4  Análisis Espectral de Series de Tiempo

Este capítulo introduce el análisis espectral y el enfoque de filtrado en series de tiempo. El objetivo es estudiar el comportamiento periódico de los datos y cómo descomponerlos en componentes de frecuencia. Se presentan los conceptos de filtros lineales invariantes en el tiempo, coherencia entre series y el uso del periodograma para identificar componentes periódicos predominantes.

4.1 Análisis espectral y filtrado

En análisis de series de tiempo, la regularidad o patrón de comportamiento repetitivo puede describirse mejor en el dominio de la frecuencia, es decir, analizando cómo se descompone la serie en componentes sinusoidales (coseno y seno).

• Frecuencia (\(\omega\)): Se mide en ciclos por observación.

Si \(\omega = 0.25\), significa que cada ciclo completo ocurre cada 4 observaciones. Cuando la unidad de tiempo es segundos, a un ciclo cada segundo se le llama 1 Hz (Hertz).

• Período: Es la inversa de la frecuencia, \(T = 1/\omega\).

Ejemplo: \(\omega = 0.1\) implica un período de 10 observaciones.

• Frecuencia de plegado (Nyquist): En datos discretos, la máxima frecuencia observable sin aliasing es \(0.5\) ciclos por observación.

Valores mayores a \(0.5\) no pueden distinguirse y se reflejan como frecuencias menores (fenómeno de aliasing).

Ejemplo clásico: en películas, una rueda girando rápido puede parecer que gira hacia atrás: ejemplo.

4.1.1 Comportamiento cíclico y periodicidad

Un proceso periódico básico está dado por:

\[ x_{t} = A \cos (2 \pi \omega t + \phi) \]

donde \(A\) es la amplitud y \(\phi\) la fase. Usando identidades trigonométricas:

\[ x_{t} = U_{1} \cos (2 \pi \omega t) + U_{2} \sin (2 \pi \omega t) \]

con \(U_{1} = A \cos \phi\) y \(U_{2} = -A \sin \phi\). Si \(U_{1}\) y \(U_{2}\) son normales independientes con varianza \(\sigma^{2}\), entonces el proceso es estacionario con:

\[ \gamma_{x}(h) = \sigma^{2} \cos(2 \pi \omega h) \]

y

\[ \operatorname{var}(x_{t}) = \gamma_{x}(0) = \sigma^{2}. \]

El proceso puede extenderse a una mezcla de \(q\) componentes periódicos:

\[ x_{t} = \sum_{k=1}^{q} \big[ U_{k1} \cos(2 \pi \omega_{k} t) + U_{k2} \sin(2 \pi \omega_{k} t) \big] \]

con función de autocovarianza

\[ \gamma_{x}(h) = \sum_{k=1}^{q} \sigma_{k}^{2} \cos(2 \pi \omega_{k} h). \]

La varianza total resulta:

\[ \gamma_{x}(0) = \sum_{k=1}^{q} \sigma_{k}^{2}. \]

4.1.1.1 Ejemplo: Una serie periódica

Se construyen tres componentes:

\[ \begin{aligned} x_{t1} &= 2 \cos(2 \pi t 6/100) + 3 \sin(2 \pi t 6/100), \\ x_{t2} &= 4 \cos(2 \pi t 10/100) + 5 \sin(2 \pi t 10/100), \\ x_{t3} &= 6 \cos(2 \pi t 40/100) + 7 \sin(2 \pi t 40/100). \end{aligned} \]

La suma \(x_{t} = x_{t1} + x_{t2} + x_{t3}\) muestra un comportamiento periódico compuesto.

library(astsa)

x1 = 2*cos(2*pi*1:100*6/100) + 3*sin(2*pi*1:100*6/100)
x2 = 4*cos(2*pi*1:100*10/100) + 5*sin(2*pi*1:100*10/100)
x3 = 6*cos(2*pi*1:100*40/100) + 7*sin(2*pi*1:100*40/100)
x = x1 + x2 + x3
par(mfrow=c(2,2))
plot.ts(x1, ylim=c(-10,10), main=expression(omega==6/100 ~~~A^2==13))
plot.ts(x2, ylim=c(-10,10), main=expression(omega==10/100 ~~~A^2==41))
plot.ts(x3, ylim=c(-10,10), main=expression(omega==40/100 ~~~A^2==85))
plot.ts(x, ylim=c(-16,16), main="sum")

4.1.2 Ejemplo: Estimación y el periodograma

Toda serie \(x_{1},\dots,x_{n}\) con \(n\) impar puede expresarse como:

\[ x_{t} = a_{0} + \sum_{j=1}^{(n-1)/2} \big[ a_{j} \cos(2 \pi t j/n) + b_{j} \sin(2 \pi t j/n) \big]. \] Si \(n\) es par se le agrega un componente adicional: \(a_{n/2}(-1)^t\).

Los coeficientes se calculan como:

\[ a_{j} = \frac{2}{n} \sum_{t=1}^{n} x_{t} \cos(2 \pi t j/n), \quad b_{j} = \frac{2}{n} \sum_{t=1}^{n} x_{t} \sin(2 \pi t j/n). \]

El periodograma escalado se define como:

\[ P(j/n) = a_{j}^{2} + b_{j}^{2}, \]

que estima la varianza en la frecuencia \(\omega_{j}=j/n\). A estas frecuencias se les llama frecuencias de Fourier o frecuencias armónicas o fundamentales.

A través de la transformada rápida de Fourier (FFT):

\[ d(j/n) = n^{-1/2} \sum_{t=1}^{n} x_{t} e^{-2 \pi i t j/n}, \]

y

\[ P(j/n) = \frac{4}{n} |d(j/n)|^{2}. \]

se puede calcular fácilmente el periodograma. En el caso de R, la función fft calcula la FFT y lo calcula sin el factor \(n^{-1/2}\) y con un factor adicional de \(e^{2\pi i \omega_j}\).

P = Mod(2*fft(x)/100)^2; Fr = 0:99/100
plot(Fr, P, type="o", xlab="frequency", ylab="scaled periodogram")

Note que el periodograma es simétrico en \(0.5\):

\[P(j/n)=P(1-j/n).\]

4.1.2.1 Ejemplo: Magnitud de una estrella

Se analizan observaciones diarias de la magnitud de una estrella durante 600 días. El periodograma revela componentes dominantes con ciclos de 29 y 24 días.

Este comportamiento puede interpretarse como una señal modulada en amplitud.

t = 1:200
plot.ts(x <- 2*cos(2*pi*.2*t)*cos(2*pi*.01*t))
lines(cos(2*pi*.19*t)+cos(2*pi*.21*t), col=2)

Px = Mod(fft(x))^2; plot(0:199/200, Px, type='o')

n = length(star)
par(mfrow=c(2,1), mar=c(3,3,1,1), mgp=c(1.6,.6,0))
plot(star, ylab="star magnitude", xlab="day")
Per = Mod(fft(star-mean(star)))^2/n
Freq = (1:n -1)/n
plot(Freq[1:50], Per[1:50], type='h', lwd=3, ylab="Periodogram",
    xlab="Frequency")
u = which.max(Per[1:50]) 
uu = which.max(Per[1:50][-u]) 
1/Freq[22]; 1/Freq[26]

[1] 28.57143

[1] 24

text(.05, 7000, "24 day cycle"); text(.027, 9000, "29 day cycle")

y = cbind(1:50, Freq[1:50], Per[1:50]); y[order(y[,3]),]

      [,1]        [,2]         [,3]
 [1,]    1 0.000000000 9.443191e-29
 [2,]    2 0.001666667 4.507982e-01
 [3,]    3 0.003333333 6.383881e-01
 [4,]    4 0.005000000 6.520257e-01
 [5,]   42 0.068333333 8.665436e-01
 [6,]    5 0.006666667 9.562050e-01
 [7,]    6 0.008333333 1.102173e+00
 [8,]    7 0.010000000 1.571938e+00
 [9,]    8 0.011666667 1.919698e+00
[10,]    9 0.013333333 2.660354e+00
[11,]   50 0.081666667 2.716201e+00
[12,]   49 0.080000000 2.950864e+00
[13,]   48 0.078333333 3.131141e+00
[14,]   10 0.015000000 3.360490e+00
[15,]   41 0.066666667 3.434992e+00
[16,]   47 0.076666667 3.436210e+00
[17,]   46 0.075000000 3.727665e+00
[18,]   45 0.073333333 4.217734e+00
[19,]   40 0.065000000 4.333342e+00
[20,]   11 0.016666667 4.602068e+00
[21,]   44 0.071666667 4.961120e+00
[22,]   39 0.063333333 5.104726e+00
[23,]   38 0.061666667 5.787418e+00
[24,]   12 0.018333333 6.000647e+00
[25,]   37 0.060000000 6.609014e+00
[26,]   36 0.058333333 7.479905e+00
[27,]   43 0.070000000 7.759837e+00
[28,]   13 0.020000000 8.299324e+00
[29,]   35 0.056666667 8.579677e+00
[30,]   34 0.055000000 9.855970e+00
[31,]   14 0.021666667 1.130510e+01
[32,]   33 0.053333333 1.149882e+01
[33,]   32 0.051666667 1.354988e+01
[34,]   15 0.023333333 1.626158e+01
[35,]   31 0.050000000 1.627495e+01
[36,]   30 0.048333333 1.994590e+01
[37,]   16 0.025000000 2.384928e+01
[38,]   29 0.046666667 2.512683e+01
[39,]   28 0.045000000 3.282879e+01
[40,]   17 0.026666667 3.760709e+01
[41,]   27 0.043333333 4.499063e+01
[42,]   18 0.028333333 6.410010e+01
[43,]   25 0.040000000 1.085316e+02
[44,]   19 0.030000000 1.276647e+02
[45,]   24 0.038333333 2.152119e+02
[46,]   20 0.031666667 3.395142e+02
[47,]   23 0.036666667 6.436224e+02
[48,]   21 0.033333333 2.136963e+03
[49,]   26 0.041666667 9.011002e+03
[50,]   22 0.035000000 1.102080e+04

4.2 Procesos periódicos y distribución espectral

En esta sección se introduce la densidad espectral como herramienta fundamental en el análisis en el dominio de la frecuencia. Al igual que la descomposición de Wold justifica el uso de la regresión en series de tiempo, los teoremas de representación espectral justifican la descomposición de series estacionarias en componentes periódicos, en proporción a sus varianzas.

4.2.0.1 Ejemplo: Proceso estacionario periódico

Consideremos el proceso

\[ x_{t} = U_{1} \cos (2 \pi \omega_{0} t) + U_{2} \sin (2 \pi \omega_{0} t), \]

donde \(U_{1}\) y \(U_{2}\) son variables aleatorias incorreladas, de media cero y varianza común \(\sigma^{2}\).

La función de autocovarianza es

\[ \gamma(h) = \sigma^{2} \cos (2 \pi \omega_{0} h) = \frac{\sigma^{2}}{2} e^{-2 \pi i \omega_{0} h} + \frac{\sigma^{2}}{2} e^{2 \pi i \omega_{0} h} = \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2 \pi i \omega h} dF(\omega), \]

donde \(F(\omega)\) es la función de distribución espectral:

\[ F(\omega)= \begin{cases} 0 & \omega < -\omega_{0} \\ \sigma^{2}/2 & -\omega_{0} \leq \omega < \omega_{0} \\ \sigma^{2} & \omega \geq \omega_{0} \end{cases} \]

\(F(\omega)\) se comporta como una función de distribución acumulada, pero en lugar de probabilidades acumula varianzas. En particular, \(F(\infty)=\sigma^{2}=\operatorname{var}(x_{t})\).

4.2.1 Representación espectral de la autocovarianza

4.2.1.1 Propiedad

Si \({x_{t}}\) es estacionaria con autocovarianza \(\gamma(h)\), entonces existe una única función \(F(\omega)\) tal que:

\[ \gamma(h)=\int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2 \pi i \omega h} dF(\omega), \]

con \(F(-1/2)=0\) y \(F(1/2)=\gamma(0)\).

4.2.2 La densidad espectral

4.2.2.1 Propiedad

Si \(\gamma(h)\) cumple la condición de suma absoluta

\[ \sum_{h=-\infty}^{\infty} |\gamma(h)| < \infty, \]

entonces:

\[ \gamma(h) = \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2 \pi i \omega h} f(\omega) d\omega, \]

donde la densidad espectral es

\[ f(\omega) = \sum_{h=-\infty}^{\infty} \gamma(h) e^{-2 \pi i \omega h}, \quad -1/2 \leq \omega \leq 1/2. \]

• \(f(\omega) \geq 0\).
• \(f(\omega)\) es par: \(f(\omega)=f(-\omega)\).
• La varianza total se obtiene como

\[ \gamma(0)=\int_{-\frac{1}{2}}^{\frac{1}{2}} f(\omega) d\omega. \]

Autocovarianza y distribución espectral contienen la misma información: la primera en términos de rezagos y la segunda en términos de ciclos.

4.2.2.2 Ejemplo: Ruido blanco

Para \(w_{t}\) con varianza \(\sigma_{w}^{2}\) y \(\gamma_{w}(h)=\sigma_{w}^{2}\) si \(h=0\) y 0 en otro caso, se cumple:

\[ f_{w}(\omega) = \sigma_{w}^{2}, \quad -1/2 \leq \omega \leq 1/2. \]

Esto implica potencia igual en todas las frecuencias (analógico a la luz blanca).

4.2.3 Procesos lineales y filtros

4.2.3.1 Propiedad: Espectro de una serie filtrada

Si

\[ y_{t} = \sum_{j=-\infty}^{\infty} a_{j} x_{t-j}, \quad \sum |a_{j}| < \infty, \]

entonces

\[ f_{y}(\omega) = |A(\omega)|^{2} f_{x}(\omega), \]

donde

\[ A(\omega) = \sum_{j=-\infty}^{\infty} a_{j} e^{-2 \pi i \omega j}. \]

4.2.4 Modelos ARMA

4.2.4.1 Propiedad: Densidad espectral de un ARMA

Si \(x_{t}\) es ARMA\((p,q)\):

\[ f_{x}(\omega) = \sigma_{w}^{2} \frac{|\theta(e^{-2 \pi i \omega})|^{2}}{|\phi(e^{-2 \pi i \omega})|^{2}}, \]

con \(\phi(z)=1-\sum_{k=1}^{p} \phi_{k} z^{k}\) y \(\theta(z)=1+\sum_{k=1}^{q} \theta_{k} z^{k}\).

4.2.4.2 Ejemplo: MA(1)

Modelo:

\[ x_{t} = w_{t} + 0.5 w_{t-1}. \]

Autocovarianzas:

\[ \gamma(0) = 1.25 \sigma_{w}^{2}, \quad \gamma(\pm 1)=0.5\sigma_{w}^{2}, \quad \gamma(\pm h)=0 ; (h>1). \]

Densidad espectral:

\[ f(\omega) = \sigma_{w}^{2}[1.25 + \cos(2 \pi \omega)]. \]

4.2.4.3 Ejemplo: AR(2)

Para \(x_{t}-x_{t-1}+0.9x_{t-2}=w_{t}\), con \(\sigma_{w}=1\):

\[ f_{x}(\omega) = \frac{1}{2.81-3.8 \cos(2 \pi \omega) + 1.8 \cos(4 \pi \omega)}. \]

La varianza se concentra en una banda estrecha de frecuencias.

par(mfrow=c(3,1))
astsa::arma.spec(main="White Noise")
astsa::arma.spec(ma=.5, main="Moving Average")
astsa::arma.spec(ar=c(1,-.9), main="Autoregression")

4.2.4.4 Ejemplo: Modelos explosivos

Para \(x_{t}=2x_{t-1}+w_{t}\):

\[ f_{x}(\omega)=\sigma_{w}^{2}|1-2e^{-2 \pi i \omega}|^{-2}. \]

Se puede reescribir como modelo causal equivalente:

\[ x_{t} = \tfrac{1}{2}x_{t-1}+v_{t}, \quad v_{t}\sim N(0,\tfrac{1}{4}\sigma_{w}^{2}). \]

4.3 Análisis espectral muestral

En esta sección se presenta la conexión entre el periodograma (concepto muestral) y la densidad espectral (concepto poblacional). Se introduce la Transformada Discreta de Fourier (DFT), sus propiedades, su relación con el análisis de varianza (ANOVA) y con el Análisis de Componentes Principales (PCA). Finalmente, se discuten propiedades asintóticas del periodograma y la necesidad de suavizarlo para reducir su varianza.

4.3.1 Transformada Discreta de Fourier (DFT)

Definición

Dado un conjunto de datos \(x_{1}, \ldots, x_{n}\), la DFT se define como

\[ d\left(\omega_{j}\right)=n^{-1/2} \sum_{t=1}^{n} x_{t} e^{-2 \pi i \omega_{j} t}, \quad j=0,1,\ldots,n-1, \]

donde \(\omega_{j}=j/n\) son las frecuencias fundamentales.

La DFT se puede calcular rápidamente usando el algoritmo FFT (Fast Fourier Transform). En R, la función fft() calcula la DFT, aunque con diferencias de escala.

La DFT es reversible:

\[ x_{t}=n^{-1/2} \sum_{j=0}^{n-1} d(\omega_{j}) e^{2 \pi i \omega_{j} t}, \quad t=1,\ldots,n. \]

(dft <- fft(1:4)/sqrt(4))

[1]  5+0i -1+1i -1+0i -1-1i

(idft <- fft(dft, inverse=TRUE)/sqrt(4))

[1] 1+0i 2+0i 3+0i 4+0i

Re(idft)

[1] 1 2 3 4

4.3.2 Periodograma

Definición

El periodograma se define como

\[ I(\omega_{j}) = |d(\omega_{j})|^{2}, \quad j=0,1,\ldots,n-1. \]

• \(I(0)=n \bar{x}^{2}\).
• Para \(j \neq 0\) (ejercicio):

\[ d(\omega_{j}) = n^{-1/2}\sum_{t=1}^{n} (x_{t}-\bar{x}) e^{-2 \pi i \omega_{j} t}. \]

Así,

\[ \begin{aligned} I(\omega_j) &= \lvert d(\omega_j)\rvert^{2} = n^{-1}\sum_{t=1}^{n}\sum_{s=1}^{n} (x_t-\bar{x})(x_s-\bar{x})\,e^{-2\pi i \omega_j (t-s)}\\[4pt] &= n^{-1}\sum_{h=-(n-1)}^{n-1}\sum_{t=1}^{\,n-|h|} (x_{t+|h|}-\bar{x})(x_t-\bar{x})\,e^{-2\pi i \omega_j h}\\[4pt] &= \sum_{h=-(n-1)}^{n-1} \hat{\gamma}(h)\,e^{-2\pi i \omega_j h}. \end{aligned} \]

lo que lo convierte en un estimador muestral de la densidad espectral.

4.3.3 Transformadas seno y coseno

Definición

• Transformada coseno:

\[ d_{c}(\omega_{j})=n^{-1/2}\sum_{t=1}^{n} x_{t} \cos(2\pi \omega_{j}t), \]

• Transformada seno:

\[ d_{s}(\omega_{j})=n^{-1/2}\sum_{t=1}^{n} x_{t} \sin(2\pi \omega_{j}t). \]

Relación:

\[ d(\omega_{j}) = d_{c}(\omega_{j}) - i d_{s}(\omega_{j}), \]

y

\[ I(\omega_{j})=d_{c}^{2}(\omega_{j})+d_{s}^{2}(\omega_{j}). \]

4.3.4 Ejemplo: ANOVA espectral

Con \(n\) impar y \(m=(n-1)/2\):

\[ x_{t}=a_{0}+\sum_{j=1}^{m}\left[a_{j}\cos(2\pi \omega_{j}t)+b_{j}\sin(2\pi \omega_{j}t)\right], \]

donde usando regresión poblacional:

\[ a_{j}=\tfrac{2}{\sqrt{n}} d_{c}(\omega_{j}), \quad b_{j}=\tfrac{2}{\sqrt{n}} d_{s}(\omega_{j}). \]

Y además \(a_0=\bar x\). Por lo tanto, podemos escribir

\[ (x_t - \bar{x}) = \frac{2}{\sqrt{n}} \sum_{j=1}^{m} \left[ d_c(\omega_j)\cos(2\pi \omega_j t) + d_s(\omega_j)\sin(2\pi \omega_j t) \right] \]

para \(t=1,\ldots,n\). Elevando ambos lados al cuadrado y sumando obtenemos

\[ \sum_{t=1}^{n}(x_t - \bar{x})^2 = 2 \sum_{j=1}^{m} \left[ d_c^2(\omega_j) + d_s^2(\omega_j) \right] = 2 \sum_{j=1}^{m} I(\omega_j) \] lo que muestra que la varianza total se descompone en la suma de las varianzas de los componentes de frecuencia.

Tabla ANOVA:

Fuente	df	SS	MS
\(\omega_{1}\)	2	\(2I(\omega_{1})\)	\(I(\omega_{1})\)
\(\omega_{2}\)	2	\(2I(\omega_{2})\)	\(I(\omega_{2})\)
\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)
\(\omega_{m}\)	2	\(2I(\omega_{m})\)	\(I(\omega_{m})\)
Total	\(n-1\)	\(\sum (x_{t}-\bar{x})^{2}\)

x <- c(1, 2, 3, 2, 1)
c1 <- cos(2*pi*1:5*1/5); s1 <- sin(2*pi*1:5*1/5)
c2 <- cos(2*pi*1:5*2/5); s2 <- sin(2*pi*1:5*2/5)
omega1 <- cbind(c1, s1); omega2 <- cbind(c2, s2)
anova(lm(x ~ omega1 + omega2))

Warning in anova.lm(lm(x ~ omega1 + omega2)): ANOVA F-tests on an essentially
perfect fit are unreliable

Analysis of Variance Table

Response: x
          Df  Sum Sq Mean Sq F value Pr(>F)
omega1     2 2.74164 1.37082     NaN    NaN
omega2     2 0.05836 0.02918     NaN    NaN
Residuals  0 0.00000     NaN               

Mod(fft(x))^2/5

[1] 16.20000000  1.37082039  0.02917961  0.02917961  1.37082039

Nota: Análisis espectral como PCA

La densidad espectral puede verse como aproximación de los autovalores de la matriz de covarianzas \(\Gamma_{n}\). Los vectores propios aproximados corresponden a senos y cosenos, y los coeficientes DFT (\(y_{j}\)) son asintóticamente no correlacionados con varianza \(f(\omega_{j})\). Ver ejemplo 4.11 en el Shumway.

4.3.5 Propiedades asintóticas del periodograma

Para un proceso estacionario con \(\gamma(h)\) absolutamente sumable:

\[ E[I(\omega_{j})] \to f(\omega), \]

cuando \(n \to \infty\).

Además, bajo ciertas condiciones:

\[ \frac{2 I(\omega_{j:n})}{f(\omega_{j})} \xrightarrow{d} \chi^{2}_{2}. \]

donde \(\omega_{j:n}\rightarrow \omega_j\). De aquí se obtiene un intervalo de confianza aproximado para \(f(\omega)\):

\[ \frac{2 I(\omega_{j:n})}{\chi^{2}_{2}(1-\alpha/2)} \leq f(\omega) \leq \frac{2 I(\omega_{j:n})}{\chi^{2}_{2}(\alpha/2)}. \]

4.3.6 Ejemplo: Periodograma del SOI y Reclutamiento

Se utilizan \(n=453\) observaciones y se completan con ceros hasta \(n'=480\).

par(mfrow=c(2,1))
soi.per <- astsa::mvspec(soi, log="no")
abline(v=1/4, lty=2)
rec.per <- astsa::mvspec(rec, log="no")
abline(v=1/4, lty=2)

Cálculo de intervalos de confianza:

soi.per$spec[40] # frecuencia 1/12

[1] 0.9722312

soi.per$spec[10] # frecuencia 1/48

[1] 0.05372962

U <- qchisq(.025,2)
L <- qchisq(.975,2)
2*soi.per$spec[10]/L

[1] 0.0145653

2*soi.per$spec[10]/U

[1] 2.122207

2*soi.per$spec[40]/L

[1] 0.2635573

2*soi.per$spec[40]/U

[1] 38.40108

Nota:

El periodograma es un estimador no consistente de la densidad espectral, ya que su varianza no tiende a cero con \(n\):

\[E[I(\omega)]\approx f(\omega)\qquad \text{Var}[I(\omega)]\approx f^2(\omega)\] y \(\text{Var}[I(\omega)] \nrightarrow 0\) cuando \(n \to \infty\).

Por ello, en la práctica es necesario aplicar suavizado del periodograma para obtener estimaciones más útiles de la densidad espectral.

4.4 Estimación espectral no paramétrica

Esta sección introduce la estimación espectral no paramétrica mediante el promedio (suavizado) del periodograma sobre una banda de frecuencias cercana a una frecuencia de interés. Se define el ancho de banda y se derivan distribuciones aproximadas para construir intervalos de confianza tanto en escala lineal como logarítmica. Se discute el ajuste de grados de libertad al realizar zero-padding, el uso de kernels (Daniell y modificado) para promedios ponderados, y la técnica de tapering para reducir leakage.

4.4.1 Banda de frecuencias y definición operacional

Sea una banda de \(L \ll n\) frecuencias fundamentales contiguas, centrada en \(\omega_j=j/n\) y cercana a \(\omega\): \[ \mathcal{B}=\left\{\omega^{*}: \omega_j-\frac{m}{n}\le \omega^{*}\le \omega_j+\frac{m}{n}\right\},\qquad L=2m+1. \]

Si \(f(\omega)\) es aproximadamente constante en \(\mathcal{B}\), entonces para \(k=-m,\ldots,m\), \[ f\left(\omega_j+\frac{k}{n}\right)\approx f(\omega). \]

Promedio (suavizado) del periodograma: \[ \bar{f}(\omega)=\frac{1}{L}\sum_{k=-m}^{m} I\left(\omega_j+\frac{k}{n}\right). \]

Bajo supuestos apropiados y \(n\) grande, \[ \frac{2L\bar{f}(\omega)}{f(\omega)}\ \dot{\sim}\ \chi^2_{2L}. \]

4.4.2 Ancho de banda y grados de libertad

El ancho de banda (en frecuencia fundamental) es \[ \mathrm{B}=\frac{L}{n}, \qquad 2L=2\mathrm{B}n. \]

Intervalo de confianza al \(100(1-\alpha)%\): \[ \frac{2L\bar{f}(\omega)}{\chi^2_{2L}(1-\alpha/2)}\le f(\omega)\le \frac{2L\bar{f}(\omega)}{\chi^2_{2L}(\alpha/2)}. \]

En escala logarítmica, \[ \big[\log \bar{f}(\omega)-a_L,\ \log \bar{f}(\omega)+b_L\big], \] donde \(a_L=-\log(2L)+\log \chi^2_{2L}(1-\alpha/2)\) y \(b_L=\log(2L)-\log \chi^2_{2L}(\alpha/2)\).

Si hay padding a \(n'\), \[ df=\frac{2Ln}{n'}\quad\text{(grados de libertad ajustados)}, \] y \[ \frac{df\cdot \bar{f}(\omega)}{\chi^2_{df}(1-\alpha/2)}\le f(\omega)\le \frac{df\cdot\bar{f}(\omega)}{\chi^2_{df}(\alpha/2)}. \]

4.4.2.1 Ejemplo: Periodograma promediado para SOI y Recruitment

Se elige \(L=9\) (i.e., \(m=4\)) para suavizar la potencia en bajas frecuencias (≈ 4 años) y en el ciclo anual:

soi.ave = mvspec(soi, kernel('daniell',4), log='no')

Bandwidth: 0.225 | Degrees of Freedom: 16.99 | split taper: 0% 

abline(v=c(.25,1,2,3), lty=2)

soi.ave$bandwidth # = 0.225

[1] 0.225

# Repetir reemplazando 'soi' por 'rec' en la tercera línea

En escala \(\log_{10}\), se visualizan intervalos genéricos como en la figura descrita:

# (La visualización logarítmica se obtiene omitiendo log="no" en mvspec)
soi.ave = mvspec(soi, kernel('daniell',4),log="yes")

Bandwidth: 0.225 | Degrees of Freedom: 16.99 | split taper: 0% 

abline(v=c(.25,1,2,3), lty=2)

Cálculo de intervalos al 95% (SOI):

df = soi.ave$df                 # 16.9875
U  = qchisq(.025, df)           # 7.555916
L  = qchisq(.975, df)           # 30.17425
soi.ave$spec[10]                # 0.0495202  (ω = 1/48)

[1] 0.04952026

soi.ave$spec[40]                # 0.1190800  (ω = 1/12)

[1] 0.11908

# Intervalos
df*soi.ave$spec[10]/L           # 0.0278789

[1] 0.02787891

df*soi.ave$spec[10]/U           # 0.1113333

[1] 0.1113333

df*soi.ave$spec[40]/L           # 0.0670396

[1] 0.06703963

df*soi.ave$spec[40]/U           # 0.2677201

[1] 0.2677201

# Repetir con 'rec' en lugar de 'soi'

4.4.3 Promedio ponderado del periodograma (kernels) y grados de libertad efectivos

Un estimador más general pondera de forma desigual: \[ \hat{f}(\omega)=\sum_{k=-m}^{m} h_kI\left(\omega_j+\frac{k}{n}\right),\quad \sum h_k=1,\ h_k>0. \]

Condiciones asintóticas (con \(m\to\infty\), \(m/n\to 0\)):

• 1. \(\mathrm{E}\hat{f}(\omega)\to f(\omega)\),
• 2. \(\big(\sum h_k^2\big)^{-1}\operatorname{cov}(\hat{f}(\omega),\hat{f}(\lambda))\to f^2(\omega)\) para \(\omega=\lambda\neq 0,1/2\).

Grados de libertad efectivos: \[ L_h=\left(\sum_{k=-m}^{m} h_k^2\right)^{-1},\qquad \frac{2L_h\hat{f}(\omega)}{f(\omega)}\sim \chi^2_{2L_h}. \]

Ancho de banda efectivo: \[ \mathrm{B}=\frac{L_h}{n}. \]

IC aproximado: \[ \frac{2L_h \hat{f}(\omega)}{\chi^2_{2L_h}(1-\alpha/2)}\le f(\omega)\le \frac{2L_h \hat{f}(\omega)}{\chi^2_{2L_h}(\alpha/2)}. \]

4.4.4 Pesos Daniell y Daniell modificado

Aplicando el kernel de Daniell (\(m=1\), \(L=3\)): \(\{1/3,1/3,1/3\}\) produce \[ \hat{u}_t=\tfrac{1}{3}u_{t-1}+\tfrac{1}{3}u_t+\tfrac{1}{3}u_{t+1},\quad \hat{\hat{u}}_t=\tfrac{1}{9}u_{t-2}+\tfrac{2}{9}u_{t-1}+\tfrac{3}{9}u_t+\tfrac{2}{9}u_{t+1}+\tfrac{1}{9}u_{t+2}. \]

Daniell modificado (\(m=1\)): \(\{1/4,1/2,1/4\}\) (poniendo la mitad del peso en cada extremo) con \[ \begin{align*} \hat{u}_t&=\tfrac{1}{4}u_{t-1}+\tfrac{2}{4}u_t+\tfrac{1}{4}u_{t+1},\\ \hat{\hat{u}}_t&=\tfrac{1}{16}u_{t-2}+\tfrac{4}{16}u_{t-1}+\tfrac{6}{16}u_t+\tfrac{4}{16}u_{t+1}+\tfrac{1}{16}u_{t+2}. \end{align*} \]

kernel("modified.daniell", c(1,1))  # devuelve coeficientes 1/16,4/16,6/16,4/16,1/16 al aplicar dos veces

mDaniell(1,1) 
coef[-2] = 0.0625
coef[-1] = 0.2500
coef[ 0] = 0.3750
coef[ 1] = 0.2500
coef[ 2] = 0.0625

4.4.4.1 Ejemplo: Periodograma suavizado (SOI y Recruitment)

Kernel modificado de Daniell aplicado dos veces con \(m=3\); se obtiene \(L_h=1/\sum h_k^2=9.232\), \(\mathrm{B}=9.232/480=0.019\) y \(df=2L_h\cdot 453/480=17.43\).

kernel("modified.daniell", c(3,3))

mDaniell(3,3) 
coef[-6] = 0.006944
coef[-5] = 0.027778
coef[-4] = 0.055556
coef[-3] = 0.083333
coef[-2] = 0.111111
coef[-1] = 0.138889
coef[ 0] = 0.152778
coef[ 1] = 0.138889
coef[ 2] = 0.111111
coef[ 3] = 0.083333
coef[ 4] = 0.055556
coef[ 5] = 0.027778
coef[ 6] = 0.006944

plot(kernel("modified.daniell", c(3,3))) # no mostrado

k = kernel("modified.daniell", c(3,3))
soi.smo = mvspec(soi, kernel = k, taper = .1, log = "no")

Bandwidth: 0.231 | Degrees of Freedom: 15.61 | split taper: 10% 

abline(v = c(.25, 1), lty = 2)

## Repetir con 'rec' en lugar de 'soi'
df_calc <- soi.smo$df      # 17.42618
bw_calc <- soi.smo$bandwidth  # 0.2308103

4.4.5 Tapering (atenuación) y ventanas espectrales

Sea la serie atenuada \(y_t=h_t x_t\), \[ y_t=h_t x_t, \qquad d_y(\omega_j)=n^{-1/2}\sum_{t=1}^{n} h_t x_t e^{-2\pi i \omega_j t}. \]

Entonces \[ \mathrm{E}\left[I_y(\omega_j)\right]=\int_{-1/2}^{1/2} W_n(\omega_j-\omega)f_x(\omega),d\omega, \quad W_n(\omega)=|H_n(\omega)|^2, \] con \[ H_n(\omega)=n^{-1/2}\sum_{t=1}^{n} h_t e^{-2\pi i \omega t}. \]

A \(W_n(\omega)\) se le llama ventana espectral. Sin atenuación (\(h_t\equiv 1\)), el kernel de Fejér: \[ W_n(\omega)=\frac{\sin^2(n\pi \omega)}{n\sin^2(\pi \omega)},\quad W_n(0)=n. \]

Para el promedio simple \(\bar f\) (i.e., \(h_k=1/L\)), \[ W_n(\omega)=\frac{1}{nL}\sum_{k=-m}^{m}\frac{\sin^2\big[n\pi(\omega+k/n)\big]}{\sin^2\big[\pi(\omega+k/n)\big]}. \]

Taper de coseno (Blackman–Tukey): \[ h_t=.5\left[1+\cos!\left(\frac{2\pi (t-\bar{t})}{n}\right)\right],\quad \bar{t}=\frac{n+1}{2}. \]

4.4.5.1 Ejemplo: Efecto del tapering (SOI)

El tapering del 50% de cada extremo separa mejor los picos anual (\(\omega=1\)) y El Niño (\(\omega=1/4\)):

s0  = mvspec(soi, spans = c(7,7), plot = FALSE)         # sin taper
s50 = mvspec(soi, spans = c(7,7), taper = .5, plot = FALSE)  # taper completo
plot(s50$freq, s50$spec, log = "y", type = "l", ylab = "spectrum",
     xlab = "frequency")  # línea sólida
lines(s0$freq, s0$spec, lty = 2)    # línea discontinua

4.5 Estimación Espectral Paramétrica

Los métodos anteriores corresponden a estimadores no paramétricos, ya que no suponen una forma funcional específica para la densidad espectral. En cambio, un estimador espectral paramétrico se obtiene ajustando un modelo \(\operatorname{AR}(p)\) o \(\operatorname{ARMA}(p,q)\) y sustituyendo sus estimaciones en la fórmula de la densidad espectral:

\[ \hat{f}_{x}(\omega) = \frac{\hat{\sigma}_{w}^{2}}{\left|\hat{\phi}\left(e^{-2\pi i \omega}\right)\right|^{2}} \]

donde

\[ \hat{\phi}(z) = 1 - \hat{\phi}_{1}z - \hat{\phi}_{2}z^{2} - \cdots - \hat{\phi}_{p}z^{p} \]

El orden \(p\) se selecciona mediante criterios de información como AIC, AICc o BIC. Este tipo de estimadores presentan una resolución superior frente a los no paramétricos cuando existen picos espectrales muy próximos.

4.5.1 Distribución Asintótica y Intervalos de Confianza

Bajo ciertas condiciones asintóticas (\(p \to \infty\), \(p^3/n \to 0\)), la distribución de \(\hat{f}_x(\omega)\) permite construir intervalos de confianza aproximados:

\[ \frac{\hat{f}_{x}(\omega)}{(1 + C z_{\alpha / 2})} \leq f_{x}(\omega) \leq \frac{\hat{f}_{x}(\omega)}{(1 - C z_{\alpha / 2})} \]

donde \(C = \sqrt{2p/n}\) y \(z_{\alpha/2}\) es el cuantil superior de la distribución normal estándar. Cuando las condiciones asintóticas no se cumplen, puede utilizarse un bootstrap espectral para estimar la distribución muestral de \(\hat{f}_x(\omega)\), especialmente implementando el modelo en forma de espacio-estado.

4.5.2 Propiedad: Aproximación Espectral AR

Sea \(g(\omega)\) la densidad espectral de un proceso estacionario. Entonces, para cualquier \(\epsilon > 0\), existe un proceso autorregresivo

\[ x_t = \sum_{k=1}^{p} \phi_k x_{t-k} + w_t \]

tal que

\[ |f_x(\omega) - g(\omega)| < \epsilon \quad \forall \omega \in [-1/2, 1/2] \]

donde los ceros del polinomio \(\phi(z) = 1 - \sum_{k=1}^p \phi_k z^k\) están fuera del círculo unitario. Esto implica que cualquier espectro puede aproximarse arbitrariamente bien por un modelo AR(p), aunque no se sabe a priori qué tan grande debe ser \(p\).

4.5.2.1 Ejemplo: Estimador Espectral AR para la Serie SOI

Para estimar el espectro paramétrico del índice SOI, se ajustaron modelos \(\operatorname{AR}(p)\) para \(p=1,\dots,30\). Los criterios AIC, AICc y BIC alcanzaron su mínimo en \(p=15\).

spaic <- spec.ar(soi, log = "no")  # estimación espectral con mínimo AIC
abline(v = frequency(soi) * 1/52, lty = 3)  # marca del ciclo El Niño

soi.ar <- ar(soi, order.max = 30)  # estimaciones y AICs
dev.new()
plot(1:30, soi.ar$aic[-1], type = "o", xlab = "p", ylab = "AIC")

Los criterios de selección se pueden calcular directamente mediante:

n <- length(soi)
AIC <- rep(0, 30); AICc <- rep(0, 30); BIC <- rep(0, 30)

for (k in 1:30) {
  sigma2 <- ar(soi, order = k, aic = FALSE)$var.pred
  BIC[k]  <- log(sigma2) + (k * log(n) / n)
  AICc[k] <- log(sigma2) + ((n + k) / (n - k - 2))
  AIC[k]  <- log(sigma2) + ((n + 2 * k) / n)
}

IC <- cbind(AIC, BIC + 1)
ts.plot(IC, type = "o", xlab = "p", ylab = "AIC / BIC")

El modelo \(\operatorname{AR}(15)\) seleccionado reproduce los picos de las periodicidades anual y cuatrienal observadas en las estimaciones no paramétricas, incluyendo los armónicos del ciclo anual.

4.6 Espectro Cruzado y Función de Coherencia

Para dos series estacionarias \(x_t\) y \(y_t\), la función de covarianza cruzada está dada por:

\[ \gamma_{xy}(h) = \mathrm{E}[(x_{t+h} - \mu_x)(y_t - \mu_y)] \]

La representación espectral de esta función es:

\[ \gamma_{xy}(h) = \int_{-\frac{1}{2}}^{\frac{1}{2}} f_{xy}(\omega) e^{2\pi i \omega h} d\omega, \quad h = 0, \pm 1, \pm 2, \ldots \]

donde el espectro cruzado se define como la transformada de Fourier:

\[ f_{xy}(\omega) = \sum_{h=-\infty}^{\infty} \gamma_{xy}(h) e^{-2\pi i \omega h}, \quad -\tfrac{1}{2} \leq \omega \leq \tfrac{1}{2}. \]

El espectro cruzado puede expresarse como una función compleja:

\[ f_{xy}(\omega) = c_{xy}(\omega) - i q_{xy}(\omega) \]

donde

• \(c_{xy}(\omega)\) es el cospectro:

\[ c_{xy}(\omega) = \sum_{h=-\infty}^{\infty} \gamma_{xy}(h) \cos(2\pi \omega h) \]

• \(q_{xy}(\omega)\) es el cuadrespectro:

\[ q_{xy}(\omega) = \sum_{h=-\infty}^{\infty} \gamma_{xy}(h) \sin(2\pi \omega h) \]

Estas funciones cumplen las simetrías:

\[ f_{yx}(\omega) = f_{xy}^*(\omega), \quad c_{yx}(\omega) = c_{xy}(\omega), \quad q_{yx}(\omega) = -q_{xy}(\omega) \]

donde \(*\) significa conjugación compleja. La coherencia cuadrada mide la intensidad de la relación lineal entre las dos series en cada frecuencia:

\[ \rho_{y \cdot x}^2(\omega) = \frac{|f_{yx}(\omega)|^2}{f_{xx}(\omega) f_{yy}(\omega)} \]

Esta expresión es análoga al coeficiente de correlación clásico:

\[ \rho_{yx}^2 = \frac{\sigma_{xy}^2}{\sigma_x^2 \sigma_y^2} \]

4.6.0.1 Ejemplo: Media Móvil de Tres Puntos

Sea \(y_t = (x_{t-1} + x_t + x_{t+1}) / 3\) y \(x_t\) un proceso estacionario con densidad espectral \(f_{xx}(\omega)\). Entonces:

\[ \begin{aligned} \gamma_{xy}(h) &= \operatorname{cov}(x_{t+h}, y_t) = \tfrac{1}{3}\,\operatorname{cov}\!\left(x_{t+h},\, x_{t-1}+x_t+x_{t+1}\right)\\[2mm] &= \tfrac{1}{3}\Big[ \gamma_{xx}(h+1) + \gamma_{xx}(h) + \gamma_{xx}(h-1) \Big]\\[2mm] &= \tfrac{1}{3}\int_{-1/2}^{1/2} \Big( e^{2\pi i \omega} + 1 + e^{-2\pi i \omega} \Big) e^{2\pi i \omega h}\, f_{xx}(\omega)\, d\omega\\[2mm] &= \tfrac{1}{3}\int_{-1/2}^{1/2} \Big[ 1 + 2\cos(2\pi \omega) \Big]\, f_{xx}(\omega)\, e^{2\pi i \omega h}\, d\omega . \end{aligned} \]

Aplicando la representación espectral:

\[ f_{xy}(\omega) = \tfrac{1}{3}[1 + 2\cos(2\pi \omega)] f_{xx}(\omega) \]

y la densidad espectral de \(y_t\) es:

\[ f_{yy}(\omega) = \tfrac{1}{9}[1 + 2\cos(2\pi \omega)]^2 f_{xx}(\omega) \]

Al sustituir en la coherencia cuadrada se obtiene:

\[ \rho_{y \cdot x}^2(\omega) = 1 \]

Esto indica coherencia perfecta entre \(x_t\) y \(y_t\) para todas las frecuencias. Si se añade ruido al promedio móvil, la coherencia deja de ser igual a 1.

4.6.0.2 Propiedad: Representación Espectral de un Proceso Vectorial

Si \(x_t = (x_{t1}, x_{t2}, \ldots, x_{tp})'\) es un proceso estacionario con matriz de autocovarianza \(\Gamma(h) = {\gamma_{jk}(h)}\) que satisface:

\[ \sum_{h=-\infty}^{\infty} |\gamma_{jk}(h)| < \infty \]

entonces tiene la representación espectral:

\[ \Gamma(h) = \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2\pi i \omega h} f(\omega) d\omega \]

donde la matriz espectral \(f(\omega) = {f_{jk}(\omega)}\) se define por:

\[ f(\omega) = \sum_{h=-\infty}^{\infty} \Gamma(h) e^{-2\pi i \omega h} \]

y es hermitiana, es decir, \(f(\omega) = f^*(\omega)\).

4.6.0.3 Ejemplo: Matriz Espectral Bivariante

Para un proceso bivariante estacionario \((x_t, y_t)\):

\[ \Gamma(h) = \begin{pmatrix} \gamma_{xx}(h) & \gamma_{xy}(h) \\ \gamma_{yx}(h) & \gamma_{yy}(h) \end{pmatrix}, \quad f(\omega) = \begin{pmatrix} f_{xx}(\omega) & f_{xy}(\omega) \\ f_{yx}(\omega) & f_{yy}(\omega) \end{pmatrix} \]

El cálculo de la matriz espectral se realiza mediante el vector de transformadas de Fourier discretas (DFT):

\[ d(\omega_j) = (d_1(\omega_j), \ldots, d_p(\omega_j))' \]

y la estimación suavizada del espectro:

\[ \bar{f}(\omega) = L^{-1} \sum_{k=-m}^{m} I(\omega_j + k/n) \]

con

\[ I(\omega_j) = d(\omega_j) d^*(\omega_j) \]

Una versión ponderada es:

\[ \hat{f}(\omega) = \sum_{k=-m}^{m} h_k I(\omega_j + k/n) \]

donde \({h_k}\) son los pesos definidos anteriormente.

La coherencia estimada entre \(y_t\) y \(x_t\) es:

\[ \hat{\rho}_{y \cdot x}^2(\omega) = \frac{|\hat{f}_{yx}(\omega)|^2}{\hat{f}_{xx}(\omega) \hat{f}_{yy}(\omega)} \]

Si se usan pesos iguales, se denota por \(\bar{\rho}_{y \cdot x}^2(\omega)\).

4.6.0.4 Intervalos y Pruebas de Coherencia

Bajo condiciones generales, si \(\rho_{y \cdot x}^2(\omega) > 0\):

\[ |\hat{\rho}_{y \cdot x}(\omega)| \sim AN(|\rho_{y \cdot x}(\omega)|, (1 - \rho_{y \cdot x}^2(\omega))^2 / 2L_h) \]

Esto permite construir intervalos de confianza aproximados para \(\rho_{y \cdot x}^2(\omega)\).

Para probar \(H_0: \rho_{y \cdot x}^2(\omega) = 0\), se utiliza:

\[ \bar{\rho}_{y \cdot x}^2(\omega) = \frac{|\bar{f}_{yx}(\omega)|^2}{\bar{f}_{xx}(\omega)\bar{f}_{yy}(\omega)} \]

y el estadístico

\[ F = \frac{\bar{\rho}_{y \cdot x}^2(\omega)}{(1 - \bar{\rho}_{y \cdot x}^2(\omega))}(L - 1) \]

que sigue una distribución \(F_{2, 2L - 2}\) aproximada. El valor crítico para un nivel de significancia \(\alpha\) es:

\[ C_\alpha = \frac{F_{2, 2L - 2}(\alpha)}{L - 1 + F_{2, 2L - 2}(\alpha)} \]

4.6.0.5 Ejemplo: Coherencia entre SOI y Reclutamiento

Se calcula la coherencia cuadrada entre las series SOI y Recruitment. Con \(L = 19\), \(n = 453\), \(n' = 480\), \(\alpha = 0.001\), se obtiene \(df \approx 36\) y:

\[ C_{.001} = 0.32 \]

Esto permite rechazar \(H_0\) para valores superiores a 0.32.

La coherencia es fuerte en la frecuencia anual y también en frecuencias más bajas asociadas al ciclo El Niño (3–7 años), con un máximo cercano a un ciclo de 9 años. Otras frecuencias muestran coherencia moderada, mientras que los armónicos estacionales presentan persistencia.

sr <- mvspec(cbind(soi, rec), kernel("daniell", 9), plot = FALSE)
sr$df  # df = 35.8625

[1] 35.8625

f <- qf(.999, 2, sr$df - 2)  # = 8.529792
C <- f / (18 + f)  # = 0.321517
plot(sr, plot.type = "coh", ci.lty = 2)
abline(h = C)

4.7 Modelos de regresión con rezagos

La regresión rezagada se define como:

\[ y_{t} = \sum_{r=-\infty}^{\infty} \beta_{r} x_{t-r} + v_{t}, \]

donde \(v_t\) es un proceso estacionario de ruido, \(x_t\) es la serie de entrada y \(y_t\) la serie de salida. El objetivo es estimar los coeficientes del filtro \(\beta_r\).

Se asume que las series tienen medias cero y son conjuntamente estacionarias, con matriz espectral:

\[ f(\omega)= \begin{pmatrix} f_{xx}(\omega) & f_{xy}(\omega) \\ f_{yx}(\omega) & f_{yy}(\omega) \end{pmatrix}. \]

4.7.1 Condiciones de ortogonalidad y ecuaciones normales

Minimizando el error cuadrático medio,

\[ MSE = \mathrm{E}\left(y_t - \sum_{r=-\infty}^{\infty} \beta_r x_{t-r}\right)^2, \]

se obtienen las condiciones de ortogonalidad:

\[ \mathrm{E}\left[\left(y_t - \sum_{r=-\infty}^{\infty} \beta_r x_{t-r}\right)x_{t-s}\right] = 0, \]

para todo \(s=0, \pm1, \pm2, \ldots\), lo que lleva al sistema de ecuaciones normales:

\[ \sum_{r=-\infty}^{\infty} \beta_r \gamma_{xx}(s-r) = \gamma_{yx}(s). \]

4.7.2 Solución en el dominio de la frecuencia

Para evitar la inversión de matrices grandes, se pasa al dominio de la frecuencia. Definiendo la transformada de Fourier de los coeficientes:

\[ B(\omega) = \sum_{r=-\infty}^{\infty} \beta_r e^{-2\pi i \omega r}, \] y se obtiene: \[ \int_{-1/2}^{1/2} \sum_{r=-\infty}^{\infty} \beta_r \, e^{2\pi i \omega (s - r)} f_{xx}(\omega) \, d\omega = \int_{-1/2}^{1/2} e^{2\pi i \omega s} B(\omega) f_{xx}(\omega) \, d\omega , \]

de donde se obtiene la relación po unicidad de la densidad espectral:

\[ B(\omega) f_{xx}(\omega) = f_{yx}(\omega), \]

y el estimador correspondiente:

\[ \hat{B}(\omega_k) = \frac{\hat{f}_{yx}(\omega_k)}{\hat{f}_{xx}(\omega_k)}. \]

La transformada inversa da los coeficientes estimados:

\[ \hat{\beta}_t = M^{-1} \sum_{k=0}^{M-1} \hat{B}(\omega_k) e^{2\pi i \omega_k t}, \]

definiendo \(\hat{\beta}_t = 0\) para \(|t| \geq M/2\).

4.7.2.1 Ejemplo: regresión rezagada entre SOI y Recruitment

Se analiza la relación entre las series SOI (entrada) y Recruitment (salida). El modelo básico es:

\[ y_t = \sum_{r=-\infty}^{\infty} a_r x_{t-r} + w_t, \]

y el modelo inverso:

\[ x_t = \sum_{r=-\infty}^{\infty} b_r y_{t-r} + v_t. \]

LagReg(soi, rec, L=15, M=32, threshold=6)

INPUT: soi OUTPUT: rec   L = 15    M = 32 

The coefficients beta(0), beta(1), beta(2) ... beta(M/2-1) are 

4.03743 2.103372 3.31812 0.01247538 0.005194443 -18.90914 -12.60978 -8.746491 
-6.670373 -4.404543 -3.748336 -3.760936 -2.991477 -1.355261 1.375379 3.955252 



The coefficients beta(0), beta(-1), beta(-2) ... beta(-M/2+1) are 

4.03743 2.987159 1.409949 2.788212 1.017324 -0.5528797 0.402843 1.389537 
4.426287 5.563582 6.315986 4.540402 3.703423 2.840445 3.798354 2.974338 

The positive lags, at which the coefficients are large
in absolute value, and the coefficients themselves, are: 
     lag s    beta(s)
[1,]     5 -18.909140
[2,]     6 -12.609781
[3,]     7  -8.746491
[4,]     8  -6.670373

The prediction equation is
rec(t) = alpha + sum_s[ beta(s)*soi(t-s) ], where alpha = 66.01941
MSE = 411.5948

$betas
        S             b
 [1,] -15   2.974338364
 [2,] -14   3.798354349
 [3,] -13   2.840445310
 [4,] -12   3.703422986
 [5,] -11   4.540402464
 [6,] -10   6.315985671
 [7,]  -9   5.563581967
 [8,]  -8   4.426286804
 [9,]  -7   1.389536617
[10,]  -6   0.402842994
[11,]  -5  -0.552879696
[12,]  -4   1.017323713
[13,]  -3   2.788212457
[14,]  -2   1.409949212
[15,]  -1   2.987158959
[16,]   0   4.037429653
[17,]   1   2.103372494
[18,]   2   3.318120131
[19,]   3   0.012475378
[20,]   4   0.005194443
[21,]   5 -18.909139762
[22,]   6 -12.609781033
[23,]   7  -8.746490655
[24,]   8  -6.670373374
[25,]   9  -4.404543150
[26,]  10  -3.748336347
[27,]  11  -3.760935651
[28,]  12  -2.991477145
[29,]  13  -1.355261381
[30,]  14   1.375378996
[31,]  15   3.955252475

$fit
             output      fit      resids
Oct 1950  50.910000 55.46485  -4.5548459
Nov 1950  44.700000 60.64946 -15.9494644
Dec 1950  42.850000 58.79339 -15.9433933
Jan 1951  39.620000 59.91178 -20.2917801
Feb 1951  44.450000 58.73152 -14.2815208
Mar 1951  38.980000 61.14768 -22.1676769
Apr 1951  42.620000 58.15309 -15.5330909
May 1951  48.270000 60.50992 -12.2399245
Jun 1951  59.390000 63.41302  -4.0230240
Jul 1951  51.660000 66.94203 -15.2820315
Aug 1951  38.550000 60.97748 -22.4274757
Sep 1951  60.330000 51.41505   8.9149504
Oct 1951  72.270000 62.93072   9.3392833
Nov 1951  68.620000 66.14905   2.4709533
Dec 1951  69.630000 62.27760   7.3523955
Jan 1952  72.200000 64.82379   7.3762147
Feb 1952  67.870000 63.19276   4.6772429
Mar 1952  64.910010 58.42621   6.4838004
Apr 1952  53.850000 58.24125  -4.3912535
May 1952  37.960000 52.97174 -15.0117449
Jun 1952  23.230000 45.49734 -22.2673382
Jul 1952  12.680000 38.74233 -26.0623253
Aug 1952   9.840000 31.30974 -21.4697427
Sep 1952   7.820000 32.54329 -24.7232898
Oct 1952  11.780000 34.64882 -22.8688156
Nov 1952  10.220000 47.43442 -37.2144233
Dec 1952  12.190000 47.90511 -35.7151132
Jan 1953  18.600000 52.33759 -33.7375868
Feb 1953  26.970000 59.79484 -32.8248419
Mar 1953  22.520000 62.14320 -39.6231987
Apr 1953  19.180000 51.84837 -32.6683732
May 1953  17.140000 41.69523 -24.5552285
Jun 1953  18.610000 32.62057 -14.0105722
Jul 1953  20.020000 32.67419 -12.6541933
Aug 1953  22.650000 35.94482 -13.2948225
Sep 1953  38.990000 41.62853  -2.6385334
Oct 1953  76.550000 57.53823  19.0117717
Nov 1953  87.990000 73.21183  14.7781738
Dec 1953  99.800000 76.34047  23.4595326
Jan 1954  96.690000 84.99525  11.6947548
Feb 1954  87.450000 86.95297   0.4970327
Mar 1954  88.570000 86.08340   2.4865996
Apr 1954  97.430000 78.27266  19.1573395
May 1954  99.990000 63.78653  36.2034743
Jun 1954  94.880000 57.96131  36.9186863
Jul 1954  86.990000 50.72819  36.2618103
Aug 1954  79.730010 51.33755  28.3924593
Sep 1954  92.350000 55.09592  37.2540850
Oct 1954  91.290000 67.62784  23.6621604
Nov 1954  94.310000 70.43674  23.8732630
Dec 1954  84.950000 75.05103   9.8989664
Jan 1955  82.970000 70.10621  12.8637920
Feb 1955  92.980010 68.12601  24.8539970
Mar 1955  81.060000 74.72645   6.3335460
Apr 1955  62.370000 66.07793  -3.7079310
May 1955  52.990000 60.02543  -7.0354343
Jun 1955  39.530000 58.19063 -18.6606334
Jul 1955  42.900000 52.19873  -9.2987326
Aug 1955  33.760000 58.19204 -24.4320407
Sep 1955  40.970000 56.65872 -15.6887187
Oct 1955  60.500000 62.78288  -2.2828791
Nov 1955  66.610000 72.24598  -5.6359753
Dec 1955  80.380000 71.97679   8.4032137
Jan 1956  95.860000 77.20286  18.6571357
Feb 1956  97.740000 82.45199  15.2880051
Mar 1956  80.240000 78.13810   2.1019018
Apr 1956  73.440000 60.05058  13.3894225
May 1956  65.670000 54.23271  11.4372871
Jun 1956  47.810000 48.94241  -1.1324118
Jul 1956  33.510000 40.98933  -7.4793340
Aug 1956  34.220000 40.30554  -6.0855361
Sep 1956  32.950000 47.59134 -14.6413361
Oct 1956  32.550000 51.40639 -18.8563899
Nov 1956  46.920000 56.00675  -9.0867484
Dec 1956  44.640000 66.83289 -22.1928908
Jan 1957  53.020000 63.73891 -10.7189126
Feb 1957  41.980000 66.98091 -25.0009097
Mar 1957  30.430000 58.44640 -28.0164031
Apr 1957  24.430000 47.63295 -23.2029500
May 1957  18.050000 43.15767 -25.1076701
Jun 1957  20.980000 35.58360 -14.6035957
Jul 1957  12.370000 43.48776 -31.1177571
Aug 1957  12.030000 34.17513 -22.1451338
Sep 1957  12.410000 37.55309 -25.1430935
Oct 1957  15.890000 42.11777 -26.2277729
Nov 1957  20.460000 46.94058 -26.4805837
Dec 1957  26.950000 54.81665 -27.8666462
Jan 1958  30.290000 59.17923 -28.8892264
Feb 1958  26.210000 58.45519 -32.2451937
Mar 1958  23.340000 47.59720 -24.2571980
Apr 1958  25.550000 38.97749 -13.4274891
May 1958  25.400000 39.83718 -14.4371818
Jun 1958  24.160000 37.09758 -12.9375812
Jul 1958  23.340000 35.86964 -12.5296450
Aug 1958  24.380000 36.84176 -12.4617579
Sep 1958  27.200000 39.09511 -11.8951100
Oct 1958  29.180000 44.12220 -14.9422020
Nov 1958  43.300000 47.15174  -3.8517439
Dec 1958  53.920000 58.78207  -4.8620695
Jan 1959  59.760000 63.55081  -3.7908082
Feb 1959  64.520000 63.33799   1.1820106
Mar 1959  65.840000 63.36501   2.4749855
Apr 1959  70.190000 58.67984  11.5101554
May 1959  75.270000 58.63878  16.6312212
Jun 1959  77.630000 60.68904  16.9409556
Jul 1959  76.960000 61.26955  15.6904477
Aug 1959  77.700000 61.08739  16.6126085
Sep 1959  85.130000 62.04792  23.0820766
Oct 1959  99.339990 67.03301  32.3069766
Nov 1959  97.700000 78.39372  19.3062829
Dec 1959  97.019990 86.00575  11.0142430
Jan 1960  98.830000 85.04694  13.7830617
Feb 1960  99.800000 80.32299  19.4770092
Mar 1960  96.740000 75.19689  21.5431066
Apr 1960  93.540000 64.93355  28.6064498
May 1960  82.080000 64.30701  17.7729939
Jun 1960  71.510000 60.21930  11.2906960
Jul 1960  49.610000 58.32862  -8.7186167
Aug 1960  41.760000 53.79374 -12.0337424
Sep 1960  59.780000 53.78001   5.9999864
Oct 1960  90.970000 65.48720  25.4828019
Nov 1960  85.260000 78.48513   6.7748654
Dec 1960 100.000000 74.44972  25.5502788
Jan 1961  98.500000 86.60843  11.8915728
Feb 1961  98.700000 75.84515  22.8548505
Mar 1961  93.100000 71.55385  21.5461492
Apr 1961  79.370000 65.39797  13.9720278
May 1961  81.970000 54.15874  27.8112631
Jun 1961  55.940000 60.71860  -4.7786048
Jul 1961  50.390000 50.21346   0.1765429
Aug 1961  48.640000 52.98966  -4.3496573
Sep 1961  40.120000 57.79170 -17.6716953
Oct 1961  63.040000 54.84918   8.1908220
Nov 1961  60.510000 69.26448  -8.7544800
Dec 1961  78.720000 67.70705  11.0129504
Jan 1962  71.370000 74.73714  -3.3671373
Feb 1962  76.430000 69.44917   6.9808335
Mar 1962  71.250000 68.43220   2.8178035
Apr 1962  56.460000 64.54049  -8.0804911
May 1962  41.840000 54.19132 -12.3513158
Jun 1962  41.240000 48.61505  -7.3750538
Jul 1962  35.280000 50.43454 -15.1545377
Aug 1962  39.860000 49.59512  -9.7351191
Sep 1962  45.130000 56.75469 -11.6246914
Oct 1962  53.800000 62.76113  -8.9611275
Nov 1962  77.540000 66.70363  10.8363728
Dec 1962  80.020000 76.35071   3.6692944
Jan 1963  81.280000 73.26362   8.0163830
Feb 1963  73.580000 70.42578   3.1542221
Mar 1963  66.060000 62.47712   3.5828826
Apr 1963  59.460000 54.97817   4.4818252
May 1963  59.490000 51.94551   7.5444920
Jun 1963  51.900000 54.18085  -2.2808515
Jul 1963  35.210000 53.31176 -18.1017632
Aug 1963  39.650000 47.52746  -7.8774617
Sep 1963  31.900000 54.92218 -23.0221791
Oct 1963  61.560000 51.33222  10.2277839
Nov 1963  88.250000 66.71338  21.5366193
Dec 1963  96.460000 79.43603  17.0239693
Jan 1964  83.940000 91.05916  -7.1191601
Feb 1964  89.050000 96.32506  -7.2750584
Mar 1964  92.970000 83.58682   9.3831780
Apr 1964  98.290000 74.85636  23.4336412
May 1964  99.790000 64.33922  35.4507829
Jun 1964  94.010000 56.89794  37.1120584
Jul 1964  87.060000 53.97972  33.0802765
Aug 1964  80.460000 54.62098  25.8390234
Sep 1964  74.860000 56.68969  18.1703106
Oct 1964  67.180000 59.15780   8.0222046
Nov 1964  73.850000 60.30013  13.5498729
Dec 1964  80.150000 66.90054  13.2494591
Jan 1965  69.450000 71.49991  -2.0499086
Feb 1965  50.390000 66.80960 -16.4195952
Mar 1965  31.680000 60.02782 -28.3478214
Apr 1965  31.250000 51.40235 -20.1523480
May 1965  23.110000 52.57145 -29.4614464
Jun 1965  11.320000 49.55029 -38.2302926
Jul 1965   8.960000 42.33047 -33.3704688
Aug 1965   6.030000 43.91011 -37.8801103
Sep 1965  11.700000 39.97366 -28.2736623
Oct 1965  34.630000 52.29710 -17.6670975
Nov 1965  58.310000 68.76890 -10.4589039
Dec 1965  58.660000 76.49923 -17.8392318
Jan 1966  72.620000 70.87028   1.7497206
Feb 1966  85.760000 72.37486  13.3851425
Mar 1966  94.290000 72.34878  21.9412232
Apr 1966  92.769990 72.00743  20.7625583
May 1966  93.180000 64.28942  28.8905761
Jun 1966  89.320000 61.97434  27.3456640
Jul 1966  81.630000 56.52645  25.1035547
Aug 1966  71.440000 51.14423  20.2957709
Sep 1966  66.420000 50.22944  16.1905611
Oct 1966  80.020000 52.25504  27.7649589
Nov 1966  76.520000 64.06465  12.4553545
Dec 1966  77.510000 65.12944  12.3805582
Jan 1967  67.730010 68.66399  -0.9339802
Feb 1967  50.520000 65.23442 -14.7144184
Mar 1967  48.970000 56.45655  -7.4865548
Apr 1967  50.640000 58.10594  -7.4659441
May 1967  38.730000 59.98826 -21.2582594
Jun 1967  30.790000 55.56071 -24.7707091
Jul 1967  23.750000 54.44341 -30.6934052
Aug 1967  26.280000 51.45704 -25.1770356
Sep 1967  36.670000 54.56903 -17.8990314
Oct 1967  68.910010 62.59309   6.3169159
Nov 1967  97.390000 75.10586  22.2841358
Dec 1967  96.100000 86.66459   9.4354112
Jan 1968  90.300000 93.33260  -3.0325991
Feb 1968  84.920000 90.71916  -5.7991554
Mar 1968  91.410010 86.57854   4.8314703
Apr 1968  92.540000 73.55996  18.9800385
May 1968  98.040000 70.78208  27.2579162
Jun 1968  99.960000 66.29868  33.6613200
Jul 1968  88.830000 62.56550  26.2645049
Aug 1968  83.070000 55.91869  27.1513051
Sep 1968  86.320000 56.64910  29.6709002
Oct 1968  99.830000 63.28617  36.5438309
Nov 1968  96.620000 76.99142  19.6285800
Dec 1968  99.940000 73.40728  26.5327224
Jan 1969  96.890000 80.97614  15.9138606
Feb 1969  85.120000 73.96207  11.1579347
Mar 1969  77.970000 63.12812  14.8418782
Apr 1969  67.380000 62.96310   4.4168958
May 1969  44.500000 58.24373 -13.7437278
Jun 1969  26.720000 51.99713 -25.2771346
Jul 1969  13.250000 47.73983 -34.4898286
Aug 1969  10.640000 41.48203 -30.8420302
Sep 1969  23.830000 43.16753 -19.3375285
Oct 1969  29.180000 58.73095 -29.5509494
Nov 1969  26.910000 63.56160 -36.6515968
Dec 1969  20.090000 62.69574 -42.6057399
Jan 1970  22.330000 56.49324 -34.1632404
Feb 1970  22.070000 54.89346 -32.8234565
Mar 1970  26.200000 51.77328 -25.5732771
Apr 1970  29.810000 54.50984 -24.6998382
May 1970  30.100000 57.05635 -26.9563467
Jun 1970  24.250000 54.17211 -29.9221057
Jul 1970  25.300000 46.09741 -20.7974071
Aug 1970  23.500000 45.96089 -22.4608900
Sep 1970  35.620000 41.81616  -6.1961613
Oct 1970  52.110000 53.02520  -0.9151953
Nov 1970  56.790000 64.00446  -7.2144572
Dec 1970  69.090000 63.10663   5.9833741
Jan 1971  86.640000 69.23918  17.4008206
Feb 1971  99.280000 74.23929  25.0407088
Mar 1971  98.480010 80.07786  18.4021521
Apr 1971  98.450000 70.99879  27.4512143
May 1971  94.769990 67.68855  27.0814360
Jun 1971  93.580000 59.22094  34.3590614
Jul 1971  78.070000 57.41414  20.6558623
Aug 1971  66.880000 50.47958  16.4004173
Sep 1971  77.040000 48.98345  28.0565479
Oct 1971  88.720000 60.83513  27.8848738
Nov 1971  94.880000 69.45507  25.4249333
Dec 1971  99.670000 77.15790  22.5120971
Jan 1972 100.000000 87.65982  12.3401757
Feb 1972  99.900000 81.49784  18.4021630
Mar 1972  96.910010 77.76249  19.1475151
Apr 1972  66.880000 70.72156  -3.8415611
May 1972  52.390000 53.05695  -0.6669514
Jun 1972  40.610000 49.49276  -8.8827581
Jul 1972  30.650000 47.13399 -16.4839892
Aug 1972  32.040000 46.29102 -14.2510178
Sep 1972  45.280000 54.82298  -9.5429841
Oct 1972  35.000000 64.96531 -29.9653113
Nov 1972  35.620000 60.53636 -24.9163608
Dec 1972  36.980000 62.49909 -25.5190922
Jan 1973  39.890000 62.40614 -22.5161444
Feb 1973  36.880000 60.98578 -24.1057829
Mar 1973  30.850000 58.93099 -28.0809866
Apr 1973  19.330000 53.60748 -34.2774766
May 1973  13.260000 43.40637 -30.1463685
Jun 1973  11.120000 35.65590 -24.5359025
Jul 1973   9.140001 33.85179 -24.7117843
Aug 1973   8.210000 31.98578 -23.7757770
Sep 1973  10.760000 33.97812 -23.2181227
Oct 1973  10.430000 43.81721 -33.3872135
Nov 1973  13.750000 43.32031 -29.5703101
Dec 1973  37.910000 50.07300 -12.1630025
Jan 1974  41.850000 66.61380 -24.7638047
Feb 1974  44.670000 61.48503 -16.8150280
Mar 1974  50.570000 58.64624  -8.0762367
Apr 1974  50.340000 58.37807  -8.0380697
May 1974  49.540000 48.06835   1.4716535
Jun 1974  56.930000 43.25709  13.6729062
Jul 1974  60.160000 50.09714  10.0628579
Aug 1974  57.470000 51.39371   6.0762896
Sep 1974  71.680000 50.00081  21.6791889
Oct 1974  97.280000 61.94669  35.3333056
Nov 1974  62.090000 81.71778 -19.6277797
Dec 1974  59.970000 96.10361 -36.1336050
Jan 1975  51.180000 96.67658 -45.4965812
Feb 1975  51.480000 99.57852 -48.0985154
Mar 1975  66.080000 94.68659 -28.6065885
Apr 1975  86.390000 83.87118   2.5188176
May 1975  93.580000 75.19548  18.3845205
Jun 1975  99.900000 70.70677  29.1932283
Jul 1975  93.860000 60.67588  33.1841220
Aug 1975  82.820000 55.08542  27.7345771
Sep 1975  84.839990 51.69383  33.1461617
Oct 1975  89.510000 56.93512  32.5748794
Nov 1975  86.890000 64.83402  22.0559820
Dec 1975  87.150000 66.40872  20.7412796
Jan 1976  78.470000 70.29792   8.1720785
Feb 1976  55.930000 66.18266 -10.2526614
Mar 1976  41.270000 56.67961 -15.4096109
Apr 1976  19.660000 52.52664 -32.8666427
May 1976   9.439999 42.28712 -32.8471187
Jun 1976   4.660000 35.73136 -31.0713619
Jul 1976   2.360000 32.08282 -29.7228202
Aug 1976   1.720000 28.42818 -26.7081774
Sep 1976   3.320000 31.02707 -27.7070685
Oct 1976  12.130000 45.06013 -32.9301264
Nov 1976  16.810000 61.61545 -44.8054510
Dec 1976  24.300000 64.91256 -40.6125616
Jan 1977  52.420000 68.38652 -15.9665208
Feb 1977  58.050000 76.48756 -18.4375600
Mar 1977  59.420000 67.70274  -8.2827366
Apr 1977  57.520000 59.44645  -1.9264479
May 1977  60.130000 49.01481  11.1151923
Jun 1977  64.680000 45.01765  19.6623462
Jul 1977  74.940000 47.82771  27.1122915
Aug 1977  69.730010 56.86515  12.8648561
Sep 1977  77.110000 53.56674  23.5432561
Oct 1977  97.930000 60.82620  37.1038048
Nov 1977  98.740000 75.18611  23.5538913
Dec 1977  98.880000 73.10698  25.7730244
Jan 1978  90.410010 74.47822  15.9317903
Feb 1978  77.860000 63.67500  14.1850041
Mar 1978  61.480000 53.85258   7.6274219
Apr 1978  47.660000 48.00796  -0.3479581
May 1978  30.740000 44.60372 -13.8637213
Jun 1978  20.110000 41.86948 -21.7594849
Jul 1978  12.080000 40.78005 -28.7000488
Aug 1978   8.970000 39.33970 -30.3697009
Sep 1978  20.030000 40.78782 -20.7578225
Oct 1978  71.540000 56.86895  14.6710475
Nov 1978  97.519990 76.26770  21.2522908
Dec 1978  95.140000 87.55752   7.5824815
Jan 1979  92.220000 96.08129  -3.8612867
Feb 1979  80.090000 88.18255  -8.0925524
Mar 1979  74.590000 86.92659 -12.3365939
Apr 1979  83.660010 82.34950   1.3105091
May 1979  87.360000 70.89798  16.4620199
Jun 1979  96.630000 72.95469  23.6753110
Jul 1979  93.360000 67.17879  26.1812130
Aug 1979  94.700000 54.14705  40.5529461
Sep 1979  99.660010 61.98312  37.6768940
Oct 1979  91.600000 70.74041  20.8595898
Nov 1979  89.980010 83.84531   6.1346966
Dec 1979  99.390000 88.22922  11.1607832
Jan 1980  99.460000 77.36771  22.0922856
Feb 1980  99.370000 71.21830  28.1517032
Mar 1980  99.519990 75.88240  23.6375896
Apr 1980  96.640000 75.76737  20.8726314
May 1980  89.550000 69.20618  20.3438245
Jun 1980  68.670000 66.75287   1.9171264
Jul 1980  65.020000 56.85298   8.1670196
Aug 1980  61.820000 57.97755   3.8424537
Sep 1980  76.920000 61.11945  15.8005472
Oct 1980  80.170000 70.04876  10.1212392
Nov 1980  77.480010 73.85689   3.6231155
Dec 1980  82.340000 72.19841  10.1415853
Jan 1981  74.110000 74.40732  -0.2973228
Feb 1981  69.030000 67.50906   1.5209367
Mar 1981  79.480010 64.55358  14.9264289
Apr 1981  78.760000 70.27149   8.4885076
May 1981  67.550000 68.52882  -0.9788172
Jun 1981  59.980000 63.87486  -3.8948558
Jul 1981  44.350000 62.05386 -17.7038635
Aug 1981  41.180000 54.54165 -13.3616549
Sep 1981  71.530000 54.95476  16.5752356
Oct 1981  95.519990 69.63976  25.8802279
Nov 1981  93.480010 80.67281  12.8071974
Dec 1981  98.180000 77.10573  21.0742660
Jan 1982  70.480010 88.73535 -18.2553450
Feb 1982  77.630000 96.54231 -18.9123080
Mar 1982  88.110000 84.31749   3.7925073
Apr 1982  93.150000 77.92560  15.2243975
May 1982  99.010000 73.83287  25.1771309
Jun 1982  93.310000 66.24079  27.0692104
Jul 1982  81.210000 57.44524  23.7647616
Aug 1982  79.630000 55.01480  24.6151969
Sep 1982  80.670000 58.02065  22.6493476
Oct 1982  85.630000 62.17996  23.4500385
Nov 1982  88.660010 69.46371  19.1962972
Dec 1982  93.650000 73.71842  19.9315810
Jan 1983  95.490000 77.16559  18.3244097
Feb 1983  98.269990 77.74186  20.5281283
Mar 1983  86.190000 79.21240   6.9776045
Apr 1983  79.690000 68.12092  11.5690763
May 1983  72.260000 64.58441   7.6755860
Jun 1983  35.060000 62.00364 -26.9436353
Jul 1983  20.980000 47.04796 -26.0679595
Aug 1983  29.670000 43.20584 -13.5358354
Sep 1983  42.090000 53.85102 -11.7610221
Oct 1983  52.960000 62.01009  -9.0500917
Nov 1983  69.450000 70.13538  -0.6853795
Dec 1983  76.860000 77.97267  -1.1126694
Jan 1984  86.190000 76.54044   9.6495642
Feb 1984  96.000000 76.50740  19.4926047
Mar 1984  96.070000 79.15992  16.9100788
Apr 1984  86.850000 73.41447  13.4355347
May 1984  76.660010 61.97078  14.6892347
Jun 1984  61.470000 54.86607   6.6039320
Jul 1984  46.260000 47.03129  -0.7712934
Aug 1984  40.150000 42.63067  -2.4806742
Sep 1984  72.590000 46.23158  26.3584185
Oct 1984  85.170000 64.88393  20.2860673
Nov 1984  91.740000 73.52080  18.2192015
Dec 1984  99.220000 78.64552  20.5744763
Jan 1985  76.550000 89.91407 -13.3640659
Feb 1985  64.170000 96.21493 -32.0449308
Mar 1985  69.200000 96.37067 -27.1706743
Apr 1985  70.370000 91.39517 -21.0251735
May 1985  79.550000 89.27045  -9.7204546
Jun 1985  74.790000 82.90792  -8.1179196
Jul 1985  70.900000 86.50457 -15.6045748
Aug 1985  78.860000 89.49937 -10.6393748
Sep 1985  84.280000 84.51239  -0.2323920
Oct 1985  83.430000 82.60754   0.8224564
Nov 1985  85.550000 82.47700   3.0729958
Dec 1985  80.170000 81.06663  -0.8966264
Jan 1986  90.820000 85.37107   5.4489314
Feb 1986  99.390000 79.20320  20.1868020
Mar 1986  99.180000 70.50652  28.6734846
Apr 1986  89.100000 65.81639  23.2836147
May 1986  82.180000 57.42544  24.7545593
Jun 1986  77.640000 57.72474  19.9152602
Jul 1986  55.930000 60.05927  -4.1292702
Aug 1986  49.730000 53.66307  -3.9330704
Sep 1986  70.120000 56.11125  14.0087499
Oct 1986  79.200000 67.69005  11.5099468
Nov 1986  87.830000 71.98548  15.8445220
Dec 1986  88.200000 77.90792  10.2920768
Jan 1987  94.830000 77.26789  17.5621145
Feb 1987  98.660010 78.06609  20.5939177
Mar 1987  94.839990 79.24883  15.5911592
Apr 1987  83.060000 71.28067  11.7793334
May 1987  61.420000 63.09806  -1.6780601
Jun 1987  47.470000 53.05231  -5.5823076
Jul 1987  31.810000 48.17713 -16.3671342
Aug 1987  22.950000 44.41360 -21.4635966
Sep 1987  17.870000 44.35441 -26.4844075

Salida estimada:

lag   s    beta(s)
5   -18.4793
6   -12.2633
7    -8.5394
8    -6.9846

Ecuación de predicción:

\[ \text{rec}(t) = 65.97 - 18.5x_{t-5} - 12.3x_{t-6} - 8.5x_{t-7} - 7x_{t-8} + w_t. \]

Modelo inverso:

LagReg(rec, soi, L=15, M=32, inverse=TRUE, threshold=.01)

INPUT: rec OUTPUT: soi   L = 15    M = 32 

The coefficients beta(0), beta(1), beta(2) ... beta(M/2-1) are 

0.00401381 0.001342919 -0.001857819 0.001637061 -0.0005376305 -0.002942707 
7.277553e-05 -0.001615343 0.001750733 0.002040213 0.001887672 -0.00182302 
0.001794064 0.0007542296 5.312944e-05 -0.003376224 


The coefficients beta(0), beta(-1), beta(-2) ... beta(-M/2+1) are 

0.00401381 -0.003704779 0.002380091 -0.0008144651 0.01355105 -0.01914634 
0.0007317915 0.001648078 4.701496e-05 -0.001410605 0.0001172051 0.0002275781 
0.001111257 0.0004196674 0.0003951922 5.630268e-05 

The negative lags, at which the coefficients are large
in absolute value, and the coefficients themselves, are: 
     lag s     beta(s)
[1,]     4  0.01355105
[2,]     5 -0.01914634

The prediction equation is
soi(t) = alpha + sum_s[ beta(s)*rec(t+s) ], where alpha = 0.4284158
MSE = 0.07168914

$betas
        S             b
 [1,] -15  5.630268e-05
 [2,] -14  3.951922e-04
 [3,] -13  4.196674e-04
 [4,] -12  1.111257e-03
 [5,] -11  2.275781e-04
 [6,] -10  1.172051e-04
 [7,]  -9 -1.410605e-03
 [8,]  -8  4.701496e-05
 [9,]  -7  1.648078e-03
[10,]  -6  7.317915e-04
[11,]  -5 -1.914634e-02
[12,]  -4  1.355105e-02
[13,]  -3 -8.144651e-04
[14,]  -2  2.380091e-03
[15,]  -1 -3.704779e-03
[16,]   0  4.013810e-03
[17,]   1  1.342919e-03
[18,]   2 -1.857819e-03
[19,]   3  1.637061e-03
[20,]   4 -5.376305e-04
[21,]   5 -2.942707e-03
[22,]   6  7.277553e-05
[23,]   7 -1.615343e-03
[24,]   8  1.750733e-03
[25,]   9  2.040213e-03
[26,]  10  1.887672e-03
[27,]  11 -1.823020e-03
[28,]  12  1.794064e-03
[29,]  13  7.542296e-04
[30,]  14  5.312944e-05
[31,]  15 -3.376224e-03

$fit
             output          fit        resids
Jan 1950  0.3770000  0.044411097  0.3325889035
Feb 1950  0.2460000  0.225726913  0.0202730872
Mar 1950  0.3110000  0.297669179  0.0133308213
Apr 1950  0.1040000  0.178134968 -0.0741349682
May 1950 -0.0160000  0.097892595 -0.1138925951
Jun 1950  0.2350000  0.262458383 -0.0274583828
Jul 1950  0.1370000  0.213727099 -0.0767270992
Aug 1950  0.1910000  0.250500331 -0.0595003306
Sep 1950 -0.0160000  0.114253635 -0.1302536353
Oct 1950  0.2900000  0.284435663  0.0055643366
Nov 1950  0.0380000  0.140618762 -0.1026187623
Dec 1950 -0.0160000  0.081767770 -0.0977677700
Jan 1951 -0.1580000 -0.054576082 -0.1034239180
Feb 1951  0.3660000  0.244112761  0.1218872393
Mar 1951  0.6070000  0.390371645  0.2166283554
Apr 1951 -0.3550000 -0.204289826 -0.1507101742
May 1951 -0.1800000 -0.137755270 -0.0422447296
Jun 1951  0.2680000  0.093928375  0.1740716254
Jul 1951  0.0930000  0.025129249  0.0678707515
Aug 1951  0.0270000 -0.010390280  0.0373902804
Sep 1951  0.2460000  0.107339554  0.1386604456
Oct 1951  0.2020000  0.105336484  0.0966635158
Nov 1951  0.4320000  0.276984202  0.1550157980
Dec 1951  0.6170000  0.431344779  0.1856552212
Jan 1952  0.7600000  0.498044178  0.2619558222
Feb 1952  0.8910000  0.500431102  0.3905688984
Mar 1952  0.6070000  0.411843144  0.1951568562
Apr 1952  0.5740000  0.412033769  0.1619662307
May 1952  0.0050000  0.308841156 -0.3038411558
Jun 1952  0.4750000  0.392371592  0.0826284076
Jul 1952  0.2020000  0.333513673 -0.1315136726
Aug 1952 -0.0270000  0.237481214 -0.2644812136
Sep 1952 -0.0380000  0.164088586 -0.2020885863
Oct 1952  0.7160000  0.362712060  0.3532879400
Nov 1952  0.8360000  0.366358664  0.4696413363
Dec 1952  0.8910000  0.360156692  0.5308433081
Jan 1953  0.5300000  0.304367438  0.2256325621
Feb 1953  0.5300000  0.297291143  0.2327088575
Mar 1953  0.3770000  0.266043253  0.1109567475
Apr 1953 -0.2350000 -0.011168646 -0.2238313537
May 1953 -0.5850000 -0.508880959 -0.0761190408
Jun 1953 -0.1800000 -0.218937698  0.0389376977
Jul 1953 -0.5300000 -0.290031955 -0.2399680453
Aug 1953 -0.4640000 -0.070448968 -0.3935510318
Sep 1953 -0.4430000  0.064319431 -0.5073194312
Oct 1953  0.0490000 -0.082336150  0.1313361503
Nov 1953  0.4540000 -0.236795527  0.6907955269
Dec 1953  0.2570000 -0.165747866  0.4227478657
Jan 1954  0.4100000 -0.033219398  0.4432193982
Feb 1954  0.2240000  0.048599350  0.1754006504
Mar 1954  0.1480000  0.080683800  0.0673162002
Apr 1954 -0.4320000 -0.259323261 -0.1726767389
May 1954 -0.0930000 -0.068014053 -0.0249859468
Jun 1954 -0.2680000 -0.140200103 -0.1277998967
Jul 1954  0.1580000  0.079933781  0.0780662192
Aug 1954 -0.0600000 -0.008994280 -0.0510057198
Sep 1954 -0.3990000 -0.227480385 -0.1715196148
Oct 1954  0.2350000  0.136390275  0.0986097246
Nov 1954  0.3660000  0.332706696  0.0332933040
Dec 1954  0.2020000  0.259030255 -0.0570302549
Jan 1955  0.3440000  0.389631128 -0.0456311276
Feb 1955 -0.0380000  0.142710864 -0.1807108642
Mar 1955  0.2900000  0.363375421 -0.0733754208
Apr 1955 -0.1260000  0.101473749 -0.2274737486
May 1955 -0.3660000 -0.174751167 -0.1912488330
Jun 1955 -0.1150000 -0.027083322 -0.0879166781
Jul 1955 -0.3010000 -0.207931486 -0.0930685137
Aug 1955 -0.4860000 -0.317718860 -0.1682811399
Sep 1955 -0.1370000 -0.143943752  0.0069437517
Oct 1955  0.7380000  0.216593125  0.5214068749
Nov 1955  0.3660000  0.109644880  0.2563551197
Dec 1955  0.3660000  0.166264796  0.1997352036
Jan 1956  0.6500000  0.402926742  0.2470732584
Feb 1956  0.6280000  0.434697651  0.1933023491
Mar 1956  0.1260000  0.227323765 -0.1013237649
Apr 1956  0.1690000  0.261260858 -0.0922608576
May 1956  0.1370000  0.251709562 -0.1147095617
Jun 1956 -0.2570000 -0.028843728 -0.2281562723
Jul 1956  0.1690000  0.209538482 -0.0405384816
Aug 1956 -0.0930000  0.018195784 -0.1111957838
Sep 1956  0.4750000  0.343129132  0.1318708678
Oct 1956  0.6390000  0.414665761  0.2243342394
Nov 1956  0.5960000  0.373029181  0.2229708186
Dec 1956  0.7490000  0.413876527  0.3351234732
Jan 1957  0.1910000  0.271322072 -0.0803220716
Feb 1957  1.0000000  0.475876608  0.5241233917
Mar 1957  0.4860000  0.365711840  0.1202881602
Apr 1957  0.4100000  0.353828875  0.0561711248
May 1957  0.1580000  0.292349019 -0.1343490189
Jun 1957  0.1260000  0.252007903 -0.1260079034
Jul 1957  0.0600000  0.189676462 -0.1296764623
Aug 1957  0.2460000  0.213673996  0.0323260035
Sep 1957  0.7380000  0.337051554  0.4009484461
Oct 1957  0.8030000  0.336713267  0.4662867333
Nov 1957  0.4210000  0.255508353  0.1654916469
Dec 1957  0.6170000  0.288328120  0.3286718802
Jan 1958  0.7050000  0.310036921  0.3949630789
Feb 1958  0.6390000  0.308933618  0.3300663817
Mar 1958  0.4540000  0.277909568  0.1760904321
Apr 1958  0.3110000  0.238009986  0.0729900139
May 1958  0.3550000  0.238314193  0.1166858069
Jun 1958 -0.1580000 -0.005201018 -0.1527989824
Jul 1958 -0.0380000 -0.017194324 -0.0208056757
Aug 1958  0.1150000  0.014903194  0.1000968057
Sep 1958  0.1370000  0.002904748  0.1340952520
Oct 1958  0.2570000  0.042134571  0.2148654291
Nov 1958  0.1150000 -0.023264614  0.1382646139
Dec 1958  0.0380000 -0.061580950  0.0995809497
Jan 1959  0.0820000 -0.037926982  0.1199269824
Feb 1959  0.1480000  0.006881537  0.1411184630
Mar 1959  0.0600000 -0.016365955  0.0763659550
Apr 1959 -0.1910000 -0.148595467 -0.0424045328
May 1959 -0.6070000 -0.319980445 -0.2870195548
Jun 1959 -0.5850000 -0.096020387 -0.4889796133
Jul 1959 -0.2680000 -0.105224269 -0.1627757311
Aug 1959 -0.0930000 -0.149094179  0.0560941794
Sep 1959 -0.0930000 -0.143138594  0.0501385944
Oct 1959  0.2570000 -0.071406285  0.3284062851
Nov 1959 -0.0050000 -0.051604212  0.0466042119
Dec 1959  0.2240000  0.124449462  0.0995505375
Jan 1960  0.1690000  0.171531240 -0.0025312400
Feb 1960  0.4320000  0.447601454 -0.0156014543
Mar 1960  0.2020000  0.301132253 -0.0991322527
Apr 1960 -0.3660000 -0.150260476 -0.2157395239
May 1960 -0.6610000 -0.503244858 -0.1577551416
Jun 1960  0.0930000  0.028737916  0.0642620842
Jul 1960 -0.7160000 -0.330855583 -0.3851444168
Aug 1960  0.1480000 -0.102393629  0.2503936295
Sep 1960 -0.0930000 -0.126549469  0.0335494690
Oct 1960  0.2790000 -0.016619769  0.2956197693
Nov 1960  0.4320000  0.170373576  0.2616264239
Dec 1960 -0.1040000 -0.065462790 -0.0385372095
Jan 1961  0.6070000  0.468149100  0.1388509001
Feb 1961  0.1800000  0.221677494 -0.0416774938
Mar 1961  0.0710000  0.179975268 -0.1089752681
Apr 1961  0.2460000  0.319387730 -0.0733877297
May 1961 -0.4320000 -0.234901255 -0.1970987449
Jun 1961  0.0600000  0.124128999 -0.0641289989
Jul 1961 -0.3880000 -0.258809959 -0.1291900412
Aug 1961  0.2020000  0.128680206  0.0733197940
Sep 1961 -0.1040000 -0.067800465 -0.0361995354
Oct 1961  0.1910000  0.099945867  0.0910541334
Nov 1961  0.4750000  0.312925770  0.1620742301
Dec 1961  0.5520000  0.392425225  0.1595747755
Jan 1962  0.1690000  0.205796705 -0.0367967053
Feb 1962  0.3330000  0.311778248  0.0212217519
Mar 1962 -0.0050000  0.143323776 -0.1483237762
Apr 1962 -0.0380000  0.104486377 -0.1424863774
May 1962 -0.1260000  0.009901654 -0.1359016540
Jun 1962 -0.4430000 -0.327144813 -0.1158551875
Jul 1962 -0.0160000 -0.052925850  0.0369258502
Aug 1962  0.0270000 -0.043443636  0.0704436364
Sep 1962  0.3330000  0.121057483  0.2119425170
Oct 1962  0.3550000  0.160694872  0.1943051285
Nov 1962  0.3440000  0.185156818  0.1588431818
Dec 1962  0.1150000  0.095145511  0.0198544887
Jan 1963  0.3110000  0.240872745  0.0701272555
Feb 1963  0.5960000  0.457572663  0.1384273366
Mar 1963  0.0050000  0.146395934 -0.1413959338
Apr 1963  0.3880000  0.354946703  0.0330532974
May 1963 -0.5080001 -0.317954290 -0.1900458099
Jun 1963 -0.5520000 -0.427045955 -0.1249540451
Jul 1963 -0.8580000 -0.222559915 -0.6354400851
Aug 1963 -0.5960000  0.128406335 -0.7244063349
Sep 1963  0.0600000 -0.139090571  0.1990905708
Oct 1963  0.0710000 -0.144898359  0.2158983586
Nov 1963  0.2240000 -0.193636766  0.4176367660
Dec 1963  0.2790000 -0.150264697  0.4292646970
Jan 1964  0.3220000 -0.019272294  0.3412722941
Feb 1964  0.1800000  0.035469694  0.1445303058
Mar 1964  0.1260000  0.067655738  0.0583442617
Apr 1964  0.0820000  0.085438312 -0.0034383116
May 1964  0.1260000  0.156596315 -0.0305963149
Jun 1964 -0.2130000 -0.075181805 -0.1378181951
Jul 1964 -0.2350000 -0.105418241 -0.1295817589
Aug 1964  0.1690000  0.184819173 -0.0158191727
Sep 1964  0.3880000  0.404752152 -0.0167521522
Oct 1964  0.5190000  0.504697152  0.0143028476
Nov 1964  0.1260000  0.259389969 -0.1333899695
Dec 1964  0.4100000  0.409414206  0.0005857938
Jan 1965  0.7380000  0.524843995  0.2131560051
Feb 1965  0.4430000  0.410262496  0.0327375044
Mar 1965  0.5960000  0.434380791  0.1616192088
Apr 1965 -0.1370000  0.286116487 -0.4231164869
May 1965 -0.4860000 -0.076074590 -0.4099254097
Jun 1965 -0.3770000 -0.218734332 -0.1582656678
Jul 1965  0.2680000  0.095453266  0.1725467337
Aug 1965 -0.1690000 -0.167086739 -0.0019132613
Sep 1965 -0.2130000 -0.229496984  0.0164969836
Oct 1965 -0.1910000 -0.214754472  0.0237544720
Nov 1965  0.3770000 -0.070061408  0.4470614081
Dec 1965  0.1260000 -0.098509326  0.2245093264
Jan 1966  0.3110000 -0.019048398  0.3300483984
Feb 1966  0.3880000  0.075879892  0.3121201082
Mar 1966  0.3880000  0.166773512  0.2212264879
Apr 1966  0.1800000  0.124802947  0.0551970527
May 1966 -0.3330000 -0.203613504 -0.1293864961
Jun 1966  0.0490000  0.047692930  0.0013070698
Jul 1966 -0.0820000 -0.018690612 -0.0633093881
Aug 1966  0.1910000  0.181975915  0.0090240850
Sep 1966  0.3990000  0.378955461  0.0200445387
Oct 1966  0.0930000  0.175418613 -0.0824186129
Nov 1966  0.0050000  0.122440105 -0.1174401049
Dec 1966  0.3660000  0.373103235 -0.0071032349
Jan 1967  0.3330000  0.363732173 -0.0307321731
Feb 1967  0.3770000  0.390927068 -0.0139270682
Mar 1967  0.0710000  0.247087456 -0.1760874563
Apr 1967 -0.1580000  0.082441161 -0.2404411610
May 1967 -0.5410000 -0.394041563 -0.1469584366
Jun 1967 -0.6830000 -0.502443142 -0.1805568585
Jul 1967 -0.6830000 -0.091810655 -0.5911893453
Aug 1967 -0.3440000  0.001757251 -0.3457572510
Sep 1967 -0.3010000  0.026168468 -0.3271684684
Oct 1967  0.2020000 -0.170996092  0.3729960917
Nov 1967 -0.1260000 -0.104684825 -0.0213151753
Dec 1967  0.0820000 -0.194677132  0.2766771323
Jan 1968  0.1150000 -0.156907336  0.2719073363
Feb 1968  0.4640000  0.082209412  0.3817905876
Mar 1968  0.1040000  0.041669152  0.0623308477
Apr 1968 -0.1800000 -0.098610481 -0.0813895189
May 1968 -0.6720000 -0.313236595 -0.3587634050
Jun 1968  0.1040000 -0.068702193  0.1727021931
Jul 1968 -0.4860000 -0.175766898 -0.3102331023
Aug 1968  0.0930000 -0.072381089  0.1653810890
Sep 1968  0.2790000  0.111640607  0.1673593928
Oct 1968  0.1040000  0.089041086  0.0149589144
Nov 1968  0.1800000  0.194910807 -0.0149108067
Dec 1968  0.4750000  0.489473411 -0.0144734108
Jan 1969  0.5300000  0.519847314  0.0101526862
Feb 1969  0.6610000  0.536810847  0.1241891533
Mar 1969  0.3550000  0.404250171 -0.0492501712
Apr 1969 -0.3440000  0.116341744 -0.4603417440
May 1969 -0.0380000  0.192647161 -0.2306471614
Jun 1969  0.2350000  0.308607454 -0.0736074543
Jul 1969  0.4860000  0.408424597  0.0775754028
Aug 1969  0.1690000  0.273118654 -0.1041186539
Sep 1969  0.3330000  0.308451049  0.0245489509
Oct 1969  0.1370000  0.225853403 -0.0888534027
Nov 1969  0.1690000  0.212700952 -0.0437009525
Dec 1969  0.3330000  0.256067798  0.0769322021
Jan 1970  0.7050000  0.372003676  0.3329963237
Feb 1970  0.3770000  0.272626391  0.1043736089
Mar 1970  0.5850000  0.321318399  0.2636816010
Apr 1970 -0.1260000  0.064872902 -0.1908729019
May 1970 -0.2130000 -0.086611502 -0.1263884980
Jun 1970  0.1480000  0.047240420  0.1007595802
Jul 1970 -0.1260000 -0.124840627 -0.0011593730
Aug 1970 -0.3440000 -0.294180960 -0.0498190399
Sep 1970 -0.5080001 -0.298369774 -0.2096303260
Oct 1970  0.2790000 -0.111767649  0.3907676488
Nov 1970  0.0820000 -0.122033770  0.2040337701
Dec 1970  0.3550000 -0.051981723  0.4069817235
Jan 1971  0.0820000 -0.079065765  0.1610657654
Feb 1971  0.5740000  0.201768318  0.3722316821
Mar 1971  0.3550000  0.205839080  0.1491609197
Apr 1971 -0.2570000 -0.140323936 -0.1166760640
May 1971 -0.3770000 -0.226274511 -0.1507254894
Jun 1971 -0.3440000 -0.185939709 -0.1580602909
Jul 1971 -0.6500000 -0.194176210 -0.4558237899
Aug 1971 -0.0930000 -0.135584982  0.0425849816
Sep 1971 -0.1370000 -0.129198502 -0.0078014980
Oct 1971  0.1150000 -0.073306249  0.1883062491
Nov 1971  0.7050000  0.461140960  0.2438590400
Dec 1971  0.3990000  0.331633284  0.0673667162
Jan 1972  0.3660000  0.360822454  0.0051775457
Feb 1972  0.3660000  0.391888631 -0.0258886306
Mar 1972  0.0380000  0.230306783 -0.1923067834
Apr 1972 -0.2680000 -0.004354769 -0.2636452314
May 1972  0.3220000  0.371885457 -0.0498854565
Jun 1972  0.0820000  0.220709954 -0.1387099539
Jul 1972  0.0820000  0.203072585 -0.1210725846
Aug 1972  0.0600000  0.165786168 -0.1057861677
Sep 1972  0.2680000  0.262850193  0.0051498067
Oct 1972  0.4210000  0.337513954  0.0834860459
Nov 1972  0.7700000  0.476366943  0.2936330571
Dec 1972  0.8030000  0.436477139  0.3665228613
Jan 1973  0.6610000  0.395195440  0.2658045604
Feb 1973  0.7160000  0.404105926  0.3118940739
Mar 1973  0.6280000  0.395080978  0.2329190224
Apr 1973  0.1910000  0.333655329 -0.1426553288
May 1973  0.5300000  0.374528793  0.1554712075
Jun 1973  0.1800000  0.306491106 -0.1264911061
Jul 1973 -0.4640000 -0.111094929 -0.3529050711
Aug 1973  0.2790000  0.140861821  0.1381381792
Sep 1973  0.3550000  0.140260278  0.2147397219
Oct 1973  0.2020000  0.065510842  0.1364891579
Nov 1973  0.5520000  0.149865683  0.4021343171
Dec 1973  0.6170000  0.162066012  0.4549339881
Jan 1974  0.1040000  0.009733739  0.0942662608
Feb 1974  0.2240000  0.048033314  0.1759666862
Mar 1974  0.4320000  0.143306847  0.2886931533
Apr 1974 -0.2130000 -0.165214929 -0.0477850715
May 1974 -0.9670000 -0.462800777 -0.5041992230
Jun 1974 -1.0000000  0.557865669 -1.5578656690
Jul 1974 -0.4320000  0.121594526 -0.5535945255
Aug 1974 -0.6830000  0.261162611 -0.9441626106
Sep 1974 -0.5080001  0.136304997 -0.6443050974
Oct 1974 -0.1370000 -0.139166216  0.0021662159
Nov 1974  0.0820000 -0.330183030  0.4121830302
Dec 1974 -0.0600000 -0.192623412  0.1326234121
Jan 1975  0.3330000 -0.216196230  0.5491962301
Feb 1975  0.3550000 -0.014909728  0.3699097282
Mar 1975  0.3880000  0.114617508  0.2733824919
Apr 1975 -0.0600000 -0.073661472  0.0136614721
May 1975 -0.2020000 -0.135702078 -0.0662979217
Jun 1975  0.0050000 -0.022255144  0.0272551444
Jul 1975 -0.1150000 -0.062736938 -0.0522630621
Aug 1975  0.1370000  0.106976544  0.0300234558
Sep 1975  0.4540000  0.420911895  0.0330881047
Oct 1975  0.3880000  0.396156581 -0.0081565815
Nov 1975  0.7380000  0.611250572  0.1267494284
Dec 1975  0.7700000  0.514088013  0.2559119870
Jan 1976  0.8030000  0.467115763  0.3358842369
Feb 1976  0.8360000  0.446378344  0.3896216565
Mar 1976  0.6500000  0.427464589  0.2225354109
Apr 1976  0.0050000  0.388157778 -0.3831577784
May 1976 -0.3660000  0.241160222 -0.6071602217
Jun 1976  0.0710000  0.270940095 -0.1999400949
Jul 1976 -0.0050000  0.190952932 -0.1959529315
Aug 1976 -0.4540000 -0.245944730 -0.2080552700
Sep 1976  0.1910000  0.027316859  0.1636831405
Oct 1976  0.4320000  0.077378777  0.3546212227
Nov 1976  0.6830000  0.132321754  0.5506782457
Dec 1976  0.3880000  0.056602822  0.3313971778
Jan 1977  0.2350000  0.004855222  0.2301447782
Feb 1977 -0.0930000 -0.129928933  0.0369289328
Mar 1977  0.4750000  0.108857047  0.3661429533
Apr 1977 -0.0820000 -0.103043557  0.0210435573
May 1977 -0.6170000 -0.401663706 -0.2153362943
Jun 1977 -0.0930000 -0.135039420  0.0420394198
Jul 1977 -0.0710000 -0.126743558  0.0557435582
Aug 1977  0.4320000  0.037322876  0.3946771242
Sep 1977  0.4210000  0.162832362  0.2581676379
Oct 1977  0.4970000  0.306383583  0.1906164172
Nov 1977  0.4540000  0.349019801  0.1049801989
Dec 1977  0.5960000  0.485700349  0.1102996514
Jan 1978  0.5520000  0.459942184  0.0920578157
Feb 1978  0.6070000  0.469639634  0.1373603656
Mar 1978  0.4640000  0.420369829  0.0436301713
Apr 1978 -0.3010000  0.166467576 -0.4674675764
May 1978 -0.7700000 -0.669885678 -0.1001143217
Jun 1978 -0.6500000 -0.469292853 -0.1807071468
Jul 1978 -0.6940000 -0.071668670 -0.6223313300
Aug 1978 -0.1370000 -0.048012723 -0.0889872769
Sep 1978 -0.4640000  0.144663291 -0.6086632908
Oct 1978 -0.2460000  0.085593935 -0.3315939350
Nov 1978  0.1690000 -0.162594302  0.3315943018
Dec 1978 -0.2020000 -0.110527418 -0.0914725817
Jan 1979  0.0820000 -0.237875225  0.3198752250
Feb 1979  0.6070000 -0.049648486  0.6566484864
Mar 1979 -0.1580000 -0.119616506 -0.0383834944
Apr 1979 -0.4540000 -0.196424127 -0.2575758731
May 1979 -0.7810000  0.025108879 -0.8061088786
Jun 1979 -0.3880000 -0.053095828 -0.3349041715
Jul 1979  0.1800000 -0.255215235  0.4352152353
Aug 1979  0.0600000 -0.129040253  0.1890402528
Sep 1979 -0.5080001 -0.126368509 -0.3816315910
Oct 1979 -0.2680000 -0.130459863 -0.1375401375
Nov 1979  0.2240000 -0.073286080  0.2972860803
Dec 1979  0.1150000  0.023434570  0.0915654300
Jan 1980  0.3990000  0.327133167  0.0718668329
Feb 1980  0.0270000  0.114071417 -0.0870714169
Mar 1980  0.0160000  0.125878372 -0.1098783718
Apr 1980 -0.3770000 -0.206594678 -0.1704053217
May 1980 -0.1800000 -0.064199451 -0.1158005495
Jun 1980 -0.0380000  0.031344912 -0.0693449119
Jul 1980 -0.2020000 -0.098158281 -0.1038417194
Aug 1980  0.1370000  0.125274035  0.0117259650
Sep 1980  0.0930000  0.111012305 -0.0180123045
Oct 1980 -0.2790000 -0.157906438 -0.1210935622
Nov 1980 -0.0380000 -0.002512296 -0.0354877037
Dec 1980  0.2350000  0.202361257  0.0326387428
Jan 1981  0.1690000  0.195391784 -0.0263917842
Feb 1981  0.3990000  0.392067606  0.0069323937
Mar 1981  0.1580000  0.240958616 -0.0829586160
Apr 1981 -0.5410000 -0.383089550 -0.1579104502
May 1981 -0.6280000 -0.431135689 -0.1968643113
Jun 1981  0.0270000 -0.066988037  0.0939880371
Jul 1981 -0.7380000 -0.184619499 -0.5533805011
Aug 1981 -0.9130000  0.409423649 -1.3224236489
Sep 1981 -0.0270000 -0.102836367  0.0758363668
Oct 1981  0.0710000 -0.206600126  0.2776001264
Nov 1981 -0.1260000 -0.161082684  0.0350826844
Dec 1981  0.0490000 -0.204982940  0.2539829404
Jan 1982  0.4540000 -0.016439675  0.4704396753
Feb 1982  0.3010000  0.137990035  0.1630099649
Mar 1982 -0.0050000  0.004273568 -0.0092735676
Apr 1982 -0.0930000 -0.037049279 -0.0559507208
May 1982 -0.2240000 -0.117922023 -0.1060779766
Jun 1982 -0.2130000 -0.108722419 -0.1042775805
Jul 1982 -0.3110000 -0.163202641 -0.1477973587
Aug 1982 -0.2350000 -0.130812308 -0.1041876917
Sep 1982 -0.3220000 -0.159105007 -0.1628949931
Oct 1982  0.3220000  0.109854337  0.2121456632
Nov 1982  0.0930000  0.070609006  0.0223909936
Dec 1982  0.1150000  0.124784482 -0.0097844822
Jan 1983  0.7700000  0.736343951  0.0336560488
Feb 1983  0.6070000  0.501825398  0.1051746020
Mar 1983 -0.1580000  0.144644969 -0.3026449692
Apr 1983 -0.2350000  0.024606064 -0.2596060643
May 1983 -0.2020000 -0.015210608 -0.1867893916
Jun 1983 -0.3330000 -0.183633822 -0.1493661777
Jul 1983 -0.1580000 -0.102051402 -0.0559485981
Aug 1983 -0.2240000 -0.180273465 -0.0437265347
Sep 1983 -0.3550000 -0.241667759 -0.1133322415
Oct 1983  0.0050000 -0.110072221  0.1150722213
Nov 1983  0.4540000  0.067405584  0.3865944160
Dec 1983  0.4100000  0.137565909  0.2724340907
Jan 1984  0.5190000  0.290313924  0.2286860759
Feb 1984  0.5410000  0.375689163  0.1653108369
Mar 1984  0.3010000  0.286561845  0.0144381546
Apr 1984 -0.5740000 -0.417342247 -0.1566577530
May 1984 -0.3440000 -0.218607176 -0.1253928241
Jun 1984 -0.2790000 -0.173926430 -0.1050735703
Jul 1984 -0.7160000 -0.228110649 -0.4878893510
Aug 1984 -0.8690000  0.307298662 -1.1762986616
Sep 1984 -0.5960000  0.237128062 -0.8331280620
Oct 1984 -0.2900000 -0.026939990 -0.2630600101
Nov 1984 -0.4540000  0.018820567 -0.4728205666
Dec 1984 -0.2460000 -0.141088086 -0.1049119143
Jan 1985 -0.6070000  0.074447102 -0.6814471015
Feb 1985 -0.5630000  0.084423366 -0.6474233660
Mar 1985 -0.2350000 -0.120695057 -0.1143049427
Apr 1985 -0.2460000 -0.116601865 -0.1293981354
May 1985 -0.3990000 -0.026880798 -0.3721192024
Jun 1985 -0.3330000 -0.078989424 -0.2540105761
Jul 1985 -0.5300000  0.052746094 -0.5827460937
Aug 1985 -0.0490000 -0.224067039  0.1750670391
Sep 1985  0.1580000 -0.243832490  0.4018324905
Oct 1985  0.1150000 -0.123679278  0.2386792783
Nov 1985  0.3220000  0.066470084  0.2555299162
Dec 1985  0.1150000  0.062368176  0.0526318244
Jan 1986  0.0490000  0.055519296 -0.0065192958
Feb 1986  0.4540000  0.409664525  0.0443354746
Mar 1986  0.1580000  0.234178566 -0.0761785661
Apr 1986 -0.4210000 -0.240231753 -0.1807682465
May 1986 -0.2680000 -0.137774630 -0.1302253705
Jun 1986 -0.3110000 -0.179964006 -0.1310359935
Jul 1986 -0.1150000 -0.070102607 -0.0448973928
Aug 1986 -0.3220000 -0.192028937 -0.1299710628
Sep 1986 -0.3220000 -0.175516153 -0.1464838469
Oct 1986  0.1260000 -0.050476112  0.1764761115
Nov 1986  0.3330000  0.123302279  0.2096977214
Dec 1986  0.5190000  0.377997813  0.1410021874
Jan 1987  0.3990000  0.351844542  0.0471554576
Feb 1987  0.5190000  0.462639068  0.0563609317
Mar 1987  0.4320000  0.420066207  0.0119337929
Apr 1987  0.3550000  0.397267316 -0.0422673165

Salida:

lag   s    beta(s)
4     0.0159
5    -0.0212

Predicción:

\[ x_t = 0.41 + 0.016y_{t+4} - 0.02y_{t+5} + v_t. \]

Tras manipular con el operador de rezago \(B\), se obtiene:

\[ (1 - .8B)y_t = 20.5 - 50B^5x_t + \epsilon_t. \]

4.7.3 Ajuste ARMAX y diagnóstico

Para verificar la autocorrelación de los residuos y ajustar un modelo ARMAX:

fish <- ts.intersect(R = rec, RL1 = lag(rec, -1), SL5 = lag(soi, -5))
u <- lm(fish[,1] ~ fish[,2:3], na.action = NULL)
acf2(resid(u))

     [,1]  [,2]  [,3]  [,4]  [,5] [,6]  [,7] [,8]  [,9] [,10] [,11] [,12] [,13]
ACF   0.4  0.10 -0.04 -0.15 -0.03 0.06 -0.02 0.00 -0.07 -0.11 -0.10 -0.10 -0.10
PACF  0.4 -0.08 -0.06 -0.13  0.11 0.05 -0.10 0.03 -0.07 -0.05 -0.06 -0.04 -0.07
     [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
ACF  -0.07 -0.08 -0.08  0.01  0.01 -0.02 -0.02  0.02  0.03  0.04  0.03  0.00
PACF -0.04 -0.05 -0.05  0.05 -0.04 -0.04 -0.03  0.05 -0.01 -0.02 -0.01 -0.02
     [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
ACF   0.03  0.09  0.07  0.08  0.05  0.00 -0.12 -0.13 -0.03  0.06  0.10  0.07
PACF  0.03  0.06 -0.01  0.05  0.01 -0.01 -0.14 -0.02  0.06  0.05  0.03  0.03
     [,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48]
ACF  -0.07 -0.13 -0.03  0.05  0.08  0.04 -0.02 -0.09 -0.08 -0.04  0.01
PACF -0.09 -0.05  0.10  0.06  0.00 -0.04  0.00 -0.07 -0.02  0.01 -0.01

sarima(fish[,1], 1, 0, 0, xreg = fish[,2:3])

initial  value 2.047606 
iter   2 value 1.958908
iter   3 value 1.957404
iter   4 value 1.952420
iter   5 value 1.952201
iter   6 value 1.951913
iter   7 value 1.951750
iter   8 value 1.951726
iter   9 value 1.951724
iter  10 value 1.951724
iter  10 value 1.951724
final  value 1.951724 
converged
initial  value 1.951922 
iter   2 value 1.951921
iter   3 value 1.951915
iter   4 value 1.951915
iter   5 value 1.951914
iter   5 value 1.951914
iter   5 value 1.951914
final  value 1.951914 
converged
<><><><><><><><><><><><><><>
 
Coefficients: 
          Estimate     SE  t.value p.value
ar1         0.4489 0.0495   9.0591       0
intercept  14.6838 1.5605   9.4098       0
RL1         0.7902 0.0229  34.4532       0
SL5       -20.9988 1.0812 -19.4218       0

sigma^2 estimated as 49.56706 on 444 degrees of freedom 
 
AIC = 6.764027  AICc = 6.764229  BIC = 6.80984 
 

Salida estimada:

ar1 intercept RL1 SL5
0.4487 12.3323 0.8005 -21.0307
s.e.  0.0503   1.5746 0.0234  1.0915
σ² = 49.93

Modelo final:

\[ y_t = 12 + 0.8y_{t-1} - 21x_{t-5} + \epsilon_t, \quad \epsilon_t = 0.45\epsilon_{t-1} + w_t. \]