Theil-Sen slope estimates and tests for trend.

## Usage

```
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,
date.format = NULL,
plot = TRUE,
silent = FALSE,
...
)
```

## Arguments

- mydata
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.- pollutant
The parameter for which a trend test is required. Mandatory.

- deseason
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 imputed using a Kalman filter and Kalman smooth.- type
`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.- avg.time
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
Statistic used for calculating monthly values. Default is “mean”, but can also be “percentile”. See

`timeAverage`

for more details.- percentile
Single percentile value to use if

`statistic = "percentile"`

is chosen.- data.thresh
The data capture threshold to use ( 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`

.- alpha
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.

- dec.place
The number of decimal places to display the trend estimate at. The default is 2.

- xlab
x-axis label, by default

`"year"`

.- lab.frac
Fraction along the y-axis that the trend information should be printed at, default 0.99.

- lab.cex
Size of text for trend information.

- x.relation
This determines how the x-axis scale is plotted. “same” ensures all panels use the same scale and “free” will use panel-specific scales. The latter is a useful setting when plotting data with very different values.

- y.relation
This determines how the y-axis scale is plotted. “same” ensures all panels use the same scale and “free” will use panel-specific scales. The latter is a useful setting when plotting data with very different values.

- data.col
Colour name for the data

- trend
list containing information on the line width, line type and line colour for the main trend line and confidence intervals respectively.

- text.col
Colour name for the slope/uncertainty numeric estimates

- slope.text
The text shown for the slope (default is ‘units/year’).

- cols
Predefined colour scheme, currently only enabled for

`"greyscale"`

.- shade
The colour used for marking alternate years. Use “white” or “transparent” to remove shading.

- auto.text
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.- autocor
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.- slope.percent
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 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 depending on whether monthly or annual values are considered.- date.breaks
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.- date.format
This option controls the date format on the x-axis. While

`TheilSen`

generally sets the date format sensibly there can be some situations where the user wishes to have more control. For format types see`strptime`

. For example, to format the date like “Jan-2012” set`date.format = "%b-%Y"`

.- plot
Should a plot be produced.

`FALSE`

can be useful when analysing data to extract trend components and plotting them in other ways.- silent
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.

## Value

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.

## Details

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. 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.

## References

Helsel, D., Hirsch, R., 2002. Statistical methods in water resources. US Geological Survey. 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 also

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.

## Examples

```
# load example data from package
data(mydata)
# trend plot for nox
TheilSen(mydata, pollutant = "nox")
#> [1] "Taking bootstrap samples. Please wait."
# trend plot for ozone with p=0.01 i.e. uncertainty in slope shown at
# 99 % confidence interval
if (FALSE) TheilSen(mydata, pollutant = "o3", ylab = "o3 (ppb)", alpha = 0.01)
# trend plot by each of 8 wind sectors
if (FALSE) TheilSen(mydata, pollutant = "o3", type = "wd", ylab = "o3 (ppb)")
# and for a subset of data (from year 2000 onwards)
if (FALSE) TheilSen(selectByDate(mydata, year = 2000:2005), pollutant = "o3", ylab = "o3 (ppb)")
```