turnpoints: 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)
## S3 method for class 'turnpoints':
print(x, \dots)
## S3 method for class 'turnpoints':
summary(object, \dots)
## S3 method for class 'summary.turnpoints':
print(x, \dots)
## 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, \dots)

Arguments

a vector or a time series for turnpoints(), a 'turnpoints' object for the methods

object

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

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

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:
dataThe dataset to which the calculation is done
nThe number of observations
pointsThe value of the points in the series, after elimination of ex-aequos
posThe position of the points on the time scale in the series (including ex-aequos)
exaequosLocation of exaequos (1), or not (0)
nturnsTotal number of tunring points in the whole time series
firstispeakIs the first turning point a peak (TRUE), or not (FALSE)
peaksLogical vector. Location of the peaks in the time series without ex-aequos
pitsLogical vector. Location of the pits in the time series without ex-aequos
tpposPosition of the turning points in the initial series (with ex-aequos)
probaProbability to find a turning point at this location (see details)
infoQuantity of information associated with this point (see details)

encoding

latin1

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))!

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).

References

Ibanez, F., 1982. Sur une nouvelle application de la theorie de l'information � 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.

Examples

Run this code

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 line")

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