Chapter_6_Preparing_summaries.qmd

# Preparing summaries
## Introduction

In this chapter and the next, we use the ***Climatic \> Prepare*** menu
to summarise data into a form ready for analysis and then present the
results as graphs and tables. Here we consider monthly and annual
summaries of rainfall, temperature and other elements. The next chapter
uses similar ideas for more specialised summaries of the rainfall data,
such as the start and length of the season.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.1a The main climatic summary menu***          ***Fig. 6.1b Presenting the summary***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.1a.png){width="2.9830041557305336in"   ![](figures/Fig6.1b.png){width="3.092963692038495in"
  height="2.510556649168854in"}                          height="2.541311242344707in"}

  ------------------------------------------------------------------------------------------------------------

In Section 6.2 data from Ghana are used to illustrate the summary of
rainfall data.

[To be continued]{.mark}

## Preparing the data

In the second tutorial (reference/link) we showed how to plot annual
temperature data, after starting from the daily records. That was
without making use of the special climatic menu. These ideas are
repeated here and extended for rainfall. The climatic menu is used, and
the example is with data from 2 stations. The first step is to prepare
the data.

Use ***File \> Open from Library \> Instat \> Browse \> Climatic \>
Ghana*** and open the file called ***ghana_two_stations.rds***, Fig.
6.2a. The data are from Saltpond, which is on the coast, with a bimodal
pattern of rainfall, and Tamale, which is further North, and with
unimodal rainfall. The rainfall starts in 1944 for each station. Other
elements, Fig. 6.2a, start later.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.2a Data from 2 stations***          ***Fig. 6.2b***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.2a.png){width="3.290274496937883in"   ![](figures/Fig6.2b.png){width="2.968218503937008in"
  height="3.2845319335083114in"}                          height="1.3167279090113735in"}

  ------------------------------------------------------------------------------------------------------------

The data is in the right shape and already has a date column.

First use ***Climatic \> Dates \> Infill***, Fig. 6.2b, to check there
are no missing dates. Some files simply omit the days when all data are
missing.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.2c Infill missing dates***          ***Fig. 6.2d Results from infilling***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.2c.png){width="3.011919291338583in"   ![](figures/Fig6.2d.png){width="3.0440813648293963in"
  height="2.6333103674540683in"}                          height="0.746661198600175in"}

  ------------------------------------------------------------------------------------------------------------

Complete the dialogue as shown in Fig. 6.2c and press Ok. The number of
rows in the data increases slightly to 53297 and the output window
states that just under 300 rows have been added.

Use ***Climatic \> Tidy and Examine \> One Variable Summarise*** and
complete the dialogue as shown in Fig. 6.2e. The results are in Fig.
6.2f. They show no missing values for the date column (which is good and
a relief), and very few missing rainfall days. Most of those were
infilled. The other variables have reasonable values. Hence we proceed.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.2e Checking the data***          ***Fig. 6.2f Results***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.2e.png){width="2.6116010498687663in"   ![](figures/Fig6.2f.png){width="3.4144444444444444in"
  height="2.139025590551181in"}                          height="1.937435476815398in"}

  ------------------------------------------------------------------------------------------------------------

Use ***Climatic \> Date \> Use Date***, Fig. 6.2g. Then it is convenient
to reorder the columns to put the date variables before the climatic
data, Fig. 6.2h.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.2g Generate further date variables***          ***Fig. 6.2h Data***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.2g.png){width="2.626457786526684in"   ![](figures/Fig6.2h.png){width="3.3682863079615046in"
  height="4.1896883202099735in"}                          height="2.546360454943132in"}

  ------------------------------------------------------------------------------------------------------------

Finally, in this preparation, use ***Climatic \> Define Climatic
Data***. The dialogue should fill automatically as shown in Fig. 6.2i.
Check that the data are unique and press Ok.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.2i***                 ***Fig. 6.2j A count column***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.2i.png){width="3.0969564741907263in"   ![](figures/Fig6.2j.png){width="2.9782939632545933in"
  height="3.6375339020122484in"}                          height="4.019535214348206in"}

  ------------------------------------------------------------------------------------------------------------

Finally, a new step, because we would like to analyse the number of rain
days as well as the rainfall totals. A new column, giving whether a day
was rainy-or-not, is generated. We show two ways this new column can be
generated.

The first way is simple, but it generates a complicated R command,
because it is a special case of a more general function. Use ***Climatic
\> Prepare \> Transform***, and complete the dialogue as shown in Fig.
6.2j. This produces a new column, which takes the value 1 for each rain
day, and 0 otherwise. We have explained in Chapter 2 why we use the
seemingly odd value of 0.85mm as a threshold for rain[^26].

Now try the second method, which generates a very simple R command. It
uses R-Instat's powerful calculator, from ***Prepare \> Column:
Calculate \> Calculations***, Fig. 6.2k

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.2k Using the calculator***          ***Fig. 6.2l Using the additional logical keyboard***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.2k.png){width="3.0777515310586177in"   ![](figures/Fig6.2l.png){width="2.968632983377078in"
  height="2.546943350831146in"}                          height="1.8799464129483814in"}

  ------------------------------------------------------------------------------------------------------------

The resulting data are shown in Fig. 6.2m. The calculator has produced a
logical column, while the transformation using Prepare \> Transform has
a column of 0 for dry and 1 for rain. There are the same in R as it
interprets TRUE as a 1 and False as a zero.

They are now not both needed, so delete one of them. We have kept the
logical column.

+-----------------------------------------------------------------------+
| ***Fig. 6.2m Data***                                                  |
|                                                                       |
| ***After Right-click \> Reorder columns***                            |
+=======================================================================+
| ![](figures/Fig6.2m.png){width="3.800120297462817in"                   |
| height="2.891173447069116in"}                                         |
+-----------------------------------------------------------------------+

## Annual summaries

The data are now ready to produce annual (or other) summaries. So, use
***Climatic \> Prepare \> Climatic Summaries***. It should initially be
as shown in Fig. 6.3a. (If not, then you might be in a different data
frame, or you may not have followed the steps in the section above.)

We are going to produce the annual totals. Fig. 6.3a also indicates it
is equally easy to produce the totals for any subset of the year.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.3a The climatic summary dialogue***          ***Fig. 6.3b The summaries sub-dialogue***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.3a.png){width="3.101706036745407in"   ![](figures/Fig6.3b.png){width="2.9698425196850393in"
  height="3.6587226596675415in"}                          height="3.166195319335083in"}

  ------------------------------------------------------------------------------------------------------------

In Fig. 6.3a add the rainfall column and then press the Summaries
button. In the sub-dialogue ***untick the N Total***, and keep the ***N
Non Missing*** and the ***Sum*** as shown in Fig. 6.3b. Then press
***Return*** and ***Ok***.

Now return to the dialogue and use the Rainday column instead of
rainfall.

Also Press on the Summaries button again and untick the N Non Missing
checkbox from Fig. 6.3b. Press Return and Ok again.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.3c***                                        ***Fig. 6.3d***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.3c.png){width="2.6389851268591427in"   ![](figures/Fig6.3d.png){width="3.254145888013998in"
  height="2.938444881889764in"}                          height="3.783175853018373in"}

  ------------------------------------------------------------------------------------------------------------

The results are in Fig. 6.3d. These data are now at the "year" levels
and there are 146 rows, i.e. years, from the 2 sites together. We see
that at Saltpond in 1944 the total rainfall was 724mm from 69 rain days.
So, the mean rain per rain day is over 10mm and sometimes considerably
so. For example, again from Fig. 6.3d in 1951 there was a total of
1428mm from 85 rain days.

There were some missing values in the data, but we defer a discussion of
this topic to the next section. Here we have been conservative in that
the annual totals have been set to missing if there were any days
missing in that year.

Graphs of the data can now be produced. The PICSA project includes
discussing time-series graphs with farmers. They must be simple to
produce, but also very clear. The special dialogue for this is
***Climatic \> PICSA \> Rainfall Graph***, Fig. 6.3e.

In Fig. 6.3e, check you are using the correct (yearly) data frame and
complete it as shown. In the sub-dialogue, opt to add the mean line, but
(at this stage) without a label.

In the sub-dialogue, click also on the Y-axis tab and set the lower
limit to 0 (zero).

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.3e PICSA-style rainfall graphs***                                        ***Fig. 6.3f Sub-dialogue to add lines***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.3e.png){width="3.3122484689413825in"   ![](figures/Fig6.3f.png){width="2.7570866141732284in"
  height="3.08748687664042in"}                          height="2.6320800524934382in"}

  ------------------------------------------------------------------------------------------------------------

  -----------------------------------------------------------------------
  ***Fig. 6.3g PICSA-style graph for 2 stations***
  -----------------------------------------------------------------------
  ![](figures/fig6.3g.png){width="6.065343394575678in"
  height="2.9547211286089237in"}

  -----------------------------------------------------------------------

It can be very useful for researchers and also intermediaries, to see
results from multiple stations. This is easy with R-Instat, where they
can be in the same data frame. Fig. 6.3h therefore shows another
example, with a facetted graph for 12 stations from xxx.

  -----------------------------------------------------------------------
  ***Fig. 6.3h PICSA-style rainfall graph for 12 stations from xxx***
  -----------------------------------------------------------------------
  
  -----------------------------------------------------------------------

[Graph to add]{.mark}

However, most farmers are particularly interested in the results from a
single station that is as close as possible to their location. Hence
once you have the appropriate graph for multiple stations, you can then
filter the data to look at each station in turn. Filtering is either
done from the right-click menu or from within each dialogue.

Return to the ***previous PICSA dialogue*** and choose ***Data
Options*** to give the sub-dialogue shown in Fig. 6.3i

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.3i Define a filter***                                        ***Fig. 6.3j Choose a single station***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.3i.png){width="2.0657567804024497in"   ![](figures/Fig6.3j.png){width="3.9102220034995625in"
  height="2.5604385389326336in"}                          height="2.5876093613298337in"}

  ------------------------------------------------------------------------------------------------------------

Choose to define a new Filter and complete the resulting sub-dialogue as
shown in Fig. 6.3j. As shown in Fig. 6.3j, give the filter the same name
as the station. That will make it easy later.

Now return to the PICSA dialogue. Choose the ***sub-dialogue*** and
change the lines to give ***terciles with labels.*** The resulting graph
for just Saltpond is in Fig. 6.3k.

  -----------------------------------------------------------------------------------------------------------
  ***Fig. 6.3k Single station with terciles***          ***Fig. 6.3l Graph for the second station***
  ----------------------------------------------------- -----------------------------------------------------
  ![](figures/Fig6.3k.png){width="2.961254374453193in"   ![](figures/Fig6.3l.png){width="2.897379702537183in"
  height="2.9208737970253718in"}                        height="2.897379702537183in"}

  -----------------------------------------------------------------------------------------------------------

Repeat the filtering exercise above to ***set a filter for Tamale***.
The resulting graph with the line for the mean is in Fig. 6.3l.

Similar graphs for the number of rain days is also sometimes needed for
PICSA. They are now very easy to produce.

Return to the Climatic \> PICSA \> Rainfall Graph dialogue, and simply
change the Y-variable,

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.3m***                                       ***Fig. 6.3n***
  ----------------------------------------------------- ------------------------------------------------------
  ![](figures/Fig6.3m.png){width="2.777380796150481in"   ![](figures/Fig6.3n.png){width="3.2896248906386703in"
  height="2.7827416885389327in"}                        height="3.2821511373578303in"}

  ------------------------------------------------------------------------------------------------------------

***Return to the dialogue*** again, press ***Data Option*** and choose
the filter you called ***Saltpond***, Fig. 6.3o. You don't need to
define the filter as that was done earlier. You can just keep using it.

  -----------------------------------------------------------------------------------------------------------
  ***Fig. 6.3o***                                       ***Fig. 6.3p***
  ----------------------------------------------------- -----------------------------------------------------
  ![](figures/Fig6.3o.png){width="2.454096675415573in"   ![](figures/Fig6.3p.png){width="3.489805336832896in"
  height="2.736826334208224in"}                         height="3.489805336832896in"}

  -----------------------------------------------------------------------------------------------------------

Finally, in this section, we stress that the idea of the climatic menu
is simply to make it even easier to do common climatic analyses. By
"even easier" we mean easier than using the main dialogues in R-instat.
If what you would like to do is not (yet) possible with the climatic
menu it may still be possible with the ordinary use of R-Instat[^27]. It
is important that you remain in charge and are not limited by the
particular dialogues. As an example, suppose you would like to fit a
trend line to the rainfall data. The PICSA graphs permit horizontal
lines, but not trend lines. Perhaps the last graph, Fig. 6.3p has a
downward slope?

  -----------------------------------------------------------------------------------------------------------
  ***Fig. 6.3q Graph with "ordinary" dialogue***                                       ***Fig. 6.3r Saltpond annual rainfall with trend line***
  ----------------------------------------------------- -----------------------------------------------------
  ![](figures/Fig6.3q.png){width="2.9803674540682414in"   ![](figures/Fig6.3r.png){width="3.053941382327209in"
  height="2.744633639545057in"}                         height="3.1782305336832897in"}

  -----------------------------------------------------------------------------------------------------------

One way to check this possibility is through the "ordinary" graphics
dialogues in R-Instat. So, use ***Describe \> Specific \> Line Plot***
and complete the dialogue as shown in Fig. 6.3q. The results are in Fig.
6.3t. They perhaps hint at a possible trend[^28]. If there is a trend

  -----------------------------------------------------------------------------------------------------------
  ***Fig. 6.3s Analysis for rainfall totals***                                       ***Fig. 6.3t Cumulative or Exceedance Graph***
  ----------------------------------------------------- -----------------------------------------------------
  ![](figures/Fig6.3s.png){width="2.8972101924759404in"   ![](figures/Fig6.3t.png){width="3.1671128608923884in"
  height="3.0358169291338584in"}                         height="3.575345581802275in"}

  -----------------------------------------------------------------------------------------------------------

If there is a trend at Saltpond, perhaps it should also be evident in an
analysis of the annual totals. Fig. 6.3s shows this is not the case.

Our aim in the above discussion is primarily to discuss the value of the
"ordinary" R-Instat dialogues, so users do not restrict all their
analysis to the climatic menu. We return to trend analysis in Section
6.6, when we process the temperature data.

Time series are not the only way to display the annual summaries. Use
***Climatic \> PICSA \> Cumulative/Exceedance Graph***, and complete the
dialogue as shown in Fig. 6.3t. If the filter is still operating, then
remove it by including ***Data Options*** and choosing ***no_filter***
in the resulting subdialogue, Fig. 6.3u. The resulting graph is in Fig.
6.3v.

  -----------------------------------------------------------------------------------------------------------
  ***Fig. 6.3u Removing a filter***                                       ***Fig. 6.3v Cumulative distributions***
  ----------------------------------------------------- -----------------------------------------------------
  ![](figures/Fig6.3u.png){width="2.9515409011373577in"   ![](figures/Fig6.3v.png){width="3.0426082677165356in"
  height="3.9029779090113736in"}                         height="3.2070734908136482in"}

  -----------------------------------------------------------------------------------------------------------

Statisticians like cumulative distributions, but many users prefer
exceedance graphs. If that is your wish, then return to the dialogue in
Fig. 6.3t and tick the box for an ***Exceedance Graph***. The result is
in Fig. 6.3w.

These are just the inverse of each other. Starting with an amount -- on
the x-axis, you can read the probability of the total rainfall being
less than this amount (cumulative graph) or greater than this amount
(exceedance graph). So, if you need 800mm for a particular crop, then
the exceedance graph informs you there is about a 75% chance of getting
this amount, or more, at Saltpond and about a 90% chance at Tamale. The
cumulate graph would show a 25% or 10% chance of failure, i.e. of
getting less than this amount.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.3w Exceedance graph***                       ***Fig. 6.3x Exceedance graph for rain days***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.3w.png){width="2.8365912073490813in"   ![](figures/Fig6.3x.png){width="2.862159886264217in"
  height="2.9799201662292214in"}                         height="3.0429265091863518in"}

  ------------------------------------------------------------------------------------------------------------

Finally change the variable in the dialogue in Fig. 6.3t to the number
of rain days, to give the graph in Fig. 6.3x. The shapes are the same.
The steeper the graphs the smaller the variability and Fig. 6.3v, w and
x all show the totals are slightly more variable at Saltpond, compared
to Tamale. This can be confirmed numerically using Describe \> Specific
\> Summary Tables. Results are in Fig. 6.3y.

  -----------------------------------------------------------------------
  ***Fig. 6.3y Numerical results***
  -----------------------------------------------------------------------
  ![](figures/Fig6.3y.png){width="3.020988626421697in"
  height="3.3265594925634296in"}

  -----------------------------------------------------------------------

The standard deviation of the annual totals is 273mm at Saltpond
compared to 193mm at Tamale, while the means are relatively close.
Similarly the standard deviation for the number of rain days is 13 days
at Saltpond compared to 9 days at Tamale.

This points to looking at the data in more detail. Hence monthly
summaries are examined in the next section.

## More detailed summaries - rainfall

For many applications it is important to know about the seasonality of
the data. In this section we therefore consider monthly (rather than
annual) totals.

In Fig. 6.2g [(link)]{.mark} we used the Climatic \> Dates \> Use Date
dialogue to add the months to the daily data and these are used in this
section. Other possibilities with this dialogue are to produce quarters,
dekads (10-day periods), pentades or weeks. Any of these periods can be
used instead.

Use ***Climatic \> Prepare \> Climatic Summaries***. It was used
initially in Fig. 6.3a to produce the annual summaries.

  ------------------------------------------------------------------------------------------------------------
  ***Fig.6.4a***                                        ***Fig. 6.4b***
  ----------------------------------------------------- ------------------------------------------------------
  ![](figures/Fig6.4a.png){width="3.604351487314086in"   ![](figures/Fig6.4b.png){width="2.3751224846894137in"
  height="4.090487751531058in"}                         height="3.6876891951006123in"}

  ------------------------------------------------------------------------------------------------------------

In Fig. 6.4a change the tab at the top to Annual + Within and complete
as shown. Click on Summaries and choose just the 2 statistics shown in
Fig. 6.4b. Then press Return and Ok.

Now change from rainfall to Rainday in Fig. 6.4a and use the Summaries
to just get the Sum, i.e. untick the N Non Missing.

The resulting data frame is shown in Fig. 6.4c. There are 1751 rows of
data, i.e. the 141 years for the 2 stations, times 12, because the data
are now monthly. For example, at Saltpond, January 1944 had a total of
41mm from 2 rain days, whie June of the same year had 22 rain days and a
total of 256mm.

  -----------------------------------------------------------------------------------------------------------
  ***Fig. 6.4c***                                       ***Fig. 6.4d***
  ----------------------------------------------------- -----------------------------------------------------
  ![](figures/Fig6.4c.png){width="3.071080489938758in"   ![](figures/Fig6.4d.png){width="2.824102143482065in"
  height="3.2467333770778652in"}                        height="3.2376738845144355in"}

  -----------------------------------------------------------------------------------------------------------

One way to show the seasonal pattern is through boxplots. Use
***Describe \> Specific \> Boxplot*** and complete the dialogue as shown
in Fig. 6.4d. Use the ***Plot Options***, Fig. 6.4e, to include the
stations as facets and give the results as in Fig 6.4f[^29].

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.4e***                                       ***Fig. 6.4f***
  ----------------------------------------------------- ------------------------------------------------------
  ![](figures/Fig6.4e.png){width="2.658153980752406in"   ![](figures/Fig6.4f.png){width="3.3286668853893264in"
  height="2.7355227471566055in"}                        height="3.287153324584427in"}

  ------------------------------------------------------------------------------------------------------------

In Fig. 6.4d, change the variable to ***sum_Rainday*** to also give the
graph in Fig.6.4g.

Both graphs show the different seasonal pattern at the 2 sites. June is
the peak month of the rainy season at Saltpond and there is one year
where the monthly total exceeded 800mm. In June, Fig. 6.4g also shows on
average about half the days are rainy at Saltpond and that is similar to
the number of rain days in Tamale in September.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.4g***                                        ***Fig. 6.4h***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.4g.png){width="3.1435793963254595in"   ![](figures/Fig6.4h.png){width="2.916351706036745in"
  height="3.0935706474190727in"}                         height="2.7955577427821523in"}

  ------------------------------------------------------------------------------------------------------------

Line plots can show the seasonal and time-series nature of the data
together. As an example, use ***Describe \> Specific \> Line*** and
complete as shown in Fig. 6.4h. In the Plot Options, use the Month as
the factor for the facets, to give the graph as shown in Fig. 6.4i[^30]

  -----------------------------------------------------------------------
  ***Fig. 6.4i***
  -----------------------------------------------------------------------
  ![](figures/Fig6.4i.png){width="6.0539687226596675in"
  height="2.929728783902012in"}

  -----------------------------------------------------------------------

The results in Fig. 6.4i show the interesting nature of the June
rainfall totals at Saltpond and that the extreme monthly total was in
1962. It also shows the way Tamale consistently has more rainfall than
Saltpond in July to September.

The same type of graph can also be produced for the number of raindays,
see Fig. 6.4j for a different layout[^31]. It also shows that the
initial analysis of rainfall trends using the annual rainfall totals may
have been over-simplistic. If trends do exist, then the next step could
be to examine whether they are consistent, or not, during the year, i.e.
for the different months. Thus, if rainfall seems to be decreasing, then
is that in all months/seasons, or just in a part of the year. This issue
is examined further in Section 6.6 when analysing the temperature
records.

  -----------------------------------------------------------------------
  ***Fig. 6.4j Time series graphs for the 2 stations by month***
  -----------------------------------------------------------------------
  ![](figures/Fig6.4j.png){width="6.067823709536308in"
  height="2.9034930008748905in"}

  -----------------------------------------------------------------------

In reports it can be useful to include the daily data for a sample of
the years. Fig. 6.2k shows the daily data for Saltpond in 1962[^32],
when June had exceptionally high rainfall.

  -----------------------------------------------------------------------
  ***Fig. 6.2k Daily data for Saltpond for 1962***
  -----------------------------------------------------------------------
  ![](figures/Fig6.4k.png){width="2.9407699037620296in"
  height="4.576652449693788in"}

  -----------------------------------------------------------------------

There is nothing obviously wrong with the June data, but they look
sufficiently curious, that a check back to the paper records and perhaps
with nearby stations would seem sensible.

## Options for Missing values

Analyses need to be able to take account of missing values in the data.
Statistical packages are usually "sensible" in their handling of missing
values and R is no exception. However, defining how they are to be
handled in each circumstance is the responsibility of the user and we
consider here the options in R and R-Instat.

To illustrate the problem Fig. 6.5a gives an inventory plot for the
Ghana data. It shows there is hardly a problem for the rainfall data.
The measurement of the other elements started later and there is a
slightly greater proportion of missing values.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.5a***                                        ***Fig. 6.5b Default annual summaries of rainfall***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.5a.png){width="3.0282392825896762in"   ![](figures/Fig6.5b.png){width="3.0918077427821524in"
  height="3.141479658792651in"}                         height="3.154325240594926in"}

  ------------------------------------------------------------------------------------------------------------

Fig. 6.5b shows some of the annual totals that were used for analysis in
Section 6.3. A column in Fig. 6.3b shows also the number of missing
values each year. It shows there were missing values in the last 2 years
and the annual summary has therefore been set to missing. This is "safe"
but it may be disappointing as the last 2 years totals have therefore
been set to missing, and have therefore been excluded from the analysis.

Repeating this point, the default in R, and hence in R-Instat, is that
when there are any missing values (even just one day in the year) then
the summary is set to missing.

The opposite approach is also simple to undertake. This is where all the
missing days are omitted, and the summary is then calculated using the
remaining data. This uses the same ***Climatic \> Prepare \> Climatic
Summaries*** dialogue, but check the box labelled ***Omit Missing
Values***.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.5c Data with both summaries***                                        ***Fig. 6.5d Years with missing values***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.5c.png){width="2.973963254593176in"   ![](figures/Fig6.5d.png){width="3.130285433070866in"
  height="3.0356014873140857in"}                         height="1.9515627734033245in"}

  ------------------------------------------------------------------------------------------------------------

The results are in the last 2 columns of Fig. 6.5c. They show the total
rainfall to be 665mm in Tamale in 2015 from 53 rain days -- quite low
compared with other years. In 2016 the values are 998mm with 67 rain
days.

R-Instat has added intermediate options described below. Before that, we
consider what more can be done with just these 2 extremes.

Fig. 6.5d shows the annual data for those years where there are missing
values. There are just 9 years overall, 5 at Saltpond and 4 at Tamale.
Hence, with data from 1944 to 2016, this leaves over 60 years of data at
each site. Hence, one option is to accept the omission of those years
and proceed, which is what was done in Section 6.3.

A second possibility results from the observation, Fig. 6.5d, that in 3
of the 9 years there was just a single day missing in the year. Perhaps
it is reasonable to accept the totals in those years and then just have
6 missing years overall.

To go further we now look in more detail at the daily data. One
coincidence is that both sites have missing data in 1949 and an
examination is that this is for the same 3 months, i.e. from October to
December. We don't like coincidences and wonder why.

More generally, the other years have just one or two months missing. If
that were between November and February -- when there is usually little
rain, then perhaps the total could be accepted. In this case that is not
the case. For example August 2015 is missing in Tamale, and this perhaps
explains why the total and number of rain days was low in that year.
Omitting it, as we did, in Section 6.3, was sensible.

A more major possibility is that Saltpond collects data every 3 hours
and Tamale collects hourly data. So perhaps the Met service has more
detailed records that could help to infill the missing daily values.

To see further options for missing values, return to the ***Climatic \>
Prepare \> Climatic Summaries*** dialogue. Choose the ***Summaries***
button and the ***Missing Options*** tab, Fig. 6.5e. The setting we
chose of 27 means that any year with a month or more missing, gives a
missing summary. In this case, as shown in Fig. 6.5f, it has just given
the annual totals for the 3 years with just a single missing day.

In some examples the third option in Fig. 6.5e becomes important.
Sometimes the data, as supplied, starts, or ends during a year. In this
instance the first and/or the last year may be incomplete. For example
the Tamale data in 1944 start in February, rather than January. This was
not an issue, because January is relatively dry, but had they started in
July 1944 that would have been different and should have been allowed
for.

In Fig. 6.5e this corresponds to setting the Option Not Missing to about
340 (days) rather than the Missing Days to 27.

  -------------------------------------------------------------------------------------------------------------
  ***Fig. 6.5e***                                        ***Fig. 6.5f***
  ------------------------------------------------------ ------------------------------------------------------
  ![](figures/Fig6.5e.png){width="2.6578444881889762in"   ![](figures/Fig6.5f.png){width="3.4301706036745405in"
  height="2.0866196412948383in"}                         height="2.027972440944882in"}

  -------------------------------------------------------------------------------------------------------------

A different, and more major, operation is to try to "infill" or complete
the data, where there are missing values. There is a wide variety of
methods, ranging from input of the mean value from that day of the year
to using estimates from a neighbouring station, or from satellite
observations. They are considered in [Chapter xxx]{.mark}.

## Processing temperature data

The ***Climatic \> Prepare \> Climatic Summaries*** dialogue applies to
any element. With the Ghana data the annual temperature summaries can
therefore be added to those of the rainfall calculated in Section 6.3.

  -------------------------------------------------------------------------------------------------------------
  ***Fig. 6.6a***                                        ***Fig. 6.6b Include temperature extremes***
  ------------------------------------------------------ ------------------------------------------------------
  ![](figures/Fig6.6a.png){width="3.0125404636920385in"   ![](figures/Fig6.6b.png){width="2.936470909886264in"
  height="3.210363079615048in"}                         height="2.6364774715660544in"}

  -------------------------------------------------------------------------------------------------------------

Complete the dialogue as shown in Fig. 6.6a and then the Summaries
sub-dialogue as shown in Fig. 6.6b. This produces the annual mean and
the annual extremes of the daily minimum temperatures.

Then use the Missing Options tab, shown in Fig. 6.6b and complete it as
shown earlier in Fig. 6.5e. This will give the annual summaries if there
are a few missing days, but not if a month or more is missing.

Once you have these summaries, return to the dialogue in Fig. 6.6a and
replace the minimum by the maximum temperatures.

The measurement of temperatures started in 1960, hence the summary data
are now filtered, prior to producing graphs.

  -------------------------------------------------------------------------------------------------------------
  ***Fig. 6.6c Annual temperature data***                                        ***Fig. 6.6d***
  ------------------------------------------------------ ------------------------------------------------------
  ![](figures/Fig6.6c.png){width="3.1401574803149606in"   ![](figures/Fig6.6d.png){width="2.8428937007874016in"
  height="2.157737314085739in"}                         height="2.385305118110236in"}

  -------------------------------------------------------------------------------------------------------------

Use the ***Describe \> Specific \> Line Plot*** dialogue, and complete
as shown in Fig. 6.6d. In the plot options, choose to ***Facet by the
station***[^33].

  -----------------------------------------------------------------------
  ***Fig. 6.6e***
  -----------------------------------------------------------------------
  ![](figures/Fig6.6e.png){width="5.985853018372703in"
  height="3.079802055993001in"}

  -----------------------------------------------------------------------

In Fig. 6.6e the data on the extremes must be treated with caution,
because they are the values on a single day each year. There does appear
to be a trend in the mean for Tmax, particularly at Tamale. This can be
confirmed using the ***Model \> Three Variables \> Fit Model***
dialogue, which is described in more detail in Chapter xxx. The results
show an estimated increase of 2.3°C for Tamale. The estimated increase
for Saltpond was just 0.4°C and that was not statistically significant.

The graph can be repeated for the minimum temperatures (not shown).
Instead, Fig. 6.6 shows Tmax and Tmin together. The estimated trend in
Tmin is an increase of 2.0°C per 100 years and is almost the same at the
two sites. Fig. 6.6f also shows clearly the much greater diurnal range
at Tamale, compared to Saltpond.

  -----------------------------------------------------------------------
  ***Fig. 6.6f***
  -----------------------------------------------------------------------
  ![](figures/Fig6.6f.png){width="6.064581146106737in"
  height="3.0027263779527558in"}

  -----------------------------------------------------------------------

In Fig. 6.6f (and earlier in Fig. 6.6e) the mean line for Saltpond looks
odd. In this Chapter the quality control steps, discussed in Chapter 5,
have been omitted and, as usual, that was not a good idea! Fortunately,
the daily data are available, so we return to these and do a simple time
series plot of the daily records, Fig. 6.6g. This indicates an oddity in
the data in about 1974.

  -------------------------------------------------------------------------------------------------------------
  ***Fig. 6.6g Tmax for Saltpond daily data by Date***                                        ***Fig. 6.6h Monthly means for Tmax***
  ------------------------------------------------------ ------------------------------------------------------
  ![](figures/Fig6.6g.png){width="3.320465879265092in"   ![](figures/Fig6.6h.png){width="2.7201159230096237in"
  height="2.6814621609798777in"}                         height="1.5in"}

  -------------------------------------------------------------------------------------------------------------

This is confirmed in Fig. 6.6h, where the monthly means for Tmax at
Saltpond are displayed for the 1970s[^34]. They show a drop of about 2
degrees from May 1974.

The next step in this small investigation is to display the daily
records, as shown in Fig. 6.6i. Looking at the daily data it became
clear they were originally recorded in degrees Fahrenheit and (at least
usually) just to the nearest degree. Hence, for clarity, the Tmax data
were transformed back into Fahrenheit[^35] and then displayed, as shown
in Fig. 6.6i.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.6i***                                       ***Fig. 6.6j***
  ----------------------------------------------------- ------------------------------------------------------
  ![](figures/Fig6.6i.png){width="2.698963254593176in"   ![](figures/Fig6.6j.png){width="3.2956649168853893in"
  height="4.114812992125985in"}                         height="3.85672353455818in"}

  ------------------------------------------------------------------------------------------------------------

Fig. 6.6i confirms the change was in May 1974, or possibly 30 April. In
most years temperatures in May are about 0.5°C lower than April, or
about 1°F. In the 1974 record it is 4 or 5 degrees Fahrenheit lower.

Fig. 6.6j therefore repeats the analysis, shown earlier in Fig. 6.6e,
but just from 1975. The results are now consistent with the data from
Tmin at Saltpond and with the Tamale data. The trend for the mean is
slightly higher at 3.2°C per 100 years.

Analyses of the temperature records, like the above, are common. There
is an immediate follow-up question that is often omitted, namely is the
trend in the temperatures consistent through the year, or is it perhaps
different in the rainy and dry seasons?

As in Section 6.4 for the rainfall, we therefore extend the analysis and
examine the monthly data.

## More detailed summaries - temperatures

We examine the possible trends in Tmin and Tmax, at the two stations,
monthly. A specific question is whether there is evidence for a
different trend in some months, compared to others.

For simplicity, given the inhomogeneity of Tmax at Saltpond, the daily
data are first filtered so only the data from 1975 are analysed, Fig.
6.7a.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.7a Filter the data (optional)***                                       ***Fig. 6.7bMonthly summaries***
  ----------------------------------------------------- ------------------------------------------------------
  ![](figures/Fig6.7a.png){width="2.7416404199475064in"   ![](figures/Fig6.7b.png){width="3.1761122047244092in"
  height="2.041117672790901in"}                         height="3.4447670603674543in"}

  ------------------------------------------------------------------------------------------------------------

Then use ***Climatic \> Prepare \> Climatic Summaries***, Fig. 9.7b. In
Fig. 9.7b click on ***Summaries*** and just choose the ***mean***.

Then repeat for Tmax, to give the data as in Fig. 6.7c.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.7c Monthly means for Tmin and Tmax***                                       ***Fig. 6.7d***
  ----------------------------------------------------- ------------------------------------------------------
  ![](figures/Fig6.7c.png){width="3.0479516622922134in"   ![](figures/Fig6.7d.png){width="2.974469597550306in"
  height="3.2115999562554682in"}                         height="2.6106496062992126in"}

  ------------------------------------------------------------------------------------------------------------

Then ***Describe \> Specific \> Line Plot***, as shown in Fig. 6.7d
indicates a reasonably consistent slope at both sites, for each of the
months, Fig. 6.7e.

  -----------------------------------------------------------------------
  ***Fig. 6.7e Trends by month for the two stations***
  -----------------------------------------------------------------------
  ![](figures/Fig6.7e.png){width="6.119820647419073in"
  height="2.979225721784777in"}

  -----------------------------------------------------------------------

However, this does not quite answer the question posed, namely that the
trend is independent of the month, i.e. it is the same in each month.

The modelling dialogues are needed to address this hypothesis. The menu
is shown in Fig. 6.7f.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.7f The Model menu***                                       ***Fig. 6.7g Filter by Station***
  ----------------------------------------------------- ------------------------------------------------------
  ![](figures/Fig6.7f.png){width="2.062606080489939in"   ![](figures/Fig6.7g.png){width="3.8126957567804025in"
  height="2.4237357830271216in"}                         height="2.2848392388451444in"}

  ------------------------------------------------------------------------------------------------------------

In the modelling menu, Fig. 6.7f, the ***One Variable*** sub-menu
permits a wide variety of distributions to be fitted to a single
variable, i.e. a single column of data.

Moving down in Fig. 6.7f the Two Variables sub-menu is designed to model
a single y (dependent) variable against one x (independent) variable. An
example would be Tmax against the year. That would be ok if we had
annual data, as in Section 6.6, but we have the monthly data.

In our case we need at least three variables. The dependent is initially
Tmax and this is modelled as a function of both the year and month, i.e.
we have a total of 3 variables.

Once you use the modelling dialogues as a routine, then the General
dialogues are usually used, or (below the line in Fig. 6.7f) the even
more general Model dialogue, where you just give an R command.

To simplify the modelling, first filter for a single station, Fig. 6.7g.
Call the filter ***Saltpond*** (rather than Filter1).

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.7h Make a new data frame***                                       ***Fig. 6.7i***
  ----------------------------------------------------- ------------------------------------------------------
  ![](figures/Fig6.7h.png){width="2.9819280402449695in"   ![](figures/Fig6.7i.png){width="3.089155730533683in"
  height="2.9965923009623796in"}                         height="2.7298272090988625in"}

  ------------------------------------------------------------------------------------------------------------

Return. to the main dialogue and opt to ***Apply as Subset***, Fig.
6.7h.

Now, for the first model. Use ***Model \> Three Variables \> Fit
Model*** with the new Saltpond data frame and complete it as shown in
Fig. 6.7i. Initially you have a '\*' between the year and month
variables. This fits a different slope for each month, as shown earlier,
for Saltpond, in Fig. 6.7e.

  ------------------------------------------------------------------------------------------------------------
  ***Fig. 6.7j***                                        ***Fig. 6.7k***
  ------------------------------------------------------ -----------------------------------------------------
  ![](figures/Fig6.7j.png){width="3.3083158355205597in"   ![](figures/Fig6.7k.png){width="2.707610454943132in"
  height="1.2852865266841644in"}                         height="1.6346784776902887in"}

  ------------------------------------------------------------------------------------------------------------

A lot of results are produced. Key information is the ANOVA table shown
in Fig. 6.7j. This shows that there is a clear ***trend (year***) and
***seasonality (month_abbr)***. It also shows that there is no evidence
of the interaction, i.e. the ***year:month_abbr*** explains very little
variation in the data, and what it explains is not statistically
significant.

Hence, the separate slopes each month are not needed. A parallel line
model is adequate.

So, return to the dialogue in Fig. 6.7i and change the '\*' into a '+'.
At the same time, click on the Display Options and choose to Save the
Fitted Values, Fig. 6.7k.

Before examining the results there is one small (optional) change that
sometimes simplifies the interpretation. With the year as given, i.e.
starting in 1975, the origin is almost 2000 years ago. Instead you could
make 1975 as the origin, using ***Prepare \> Column: Calculations \>
Calculate*** and making a new column, say ***yr \<- year -1975***. Then
use ***yr*** instead of year in the model.

  -------------------------------------------------------------------------------------------------------------
  ***Fig. 6.7l***                                        ***Fig. 6.7m***
  ------------------------------------------------------ ------------------------------------------------------
  ![](figures/Fig6.7l.png){width="2.4777220034995624in"   ![](figures/Fig6.7m.png){width="3.5530839895013124in"
  height="2.535048118985127in"}                          height="2.508620953630796in"}

  -------------------------------------------------------------------------------------------------------------

Interpreting the model in Fig. 6.7l the trend (yr coefficient) is a
possibly disturbing 3.4 degrees per 100 years. For the seasonality, the
mean temperature in January 1975 was estimated as 29.9°C. February and
March were each estimated to be an average of 0.6°C higher, i.e. about
30.5 degrees, while August had the lowest average temperatures.

In some stations there is "local warming" where the station surroundings
are more built up. Hence this should be checked, before assuming the
large trend per year is a feature of global warming.

A similar analysis for Tmin, again shows no evidence that a different
trend is needed each month. Saving those fitted values also, as shown in
Fig. 6.7m permits the parallel lines to be plotted, using ***Describe \>
Specific \> Line Plot***, as shown in Fig. 6.7n

  -----------------------------------------------------------------------
  ***Fig. 6.7n Observed and fitted temperatures at Saltpond***
  -----------------------------------------------------------------------
  ![](figures/Fig6.7n.png){width="6.061395450568679in"
  height="2.9729440069991253in"}

  -----------------------------------------------------------------------