| Nächste Überarbeitung | Vorhergehende Überarbeitung |
| p:ki:zeitreihe [2023/08/28 22:33] – angelegt Ralf Kretzschmar | p:ki:zeitreihe [2024/05/02 08:57] (aktuell) – Tscherter, Vincent |
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| 💡 Dieses Thema setzt Grundwissen bezüglich neuronaler Netze voraus (dieses Grundwissen wird z.B. in [[:p:ki:fische1|Künstliche Intelligenz für echte Fische]] vermittelt). | 💡 Dieses Thema setzt Grundwissen bezüglich neuronaler Netze voraus (dieses Grundwissen wird z.B. in [[:p:ki:fische1|Künstliche Intelligenz für echte Fische]] vermittelt). |
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| ~~NOTOC~~ {{INLINETOC 2-3}} \\ | ~~INTOC~~ |
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| ===== - Einfache Zeitreihenvorhersage ===== | ===== - Einfache Zeitreihenvorhersage ===== |
| [{{ :p:pasted:zeitreihe0.png?nolink&360|Zeitreihe der stündlichen Messgrösse X. Horizontal sind die Tagesstunden ersichtlich, vertikal die Werte der Messgrösse X. Orange: beobachtete Vergangenheit, Grün: vorherzusagende Zukunft ((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | [{{ :p:pasted:zeitreihe0.png|Zeitreihe der stündlichen Messgrösse X. Horizontal sind die Tagesstunden ersichtlich, vertikal die Werte der Messgrösse X. Orange: beobachtete Vergangenheit, Grün: vorherzusagende Zukunft ((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] |
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| 💡 Eine Zeitreihe vorherzusagen kann für vieles nützlich sein, z.B. eine Vorhersage von Wetter- oder Klimamesswerten, von Börsenkursen, Hochwasserständen etc. | 💡 Eine Zeitreihe vorherzusagen kann für vieles nützlich sein, z.B. eine Vorhersage von Wetter- oder Klimamesswerten, von Börsenkursen, Hochwasserständen etc. |
| 🤔 Wie lässt sich nun eine solche digitale Zeitreihe vorhersagen? | 🤔 Wie lässt sich nun eine solche digitale Zeitreihe vorhersagen? |
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| 💡 Eine einfache Idee ist z.B. die Folgende. Wir verwenden jeweils die letzten vier Werte der Zeitreihe als Eingangswerte für ein neuronales Netz (die letzten vier Werte können gemessen oder auch vorhergesagte Werte sein). Diese vier Werte sind in den Abbildungen blau eingezeichnet. Das neuronale Netz wird zuvor so trainiert, dass es immer nur den nächsten, unbekannten Wert voraussagt. D.h. das neuronale Netz sagt nur einen Zeitschritt (+1 Stunde) voraus. Dieser vorhergesagte Wert ist grün eingezeichnet. Ist ein grüner Wert berechnet worden, so kann auch dieser für die Vorhersage des nächsten Zeitschritts ebenfalls als Eingangswert für das neuronale Netz genommen werden. D.h. das neuronale Netz sagt einen Zeitpunkt voraus, rutscht eine Position weiter und sagt den nächsten Zeitpunkt voraus usw. (dies ist in den drei Abbildungen von links nach rechts ersichtlich). | 💡 Eine einfache Idee ist z.B. die Folgende. Wir verwenden jeweils die letzten vier Werte der Zeitreihe als Eingangswerte für ein neuronales Netz (die letzten vier Werte können gemessen oder auch vorhergesagte Werte sein). Diese vier Werte sind in den Abbildungen dunkelblau eingezeichnet. Das neuronale Netz wird zuvor so trainiert, dass es immer nur den nächsten, unbekannten Wert voraussagt. D.h. das neuronale Netz sagt nur einen Zeitschritt (+1 Stunde) voraus. Dieser vorhergesagte Wert ist grün eingezeichnet. Ist ein grüner Wert berechnet worden, so kann auch dieser für die Vorhersage des nächsten Zeitschritts ebenfalls als Eingangswert für das neuronale Netz genommen werden. D.h. das neuronale Netz sagt einen Zeitpunkt voraus, rutscht eine Position weiter und sagt den nächsten Zeitpunkt voraus usw. (dies ist in den drei Abbildungen von links nach rechts ersichtlich). |
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| | [{{:p:pasted:zeitreihe1.png?nolink&272|Zeitreihenvorhersage für Zeitschritt 34((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | [{{:p:pasted:zeitreihe2.png?nolink&272|Zeitreihenvorhersage für Zeitschritt 35((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | [{{:p:pasted:zeitreihe3.png?nolink&272|Zeitreihenvorhersage für Zeitschritt 36((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | | | [{{:p:pasted:zeitreihe1.png|Zeitreihenvorhersage für Zeitschritt 34((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | [{{:p:pasted:zeitreihe2.png|Zeitreihenvorhersage für Zeitschritt 35((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | [{{:p:pasted:zeitreihe3.png|Zeitreihenvorhersage für Zeitschritt 36((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | |
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| 🤔 Wie kann ein neuronales Netz aber für so was trainiert werden? | 🤔 Wie kann ein neuronales Netz aber für so was trainiert werden? |
| - Starte das untenstehende Programm und studiere die folgenden Punkte (es dauert einen kleinen Moment, bis die Zeitreihe angezeigt wird). | - Starte das untenstehende Programm und studiere die folgenden Punkte (es dauert einen kleinen Moment, bis die Zeitreihe angezeigt wird). |
| * Während dem Training wird der Trainingsfehler "loss" (MSE für die +1 Stundenprognose) angezeigt. | * Während dem Training wird der Trainingsfehler "loss" (MSE für die +1 Stundenprognose) angezeigt. |
| * Danach wird eine Grafik erzeugt mit | * Danach wird eine Grafik für eine Prognosezeitreihe ausgehend von einem Startpunkt erzeugt mit |
| * der wahren, beobachteten Zeitreihe (orange) | * der wahren, beobachteten Zeitreihe (orange) |
| * der vorhergesagten Zeitreihe (grün) | * der vorhergesagten Zeitreihe (grün) |
| * dem MAE für die +1, +2, +3, ... + 120 Stundenprognosen (blau) | * Zusätzlich wird der MAE für die +1, +2, +3, ... + 120 Stundenprognosen (violett) berechnet. Der MAE wird über mehrere Prognosenzeitreihen ausgehend von mehreren, verschiedenen Startpunkten aus gemittelt. |
| * Du kannst nach dem Training zwischen einer Beschreibung des neuronalen Netzes, dem Trainingsfehler und der Vorhersage über die Register ''Modell'', ''Training'' und ''Resultat'' hin und her wechseln. | * Du kannst nach dem Training zwischen einer Beschreibung des neuronalen Netzes, dem Trainingsfehler und der Vorhersage über die Register ''Modell'', ''Training'' und ''Resultat'' hin und her wechseln. |
| - Notiere den MAE für die +72 Stundenprognose, fahre dazu einfach mit der Maus auf die Grafik und suche den MAE für ''x = 72''. {{gem/plain?0=N4XyA#a71e8c87a540f361}} | - Notiere den MAE für die +72 Stundenprognose, fahre dazu einfach mit der Maus auf die Grafik und suche den MAE für ''x = 72''. {{gem/plain?0=N4XyA#a71e8c87a540f361}} |
| - Warum wird der MAE tentenziell immer grösser mit fortschreitender Prognosezeit? Schreibe deine Vermutung in das Textfeld. {{gem/plain?0=N4XyA#5daa8fe086e55930}} | - Warum wird der MAE tentenziell immer grösser mit fortschreitender Prognosezeit? Schreibe deine Vermutung in das Textfeld. {{gem/plain?0=N4XyA#5daa8fe086e55930}} |
| - Vergleiche deine Antwort mit ++unserer Antwort|\\ \\ Für die Prognosezeit +1 Stunde verwendet das neuronale Netz nur gemessene (beobachtete) Daten. Für die Prognosezeit +2 Stunden wird bereits ein vorhergesagter Wert verwendet, für +3 Stunden zwei etc. Irgendwann basiert die Prognose nur noch auf vorhergesagten Daten und bald einmal auf vorhergesagten Daten, die auf vorhergesagten Daten basieren etc. Daher wird mit fortschreitender Prognosedauer der Fehler immer grösser.\\ \\ ++ | - Vergleiche deine Antwort mit ++unserer Antwort|\\ \\ Für die Prognosezeit +1 Stunde verwendet das neuronale Netz nur gemessene (beobachtete) Daten. Für die Prognosezeit +2 Stunden wird bereits ein vorhergesagter Wert verwendet, für +3 Stunden zwei etc. Irgendwann basiert die Prognose nur noch auf vorhergesagten Daten und bald einmal auf vorhergesagten Daten, die auf vorhergesagten Daten basieren etc. Daher wird mit fortschreitender Prognosedauer der Fehler immer grösser.\\ \\ ++ |
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| == ✍ Auftrag 1 – Teil 2 == | == ✍ Auftrag 1 – Teil 2 == |
| - Ändere in der ersten Sektion im obigen Programmcode die Parameter ab und wiederhole das Training ein paar Male mit verschiedenen Einstellungen. Die Parameter sind die Folgenden: | - Ändere in der ersten Sektion im obigen Programmcode die Parameter ab und wiederhole das Training ein paar Male mit verschiedenen Einstellungen. Die Parameter sind die Folgenden: |
| * ''TIMESERIES_FEATURES'': Bestimmt, wie viele vergangene Zeitpunkte als Eingangsgrössen verwendet werden. In den einleitenden Grafiken werden vier vergangene Zeitpunkte verwendet (die blauen Punkte). | * ''TIMESERIES_FEATURES'': Bestimmt, wie viele vergangene Zeitpunkte als Eingangsgrössen verwendet werden. In den einleitenden Grafiken werden vier vergangene Zeitpunkte verwendet (die dunkelblauen Punkte). |
| * ''LERNRATE'': Legt fest, wie stark sich die Gewichte des neuronalen Netzes pro Schritt ändern dürfen. Ist der Wert zu hoch, springt der Fehler (loss) während des Trainings hin und her, ist der Wert zu tief, brauchst du viel Geduld…\\ ⚠️ Achtung! Bei einer viel zu hohen Lernrate kann es sogar sein, dass das neuronale Netz derart stark springt, dass die Zahlen grösser werden, als der Computer diese anzeigen kann. Dann wird keine grüne Linie gezeichnet und als Loss "NaN" (not a number) ausgegeben. Dann ist einfach eine kleinere Lernrate zu wählen. | * ''LERNRATE'': Legt fest, wie stark sich die Gewichte des neuronalen Netzes pro Schritt ändern dürfen. Ist der Wert zu hoch, springt der Fehler (loss) während des Trainings hin und her, ist der Wert zu tief, brauchst du viel Geduld…\\ ⚠️ Achtung! Bei einer viel zu hohen Lernrate kann es sogar sein, dass das neuronale Netz derart stark springt, dass die Zahlen grösser werden, als der Computer diese anzeigen kann. Dann wird keine grüne Linie gezeichnet und als Loss "NaN" (not a number) ausgegeben. Dann ist einfach eine kleinere Lernrate zu wählen. |
| * ''ANZAHL_HIDDEN_NEURONS'': Bestimmt die Komplexität des neuronalen Netzes. Ist der Wert zu hoch, kann Overfitting auftreten (was für dieses Beispiel, weniger ein Problem darstellt), ist die Zahl zu klein, kann die Zeitreihe nicht genügend vorhergesagt werden. | * ''ANZAHL_HIDDEN_NEURONS'': Bestimmt die Komplexität des neuronalen Netzes. Ist der Wert zu hoch, kann Overfitting auftreten (was für dieses Beispiel, weniger ein Problem darstellt), ist die Zahl zu klein, kann die Zeitreihe nicht genügend vorhergesagt werden. |
| * ''ANZAHL_EPOCHEN'': Legt die Anzahl der "Lerndurchgänge" fest. Je grösser dieser Wert ist, desto länger wird trainiert. | * ''ANZAHL_EPOCHEN'': Legt die Anzahl der "Lerndurchgänge" fest. Je grösser dieser Wert ist, desto länger wird trainiert. |
| - Notiere, mit welchen Parametereinstellungen du den tiefsten MAE für die +72 Stundenprognose erzielen konntest. {{gem/plain?0=N4XyA#a71e8c87a540f361}} | - Notiere, mit welchen Parametereinstellungen du den tiefsten MAE für die +72 Stundenprognose erzielen konntest. {{gem/plain?0=N4XyA#fb283d4c707cb080}} |
| </WRAP> | </WRAP> |
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| 🤔 Vermutlich waren deine Zeitreihenvorhersagen im vorangehenden Auftrag nicht ganz perfekt oder du musstest viele Eingangsgrössen (input features) verwenden. | 🤔 Vermutlich waren deine Zeitreihenvorhersagen im vorangehenden Auftrag nicht ganz perfekt oder du musstest viele Eingangsgrössen (input features) verwenden. |
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| ⚠️ Für die Wahl der **Eingangsgrössen** ist es immer eine Gute Idee **Fachwissen**, d.h. Wissen über die zu vorhersagende Grösse zu berücksichtigen. Interessant ist, dass sich die Zeitreihe alle 24 Stunden wiederholt. | ⚠️ Für die Wahl der **Eingangsgrössen** ist es immer eine gute Idee, **Fachwissen**, d.h. Wissen über die zu vorhersagende Grösse, zu berücksichtigen. Interessant ist, dass sich die Zeitreihe alle 24 Stunden wiederholt. |
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| 💡 Also bauen wir die Tagesstunde 0 - 23 als zusätzliche Eingangsgrösse mit ein (24 Uhr wäre ja wieder 0 Uhr). | 💡 Also bauen wir die Tagesstunde 0 - 23 als zusätzliche Eingangsgrösse mit ein (24 Uhr wäre ja wieder 0 Uhr). |
| - Verglichen mit dem MAE in "Auftrag 1 - Teil 1" ist das Resultat besser oder schlechter?{{gem/plain?0=N4XyA#21a4e7574c0199e5}} | - Verglichen mit dem MAE in "Auftrag 1 - Teil 1" ist das Resultat besser oder schlechter?{{gem/plain?0=N4XyA#21a4e7574c0199e5}} |
| - Wenn du das Programm mehrfach startest, sind die Resultate in etwa gleich gut oder stellst du grössere Unterschiede fest? | - Wenn du das Programm mehrfach startest, sind die Resultate in etwa gleich gut oder stellst du grössere Unterschiede fest? |
| 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| </WRAP> | </WRAP> |
| ===== - Dem neuronalen Netzwerk helfen ===== | ===== - Dem neuronalen Netzwerk helfen ===== |
| ⚠️ Das **Problem** ist der **Sprung** in der Stundenangabe (siehe unten). Für das neuronale Netz ist der Minimalwert 0 Uhr und der Maximalwert 23 etwas komplett anderes. Für uns (und in der Zeitreihe) sind 0 Uhr und 23 Uhr fast dasselbe. | ⚠️ Das **Problem** ist der **Sprung** in der Stundenangabe (siehe unten). Für das neuronale Netz ist der Minimalwert 0 Uhr und der Maximalwert 23 etwas komplett anderes. Für uns (und in der Zeitreihe) sind 0 Uhr und 23 Uhr fast dasselbe. |
| |
| [{{::p:pasted:time1a.png?nolink&334|Blau: Tagesstunde((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | [{{::p:pasted:time1a.png?nolink|Dunkelblau: Tagesstunde((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] |
| |
| \\ 🤔 Aber wie können wir 24 Stunden mit einer Zahl so darstellen, dass es keinen Sprung gibt, jedoch jede Tagesstunde einen anderen Wert bekommt? | \\ 🤔 Aber wie können wir 24 Stunden mit einer Zahl so darstellen, dass es keinen Sprung gibt, jedoch jede Tagesstunde einen anderen Wert bekommt? |
| 💡 Bevor du dir den Kopf zerbrichst, das ist mit einer Zahl nicht möglich, jedoch mit zwei Zahlen schon! Die Idee besteht darin, die Tagesstunde durch den Sinus UND den Cosinus der Tagesstunde zu ersetzen, d.h. durch zwei Eingangsgrössen.((Genauer gesagt verwenden wir ''sin(2*π*hour/24)'' und ''cos(2*π*hour/24)'' als zusätzliche Eingangsgrössen. Das ''2*π/24'' sorgt dafür, dass der Sinus und der Cosinus nach 24 Stunden wieder die Anfangswerte einnimmt.)). So wird jede Tagesstunde durch ein anderes Sinus / Cosinus Wertepaar dargestellt (siehe unten). | 💡 Bevor du dir den Kopf zerbrichst, das ist mit einer Zahl nicht möglich, jedoch mit zwei Zahlen schon! Die Idee besteht darin, die Tagesstunde durch den Sinus UND den Cosinus der Tagesstunde zu ersetzen, d.h. durch zwei Eingangsgrössen.((Genauer gesagt verwenden wir ''sin(2*π*hour/24)'' und ''cos(2*π*hour/24)'' als zusätzliche Eingangsgrössen. Das ''2*π/24'' sorgt dafür, dass der Sinus und der Cosinus nach 24 Stunden wieder die Anfangswerte einnimmt.)). So wird jede Tagesstunde durch ein anderes Sinus / Cosinus Wertepaar dargestellt (siehe unten). |
| |
| [{{::p:pasted:time1b.png?nolink&334|Blau: Tagesstunde; Orange: Sinus der Tagesstunde; Grau: Cosinus der Tagesstunde((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] | [{{::p:pasted:time1b.png|Dunkelblau: Tagesstunde; Blaugrau: Sinus der Tagesstunde; Grau: Cosinus der Tagesstunde((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}] |
| |
| \\ 💡 Also bauen wir statt der Tagesstunde, die Sinus- und Cosinuswerte der Tagesstunde als zusätzliche Eingangsgrössen mit ein. | \\ 💡 Also bauen wir statt der Tagesstunde, die Sinus- und Cosinuswerte der Tagesstunde als zusätzliche Eingangsgrössen mit ein. |
| <WRAP center round box > | <WRAP center round box > |
| == ✍ Auftrag 3 == | == ✍ Auftrag 3 == |
| Neben den "timeseries features" werden hier zusätzlich Sinus und Cosinus der Tagesstunde verwendet. Bei der Wahl ''TIMESERIES_FEATURES = 12'' besitzt das neuronale Netz somit 14 Eingangsgrössen. | Neben vergangenen Werten der Zeitreihe werden hier zusätzlich Sinus und Cosinus der Tagesstunde verwendet. Bei der Wahl ''TIMESERIES_FEATURES = 12'' besitzt das neuronale Netz somit 14 Eingangsgrössen. |
| - Starte das Programm. Ist das Resultat nun besser oder schlechter in den vorangehenden Aufträgen? {{gem/plain?0=N4XyA#d94fcc4114ff694a}} | - Starte das Programm. Ist das Resultat nun besser oder schlechter in den vorangehenden Aufträgen? {{gem/plain?0=N4XyA#d94fcc4114ff694a}} |
| - Spiele mit den Parametern etwas herum. Mit wie weinigen ''TIMESERIES_FEATURES'' bekommst du eine MAE für die +72 Stundenprognose von weniger als 0.1 hin? Halte die MAE zusammen mit deiner neuronalen Netz Einstellung im Textfeld fest. {{gem/plain?0=N4XyA#344040d14bb98ac5}} | - Spiele mit den Parametern etwas herum. Mit wie weinigen ''TIMESERIES_FEATURES'' bekommst du eine MAE für die +72 Stundenprognose von weniger als 0.1 hin? Halte die MAE zusammen mit deiner neuronalen Netz Einstellung im Textfeld fest. {{gem/plain?0=N4XyA#344040d14bb98ac5}} |
| 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|
| </WRAP> | </WRAP> |
| |
| <WRAP center round box > | <WRAP center round box > |
| == ✍ Auftrag 4 == | == ✍ Auftrag 4 == |
| In der Defaulteinstellung nutzt das neuronale Netz nur ein einziges timeseries feature sowie Cosinus und Sinus der Tagesstunde, d.h. nur 3 Eingangsgrössen. | In der Defaulteinstellung nutzt das neuronale Netz nur einen einzigen vergangenen Wert der Zeitreihe sowie Cosinus und Sinus der Tagesstunde, d.h. nur 3 Eingangsgrössen. |
| - Starte das Programm zwei oder dreimal. Ist das Resultat nun besser oder schlechter in den vorangehenden Aufträgen? Wie vergleicht sich das hier genutzte neuronale Netz mit den zuvor genutzten neuronalen Netzen? {{gem/plain?0=N4XyA#ab0d4297a1770c97}} | - Starte das Programm zwei oder dreimal. Ist das Resultat nun besser oder schlechter in den vorangehenden Aufträgen? Wie vergleicht sich das hier genutzte neuronale Netz mit den zuvor genutzten neuronalen Netzen? {{gem/plain?0=N4XyA#ab0d4297a1770c97}} |
| 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|
| </WRAP> | </WRAP> |
| |
| <WRAP center round box> | <WRAP center round box> |
| == Auftrag 5 == | == Auftrag 5 == |
| Das neuronale Netz verwendet ausschliesslich timeseries features als Eingangsgrössen. Obwohl der verwendete Lernalgorithmus ohne Lernrate auskommt, kann dieser optional mit einer Lernrate "getuned" werden. | Das neuronale Netz verwendet ausschliesslich vergangene Werte der Zeitreihe als Eingangsgrössen. Obwohl der verwendete Lernalgorithmus ohne Lernrate auskommt, kann dieser in diesem Auftrag zusätzlich mit einer Lernrate "getuned" werden. |
| - Starte das Programm zwei oder dreimal. Spiele mit Parametern etwas herum. Versuche, mit möglichst wenigen ''TIMESERIES_FEATURES'' auszukommen. Halte fest, mit welchen Einstellungen du dein bestes Resultat erzielen konntest.{{gem/plain?0=N4XyA#4b7eb6d8feb6e66f}} | - Starte das Programm zwei oder dreimal. Spiele mit Parametern etwas herum. Versuche, mit möglichst wenigen ''TIMESERIES_FEATURES'' auszukommen. Halte fest, mit welchen Einstellungen du dein bestes Resultat erzielen konntest.{{gem/plain?0=N4XyA#4b7eb6d8feb6e66f}} |
| 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|
| </WRAP> | |
| |
| === Eigene Notizen === | === Eigene Notizen === |
| {{gem/quill#310ef8c289347e7f}} | {{gem/quill#310ef8c289347e7f}} |
