# ngsmth

0th

Percentile

##### Non-Gaussian Smoothing

Trend estimation by non-Gaussian smoothing.

Keywords
ts
##### Usage
ngsmth(y, noisev = 2, tau2, bv = 1.0, noisew = 1, sigma2, bw = 1.0,
initd = 1, k = 200, plot = TRUE, …)
##### Arguments
y

a univariate time series.

noisev

type of system noise density.

 1 : Gaussian (normal) 2 : Pearson family

tau2

variance of dispersion of system noise.

bv

shape parameter of system noise (for noisev = 2).

noisew

type of observation noise density

 1 : Gaussian (normal) 2 : Pearson family 3 : two-sided exponential

sigma2

variance of dispersion of observation noise.

bw

shape parameter of observation noise (for noisew = 2).

initd

type of density function.

 1 : Gaussian (normal) 2 : uniform

k

number of intervals.

plot

logical. If TRUE (default), 'trend' and 'smt' are plotted.

further arguments to be passed to plot.ngsmth.

##### Details

Consider a one-dimensional state space model $$x_n = x_{n-1} + v_n,$$ $$y_n = x_n + w_n,$$ where the observation noise $w_n$ is assumed to be Gaussian distributed and the system noise $v_n$ is assumed to be distributed as the Pearson system $$q(v_n) = c/(\tau^2 + v_n^2)^b$$ with $\frac{1}{2} < b < \infty$ and $c = \tau^{2b-1}\Gamma(b) \bigm/ \Gamma(\frac{1}{2})\Gamma(b-\frac{1}{2})$.

This broad family of distributions includes the Cauchy distribution ($b = 1$) and $t$-distribution ($b = (k+1)/2$).

##### Value

An object of class "ngsmth". It contains the following components:

trend

trend.

smt

smoothed density.

##### References

Kitagawa, G. (2010) Introduction to Time Series Modeling. Chapman & Hall/CRC.

Kitagawa, G. and Gersch, W. (1996) Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics, No.116, Springer-Verlag.

• ngsmth
##### Examples
# NOT RUN {
## trend model
x <- rep(0, 400)
x[101:200] <- 1
x[201:300] <- -1
y <- x + rnorm(400, mean = 0, sd = 1.0)

# system noise density : Gaussian (normal)
s1 <- ngsmth(y, noisev = 1, tau2 = 1.4e-02, noisew = 2, sigma2 = 1.048)

plot(s1, "smt", theta = 20, phi = 60, expand = 0.3)

# system noise density : Pearson family
s2 <- ngsmth(y, noisev = 2, tau2 = 2.11e-10, bv = 0.6, noisew = 2,
sigma2 = 1.042)

plot(s2, "smt", theta = 25, phi = 30, expand = 0.25)

## seismic data
data(MYE1F)
n <- length(MYE1F)
yy <- rep(0, n)
for (i in 2:n) yy[i] <- MYE1F[i] - 0.5 * MYE1F[i-1]
m <- seq(1, n, by = 2)
y <- yy[m]
z <- tvvar(y, trend.order = 2, tau2.ini = 4.909e-02, delta = 1.0e-06)

# system noise density : Gaussian (normal)
s3 <- ngsmth(z$sm, noisev = 1, tau2 = z$tau2, noisew = 2, sigma2 = pi*pi/6,
k = 190)

plot(s3, "smt", phi = 50, expand = 0.5, col = 8)
# }

Documentation reproduced from package TSSS, version 1.2.3, License: GPL (>= 2)

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