Some New Methods for Prediction of Time Series by Wavelets

Abstract

Žŝžƪ ɋŞƾźƻɍŝžƦ ſŝ ɌƃʧȎ ŽŹ ʼnžƧƋ ƹɎȜƺŞɎƶ Şš X t,T ɌƦŹŞƌŧ ɋŞƾźƻɍŝžƦ ſŝ ƽƲŞŢƺŹ ȍɍ NJ ƥɍžƜŧ ŹŝŹ ƆɍŞƷƺ žɍſ ƱȏƇ ƽš ŝŽ Şƾ Ƹő ƸŝǁŨš žțŝ źƺŽŝŹɋŝſŝ ƽš ɋŞƾ ȍůǁƶ ſŝ ƼŹŝǁƺŞŷ ȍɍ {ψ jk (t)} jk ƽȎŞš ɌƦŹŞƌŧ ɋŞƾžɎƠŨƶ ſŝ ɋŝ ƽƲŞŢƺŹ ʼnŞƾξ jk ƹɎƻDZƷƾ źƇŞš Ɍƶ ψ jǓ (t) ƼŞț ƽɎȏŧ ưǁƓ j Ƹő ŽŹ ƽȎ źƻŨƄƾ ƂƺŞɍŽŝ|ǁȎ | žƧƋ ƹɎȜƺŞɎƶ ɋŽŝǁƷƾ ƑžƇ ƹɍŝ ƽȎ ƼźƇ ưŞƷƛŝ W j (z) ɋ|Ž ɋŽŝǁƷƾ ƑžƇ ʼnŠɍžƫŧ ƹɍŝ ŽŹ źƻŨƄƾ ɌŨšŞū žɍŹŞƫƶ ŞƾC j ƽȎ ʼnƸǁƄɎƺ źƻȎ Ɍƶ ưžŨƻȎ Ƶƾ ŝŽ w j,k;T ƱƶŞȏŧ Ŧƛžƃ ƹɎƻDZƷƾ | ŹǁƇ Ɍƶ Ƹő ɋ Ƽſŝźƺŝ ſŝ ƆɎš ɋŽŝǁƷƾŞƺ ſŝ ƚƺŞƶ NjǓǓǑ źɍő Ɍƶ ŦƃŹ ƽš žɍſ ɋ ƽƔšŝŽ ſŝ ƀɎƺ Ɍȏůǁƶ ƥɎƓŞƾ ȍůǁƶ ſŝ ƼŹŞƧŨƃŝ Şš ɌƺŞƶſ ɋŞƾ ɋžƃ ɌƻɎš ƆɎǤ ɋŝžš źɍźů ɌɍŞƾ ƅ|Ž ɋ ƽɍŝŽŝ ǐǐ ɋŞƾźƻɍŝžƦ ɌƜƏǁƶ ƂƺŞɍŽŝ|ǁȎǁŧŝ ƚšŞŧ NjǕNJǕNJ ƹɍŝ ɋ ƽƔšŝŽ ƸŞɎš ɋŝžš źƻŨƄƾ žȜɍźƷƾ ɋ ƽɍŽǁƦ ƱɍźŢŧ ʼnƂƺŞɍŽŝ|ǁȎǁŧŝ | ɌƧɎƓ ƚšŝǁŧ ʼnŞŨƄɍŝ ɋŞƾ ɋžƃ ŽŹ ƵɎƻȎ Ɍƶ ɌƦžƜƶ žɍſ ťŽǁƋ ƽš ŝŽ źƻɍŝžƦ ȍɍ ƂƺŞɍŽŝ|ǁȎǁŧŝ ƚšŞŧ ŝźŨšŝ ʼnŞŨƄɍŝŞƺ ɋŞƾ ɋžƃ ŽŹ |ŹźƺǁƇ Ɍƶ ƥɍžƜŧ žɍſ ƱȏƇ ƽš | źƻŨƄƾ ɌȜŨƄŢƷƾŹǁŷ ɋŞƾ ȍůǁƶ Şƾ{Ψ j } j<Ǔ ʼnƸő ŽŹ ƽȎ źƇŞš ɌƶƼŞț Ƹő T → ∞ žțŝ ʼnNJ ƥɍžƜŧ ɋŞƾ ƍžƦ ŦŴŧ ƽȎ źƺŝ ƼŹŝŹ ƸŞƈƺ NjǓǓǓ ƸŝŽŞȏƷƾ | ƸǁƄɎƺ źƺǁƇ Ɍƶ ƽƋʧŷ žɍſ ťŽǁƋ ƽš ʼnNj ưǁƶžƦ ſŝ ƼŹŞƧŨƃŝ Şš | źƻɍǁț Ɍƶ ɌƻɎš ƆɎǤ ťʦŹŞƜƶ ʼnťʦŹŞƜƶ ƹɍŝ ƽšŞƾ ȍůǁƶ ſŝ ƼŹŞƧŨƃŝ Şš ɌƺŞƶſ ɋŞƾ ɋžƃ ɌƻɎš ƆɎǤ ɋŝžš źɍźů ɌɍŞƾ ƅ|Ž ɋ ƽɍŝŽŝ ǐǒ ɋŞƾ ɋžƃ ŽŹ ƽŰɎŨƺ ŽŹ ŎŦƄɎƺ ƽŨƄšŝ| ƸŞƶſ ƽš ŞŨƄɍŝ ɋŞƾ ɋžƃ ƴ|Ź ɋ ƽŢŧžƶ Ɖŝǁŷ ƵɎƺŝŹ Ɍƶ ƽȎ ŽǁƓ ƸŞƷƾ ưǁƶžƦ ŽŹ žɎɎƠŧ ƹɍŝ ưŞƷƛŝ Şš źƇŞš Ɍƶ ƸŞƶſ žȜƺŞɎš ư|ŝ ɋ ƽƧƲŖƶ ŝžɍſ ŹŹžț Ɍƶ ƱɍźŢŧ c(τ ) ƽš c(z, τ ) ŞŨƄɍŝ ƵɎƃŽ Ɍƶ žɍſ ưǁƶžƦ ƽš ʼnǍ Ŧƃŝ žɍſ ťŽǁƋ ƽš ƴŝj ŲƔƃ ŽŹ ƽƳƋŞƦ Ƹő ŽŹ ƽȎɋŝ ƼſŞš Šɍžƫŧ ŠɍŝžƏ | ŹǁƇ Ɍƶ ƼŹŞƧŨƃŝ ȍůǁƶƱƺžȎ ɌƜšŞŧ ɋſŞƃŽŝǁƷƾ ƅ|Ž ſŝ ʼnƽƳƋŞƦ ɋžɎț Ƽſŝźƺŝ ſŝ źƜš ƵɍŽ|ő Ɍƶ ŦƃŹ ƽš ʼnƵɎƻȎ ɌƻɎš ƆɎǤ ƵɎƾŝǁŷ Ɍƶ ƽȎ ŝŽ