turnpoints: Analyze turning points (peaks or pits)

In pastecs: Package for Analysis of Space-Time Ecological Series

Analyze turning points (peaks or pits)

Description

Determine the number and the position of extrema (turning points, either peaks or pits) in a regular time series. Calculate the quantity of information associated to the observations in this series, according to Kendall's information theory

Usage

turnpoints(x, calc.proba = TRUE)
## S3 method for class 'turnpoints'
print(x, ...)
## S3 method for class 'turnpoints'
summary(object, ...)
## S3 method for class 'summary.turnpoints'
print(x, ...)
## S3 method for class 'turnpoints'
plot(x, level = 0.05, lhorz = TRUE, lcol = 2, llty = 2,
    type = "l", xlab = "data number", ylab = paste("I (bits), level = ",
    level * 100, "%", sep = ""), main = paste("Information (turning points) for:",
    x$data), ...)
## S3 method for class 'turnpoints'
lines(x, max = TRUE, min = TRUE, median = TRUE,
    col = c(4, 4, 2), lty = c(2, 2, 1), ...)
## S3 method for class 'turnpoints'
extract(e, n, no.tp = 0, peak = 1, pit = -1, ...)

Arguments

x	a vector or a time series for turnpoints(), a 'turnpoints' object for the methods
calc.proba	are the probabilities associated with each turning point also calculated? The default, TRUE, should be correct unless you really do not need these. In this case, the plot() method is not usable
object	a 'turnpoints' object, as returned by the function turnpoints()
e	a 'turnpoints' object, as returned by the function turnpoints()
level	the significant level to draw on the graph if lhorz=TRUE. By default, level=0.05, which corresponds to a 5% p-value for the test
lhorz	if lhorz=TRUE (by default), an horizontal line indicating significant level is drawn on the graph
lcol	the color to use to draw the significant level line, by default, color 2 is used
llty	the style to use for the significant level line. By default, style 2 is used (dashed line)
type	the type of plot, as usual meaning for this graph parameter
xlab	the label of the x-axis
ylab	the label of the y-axis
main	the main title of the graph
max	do we plot the maximum envelope line (by default, yes)
min	do we plot the minimum envelope line (by default, yes)
median	do we plot the median line inside the envelope (by default, yes)
col	a vector of three values for the color of the max, min, median lines, respectively. By default col=c(4,4,2)
lty	a vector of three values for the style of the max, min, median lines, respectively. By default lty=c(2,2,1), that is: dashed, dashed and plain lines
n	the number of points to extract. By default n=length(turnp), all points are extracted
no.tp	extract gives a vector representing the position of extrema in the original series. no.tp represents the code to use for points that are not an extremum, by default '0'
peak	the code to use to flag a peak, by default '1'
pit	the code to use to flag a pit, by default '-1'
...	Additional parameters

Details

This function tests if the time series is purely random or not. Kendall (1976) proposed a series of tests for this. Moreover, graphical methods using the position of the turning points to draw automatically envelopes around the data are implemented, and also the drawing of median points between these envelopes.

With a purely random time series, one expect to find, on average, a turning point (peak or pit that is, an observation that is preceeded and followed by, respectively, lower or higher observations) every 1.5 observation. Given it is impossible to determine if first and last observation are turning point, it gives:

E(p) = 2/3*(n-2)

with p, the number of observed turning points and n the number of observations. The variance of p is:

var(p) = (16*n - 29)/90

Ibanez (1982) demonstrated that P(t), the probability to observe a turning point at time t is:

P(t) = 2*(1/n(t-1)! * (n-1)!)

where P is the probability to observe a turning point at time t under the null hypothesis that the time series is purely random, and thus, the distribution of turning points follows a normal distribution.

The quantity of information I associated with this probability is:

I = -log2 P(t)

It can be interpreted as follows. If I is larger, there are less turning points than expected in a purely random series. There are, thus, longer sequence of increasing or decreasing values along the time scale. This is considered to be more informative.

As you can easily imagine, from this point on, it is straightforward to construct a test to determine if the series is random (regarding the distribution of the turning points), more or less monotonic (more or less turning points than expected).

Value

An object of type 'turnpoints' is returned. It has methods print(), summary(), plot(), lines() and extract(). Regarding your specific question, 'info' is the quantity of information I associated with the turning points:

data	The dataset to which the calculation is done
n	The number of observations
points	The value of the points in the series, after elimination of ex-aequos
pos	The position of the points on the time scale in the series (including ex-aequos)
exaequos	Location of exaequos (1), or not (0)
nturns	Total number of tunring points in the whole time series
firstispeak	Is the first turning point a peak (TRUE), or not (FALSE)
peaks	Logical vector. Location of the peaks in the time series without ex-aequos
pits	Logical vector. Location of the pits in the time series without ex-aequos
tppos	Position of the turning points in the initial series (with ex-aequos)
proba	Probability to find a turning point at this location (see details)
info	Quantity of information associated with this point (see details)

WARNING

the lines() method should be used to draw lines on the graph of the original dataset (plot(data, type="l") for instance), not on the graph of turning points (plot(turnp))!

Author(s)

Frederic Ibanez ([e-mail]), Philippe Grosjean ([e-mail])

References

Ibanez, F., 1982. Sur une nouvelle application de la theorie de l'information a la description des series chronologiques planctoniques. J. Exp. Mar. Biol. Ecol., 4:619-632

Kendall, M.G., 1976. Time-series, 2nd ed. Charles Griffin & Co, London.

See Also

turnogram, stat.slide

Examples

data(marbio)
plot(marbio[, "Nauplii"], type = "l")
# Calculate turning points for this series
Nauplii.tp <- turnpoints(marbio[, "Nauplii"])
summary(Nauplii.tp)
plot(Nauplii.tp)
# Add envelope and median line to original data
plot(marbio[, "Nauplii"], type = "l")
lines(Nauplii.tp)
# Note that lines() applies to the graph of original dataset
title("Raw data, envelope maxi., mini. and median lines")

pastecs documentation built on May 29, 2024, 5:56 a.m.