Predicción digital

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 15
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

El prototipo de señal digital de mínimo retardo Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k es la señal geométrica amortiguada causal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k=a^k para k = 0, 1, 2, …, donde $ |a|<1 $ (Tabla 3, primera fila debajo de los encabezados de columna). Queremos que esta señal sea la entrada a un sistema causal, lineal e invariante en el tiempo, al que llamamos el sistema de extrapolación (o predicción) E(Z). Para la salida deseada, elegimos la señal avanzada (por ejemplo, para el avance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon =2) f_{k+2}=a^{k+2} para k = ..., –3, –2, –1, 0, 1, 2, … (Tabla 3, segunda fila). El extremo delantero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^0 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^1 (Tabla 3, tercera fila) de la señal avanzada es anticausal, mientras que el extremo trasero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^{2},\ a^{3},\ a^{4},\ a^{5},\ a^{6},\ldots (Tabla 3, cuarta fila) es causal. Debido a que el sistema de predicción es causal, no puede alcanzar el extremo delantero anticausal; por lo tanto, solo puede predecir el extremo trasero causal. Para el extremo trasero, exigimos una predicción perfecta. No permitiremos ningún error de predicción.

Tabla 3. Señal de retardo mínimo, señal avanzada, extremo frontal de la señal avanzada y extremo posterior de la señal avanzada.
Tiempo k ... –3 –2 –1 0 1 2 3 4 ...
Señal de retardo mínimo Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k ... 0 0 0 1 a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^3 $ a^{4} $ ...
Señal avanzada Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{k + 2} ... 0 1 a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^5 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^6 ...
Parte delantera de Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{k + 2} ... 0 1 a 0 0 0 0 0 ...
Portón trasero de Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{k + 2} ... 0 0 0 $ a^{2} $ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^5 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^6 ...

El problema es encontrar una expresión para el filtro de predicción causal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E\left(Z\right) . Una persona ingenua procedería de esta manera. Tanto la entrada como la salida deseada (es decir, el portón trasero) son causales, por lo que podemos utilizar la transformada Z unilateral. Recordamos que la transformada Z unilateral se denota por Z (ecuación 27). La transformada Z unilateral de la entrada es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\bf Z}(f_k ) = \sum\limits_{k = 0}^\infty {a^k Z^k = \frac{1}{{1 - aZ}} ,} \end{align} (99)

mientras que la transformada "Z" unilateral de la compuerta trasera de la salida deseada es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\rm Z}\left(f_{k+\varepsilon }\right)=\sum^{\infty }_{k=0}{a^{k+\varepsilon }} Z^k=a^{\varepsilon }{\rm \ }\sum^{\infty }_{k=0}{a^k}Z^k=\frac{a^{\varepsilon }}{1-aZ}. \end{align} (100)

La distancia de predicción, o avance (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon ), es siempre un entero positivo. El filtro de predicción requerido tiene una función de transferencia dada por la relación entre la transformada Z de salida y la transformada Z de entrada; es decir,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E\left(Z\right)=\frac{{\rm Z}\left(f_{k+\varepsilon }\right)}{{\rm Z}\left(f_k\right)}=\frac{a^{\varepsilon }{\left(1-aZ\right)}^{-1}} {{\left(1-aZ\right)}^{-1}}=a^{\varepsilon }. \end{align} (101)

Esta fórmula es correcta, porque si multiplicamos la señal de entrada $ a^{k} $ por Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^{\varepsilon } , obtenemos el valor avanzado Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^{k+\varepsilon } . Si pensamos un poco, veremos que en todas partes la señal geométrica tiene la misma forma, por lo que, si utilizamos una atenuación constante Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^{\varepsilon } , podemos cambiar la forma actual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^k por la forma futura Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^{k+\varepsilon } . (Nota: se dice que dos curvas tienen la misma forma si una es un factor constante multiplicado por la otra).

La fórmula Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E\left(Z\right)=Z\left(f_{k+\varepsilon }\right){\rm /Z}\left(f_k\right) funciona sin duda en el caso en el que Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k es una señal geométrica. ¿Por qué nos molestamos en definir el retardo mínimo? Una razón es que esta fórmula para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E\left(Z\right) funciona no solo para una señal geométrica sino también para cualquier señal de retardo mínimo. De hecho, podemos decir lo siguiente: la predicción perfecta en el sentido definido anteriormente es posible si y solo si la señal de entrada tiene un retardo mínimo. Para una señal de retardo mínimo, $ E\left(Z\right) $ proporciona el sistema de predicción causal para la distancia de predicción Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon . La predicción de una señal de retardo no mínimo se tratará en la siguiente sección.

Veamos el sistema AR(2)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} y_k+{\alpha }_{1}y_{k-1}+{\alpha }_{2}y_{k-2}=u_k. \end{align} (102)

Queremos encontrar el sistema de predicción que predice la respuesta al impulso Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k . En primer lugar, sabemos que la respuesta al impulso de un sistema AR es necesariamente un retardo mínimo. La respuesta al impulso satisface la ecuación diferencial


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_k+{\alpha }_{1}f_{k-1}+{\alpha }_{2}f_{k-2}={\delta }_k \mathrm{\;\;for\;} k=0, 1, 2,\ldots . \end{align} (103)

La transformada Z de la respuesta al impulso es la función de transferencia


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F\left(Z\right)=\frac{1}{{\rm l+}{\alpha }_{1}Z+{\alpha }_{2}Z^{2}} =\frac{1}{\left(1-a_{1}Z\right)\left(1-a_{2}Z\right)}, \end{align} (104)

donde los polos Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^{-1}_{1} y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a^{-1}_{2} se encuentran fuera del círculo unitario; es decir, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |a_{1}|<1 y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |a_{2}|<1 . La respuesta al impulso $ f_{k} $ se puede encontrar desarrollando Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F\left(Z\right) en fracciones parciales. Tenemos


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F\left(z\right)=\frac{A_{1}} {1-a_{1}Z}+\frac{A_{2}}{1-a_{2}Z}, \end{align} (105)

dónde


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} A_1 = \frac{{a_1 }} {{a_1 - a_2 }},\;\;\;\;A_2 = \frac{{a_2 }}{{a_2 - a_1 }}. \end{align} (106)

Por lo tanto, la respuesta al impulso es la señal causal de retardo mínimo.


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_k=A_{1}a^k_{1}+A_{2}a^k_{2} \mathrm{\;\;\; for\;} k=0, 1, 2,\ldots . \end{align} (107)

Es decir, la respuesta al impulso AR(2) es un promedio ponderado de dos señales geométricas con los pesos Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A_{1} y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A_{2} indicados anteriormente. Queremos introducir Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k en el filtro de predicción causal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E\left(Z\right) y obtener como salida la señal avanzada Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{k+\varepsilon } . Como hemos indicado anteriormente, E(Z) es la relación entre la transformada Z unilateral de la salida y la transformada Z unilateral de la entrada. Por supuesto, sabemos que la transformada Z de entrada es F(Z). Podemos obtener simetría en nuestra expresión para E(Z) si escribimos la transformada Z de entrada F(Z) como


$ {\begin{aligned}{\rm {Z}}\left(f_{k}\right)=Z\left(A_{1}a_{1}^{k}+A_{2}a_{2}^{k}\right)=A_{1}{\rm {Z}}\left(a_{1}^{k}\right)+A_{2}{\rm {Z}}\left(a_{2}^{k}\right).\end{aligned}} $ (108)

La transformada Z de salida es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\rm Z}\left(f_{k+\varepsilon }\right)={\rm Z}\left(A_{1}a^{k+\varepsilon }_{1}+A_{2}a^{k+\varepsilon }_{2}\right)=A_{1}a^{\varepsilon }_{1}{\rm Z}\left(a^k_{1}\right)+A_{2}a^{\varepsilon }_{2}{\rm Z}\left(a^k_{2}\right) . \end{align} (109)

Por lo tanto, el sistema de predicción requerido es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E\left(Z\right)=\frac{a^{\varepsilon }_{1}A_{1}{\rm Z}\left(a^k_{1}\right)+a^{\varepsilon }_{2}A_{2}{\rm Z}\left(a^k_{2}\right)}{A_{1}{\rm Z}\left(a^k_{1}\right)+A_{2}{\rm Z}\left(a^k_{2}\right)} , \end{align} (110)

cual es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E\left(Z\right)=\frac{a^{\varepsilon }_{1}a_{1}{\left(a_{1}-a_{2}\right)}^{-1}{\left(1-a_{1}Z\right)}^{-1}+a^{\varepsilon }_{2}a_{2}{\left(a_{2}-a_{1}\right)}^{-1}{\left(1-a_{2}Z\right)}^{-1}} {a_{1}{\left(a_{1}-a_{2}\right)}^{-1}{\left(1-a_{1}Z\right)}^{-1}+a_{2}{\left(1-a_{1}\right)}^{-1}{\left(1-a_{2}Z\right)}^{-1}}. \end{align} (111)

Si simplificamos esta expresión obtenemos


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E\left(Z\right)=\frac{a^{\varepsilon +1}_{1}-a^{\varepsilon +1}_{2}} {a_{1}-a_{2}}-a_{1}a_{2}\frac{a^{\varepsilon }_{1}-a^{\varepsilon }_{2}}{a_{1}-a_{2}} Z. \end{align} (112)

Esta es la expresión general para el sistema de predicción con distancia de predicción Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon . Por lo tanto, para la distancia de predicción unitaria Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon = 1 , el sistema de predicción es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E_1 (Z) = \frac{{a_1^2 - a_2^2 }} {{a_1 - a_2 }} - a_1 a_2 Z = a_1 + a_2 - a_1 a_2 Z. \end{align} (113)

Porque la relación entre los coeficientes AR Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _1 y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _2 y las raíces $ a_{1} $ y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a_{2} es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 1+{\alpha }_{1}Z+{\alpha }_{2}Z^{2}=\left(1-a_{1}Z\right)\left(1-a_{2}Z\right) , \end{align} (114)

Nosotras tenemos


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \alpha _1 = - (a_1 + a_2 ),\;\;\;\;\alpha _2 = a_1 a_2 . \end{align} (115)

Por lo tanto, el sistema de predicción para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon =1 es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E_1 (Z) = - \alpha _1 - \alpha _2 Z. \end{align} (116)

Si Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k es la entrada (en el índice de tiempo k) al sistema de predicción (con distancia de predicción Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon ), entonces definimos Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \hat{f_k}\left(\varepsilon \right) como la salida (en el mismo índice de tiempo k). La notación Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \hat{f_k}\left(\varepsilon \right) debe leerse como “el valor predicho (que se obtiene en el tiempo presente k) del valor futuro $ f_{k+\varepsilon } $ (que no se conocerá hasta el tiempo futuro Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k + \varepsilon )”. En otras palabras, el entero positivo Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon es la distancia de predicción. En el símbolo Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \hat{f_k}\left(\varepsilon \right) , el signo de intercalación representa el valor predicho, el subíndice k representa el momento en el que se produce el valor predicho y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon representa la distancia de predicción. Es decir, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \hat{f}\left(\varepsilon \right) es la predicción del valor futuro Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm value}f_{k+\varepsilon } , la predicción se realiza en el momento actual k. Como aquí Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon = 1 , podemos escribir


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hat{f_k}\left(\varepsilon \right)=\hat{f_k}\left(1\right)=\left(-{\alpha }_{1}-{\alpha }_{2}Z\right)f_k, \end{align} (117)

cual es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hat{f_k}\left(1\right)=-{\alpha }_{1}f_k-{\alpha }_{2}f_{k-1}. \end{align} (118)

Como $ f_{k} $ es causal, es cero para "k" negativo. Por lo tanto, esta ecuación da


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hat{f}\left(1\right)=-{\alpha }_{1}f_0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \hat{f}\left(1\right)=-{\alpha }_{1}f_{1}-{\alpha }_{2}f_0\;\;\\ \hat{f}\left(1\right)=-{\alpha }_{1}f_{2}-{\alpha }_{2}f_{1}. \\ \ldots. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{align} (119)

Con la condición inicial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_0={\delta }_0=1 , las ecuaciones anteriores 119 son equivalentes a las ecuaciones


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_k+{\alpha }_{1}f_{k-1}+{\alpha }_{2}f_{k-2}={\delta }_k \mathrm{\;\; for\; } k=0, 1, 2,\ldots , \end{align} (120)

which generate the impulse response Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k . Therefore, it follows that


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hat f_{k - 1} (1) = f_k , \end{align} (121)

lo que demuestra que el sistema de predicción predice perfectamente la respuesta al impulso Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{2},f_{3} , ... de la compuerta trasera. En pocas palabras, podemos encontrar el sistema de predicción con Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon = 1 para la respuesta al impulso de un sistema AR(2) por inspección. Si el sistema AR(2) es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} y_k+{\alpha }_{1}y_{k-1}+{\alpha }_{2}y_{k-2} , \end{align} (122)

entonces el sistema de predicción es el sistema MA(1) dado por


$ {\begin{aligned}E_{1}\left(Z\right)=-{\alpha }_{1}-{\alpha }_{2}Z.\end{aligned}} $ (123)

A modo de ejemplo, busquemos el sistema de predicción que (para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon =1 ) predice la respuesta al impulso del sistema AR(p) dada por


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} y_k+{\alpha }_{1}y_{k-1}+\ldots +{\alpha }_py_{k-p}=u_k. \end{align} (124)

Por inspección, el sistema de predicción es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E_{1}\left(Z\right)=-{\alpha }_{1}-{\alpha }_{2}Z-\ldots -{\alpha }_pZ^p. \end{align} (125)

Veamos el sistema MA(1) de retardo mínimo


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F\left(Z\right)={\rm Z} \left(f_k\right)=1-bZ\;\;\; (\text{where}\ |b{\rm |<l}). \end{align} (126)

Al inspeccionarlo, vemos que la respuesta al impulso causal es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_k={\delta }_k-b{\delta }_{k-1}\;\;\; \text{for}\ k=0, 1, 2,..., \end{align} (127)

que en términos generales es: Al inspeccionar, vemos que la respuesta al impulso causal es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_0{\rm =1\ ,\ }f_{1}=-b, \ f_{2}{\rm \ =0,\ }f_{3}{\rm \ =0,\ldots} . \end{align} (128)

De este modo


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\rm Z}\left(f_{k+1}\right)=f_{1}+f_{2}Z+f_{3}Z^{2}{\rm +\ldots\ =}-b\ \\ {\rm Z}\left(f_{k+2}\right)=f_{2}+f_{3}Z+f_{4}Z^{2}{\rm +\ldots\ =0},\;\;\; \end{align} (129)

y en general,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\rm Z}\left(f_{k+\varepsilon }\right)=f_{\varepsilon }+f_{\varepsilon +1}Z+f_{\varepsilon +2}Z^{2}+\dots {\rm =\ 0\;\; for\;}\varepsilon {\rm >l}. \end{align} (130)

Por lo tanto, el sistema de predicción para la distancia de predicción Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon es


$ {\begin{aligned}E_{\varepsilon }\left(Z\right)={\frac {{\rm {Z}}\left(f_{k+\varepsilon }\right)}{{\rm {Z}}\left(f_{k}\right)}}={\frac {-b}{1-bZ}}\;\;{\text{for}}\;\varepsilon =1\;\;{\rm {and}}=0\;\;{\rm {for}}\;\varepsilon \;{\rm {2}},3,4,....\end{aligned}} $ (131)

Por lo tanto, si la distancia de predicción Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon es mayor que uno, solo se puede obtener la predicción trivial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \hat{f}\left(\varepsilon \right)= 0 . El sistema de predicción de un paso es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E_{1}\left(z\right)=\frac{-b}{1-bZ}=-b\left(1+bZ+b^{2}Z^{2}+b^{3}Z^{3}+\dots \right) . \end{align} (132)

Por lo tanto, la predicción es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hat{f}\left(1\right)=\left(-b-b^{2}Z-b^{3}Z^{2}-b^{4}Z^{3}{\rm +\ldots }\right){\rm \ }f_k , \end{align} (133)

cual es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hat{f}\left(1\right)=-bf_k-b^{2}f_{k-1}-b^{3}f_{k-2}-{\rm \ }b^{4}f_{k-3}-\ldots . \end{align} (134)

Como f(k) es causal, tenemos el sistema de ecuaciones


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hat{f}\left(1\right)=-bf_0 \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\; \\ \hat{f}\left(1\right)=-bf_{1}-b^{2}f_0\ \;\;\;\;\;\;\;\;\; \\ \hat{f}\left(1\right)=-bf_{2}-b^{2}f_{1}-b^{3}f_0 \ \\ ... \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\; \end{align} (135)

Como la respuesta al impulso Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_k tiene un retardo mínimo, se obtiene una predicción perfecta. Por lo tanto,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hat{f_0}\left(1\right)=f_{1}\ , \hat{f}\left(1\right)=f_{2}\ , \hat{f_2}\left(1\right)=f_{3},\ldots, \end{align} (136)

Por lo que el sistema de ecuaciones anterior se convierte en


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_{1}=-bf_0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ f_{2}=-bf_{1}-b^{2}f_0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ f_{3}=-bf_{2}-b^{2}f_{1}-b^{3}f_0\;\;\;\;\;\; \\ ... \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{align} (137)

Con la condició


$ {\begin{aligned}f_{1}=-b\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\f_{2}=-b\left(-b\right)-b^{2}=0\;\;\;\;\;\;\;\;\;\\f_{3}=-b^{2}\left(-b\right)-b^{3}=0\;\;\;\;\;\;\;\\...\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{aligned}} $ (138)

que de hecho es la respuesta al impulso del sistema MA(1) dado.

Veamos a continuación el sistema digital de retardo mínimo ARMA(1,1)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} y_k-ay_{k-1}=x_k-bx_{k-1} , \end{align} (139)

Entonces, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |a|< 1 y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |b|< 1 . La función de transferencia es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F\left(z\right)={\rm Z}\left\{f_k\right\}=\frac{1-bZ}{1-aZ}=1+\frac{\left(a-b\right)Z}{1-aZ}. \end{align} (140)

La respuesta al impulso de retardo mínimo es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_0{\rm =1,\ }f_{1}=a-b, \ f_{2}=\left(a-b\right)a, \ f_{3}=\left(a-b\right)a^{2}{\rm \ ,\ldots ,\ }f_{k+\varepsilon }=\left(a-b\right)a^{k+\varepsilon -1},\ldots, \end{align} (141)

por consiguiente


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\rm Z}\left(f_{k+\varepsilon }\right)=\left(a-b\right)a^{\varepsilon -1}\sum^{\infty }_{k=0}{a^k}Z^k=\left(a-b\right)a^{\varepsilon -1}{\left(1-aZ\right)}^{-1}. \end{align} (142)

Por lo tanto, el sistema de predicción es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E\left(Z\right)=\frac{1-aZ}{1-bZ}{\rm \ }\frac{a-b}{1-aZ}{\rm \ }a^{\varepsilon -1}=\frac{a-b}{1-bZ}a^{\varepsilon -1}, \end{align} (143)

que es un sistema AR(1).

A continuación, consideremos el sistema ARMA(2,1) de retardo mínimo.


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F\left(Z\right)={\rm Z}\left(f_k\right)=\frac{1-bZ}{\left(1-a_{1}Z\right)\left(1-a_{2}Z\right)}, \end{align} (144)

donde Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |b|< 1 , $ |a_{1}|<1 $ y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |a_{2}|< 1 . La expansión en fracciones parciales es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F\left(Z\right)=\frac{A-{1}} {1-a_{1}Z}+\frac{A_{2}}{1-a_{2}Z}, \end{align} (145)

dónde


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} A_1 = (a_1 - b)(a_1 - a_2 )^{ - 1} ,\;\;\;\;\;A_2 = (a_2 - b)(a_2 - a_1 )^{ - 1} . \end{align} (146)

La expansión de las fracciones da


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F(z) = A_1 \sum\limits_{k = 0}^\infty {a_1^k Z^k + A_2 \sum\limits_{k = 0}^\infty {a_2^k Z^k ,} } \end{align} (147)

Entonces la respuesta al impulso es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_k=A_{1}a^k_{1}+A_{2}a^k_{2}. \end{align} (148)

La respuesta de impulso avanzada es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f_{k+\varepsilon }=A_{1}a^{k+\varepsilon }_{1}+A_{2}a^{k+\varepsilon }_{2}. \end{align} (149)

Por lo tanto, el sistema de predicción es


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \begin{array}{l} E(Z) = \frac{{{\bf Z}(f_{k + \varepsilon } )}} {{{\bf Z}(f_k )}} = \frac{{a_1^\varepsilon A_1 {\bf Z}(a_1^k ) + a_2^\varepsilon A_2 {\bf Z}(a_2^k )}}{{A_1 {\bf Z}(a_1^k ) + A_2 {\bf Z}(a_2^k )}}, \\ \;\;\;\;\;\;\;\; = \frac{{a_1^\varepsilon (a_1 - b)(a_1 - a_2 )^{ - 1} (1 - a_1 Z)^{ - 1} + a_2^\varepsilon (a_2 - b)(a_2 - a_1 )^{ - 1} (1 - a_2 Z)^{ - 1} }}{{(a_1 - b)(a_1 - a_2 )^{ - 1} (1 - a_1 Z)^{ - 1} + (a_2 - b)(a_2 - a_1 )^{ - 1} (1 - a_2 Z)^{ - 1} }} \\ \end{array} \end{align} (150)

which, simplified, is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E\left(Z\right)=\frac{a^{\varepsilon }_{1}\left(a_{1}-b\right)\left(1-a_{2}Z\right)-a^{\varepsilon }_{2}\left(a_{2}-b\right)\left(1-a_{1}Z\right)}{\left(a_{1}-a_{2}\right)\left(1-bZ\right)}. \end{align} (151)

We see that the prediction system is an ARMA(1,1) system.


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