Prelims Time Series and Their Models Vladas Pipiras STOR UNC CH January 11 2022 Time series TS data Univariate time series data are data xt t 1 T where xt is a real valued observation at time t and T is the sample size We may also write xt t 1 T We shall focus on the situations where there is some dependence across times t Typically there should be some physical reason for such temporal dependence We shall also be interested in the situations where the data appear irregular Examples of time series arise in numerous elds Examples Social Sciences Economics Monthly money supply M1 in the US January 1951 February 1992 m1 scan usm1 dat as time series object m1 ts data m1 start 1959 end 1992 frequency 12 m1 Jan Jun Oct Aug Apr Sep May Nov Jul Mar Feb Dec 1959 143 1 140 3 139 4 140 7 139 6 140 4 141 2 140 9 141 3 141 7 142 8 144 7 1960 144 4 140 9 139 5 140 8 138 7 139 0 140 0 140 4 141 6 142 3 143 4 145 7 1961 145 7 142 8 141 8 143 5 141 8 142 4 142 8 142 7 144 3 145 7 147 6 150 5 1962 150 2 146 9 146 0 148 0 145 8 146 2 146 4 145 8 146 9 148 4 150 2 153 3 1963 153 6 150 1 149 3 151 5 149 3 151 4 151 3 150 9 152 5 154 4 156 7 159 0 1964 159 4 155 4 154 6 156 8 154 2 155 5 157 1 157 0 159 4 161 3 163 1 166 4 1965 166 9 161 9 161 5 164 2 160 3 162 2 163 5 162 8 165 6 168 2 169 9 174 4 1966 175 6 170 3 170 4 174 1 169 6 171 7 171 0 170 0 172 7 173 4 174 6 178 6 1967 178 4 173 4 174 6 176 6 174 1 177 4 179 1 179 0 181 7 183 9 185 7 190 3 1968 189 0 184 9 185 4 189 3 186 5 190 2 191 9 191 4 193 9 196 3 199 6 204 8 1969 205 9 199 3 199 8 203 6 199 4 202 3 203 3 201 5 203 2 205 0 207 0 211 4 1970 212 9 204 0 205 5 210 1 206 2 208 9 210 1 210 0 212 8 214 4 216 7 222 2 1971 222 6 216 6 218 6 223 7 221 1 225 2 227 5 225 9 227 7 229 1 231 2 236 9 1972 237 5 231 4 234 2 239 5 234 7 238 8 241 8 241 3 244 5 247 0 250 5 258 9 1973 259 4 251 2 251 6 257 0 253 6 259 3 261 1 258 6 259 5 261 4 265 6 273 3 1974 271 8 264 1 266 5 271 6 266 3 271 5 273 5 271 0 272 6 274 8 278 8 285 2 1975 281 8 273 3 276 4 281 4 278 1 286 0 288 0 286 3 287 8 288 5 293 5 299 0 1976 296 8 289 0 291 4 299 9 295 1 299 4 302 3 301 0 302 5 307 0 309 7 318 6 1977 317 7 309 0 312 2 322 7 315 6 321 7 326 3 324 3 327 7 332 0 335 4 344 1 1978 343 4 332 0 334 9 347 5 342 4 349 4 353 9 351 7 357 0 359 4 362 9 372 5 1979 367 8 356 4 360 8 376 2 367 1 376 7 383 3 381 9 385 6 387 7 389 8 398 6 1 1980 390 7 380 9 382 4 387 1 377 8 387 6 394 8 398 5 404 9 411 0 416 1 419 8 1981 416 5 405 7 412 5 431 3 418 6 423 0 427 9 426 1 427 3 429 8 435 2 447 2 1982 448 7 432 6 435 8 451 3 441 1 446 5 449 6 450 0 456 4 466 0 474 5 486 0 1983 483 0 474 2 482 9 498 7 494 1 503 7 510 7 508 5 511 5 517 4 522 1 533 4 1984 530 4 517 6 524 2 539 2 530 8 541 4 543 3 539 0 542 5 542 1 549 6 564 5 1985 561 1 551 9 558 3 575 0 569 4 585 2 592 0 594 8 602 2 605 5 615 1 633 5 1986 626 8 613 1 624 6 647 2 645 7 663 5 674 0 679 1 685 2 692 8 709 5 740 6 1987 737 5 717 1 723 5 752 5 739 9 744 4 746 8 745 0 745 2 753 7 756 0 765 9 1988 764 7 745 0 752 1 778 3 763 8 778 8 785 6 781 3 780 0 780 8 787 1 803 2 1989 793 0 772 3 775 2 791 3 767 2 773 8 781 7 777 4 778 5 784 5 791 4 811 9 1990 802 4 788 3 796 2 818 0 797 3 810 8 812 9 814 5 818 9 817 6 826 1 844 3 1991 833 2 823 4 835 0 852 9 841 9 857 8 861 9 864 2 867 3 875 0 893 4 916 8 1992 918 1 time plot plot ts m1 ylab M1 xlab Month Finance Daily detrended logarithmic changes in the S P index 1925 1987 ts read table sppri dat plot ts ts ylab x t xlab t 2 MonthM11960196519701975198019851990200400600800 Sales Marketing Monthly sales of red wine in Australia in thousands of liters January 1980 July 1995 ts read table redwine dat plot ts ts ylab x t xlab t Physical Sciences Physics Astronomy Monthly number of sunspots 1749 1983 ts scan sunsptmo dat plot ts ts ylab x t xlab t 3 tx t050001000015000 15 10 50510tx t050100150500150025003500 Hydrology Geophysics Annual minima of the Nile River in Egypt 520 1280 ts scan nile dat plot ts ts ylab x t xlab t Environment Climatology Sizes of annual tree rings of a tree in Utah 286 1985 ts scan ut dat plot ts ts ylab x t xlab t 4 tx t05001000150020002500050100150200250tx t0100200300400500600100012001400 TS databases E g see Time Series Data at CRAN Task View Time Series Analysis E g tsdl Time Series Data Library with an introduction H Homework 1 will ask you to nd a few time series and produce their time plots Objectives of TS analysis Description trend seasonality etc Modeling Prediction Forecasting Simulation Generation Explaining one TS by another TS Hypothesis testing Clustering Control Examples TS models A time series model is a collection of random variables …