2 1. Stationary Time Series ÉnÁ'fátáííi)1 1.1 (Current Through a Resistor). If a sinusoidal voltage v(t) = acos(vr + 9) is applied to a resistor of resistance r and the current recorded continuously we obtain a continuous time series x(t) = r~1acos(vt + 6). If observations are made only at times 1,2,..., the resulting time series will be discrete. Time series of this particularly simple type will play a fundamental role in our later study of stationary time series. 1 .5 1 - 0.5 - 0 -0.5 -1 -1 .5 - 0 ■TT'ľTTTTI p T tT'TVITTp1 ľl I "ľ rT'"lTTT II T r'ľľl ľ"] "IT 10 20 '30 40 50 1' i''''''' " i' " '' " '' i " " ' " '' i''' " " 60 70 80 90 100 »1.1. 100 observations of the series x(í)-= cos(.2r + tt/3). §1.1. Examples of Time Series tiÚmšlM (Population xt of the U.S.A., 1790-1980). t xt t xt 1790 3,929,214 1890 62,979,766 1800 5,308,483 1900 76,212,168 1810 7,239,881 1910 92,228,496 1820 9,638,453 1920 106,021,537 1830 12,860,702 1930 123,202,624 1840 17,063,353 1940 132,164,569 1850 23,191,876 1950 151,325,798 1860 31,443,321 1960 179,323,175 1870 38,558,371 1970 203,302,031 1880 50,189,209 1980 226,545,805 260 ^780 1830 1 880 1 930 1 980 i-w$m> 1.2. Population of the U.S.A. at ten-year intervals, 1790-1980 (U.S. Bureau of the Census). (Thousands) l«WS»«liffli»ll>^^ IT \0O\UO\UW-JOi0\00WAOhvJ UWMrvlWOO^UlUiOOOHvl-g ft) Vi OOOOKit^iC^OOUJOH--~j-JO'Vi-r>- UivlOO\OOf*WOOOO\OWWUi o oo p *o' 1-1 H 3' CO 00 CD 3' mm ■HR §1.1. Examples of Time Series i§MM 1.4 (All Star Baseball Games, 1933-1980). xt = 1 if the National League won in year t, 1 if the American League won in year ŕ. t - 1900 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 xt -1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 t -1 -1 -1 t - 1900 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Xt -1 1 1 1 1 -1 1 1 -1 -1 * * * * 1 1 t - 1900 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 xt 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 f = no game. * = two games scheduled. O l-B-B DDDODDO1 IDDDQDDOn -1 - -2 -3 DDDI 1 ' ' I ' ' ' ' I ' ' ■ ■ I ■ ' ' ' [ ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' 1930 1935 1940 1945__1_950 1955 1960 1965 1970 1975 1980 wo 1.4. Results xi,,'Fi:- of All-star baseball games, 1933-19B0. 6 1. Stationary Time Series flftfctP 1.5 (Wölfer Sunspot Numbers, 1770-1869). 1770 101 1790 90 1810 0 1830 71 1850 66 1771 82 1791 67 1811 1 1831 48 1851 64 1772 66 1792 60 1812 5 1832 28 1852 54 1773 35 1793 47 1813 12 1833 8 1853 39 1774 31 1794 41 1814 14 1834 13 1854 21 1775 7 1795 21 1815 35 1835 57 1855 7 1776 20 1796 16 1816 46 1836 122 1856 4 1777 92 1797 6 1817 41 1837 138 1857 23 1778 154 1798 4 1818 30 1838 103 1858 55 1779 125 1799 7 1819 24 1839 86 1859 94 1780 85 1800 14 1820 16 1840 63 1860 96 1781 68 1801 34 1821 7 1841 37 1861 77 1782 38 1802 45 1822 4 1842 24 1862 59 1783 23 1803 43 1823 2 1843 11 1863 44 1784 10 1804 48 1824 8 1844 15 1864 47 1785 24 1805 42 1825 17 1845 40 1865 30 1786 83 1806 28 1826 36 1846 62 1866 16 1787 132 1807 10 1827 50 1847 98 1867 7 1788 131 1808 8 1828 62 1848 124 1868 37 1789 118 1809 2 1829 67 1849 96 1869 74 160 \ 1 ' ' " ' ' I ' " ' " ' ' ' I ' ' ' " ' ' '^V........| I IT. I , I I I | I I I I M I I , | I I I I I I I I I | I........| I...... 1770 1780 1790 1800 1810 1820 1830 1840 1850 1860 1870 áU, 1.5. The Wölfer sunspot numbers, 1770-1869. r^ iĚm a Q Ö OJ O o < -4-> Ö o 1 "-ÍÍ, Si oo r— OS tr~~ tras tras os Os m Os socN^HOsun-ri-Tt-r—OOmo mososcN^-HcnoocNiT-Hr-m-sr r^VOt^-OOOsOsOOSOSOSOOOS. csi~-vcivooos*r>cNTi-omvo o\viNO»o\NOHin«a\ t--Osr--T-ioocNv©mmoocNiC-- r-T-ivo^^ši-^ooost^-ooTj-r-- r-r--t--t^oooooosoooot--oo fSvo^-ot^vomo'omOTt-voocNr-ootnoscNoomvom m h m m « oo t-- oo r- os os O H M N ^ (N >/•> oo m cn -h T-« r~ O O rf t*- un r^ vo oo oo oo os O SO CN tJ- ^h O O OS oo oo oo oo O en en os O O (N (N -ť M ^ oo -h oo r- -j r- ^5 O Os oo Os oo oo o^ost—lOoomr-r-os'^-ios OSOOOOOsOO^-hOOSOSOSOO c3 bß a > o o ^««aiM.tesiam»^^ (spuDsnoLLL) I u a) 1960 1970 1980 eden €ÜMP» &.7>! Časová řada vykazující výraznou a) trendovou, b) sezónní, c) cyklickou složku äßM< éé 'Aditivní dekompozice časové řady §9.2. Identification Techniques 277 700 TjTTTTI I I MI ľj II 12 24 36 TTTTjT TTTTTT TTI' T JTTT 48 60 72 T r T T M MIH TI TI T'T TT TTTTTTTn 1 ) I 1 11 M 1 II1 rTTTTTTTTH TTT'H I" 84 96 108 120 132 144 International airline passengers; monthly totals in thousands of passengers [Ut, t = 1,..., 144} from January 1949 to December 1960 (Box and Jenkins (1970)). 6.5 TP II II I I) T'l 1 I'M II IT MT ITT TTiriTľTTTTpTTTT'TTTT'rrj-rTT'TTrTrT'nj'TľTl'T'ľrTtT'ľ1] MTTTI Mil I^'ITHTT! "i" 24 36 48 60 72 84 96 108 120 132 144 , Natural logarithms, Vt = \nUt,t= 1,..., 144, of the data in Figure 9.7. 0U.13