassengers)

#ARIMA Autoregressive Integrated Moving Averages

Build the ARIMA model

arimaModel <- auto.arima(AirPassengers)

Predict 12 months into the future

arimaForecast <- forecast(arimaModel, h = 12) arimaForecast

Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 Jan 1961 445.6349 430.8903 460.3795 423.0851 468.1847 Feb 1961 420.3950 403.0907 437.6993 393.9304 446.8596 Mar 1961 449.1983 429.7726 468.6241 419.4892 478.9074 Apr 1961 491.8399 471.0270 512.6529 460.0092 523.6707 May 1961 503.3945 481.5559 525.2330 469.9953 536.7937 Jun 1961 566.8625 544.2637 589.4612 532.3007 601.4242 Jul 1961 654.2602 631.0820 677.4384 618.8122 689.7081 Aug 1961 638.5975 614.9704 662.2246 602.4630 674.7320 Sep 1961 540.8837 516.9028 564.8647 504.2081 577.5594 Oct 1961 494.1266 469.8624 518.3909 457.0177 531.2356 1-10 of 12 rows Visualize the forecast plot(arimaForecast)

Error detection

arimaForecast

Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 Jan 1961 445.6349 430.8903 460.3795 423.0851 468.1847 Feb 1961 420.3950 403.0907 437.6993 393.9304 446.8596 Mar 1961 449.1983 429.7726 468.6241 419.4892 478.9074 Apr 1961 491.8399 471.0270 512.6529 460.0092 523.6707 May 1961 503.3945 481.5559 525.2330 469.9953 536.7937 Jun 1961 566.8625 544.2637 589.4612 532.3007 601.4242 Jul 1961 654.2602 631.0820 677.4384 618.8122 689.7081 Aug 1961 638.5975 614.9704 662.2246 602.4630 674.7320 Sep 1961 540.8837 516.9028 564.8647 504.2081 577.5594 Oct 1961 494.1266 469.8624 518.3909 457.0177 531.2356 1-10 of 12 rows #Exponential Smoothing (ETS) Forecasting

Build the ETS model

etsModel <- ets(AirPassengers)

Predict 12 months ahead

etsForecast <- forecast(etsModel, h=12)

etsForecast

Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 Jan 1961 441.8018 419.6256 463.9780 407.8863 475.7174 Feb 1961 434.1186 407.1668 461.0704 392.8994 475.3379 Mar 1961 496.6300 460.6291 532.6310 441.5714 551.6887 Apr 1961 483.2375 443.6210 522.8539 422.6493 543.8256 May 1961 483.9914 440.0236 527.9591 416.7484 551.2343 Jun 1961 551.0244 496.3368 605.7120 467.3869 634.6619 Jul 1961 613.1797 547.3865 678.9728 512.5577 713.8016 Aug 1961 609.3648 539.2447 679.4850 502.1253 716.6044 Sep 1961 530.5408 465.4872 595.5944 431.0500 630.0317 Oct 1961 463.0332 402.8496 523.2168 370.9904 555.0761 1-10 of 12 rows

Visualize the forecast

plot(etsForecast)

TSA Example: https://www.datacamp.com/community/tutorials/time-series-r

install.packages("TSA") Installing package into ‘/cloud/lib/x86_64-pc-linux-gnu-library/4.3’ (as ‘lib’ is unspecified) trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/TSA_1.3.1.tar.gz' Content type 'application/x-gzip' length 502593 bytes (490 KB)

downloaded 490 KB

• installing binary package ‘TSA’ ...
• DONE (TSA)

The downloaded source packages are in ‘/tmp/RtmpG0TbA4/downloaded_packages’ library(TSA) Registered S3 methods overwritten by 'TSA': method from
fitted.Arima forecast plot.Arima forecast

Attaching package: ‘TSA’

The following object is masked from ‘package:readr’:

spec

The following objects are masked from ‘package:stats’:

acf, arima

The following object is masked from ‘package:utils’:

tar

View data set

data("co2") co2 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov 1994 363.05 364.18 364.87 364.47 364.32 362.13 356.72 350.88 350.69 356.06 360.09 1995 363.49 364.94 366.72 366.33 365.75 364.32 358.59 352.06 353.45 357.27 362.34 1996 366.93 366.71 367.63 368.15 369.14 367.33 361.53 356.11 354.51 360.12 363.85 1997 367.72 369.08 368.17 368.83 369.49 367.57 360.79 355.16 356.01 360.71 364.77 1998 369.40 370.12 370.88 370.53 371.56 369.28 364.50 357.46 360.54 364.04 368.74 1999 372.60 373.85 373.75 374.10 374.50 372.04 364.81 359.11 359.65 364.94 369.82 2000 373.23 375.13 374.83 375.42 376.18 374.01 366.54 360.78 361.77 367.51 370.58 2001 375.49 375.94 376.42 377.48 377.67 374.78 367.38 361.67 363.39 367.74 373.18 2002 376.68 377.42 378.27 378.73 379.01 375.95 370.78 364.07 365.36 370.25 374.04 2003 379.03 379.36 380.90 381.39 382.38 381.02 373.78 367.97 368.55 372.28 377.75 2004 382.44 382.36 381.58 383.21 383.58 382.59 374.58 368.69 368.55 373.39 378.49 Dec 1994 363.27 1995 365.65 1996 365.52 1997 367.81 1998 371.58 1999 372.62 2000 373.37 2001 374.41 2002 377.99 2003 379.99 2004 381.62

fitting

fit <- auto.arima(co2)

Time series plot

plot(fc <- forecast(fit, h = 15))

View data

data("boardings")

fitting

fit2 <- auto.arima(boardings[,"log.price"])

forecasting

plot(fc2 <- forecast(fit2, h = 15))

#Arima = autoregressive integrated moving averages

arima is forecasting algorithms based on the assumption that previous values carry internet information and can be used to predict future values

#Arima models are applied in the cases where the data shows evidence of non stationary data ##It allows us to forecatse or predict future outcomes based on a historical time series. It is based on the statistical concept of serial correlation where past data points influence future data points. gas_prod_input <- as.data.frame(read.csv("gas_prod.csv")) gas_prod_input Month Gas_prod 1 384.2611 2 380.1073 3 392.9674 4 402.1147 5 393.5196 6 384.9178 7 387.0473 8 395.2735 9 387.5905 10 365.1166 ... 1-10 of 240 rows

#create a time series object gas_prod <- ts(gas_prod_input[,2]) gas_prod Time Series: Start = 1 End = 240 Frequency = 1 [1] 384.2611 380.1073 392.9674 402.1147 393.5196 384.9178 387.0473 395.2735 [9] 387.5905 365.1166 381.2301 385.3283 384.5220 376.9251 389.6017 398.5708 [17] 381.3281 383.9691 377.4357 390.1675 367.8366 359.8777 370.1539 376.5482 [25] 373.6261 376.4238 384.3008 393.8535 368.1156 372.9807 368.8520 371.8966 [33] 349.3898 348.8001 351.5719 368.5909 361.8514 363.4106 380.7292 388.7903 [41] 365.0580 368.1457 363.4552 364.5328 349.3743 346.7684 345.4838 366.8078 [49] 358.0384 354.8464 370.9050 371.2289 362.4159 362.7829 362.6374 354.7984 [57] 344.7691 349.8382 348.9582 367.1169 357.4586 356.2123 368.7638 366.4915 [65] 361.5556 360.7517 371.7317 357.4217 345.2024 344.7793 351.5371 364.4690 [73] 365.5526 354.1007 367.7785 371.4241 362.1323 362.4000 369.9742 361.8811 [81] 343.3916 342.4114 350.3186 361.2015 353.7961 349.4083 377.3817 365.1934 [89] 365.5990 364.2395 367.2532 366.5127 341.0385 345.5748 343.9211 352.9116 [97] 354.6275 353.5596 380.7902 359.7584 366.0491 369.1072 360.5079 366.5216 [105] 341.2924 354.9480 343.4257 359.0446 362.2595 362.0339 378.7503 370.5111 [113] 373.1233 377.2825 367.3448 382.8817 354.8925 367.5603 362.0913 367.1718 [121] 366.7558 368.3599 391.9210 371.5101 380.0478 379.1329 373.5286 382.9236 [129] 362.2642 371.5335 362.2826 373.7731 374.5527 370.6616 394.6760 382.9170 [137] 380.9585 387.5343 379.7269 381.9066 368.1905 372.3537 369.7750 379.1830 [145] 374.7539 379.5710 396.1087 394.2388 383.7402 389.3859 381.8750 380.3991 [153] 375.3605 385.6053 382.3354 390.2775 374.1852 384.6754 399.8546 401.1613 [161] 392.4891 388.0526 390.8398 392.5266 384.0206 397.0242 389.7558 401.3772 [169] 385.6510 394.4542 413.1976 412.7181 402.7693 395.8044 401.7768 397.6990 [177] 392.1938 401.7197 395.6571 400.9156 383.0953 390.5547 413.5578 411.5326 [185] 399.3026 399.3511 414.2567 403.2949 398.5850 401.3869 391.8897 414.0845 [193] 393.2940 392.7363 414.0253 404.0706 397.4976 391.9574 411.3324 394.1259 [201] 387.4905 397.9629 381.6492 404.7980 391.0838 382.6579 408.3820 400.7036 [209] 402.0134 389.9063 395.4429 389.0535 385.7513 394.4641 381.9485 403.2101 [217] 395.7453 377.4446 397.5944 399.7648 398.9752 395.1845 397.3971 392.9125 [225] 381.6154 387.9312 385.1374 401.6251 401.0117 382.4033 401.9923 414.1446 [233] 406.3315 404.6103 405.5817 395.3791 395.3310 396.5201 391.4281 400.0000 #examine the time series plot(gas_prod, xlab = "Time (months)", ylab = "Gasoline production (millions of barrels)")

#check for conditions of a stationary time series plot(diff (gas_prod)) abline(a = 0, b = 0)

examine ACF and PACF of differenced series

acf(diff(gas_prod), xaxp = c(0, 48, 4), lag.max=48, main="")

pacf(diff(gas_prod), xaxp = c(0, 48, 4), lag.max=48, main="")

fit a (0,1,0)x(1,0,0)12 ARIMA model

arima_1 <- arima (gas_prod, order=c(0,1,0), seasonal = list(order=c(1,0,0),period=12)) arima_1

Call: arima(x = gas_prod, order = c(0, 1, 0), seasonal = list(order = c(1, 0, 0), period = 12))

Coefficients: sar1 0.8335 s.e. 0.0324

sigma^2 estimated as 37.29: log likelihood = -778.69, aic = 1559.38

it may be necessary to calculate AICc and BIC

http://stats.stackexchange.com/questions/76761/extract-bic-and-aicc-from-arima-object

AIC(arima_1,k = log(length(gas_prod))) #BIC [1] 1568.34

examine ACF and PACF of the (0,1,0)x(1,0,0)12 residuals

acf(arima_1$residuals, xaxp = c(0, 48, 4), lag.max=48, main="")

pacf(arima_1$residuals, xaxp = c(0, 48, 4), lag.max=48, main="")

fit a (0,1,1)x(1,0,0)12 ARIMA model

arima_2 <- arima (gas_prod, order=c(0,1,1), seasonal = list(order=c(1,0,0),period=12)) arima_2

Call: arima(x = gas_prod, order = c(0, 1, 1), seasonal = list(order = c(1, 0, 0), period = 12))

Coefficients: ma1 sar1 -0.7065 0.8566 s.e. 0.0526 0.0298

sigma^2 estimated as 25.24: log likelihood = -733.22, aic = 1470.43

it may be necessary to calculate AICc and BIC

http://stats.stackexchange.com/questions/76761/extract-bic-and-aicc-from-arima-object

AIC(arima_2,k = log(length(gas_prod))) #BIC [1] 1482.874

examine ACF and PACF of the (0,1,1)x(1,0,0)12 residuals

acf(arima_2$residuals, xaxp = c(0, 48, 4), lag.max=48, main="")

pacf(arima_2$residuals, xaxp = c(0, 48,4), lag.max=48, main="")

Normality and Constant Variance

plot(arima_2$residuals, ylab = "Residuals") abline(a=0, b=0)

hist(arima_2$residuals, xlab="Residuals", xlim=c(-20,20))

qqnorm(arima_2$residuals, main="") qqline(arima_2$residuals)

Forecasting

#predict the next 12 months arima_2.predict <- predict(arima_2,n.ahead=12) matrix(c(arima_2.predict$pred-1.96arima_2.predict$se, arima_2.predict$pred, arima_2.predict$pred+1.96arima_2.predict$se), 12,3, dimnames=list( c(241:252) ,c("LB","Pred","UB")) ) LB Pred UB 241 394.9689 404.8167 414.6645 242 378.6142 388.8773 399.1404 243 394.9943 405.6566 416.3189 244 405.0188 416.0658 427.1128 245 397.9545 409.3733 420.7922 246 396.1202 407.8991 419.6780 247 396.6028 408.7311 420.8594 248 387.5241 399.9920 412.4598 249 387.1523 399.9507 412.7492 250 387.8486 400.9693 414.0900 251 383.1724 396.6076 410.0428 252 390.2075 403.9500 417.6926 plot(gas_prod, xlim=c(145,252), xlab = "Time (months)", ylab = "Gasoline production (millions of barrels)", ylim=c(360,440)) lines(arima_2.predict$pred) lines(arima_2.predict$pred+1.96arima_2.predict$se, col=4, lty=2) lines(arima_2.predict$pred-1.96arima_2.predict$se, col=4, lty=2)