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Periodische Schwankungen - Versionsgeschichte
2024-03-28T14:58:15Z
Versionsgeschichte dieser Seite in MM*Stat
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H0130wij am 7. April 2019 um 13:34 Uhr
2019-04-07T13:34:05Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 7. April 2019, 14:34 Uhr</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=={{Vorlage:Überschrift}}==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=={{Vorlage:Überschrift}}==</div></td></tr>
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H0130wij
https://wikis.hu-berlin.de/mmstat/w/index.php?title=Periodische_Schwankungen&diff=1660&oldid=prev
H0130wij: /* {{Vorlage:Beispiele}} */
2019-04-07T13:32:16Z
<p><span dir="auto"><span class="autocomment">{{Vorlage:Beispiele}}</span></span></p>
<a href="https://wikis.hu-berlin.de/mmstat/w/index.php?title=Periodische_Schwankungen&diff=1660&oldid=1258">Änderungen zeigen</a>
H0130wij
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Jacobdan am 30. Mai 2018 um 14:33 Uhr
2018-05-30T14:33:32Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 30. Mai 2018, 15:33 Uhr</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Schließlich soll das Resultat der Zerlegung grafisch veranschaulicht werden. Beachten Sie, dass die [[Schätzung|geschätzt]]e [[Trend]]reihe <math>T(t)</math> tatsächlich keine [[Saisonschwankung]]en mehr enthält. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Schließlich soll das Resultat der Zerlegung grafisch veranschaulicht werden. Beachten Sie, dass die [[Schätzung|geschätzt]]e [[Trend]]reihe <math>T(t)</math> tatsächlich keine [[Saisonschwankung]]en mehr enthält. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><iframe k="wiwi" p="examples/<del style="font-weight: bold; text-decoration: none;">stat_PeriodischeSchwankung_ZulassungPKW_R00480004800000000000000_plot</del>.html" /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><iframe k="wiwi" p="examples/<ins style="font-weight: bold; text-decoration: none;">stat_PeriodischeSchwankung_ZulassungszahlPKW_R00480004800000000000000_plot</ins>.html" /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dies bestätigt die Wahl des [[Filter]]s <math>\left[\frac{1}{8}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{8}\right]</math> zur Glättung einer [[Zeitreihe]] mit Quartalsdaten.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dies bestätigt die Wahl des [[Filter]]s <math>\left[\frac{1}{8}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{8}\right]</math> zur Glättung einer [[Zeitreihe]] mit Quartalsdaten.</div></td></tr>
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Jacobdan
https://wikis.hu-berlin.de/mmstat/w/index.php?title=Periodische_Schwankungen&diff=1257&oldid=prev
Jacobdan am 30. Mai 2018 um 14:33 Uhr
2018-05-30T14:33:00Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 30. Mai 2018, 15:33 Uhr</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{|</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins style="font-weight: bold; text-decoration: none;">iframe k</ins>="<ins style="font-weight: bold; text-decoration: none;">wiwi</ins>" <ins style="font-weight: bold; text-decoration: none;">p</ins>="<ins style="font-weight: bold; text-decoration: none;">examples/stat_PeriodischeSchwankung_Decomposition_PKW_R00480004800000000000000_plot</ins>.<ins style="font-weight: bold; text-decoration: none;">html</ins>" /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|</del><<del style="font-weight: bold; text-decoration: none;">R output</del>="<del style="font-weight: bold; text-decoration: none;">display</del>"<del style="font-weight: bold; text-decoration: none;">></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">pdf(rpdf, width=10, height=7)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> a = c(15222, 17456, 12988, 13833, 15407, 19110, 13479, 13139, 16407, 18738, 11923, 11853, 15869, 16109, 12883, 11712, 14495, 15373, 10341, 11111, 12985, 13397, 9474, 10043, 13431, 15968, 11246, 11261, 14908, 14581, 10498, 10657, 11078, 14858, 11473, 12384, 13801, 17143, 14249, 14712, 12603, 16799, 15611, 15568, 13077, 17098, 14159, 13085, 14093, 16344, 12044, 13762)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> t = ts(a, start=c(1977,1), end=c(1989,4), fr=4)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> plot(t, ylim=c(8000,20000), xlab</del>="<del style="font-weight: bold; text-decoration: none;">Zeit", ylab="Zulassungszahl neuer Pkw (Tsd</del>.<del style="font-weight: bold; text-decoration: none;">)", col="red</del>"<del style="font-weight: bold; text-decoration: none;">, lwd=2, axes=F)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> axis(1, at=seq(1977.1, 1989.4, by=1.0), label=seq(77.1, 89.4, by=1.0), lwd=2, tck=-0.02)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> axis(2, at=seq(8000, 20000, by=2000), label=seq(8, 20, by=2), las=2, lwd=2, tck=-0.02)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> lines(decompose(t)$trend, lwd=2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> lines(decompose(t)$trend+decompose(t)$seasonal, lwd=2, col="blue")</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><</del>/<del style="font-weight: bold; text-decoration: none;">R</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Pkw-Zulassungen===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Pkw-Zulassungen===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Schließlich soll das Resultat der Zerlegung grafisch veranschaulicht werden. Beachten Sie, dass die [[Schätzung|geschätzt]]e [[Trend]]reihe <math>T(t)</math> tatsächlich keine [[Saisonschwankung]]en mehr enthält. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Schließlich soll das Resultat der Zerlegung grafisch veranschaulicht werden. Beachten Sie, dass die [[Schätzung|geschätzt]]e [[Trend]]reihe <math>T(t)</math> tatsächlich keine [[Saisonschwankung]]en mehr enthält. </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><iframe k="wiwi" p="examples/stat_PeriodischeSchwankung_ZulassungPKW_R00480004800000000000000_plot.html" /></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dies bestätigt die Wahl des [[Filter]]s <math>\left[\frac{1}{8}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{8}\right]</math> zur Glättung einer [[Zeitreihe]] mit Quartalsdaten.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dies bestätigt die Wahl des [[Filter]]s <math>\left[\frac{1}{8}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{8}\right]</math> zur Glättung einer [[Zeitreihe]] mit Quartalsdaten.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{|</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|<R output="display"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">pdf(rpdf, width=10, height=7)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> a = c(15222, 17456, 12988, 13833, 15407, 19110, 13479, 13139, 16407, 18738, 11923, 11853, 15869, 16109, 12883, 11712, 14495, 15373, 10341, 11111, 12985, 13397, 9474, 10043, 13431, 15968, 11246, 11261, 14908, 14581, 10498, 10657, 11078, 14858, 11473, 12384, 13801, 17143, 14249, 14712, 12603, 16799, 15611, 15568, 13077, 17098, 14159, 13085, 14093, 16344, 12044, 13762)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> t = ts(a, start=c(1977,1), end=c(1989,4), fr=4)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> plot(t, xlab="Period", ylab="amount of cars*E3", type="l", ylim=c(-5000, 20000), lwd=2, yaxt="n")</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> axis(2, at=seq(-5000, 20000, by=5000), label=seq(-5, 20, by=5))</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> lines(decompose(t)$trend, col="green", lwd=2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> lines(decompose(t)$seasonal, col="blue", lwd=2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> lines(decompose(t)$random, col="red", lwd=2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></R></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>
Jacobdan
https://wikis.hu-berlin.de/mmstat/w/index.php?title=Periodische_Schwankungen&diff=309&oldid=prev
Haberema: Die Seite wurde neu angelegt: „=={{Vorlage:Überschrift}}== ===Periodische Schwankung, Saisonschwankung bzw. Saisonkomponente=== Bisher wurde aus der Originalzeitreihe nur ein Trend er…“
2018-05-15T15:23:47Z
<p>Die Seite wurde neu angelegt: „=={{Vorlage:Überschrift}}== ===Periodische Schwankung, Saisonschwankung bzw. Saisonkomponente=== Bisher wurde aus der Original<a href="/mmstat/Zeitreihe" class="mw-redirect" title="Zeitreihe">zeitreihe</a> nur ein Trend er…“</p>
<p><b>Neue Seite</b></p><div>=={{Vorlage:Überschrift}}==<br />
<br />
===Periodische Schwankung, Saisonschwankung bzw. Saisonkomponente===<br />
<br />
Bisher wurde aus der Original[[zeitreihe]] nur ein Trend ermittelt. Dabei fanden Informationen über saisonale Erscheinungen Beachtung in der Wahl eines geeigneten [[Filter]]s. <br />
<br />
Nun sollen auch die ''Saisonschwankungen'' (Saisonkomponenten) berechnet werden. Einige nützliche Definitionen vorab<br />
erleichtern das Verständnis:<br />
<br />
* Perioden: <math>p_{i},\; i=1,\ldots,\; P</math><br />
<br />
: Anzahl der Wiederholungen einer Saison<br />
<br />
: Beispiel: Quartalsdaten über 10 Jahre: <math>P = 10</math><br />
<br />
* Unterzeiträume <math>k_{j},\; j=1,\ldots,\;k</math><br />
<br />
: Anzahl der Perioden in einem Zyklus<br />
<br />
: Beispiel: Quartalsdaten: <math>k = 4</math><br />
<br />
* Anzahl der Zeiträume: <math>T=k\cdot P</math><br />
<br />
* Trendwerte: <math>\widehat{x}_{i,j}</math><br />
<br />
* Beobachtungswerte: <math>x_{i,j}</math><br />
<br />
* Schwankungskomponente: <math>s_{i,j}</math><br />
<br />
Man unterscheidet zwischen [[additives Zeitreihenmodell|additiven]] und [[multiplikatives Zeitreihenmodell|multiplikativen Zeitreihenmodellen]]: <br />
<br />
Bei ersteren wird ein additiver und bei letzeren ein multiplikativer Zusammenhang zwischen [[Trend]], Saisonkomponente und [[Residuum|Residuen]] unterstellt. Entsprechend unterscheidet sich die Berechnung der Saisonkomponente:<br />
<br />
===Additives Zeitreihenmodell===<br />
<br />
: <math>s_{i,j}=x_{i,j}-\widehat{x}_{i,j};\quad \bar{s_{j}}=\frac{1}{P}\cdot \sum\limits_{i=1}^{P}s_{i,j}</math><br />
<br />
: <math>\widehat{x}_{i,j}^{ZRM}=\widehat{x}_{i,j}+\bar{s_{j}}</math> für <math>i=1,\ldots ,P\;\; j=1,\ldots, k</math><br />
<br />
Der prognostizierte Wert der [[Variable]]n <math>X\;</math> aufgrund des [[Zeitreihe]]nmodells (ZRM) setzt sich ''additiv'' aus dem Trendwert <math>\widehat{x}_{i,j}</math> und dem mittleren Saisonkoeffizienten <math>\bar{s_{j}}</math> zusammen.<br />
<br />
===Multiplikatives Zeitreihenmodell===<br />
<br />
<math>s_{i,j}=\frac{x_{i,j}}{\widehat{x}_{i,j}},\qquad \bar{s_{j}}=\frac{1}{P}\sum\limits_{i=1}^{P}s_{i,j}</math><br />
<br />
<math>\widehat{x}_{i,j}^{ZRM}=\widehat{x}_{i,j}\cdot \bar{s_{j}}</math> für <math>i=1,\ldots, P\quad j=1,\ldots,k</math><br />
<br />
Der prognostizierte Wert der [[Variable]]n <math>X\;</math> aufgrund des [[Zeitreihe]]nmodells (ZRM) setzt sich ''multiplikativ'' aus dem Trendwert <math>\widehat{x}_{i,j}</math> und dem mittleren Saisonkoeffizienten <math>\bar{s_{j}}</math> zusammen.<br />
<br />
=={{Vorlage:Beispiele}}==<br />
<br />
===PKW Zulassungen (Additives Zeitreihenmodell)===<br />
<br />
Zulassungszahl neuer PKW in Berlin - 1. Quartal 1977 - 4. Quartal 1989<br />
<br />
[[Additives Zeitreihenmodell]]:<br />
<br />
[[Filter]]: <math>\left[\frac{1}{8}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{8}\right]</math><br />
<br />
rot = Originalzeitreihe<br />
<br />
schwarz = geglättete Reihe ([[Trend]])<br />
<br />
blau = [[Trend]] und [[Saisonkomponente]]<br />
<br />
{| border="1" cellpadding="3" style="text-align:center;margin:1em 1em 1em 0; background:#f9f9f9; border:1px #AAA solid; border-collapse:collapse; empty-cells:show;"<br />
|align="center"|<math>j\;</math><br />
|align="center"|Summe<br />
|align="center"|<math>\bar{s_{j}}\;</math><br />
|align="center"|<math>P\;</math><br />
|-<br />
|align="center"|1<br />
|align="center"|2,934<br />
|align="center"|0,244<br />
|align="center"|12<br />
|-<br />
|align="center"|2<br />
|align="center"|30,424<br />
|align="center"|2,535<br />
|align="center"|12<br />
|-<br />
|align="center"|3<br />
|align="center"|-17,434<br />
|align="center"|-1,453<br />
|align="center"|12<br />
|-<br />
|align="center"|4<br />
|align="center"|-16,120<br />
|align="center"|-1,343<br />
|align="center"|12<br />
|}<br />
<br />
{|<br />
|<R output="display"><br />
<br />
pdf(rpdf, width=10, height=7)<br />
<br />
a = c(15222, 17456, 12988, 13833, 15407, 19110, 13479, 13139, 16407, 18738, 11923, 11853, 15869, 16109, 12883, 11712, 14495, 15373, 10341, 11111, 12985, 13397, 9474, 10043, 13431, 15968, 11246, 11261, 14908, 14581, 10498, 10657, 11078, 14858, 11473, 12384, 13801, 17143, 14249, 14712, 12603, 16799, 15611, 15568, 13077, 17098, 14159, 13085, 14093, 16344, 12044, 13762)<br />
t = ts(a, start=c(1977,1), end=c(1989,4), fr=4)<br />
plot(t, ylim=c(8000,20000), xlab="Zeit", ylab="Zulassungszahl neuer Pkw (Tsd.)", col="red", lwd=2, axes=F)<br />
axis(1, at=seq(1977.1, 1989.4, by=1.0), label=seq(77.1, 89.4, by=1.0), lwd=2, tck=-0.02)<br />
axis(2, at=seq(8000, 20000, by=2000), label=seq(8, 20, by=2), las=2, lwd=2, tck=-0.02)<br />
lines(decompose(t)$trend, lwd=2)<br />
lines(decompose(t)$trend+decompose(t)$seasonal, lwd=2, col="blue")<br />
<br />
</R><br />
|}<br />
<br />
===Pkw-Zulassungen===<br />
<br />
Dieses Beispiel soll zeigen, wie man eine saisonale [[Zeitreihe]] <math>x(t)</math> additiv in einen [[Trend]] <math>T(t)</math>, eine [[Saisonschwankung]] <math>S(t)</math> und [[Residuum|Residuen]] <math>e(t)</math> zerlegt. <br />
<br />
Unterstellt wird also ein geschätztes Modell in der Form <br />
<br />
<math>x(t)= T(t)+S(t)+e(t)</math>. <br />
<br />
Als Beispiel dienen Quartalsdaten über Pkw-Zulassungen in Berlin.<br />
<br />
====Trend====<br />
<br />
Zwei verschiedene Verfahren zur [[Trend]][[schätzung]] wurden oben eingeführt: Die [[Methode der kleinsten Quadrate]] und die [[Methode der gleitenden Durchschnitte]]. <br />
<br />
Hier soll letztere zur Anwendung kommen, bei der der [[Trend]] nach der Formel<br />
<br />
<math>T(t)=\sum_{i=-a}^{b}\lambda _{i}\cdot x_{t+i}</math> mit <math>\sum_{i=-a}^{b}\lambda _{i}=1</math><br />
<br />
ermittelt wird.<br />
<br />
Damit die geglättete Reihe keine [[Saisonschwankung]]en mehr enthält, verwendet man bei Quartalsdaten den [[Filter]] <math>\left[\frac{1}{8}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{8}\right]</math>. <br />
<br />
Er sichert sowohl eine gleichmäßige Berücksichtigung von Vergangenheits- und Zukunftsdaten <math>(a = b)</math> als auch die gleiche Gewichtung aller Saisonarten (jeweils mit <math>\frac{1}{4}</math>).<br />
<br />
Beispiel:<br />
<br />
<math>T(3) = \frac{1}{8} \cdot x(1) + \frac{1}{4} \cdot x(2) + \frac{1}{4} \cdot x(3) + \frac{1}{4} \cdot x(4) + \frac{1}{8} \cdot x(5)</math><br />
<br />
====Saisonschwankung====<br />
<br />
Aus dem Modell <br />
<br />
<math>x(t) = T(t) + S(t) + e(t)</math> <br />
<br />
ergibt sich <br />
<br />
<math>x(t) - T(t) = S(t) + e(t)</math>. <br />
<br />
Die linke Seite dieser Gleichung ist nach der [[Trend]][[schätzung]] bekannt. <br />
<br />
Unter der Annahme, dass die [[Saisonschwankung]] in den jeweiligen Quartalen denselben Wert hat (also z.B.: <math>S(3) = S(7) =\ldots=S(51)</math>), ist ein naheliegendes Verfahren zur Saisonbestimmung die Bildung des [[Arithmetisches Mittel|arithmetischen Mittels]] über alle Differenzen <math>x(t) - T(t)</math>, die zu einer Saison gehören.<br />
<br />
Beispiel:<br />
<br />
<math>S(3) = S(7) = \ldots = S(51) = \frac{(x(3) - T(3)) + (x(7) - T(7)) + \ldots + (x(51)<br />
- T(51))}{12}</math><br />
<br />
Für dieses Vorgehen ist es unerheblich, mit welcher Methode der [[Trend]] [[Schätzung|geschätzt]] wurde.<br />
<br />
====Residuen====<br />
<br />
Die [[Schätzung|geschätzt]]en [[Residuum|Residuen]] berechnet man mit <math>e(t)=x(t)-T(t)-S(t)</math>.<br />
<br />
====Ergebnisse der Zerlegung der Zeitreihe====<br />
<br />
Sie sollten anhand der Rechenergebnisse zu mindestens einer Periode überprüfen, ob Sie das oben beschriebene Verfahren nachvollziehen können.<br />
<br />
{| border="1" cellpadding="3" style="text-align:center;margin:1em 1em 1em 0; background:#f9f9f9; border:1px #AAA solid; border-collapse:collapse; empty-cells:show;"<br />
|align="center"|Quartal<br />
|align="center"|<math>\,t</math><br />
|align="center"|<math>\,x(t)</math><br />
|align="center"|<math>\,T(t)</math><br />
|align="center"|<math>\,x(t)-T(t)</math><br />
|align="center"|<math>\,S(t)</math><br />
|align="center"|<math>\,e(t)</math><br />
|-<br />
|align="center"|1977.1<br />
|align="center"|1<br />
|align="center"|15222<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|1977.2<br />
|align="center"|2<br />
|align="center"|17456<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|1977.3<br />
|align="center"|3<br />
|align="center"|12988<br />
|align="center"|14897,9<br />
|align="center"|-1909,9<br />
|align="center"|-1452,8<br />
|align="center"|-457,1<br />
|-<br />
|align="center"|1977.4<br />
|align="center"|4<br />
|align="center"|13833<br />
|align="center"|15127,8<br />
|align="center"|-1294,8<br />
|align="center"|-1343,3<br />
|align="center"|48,5<br />
|-<br />
|align="center"|1978.1<br />
|align="center"|5<br />
|align="center"|15407<br />
|align="center"|15395,9<br />
|align="center"|11,1<br />
|align="center"|244,5<br />
|align="center"|-233,4<br />
|-<br />
|align="center"|1978.2<br />
|align="center"|6<br />
|align="center"|19110<br />
|align="center"|15370,5<br />
|align="center"|3739,5<br />
|align="center"|2535,4<br />
|align="center"|1204,1<br />
|-<br />
|align="center"|1978.3<br />
|align="center"|7<br />
|align="center"|13479<br />
|align="center"|15408,8<br />
|align="center"|-1929,8<br />
|align="center"|-1452,8<br />
|align="center"|-477<br />
|-<br />
|align="center"|1978.4<br />
|align="center"|8<br />
|align="center"|13139<br />
|align="center"|15487,3<br />
|align="center"|-2348,3<br />
|align="center"|-1343,3<br />
|align="center"|-1005<br />
|-<br />
|align="center"|1979.1<br />
|align="center"|9<br />
|align="center"|16407<br />
|align="center"|15246,3<br />
|align="center"|1160,7<br />
|align="center"|244,5<br />
|align="center"|916,2<br />
|-<br />
|align="center"|1979.2<br />
|align="center"|10<br />
|align="center"|18738<br />
|align="center"|14891<br />
|align="center"|3847<br />
|align="center"|2535,4<br />
|align="center"|1311,6<br />
|-<br />
|align="center"|1979.3<br />
|align="center"|11<br />
|align="center"|11923<br />
|align="center"|14663<br />
|align="center"|-2740<br />
|align="center"|-1452,8<br />
|align="center"|-1287,2<br />
|-<br />
|align="center"|1979.4<br />
|align="center"|12<br />
|align="center"|11853<br />
|align="center"|14267,1<br />
|align="center"|-2414,1<br />
|align="center"|-1343,3<br />
|align="center"|-1070,8<br />
|-<br />
|align="center"|1980.1<br />
|align="center"|13<br />
|align="center"|15869<br />
|align="center"|14058,5<br />
|align="center"|1810,5<br />
|align="center"|244,5<br />
|align="center"|1566<br />
|-<br />
|align="center"|1980.2<br />
|align="center"|14<br />
|align="center"|16109<br />
|align="center"|14160,9<br />
|align="center"|1948,1<br />
|align="center"|2535,4<br />
|align="center"|-587,3<br />
|-<br />
|align="center"|1980.3<br />
|align="center"|15<br />
|align="center"|12883<br />
|align="center"|13971,5<br />
|align="center"|-1088,5<br />
|align="center"|-1452,8<br />
|align="center"|364,3<br />
|-<br />
|align="center"|1980.4<br />
|align="center"|16<br />
|align="center"|11712<br />
|align="center"|13707,8<br />
|align="center"|-1995,8<br />
|align="center"|-1343,3<br />
|align="center"|-652,5<br />
|-<br />
|align="center"|1981.1<br />
|align="center"|17<br />
|align="center"|14495<br />
|align="center"|13298<br />
|align="center"|1197<br />
|align="center"|244,5<br />
|align="center"|952,5<br />
|-<br />
|align="center"|1981.2<br />
|align="center"|18<br />
|align="center"|15373<br />
|align="center"|12905,1<br />
|align="center"|2467,9<br />
|align="center"|2535,4<br />
|align="center"|-67,5<br />
|-<br />
|align="center"|1981.3<br />
|align="center"|19<br />
|align="center"|10341<br />
|align="center"|12641,3<br />
|align="center"|-2300,3<br />
|align="center"|-1452,8<br />
|align="center"|-847,5<br />
|-<br />
|align="center"|1981.4<br />
|align="center"|20<br />
|align="center"|11111<br />
|align="center"|12205,5<br />
|align="center"|-1094,5<br />
|align="center"|-1343,3<br />
|align="center"|248,8<br />
|-<br />
|align="center"|1982.1<br />
|align="center"|21<br />
|align="center"|12985<br />
|align="center"|11850,1<br />
|align="center"|1134,9<br />
|align="center"|244,5<br />
|align="center"|890,4<br />
|-<br />
|align="center"|1982.2<br />
|align="center"|22<br />
|align="center"|13397<br />
|align="center"|11608,3<br />
|align="center"|1788,7<br />
|align="center"|2535,4<br />
|align="center"|-746,7<br />
|-<br />
|align="center"|1982.3<br />
|align="center"|23<br />
|align="center"|9474<br />
|align="center"|11530,5<br />
|align="center"|-2056,5<br />
|align="center"|-1452,8<br />
|align="center"|-603,7<br />
|-<br />
|align="center"|1982.4<br />
|align="center"|24<br />
|align="center"|10043<br />
|align="center"|11907,6<br />
|align="center"|-1864,6<br />
|align="center"|-1343,3<br />
|align="center"|-521,3<br />
|-<br />
|align="center"|1983.1<br />
|align="center"|25<br />
|align="center"|13431<br />
|align="center"|12450,5<br />
|align="center"|980,5<br />
|align="center"|244,5<br />
|align="center"|736<br />
|-<br />
|align="center"|1983.2<br />
|align="center"|26<br />
|align="center"|15968<br />
|align="center"|12824,3<br />
|align="center"|3143,7<br />
|align="center"|2535,4<br />
|align="center"|608,3<br />
|-<br />
|align="center"|1983.3<br />
|align="center"|27<br />
|align="center"|11246<br />
|align="center"|13161,1<br />
|align="center"|-1915,1<br />
|align="center"|-1452,8<br />
|align="center"|-462,3<br />
|-<br />
|align="center"|1983.4<br />
|align="center"|28<br />
|align="center"|11261<br />
|align="center"|13172,4<br />
|align="center"|-1911,4<br />
|align="center"|-1343,3<br />
|align="center"|-568,1<br />
|-<br />
|align="center"|1984.1<br />
|align="center"|29<br />
|align="center"|14908<br />
|align="center"|12905,5<br />
|align="center"|2002,5<br />
|align="center"|244,5<br />
|align="center"|1758<br />
|-<br />
|align="center"|1984.2<br />
|align="center"|30<br />
|align="center"|14581<br />
|align="center"|12736,5<br />
|align="center"|1844,5<br />
|align="center"|2535,4<br />
|align="center"|-690,9<br />
|-<br />
|align="center"|1984.3<br />
|align="center"|31<br />
|align="center"|10498<br />
|align="center"|12182,3<br />
|align="center"|-1684,3<br />
|align="center"|-1452,8<br />
|align="center"|-231,5<br />
|-<br />
|align="center"|1984.4<br />
|align="center"|32<br />
|align="center"|10657<br />
|align="center"|11738,1<br />
|align="center"|-1081,1<br />
|align="center"|-1343,3<br />
|align="center"|262,2<br />
|-<br />
|align="center"|1985.1<br />
|align="center"|33<br />
|align="center"|11078<br />
|align="center"|11894,6<br />
|align="center"|-816,6<br />
|align="center"|244,5<br />
|align="center"|-1061,1<br />
|-<br />
|align="center"|1985.2<br />
|align="center"|34<br />
|align="center"|14858<br />
|align="center"|12232,4<br />
|align="center"|2625,6<br />
|align="center"|2535,4<br />
|align="center"|90,2<br />
|-<br />
|align="center"|1985.3<br />
|align="center"|35<br />
|align="center"|11473<br />
|align="center"|12788,6<br />
|align="center"|-1315,6<br />
|align="center"|-1452,8<br />
|align="center"|137,2<br />
|-<br />
|align="center"|1985.4<br />
|align="center"|36<br />
|align="center"|12384<br />
|align="center"|13414,6<br />
|align="center"|-1030,6<br />
|align="center"|-1343,3<br />
|align="center"|312,7<br />
|-<br />
|align="center"|1986.1<br />
|align="center"|37<br />
|align="center"|13801<br />
|align="center"|14047,3<br />
|align="center"|-246,3<br />
|align="center"|244,5<br />
|align="center"|-490,8<br />
|-<br />
|align="center"|1986.2<br />
|align="center"|38<br />
|align="center"|17143<br />
|align="center"|14685,3<br />
|align="center"|2457,7<br />
|align="center"|2535,4<br />
|align="center"|-77,7<br />
|-<br />
|align="center"|1986.3<br />
|align="center"|39<br />
|align="center"|14249<br />
|align="center"|14826,5<br />
|align="center"|-577,5<br />
|align="center"|-1452,8<br />
|align="center"|875,3<br />
|-<br />
|align="center"|1986.4<br />
|align="center"|40<br />
|align="center"|14712<br />
|align="center"|14633,8<br />
|align="center"|78,2<br />
|align="center"|-1343,3<br />
|align="center"|1421,5<br />
|-<br />
|align="center"|1987.1<br />
|align="center"|41<br />
|align="center"|12603<br />
|align="center"|14761<br />
|align="center"|-2158<br />
|align="center"|244,5<br />
|align="center"|-2402,5<br />
|-<br />
|align="center"|1987.2<br />
|align="center"|42<br />
|align="center"|16799<br />
|align="center"|15038,3<br />
|align="center"|1760,7<br />
|align="center"|2535,4<br />
|align="center"|-774,7<br />
|-<br />
|align="center"|1987.3<br />
|align="center"|43<br />
|align="center"|15611<br />
|align="center"|15204,5<br />
|align="center"|406,5<br />
|align="center"|-1452,8<br />
|align="center"|1859,3<br />
|-<br />
|align="center"|1987.4<br />
|align="center"|44<br />
|align="center"|15568<br />
|align="center"|15301,1<br />
|align="center"|266,9<br />
|align="center"|-1343,3<br />
|align="center"|1610,2<br />
|-<br />
|align="center"|1988.1<br />
|align="center"|45<br />
|align="center"|13077<br />
|align="center"|15157<br />
|align="center"|-2080<br />
|align="center"|244,5<br />
|align="center"|-2324,5<br />
|-<br />
|align="center"|1988.2<br />
|align="center"|46<br />
|align="center"|17098<br />
|align="center"|14665,1<br />
|align="center"|2432,9<br />
|align="center"|2535,4<br />
|align="center"|-102,5<br />
|-<br />
|align="center"|1988.3<br />
|align="center"|47<br />
|align="center"|14159<br />
|align="center"|14481,8<br />
|align="center"|-322,8<br />
|align="center"|-1452,8<br />
|align="center"|1130<br />
|-<br />
|align="center"|1988.4<br />
|align="center"|48<br />
|align="center"|13085<br />
|align="center"|14514,5<br />
|align="center"|-1429,5<br />
|align="center"|-1343,3<br />
|align="center"|-86,2<br />
|-<br />
|align="center"|1989.1<br />
|align="center"|49<br />
|align="center"|14093<br />
|align="center"|14155,9<br />
|align="center"|-62,9<br />
|align="center"|244,5<br />
|align="center"|-307,4<br />
|-<br />
|align="center"|1989.2<br />
|align="center"|50<br />
|align="center"|16344<br />
|align="center"|13976,1<br />
|align="center"|2367,9<br />
|align="center"|2535,4<br />
|align="center"|-167,5<br />
|-<br />
|align="center"|1989.3<br />
|align="center"|51<br />
|align="center"|12044<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|1989.4<br />
|align="center"|52<br />
|align="center"|13762<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|}<br />
<br />
Schließlich soll das Resultat der Zerlegung grafisch veranschaulicht werden. Beachten Sie, dass die [[Schätzung|geschätzt]]e [[Trend]]reihe <math>T(t)</math> tatsächlich keine [[Saisonschwankung]]en mehr enthält. <br />
<br />
Dies bestätigt die Wahl des [[Filter]]s <math>\left[\frac{1}{8}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{8}\right]</math> zur Glättung einer [[Zeitreihe]] mit Quartalsdaten.<br />
<br />
{|<br />
|<R output="display"><br />
<br />
pdf(rpdf, width=10, height=7)<br />
<br />
a = c(15222, 17456, 12988, 13833, 15407, 19110, 13479, 13139, 16407, 18738, 11923, 11853, 15869, 16109, 12883, 11712, 14495, 15373, 10341, 11111, 12985, 13397, 9474, 10043, 13431, 15968, 11246, 11261, 14908, 14581, 10498, 10657, 11078, 14858, 11473, 12384, 13801, 17143, 14249, 14712, 12603, 16799, 15611, 15568, 13077, 17098, 14159, 13085, 14093, 16344, 12044, 13762)<br />
t = ts(a, start=c(1977,1), end=c(1989,4), fr=4)<br />
plot(t, xlab="Period", ylab="amount of cars*E3", type="l", ylim=c(-5000, 20000), lwd=2, yaxt="n")<br />
axis(2, at=seq(-5000, 20000, by=5000), label=seq(-5, 20, by=5))<br />
lines(decompose(t)$trend, col="green", lwd=2)<br />
lines(decompose(t)$seasonal, col="blue", lwd=2)<br />
lines(decompose(t)$random, col="red", lwd=2)<br />
<br />
</R><br />
|}</div>
Haberema