Theil-Sen slope estimates and tests for trend.
TheilSen(mydata, pollutant = "nox", deseason = FALSE, type = "default",
avg.time = "month", statistic = "mean", percentile = NA,
data.thresh = 0, alpha = 0.05, dec.place = 2, xlab = "year",
lab.frac = 0.99, lab.cex = 0.8, x.relation = "same",
y.relation = "same", data.col = "cornflowerblue", trend = list(lty =
c(1, 5), lwd = c(2, 1), col = c("red", "red")), text.col = "darkgreen",
slope.text = NULL, cols = NULL, shade = "grey95", auto.text = TRUE,
autocor = FALSE, slope.percent = FALSE, date.breaks = 7, plot = TRUE,
silent = FALSE, ...)A data frame containing the field date and at
least one other parameter for which a trend test is required;
typically (but not necessarily) a pollutant.
The parameter for which a trend test is required. Mandatory.
Should the data be de-deasonalized first? If
TRUE the function stl is used (seasonal trend
decomposition using loess). Note that if TRUE missing
data are first linearly interpolated because stl cannot
handle missing data.
type determines how the data are split i.e.
conditioned, and then plotted. The default is will produce a
single plot using the entire data. Type can be one of the
built-in types as detailed in cutData e.g.
“season”, “year”, “weekday” and so on. For
example, type = "season" will produce four plots --- one
for each season.
It is also possible to choose type as another variable in
the data frame. If that variable is numeric, then the data will
be split into four quantiles (if possible) and labelled
accordingly. If type is an existing character or factor
variable, then those categories/levels will be used directly.
This offers great flexibility for understanding the variation of
different variables and how they depend on one another.
Type can be up length two e.g. type = c("season",
"weekday") will produce a 2x2 plot split by season and day of
the week. Note, when two types are provided the first forms the
columns and the second the rows.
Can be “month” (the default), “season” or “year”. Determines the time over which data should be averaged. Note that for “year”, six or more years are required. For “season” the data are split up into spring: March, April, May etc. Note that December is considered as belonging to winter of the following year.
Statistic used for calculating monthly values.
Default is “mean”, but can also be “percentile”.
See timeAverage for more details.
Single percentile value to use if
statistic = "percentile" is chosen.
The data capture threshold to use (
aggregating the data using avg.time. A value of zero
means that all available data will be used in a particular
period regardless if of the number of values available.
Conversely, a value of 100 will mean that all data will need to
be present for the average to be calculated, else it is recorded
as NA.
For the confidence interval calculations of the slope. The default is 0.05. To show 99% confidence intervals for the value of the trend, choose alpha = 0.01 etc.
The number of decimal places to display the trend estimate at. The default is 2.
x-axis label, by default "year".
Fraction along the y-axis that the trend information should be printed at, default 0.99.
Size of text for trend information.
This determines how the x-axis scale is plotted. “same” ensures all panels use the same scale and “free” will use panel-specfic scales. The latter is a useful setting when plotting data with very different values.
This determines how the y-axis scale is plotted. “same” ensures all panels use the same scale and “free” will use panel-specfic scales. The latter is a useful setting when plotting data with very different values.
Colour name for the data
list containing information on the line width, line type and line colour for the main trend line and confidence intervals respectively.
Colour name for the slope/uncertainty numeric estimates
The text shown for the slope (default is ‘units/year’).
Predefined colour scheme, currently only enabled for
"greyscale".
The colour used for marking alternate years. Use “white” or “transparent” to remove shading.
Either TRUE (default) or FALSE. If
TRUE titles and axis labels will automatically try and
format pollutant names and units properly e.g. by subscripting
the ‘2’ in NO2.
Should autocorrelation be considered in the trend
uncertainty estimates? The default is FALSE. Generally,
accounting for autocorrelation increases the uncertainty of the
trend estimate --- sometimes by a large amount.
Should the slope and the slope uncertainties
be expressed as a percentage change per year? The default is
FALSE and the slope is expressed as an average units/year
change e.g. ppb. Percentage changes can often be confusing and
should be clearly defined. Here the percentage change is
expressed as 100 * (C.end/C.start - 1) / (end.year -
start.year). Where C.start is the concentration at the start
date and C.end is the concentration at the end date.
For avg.time = "year" (end.year - start.year) will be the
total number of years - 1. For example, given a concentration in
year 1 of 100 units and a percentage reduction of 5
years there will be 75 units but the actual time span will be 6
years i.e. year 1 is used as a reference year. Things are
slightly different for monthly values e.g. avg.time =
"month", which will use the total number of months as a basis
of the time span and is therefore able to deal with partial
years. There can be slight differences in the
estimate therefore, depending on whether monthly or annual
values are considered.
Number of major x-axis intervals to use. The
function will try and choose a sensible number of dates/times as
well as formatting the date/time appropriately to the range
being considered. This does not always work as desired
automatically. The user can therefore increase or decrease the
number of intervals by adjusting the value of date.breaks
up or down.
Should a plot be produced. FALSE can be useful when
analysing data to extract trend components and plotting them in other
ways.
When FALSE the function will give updates on trend-fitting progress.
Other graphical parameters passed onto cutData
and lattice:xyplot. For example, TheilSen passes
the option hemisphere = "southern" on to cutData
to provide southern (rather than default northern) hemisphere
handling of type = "season". Similarly, common axis and
title labelling options (such as xlab, ylab,
main) are passed to xyplot via quickText to
handle routine formatting.
As well as generating the plot itself, TheilSen
also returns an object of class ``openair''. The object includes
three main components: call, the command used to generate
the plot; data, the data frame of summarised information
used to make the plot; and plot, the plot itself. If
retained, e.g. using output <- TheilSen(mydata, "nox"),
this output can be used to recover the data, reproduce or rework
the original plot or undertake further analysis.
An openair output can be manipulated using a number of generic
operations, including print, plot and
summary.
The data component of the TheilSen output includes
two subsets: main.data, the monthly data res2 the
trend statistics. For output <- TheilSen(mydata, "nox"),
these can be extracted as object$data$main.data and
object$data$res2, respectively.
Note: In the case of the intercept, it is assumed the y-axis crosses the x-axis on 1/1/1970.
The TheilSen function provides a collection of functions to
analyse trends in air pollution data. The TheilSen function
is flexible in the sense that it can be applied to data in many
ways e.g. by day of the week, hour of day and wind direction. This
flexibility makes it much easier to draw inferences from data
e.g. why is there a strong downward trend in concentration from
one wind sector and not another, or why trends on one day of the
week or a certain time of day are unexpected.
For data that are strongly seasonal, perhaps from a background
site, or a pollutant such as ozone, it will be important to
deseasonalise the data (using the option deseason =
TRUE.Similarly, for data that increase, then decrease, or show
sharp changes it may be better to use smoothTrend.
A minimum of 6 points are required for trend estimates to be made.
Note! that since version 0.5-11 openair uses Theil-Sen to derive the p values also for the slope. This is to ensure there is consistency between the calculated p value and other trend parameters i.e. slope estimates and uncertainties. The p value and all uncertainties are calculated through bootstrap simulations.
Note that the symbols shown next to each trend estimate relate to how statistically significant the trend estimate is: p $<$ 0.001 = ***, p $<$ 0.01 = **, p $<$ 0.05 = * and p $<$ 0.1 = $+$.
Some of the code used in TheilSen is based on that from
Rand Wilcox http://www-rcf.usc.edu/~rwilcox/. This mostly
relates to the Theil-Sen slope estimates and uncertainties.
Further modifications have been made to take account of correlated
data based on Kunsch (1989). The basic function has been adapted
to take account of auto-correlated data using block bootstrap
simulations if autocor = TRUE (Kunsch, 1989). We follow the
suggestion of Kunsch (1989) of setting the block length to n(1/3)
where n is the length of the time series.
The slope estimate and confidence intervals in the slope are plotted and numerical information presented.
Helsel, D., Hirsch, R., 2002. Statistical methods in water resources. US Geological Survey. http://pubs.usgs.gov/twri/twri4a3/. Note that this is a very good resource for statistics as applied to environmental data.
Hirsch, R. M., Slack, J. R., Smith, R. A., 1982. Techniques of trend analysis for monthly water-quality data. Water Resources Research 18 (1), 107-121.
Kunsch, H. R., 1989. The jackknife and the bootstrap for general stationary observations. Annals of Statistics 17 (3), 1217-1241.
Sen, P. K., 1968. Estimates of regression coefficient based on Kendall's tau. Journal of the American Statistical Association 63(324).
Theil, H., 1950. A rank invariant method of linear and polynomial regression analysis, i, ii, iii. Proceedings of the Koninklijke Nederlandse Akademie Wetenschappen, Series A - Mathematical Sciences 53, 386-392, 521-525, 1397-1412.
… see also several of the Air Quality Expert Group (AQEG) reports for the use of similar tests applied to UK/European air quality data, see http://uk-air.defra.gov.uk/library/aqeg/.
See smoothTrend for a flexible approach to
estimating trends using nonparametric regression. The smoothTrend
function is suitable for cases where trends are not monotonic and is
probably better for exploring the shape of trends.
# NOT RUN {
# load example data from package
data(mydata)
# trend plot for nox
TheilSen(mydata, pollutant = "nox")
# trend plot for ozone with p=0.01 i.e. uncertainty in slope shown at
# 99 % confidence interval
# }
# NOT RUN {
TheilSen(mydata, pollutant = "o3", ylab = "o3 (ppb)", alpha = 0.01)
# }
# NOT RUN {
# trend plot by each of 8 wind sectors
# }
# NOT RUN {
TheilSen(mydata, pollutant = "o3", type = "wd", ylab = "o3 (ppb)")
# }
# NOT RUN {
# and for a subset of data (from year 2000 onwards)
# }
# NOT RUN {
TheilSen(selectByDate(mydata, year = 2000:2005), pollutant = "o3", ylab = "o3 (ppb)")
# }
# NOT RUN {
# }
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