The following is a document-style version of a presentation I gave at work a couple weeks ago. It's a little less useful for a general audience because you don't have access to the same database I have, but I figured it might be useful for someone who is looking at using dplyr or in manipulating the GHCND data from NCDC.
Today we’re going to briefly take a look at the GHCND climate database and a couple new R packages (dplyr and tidyr) that make data import and manipulation a lot easier than using the standard library.
For further reading, consult the vignettes for dplyr and tidyr, and download the cheat sheet:
The GHCND database contains daily observation data from locations around the world. The README linked above describes the data set and the way the data is formatted. I have written scripts that process the station data and the yearly download files and insert it into a PostgreSQL database (noaa).
The script for inserting a yearly file (downloaded from http://www1.ncdc.noaa.gov/pub/data/ghcn/daily/by_year/) is here: ghcn-daily-by_year_process.py
Without going into too much detail on the subject (read Hadley Wickham’s paper) for more information, but the basic idea is that it is much easier to analyze data when it is in a particular, “tidy”, form. A Tidy dataset has a single table for each type of real world object or type of data, and each table has one column per variable measured and one row per observation.
For example, here’s a tidy table representing daily weather observations with station × date as rows and the various variables as columns.
Getting raw data into this format is what we’ll look at today.
R libraries & data import
First, let’s load the libraries we’ll need:
library(dplyr) # data import library(tidyr) # column / row manipulation library(knitr) # tabular export library(ggplot2) # plotting library(scales) # “pretty” scaling library(lubridate) # date / time manipulations
dplyr and tidyr are the data import and manipulation libraries we will use, knitr is used to produce tabular data in report-quality forms, ggplot2 and scales are plotting libraries, and lubridate is a library that makes date and time manipulation easier.
Also note the warnings about how several R functions have been “masked” when we imported dplyr. This just means we'll be getting the dplyr versions instead of those we might be used to. In cases where we need both, you can preface the function with it's package: base::filter would us the normal filter function instead of the one from dplyr.
Next, connect to the database and the three tables we will need:
noaa_db <- src_postgres(host="mason", dbname="noaa") ghcnd_obs <- tbl(noaa_db, "ghcnd_obs") ghcnd_vars <- tbl(noaa_db, "ghcnd_variables")
The first statement connects us to the database and the next two create table links to the observation table and the variables table.
Here’s what those two tables look like:
## Observations: 29404870 ## Variables: ## $ station_id (chr) "USW00027502", "USW00027502", "USW00027502", "USW0... ## $ dte (date) 2011-05-01, 2011-05-01, 2011-05-01, 2011-05-01, 2... ## $ variable (chr) "AWND", "FMTM", "PRCP", "SNOW", "SNWD", "TMAX", "T... ## $ raw_value (dbl) 32, 631, 0, 0, 229, -100, -156, 90, 90, 54, 67, 1,... ## $ meas_flag (chr) "", "", "T", "T", "", "", "", "", "", "", "", "", ... ## $ qual_flag (chr) "", "", "", "", "", "", "", "", "", "", "", "", ""... ## $ source_flag (chr) "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", ... ## $ time_of_obs (int) NA, NA, 0, NA, NA, 0, 0, NA, NA, NA, NA, NA, NA, N...
## Observations: 82 ## Variables: ## $ variable (chr) "AWND", "EVAP", "MDEV", "MDPR", "MNPN", "MXPN",... ## $ description (chr) "Average daily wind speed (tenths of meters per... ## $ raw_multiplier (dbl) 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0....
Each row in the observation table rows contain the station_id, date, a variable code, the raw value for that variable, and a series of flags indicating data quality, source, and special measurements such as the “trace” value used for precipitation under the minimum measurable value.
Each row in the variables table contains a variable code, description and the multiplier used to convert the raw value from the observation table into an actual value.
This is an example of completely “normalized” data, and it’s stored this way because not all weather stations record all possible variables, and rather than having a single row for each station × date with a whole bunch of empty columns for those variables not measured, each row contains the station × data × variable data.
We are also missing information about the stations, so let’s load that data:
fai_stations <- tbl(noaa_db, "ghcnd_stations") %>% filter(station_name %in% c("FAIRBANKS INTL AP", "UNIVERSITY EXP STN", "COLLEGE OBSY")) glimpse(fai_stations)
## Observations: 3 ## Variables: ## $ station_id (chr) "USC00502107", "USW00026411", "USC00509641" ## $ station_name (chr) "COLLEGE OBSY", "FAIRBANKS INTL AP", "UNIVERSITY ... ## $ latitude (dbl) 64.86030, 64.80389, 64.85690 ## $ longitude (dbl) -147.8484, -147.8761, -147.8610 ## $ elevation (dbl) 181.9656, 131.6736, 144.7800 ## $ coverage (dbl) 0.96, 1.00, 0.98 ## $ start_date (date) 1948-05-16, 1904-09-04, 1904-09-01 ## $ end_date (date) 2015-04-03, 2015-04-02, 2015-03-13 ## $ variables (chr) "TMIN TOBS WT11 SNWD SNOW WT04 WT14 TMAX WT05 DAP... ## $ the_geom (chr) "0101000020E6100000A5BDC117267B62C0EC2FBB270F3750...
The first part is the same as before, loading the ghcnd_stations table, but we are filtering that data down to just the Fairbanks area stations with long term records. To do this, we use the pipe operator %>% which takes the data from the left side and passes it to the function on the right side, the filter function in this case.
filter requires one or more conditional statements with variable names on the left side and the condition on the right. Multiple conditions can be separated by commas if you want all the conditions to be required (AND) or separated by a logic operator (& for AND, | for OR). For example: filter(latitude > 70, longitude < -140).
When used on database tables, filter can also use conditionals that are built into the database which are passed directly as part of a WHERE clause. In our code above, we’re using the %in% operator here to select the stations from a list.
Now we have the station_ids we need to get just the data we want from the observation table and combine it with the other tables.
Here’s how we do it:
fai_raw <- ghcnd_obs %>% inner_join(fai_stations, by="station_id") %>% inner_join(ghcnd_vars, by="variable") %>% mutate(value=raw_value*raw_multiplier) %>% filter(qual_flag=='') %>% select(station_name, dte, variable, value) %>% collect() glimpse(fai_raw)
In order, here’s what we’re doing:
- Assign the result to fai_raw
- Join the observation table with the filtered station data, using station_id as the variable to combine against. Because this is an “inner” join, we only get results where station_id matches in both the observation and the filtered station data. At this point we only have observation data from our long-term Fairbanks stations.
- Join the variable table with the Fairbanks area observation data, using variable to link the tables.
- Add a new variable called value which is calculated by multiplying raw_value (coming from the observation table) by raw_multiplier (coming from the variable table).
- Remove rows where the quality flag is not an empty space.
- Select only the station name, date, variable and actual value columns from the data. Before we did this, each row would contain every column from all three tables, and most of that information is not necessary.
- Finally, we “collect” the results. dplyr doesn’t actually perform the full SQL until it absolutely has to. Instead it’s retrieving a small subset so that we can test our operations quickly. When we are happy with the results, we use collect() to grab the full data.
The data is still in a format that makes it difficult to analyze, with each row in the result containing a single station × date × variable observation. A tidy version of this data requires each variable be a column in the table, each row being a single date at each station.
To “pivot” the data, we use the spread function, and we'll also calculate a new variable and reduce the number of columns in the result.
fai_pivot <- fai_raw %>% spread(variable, value) %>% mutate(TAVG=(TMIN+TMAX)/2.0) %>% select(station_name, dte, TAVG, TMIN, TMAX, TOBS, PRCP, SNOW, SNWD, WSF1, WDF1, WSF2, WDF2, WSF5, WDF5, WSFG, WDFG, TSUN) head(fai_pivot)
## Source: local data frame [6 x 18] ## ## station_name dte TAVG TMIN TMAX TOBS PRCP SNOW SNWD WSF1 WDF1 ## 1 COLLEGE OBSY 1948-05-16 11.70 5.6 17.8 16.1 NA NA NA NA NA ## 2 COLLEGE OBSY 1948-05-17 15.55 12.2 18.9 17.8 NA NA NA NA NA ## 3 COLLEGE OBSY 1948-05-18 14.40 9.4 19.4 16.1 NA NA NA NA NA ## 4 COLLEGE OBSY 1948-05-19 14.15 9.4 18.9 12.2 NA NA NA NA NA ## 5 COLLEGE OBSY 1948-05-20 10.25 6.1 14.4 14.4 NA NA NA NA NA ## 6 COLLEGE OBSY 1948-05-21 9.75 1.7 17.8 17.8 NA NA NA NA NA ## Variables not shown: WSF2 (dbl), WDF2 (dbl), WSF5 (dbl), WDF5 (dbl), WSFG ## (dbl), WDFG (dbl), TSUN (dbl)
spread takes two parameters, the variable we want to spread across the columns, and the variable we want to use as the data value for each row × column intersection.
Now that we've got the data in a format we can work with, let's look at a few examples.
Find the coldest temperatures by winter year
First, let’s find the coldest winter temperatures from each station, by winter year. “Winter year” is just a way of grouping winters into a single value. Instead of the 2014–2015 winter, it’s the 2014 winter year. We get this by subtracting 92 days (the days in January, February, March) from the date, then pulling off the year.
Here’s the code.
fai_winter_year_minimum <- fai_pivot %>% mutate(winter_year=year(dte - days(92))) %>% filter(winter_year < 2014) %>% group_by(station_name, winter_year) %>% select(station_name, winter_year, TMIN) %>% summarize(tmin=min(TMIN*9/5+32, na.rm=TRUE), n=n()) %>% filter(n>350) %>% select(station_name, winter_year, tmin) %>% spread(station_name, tmin) last_twenty <- fai_winter_year_minimum %>% filter(winter_year > 1993) last_twenty
## Source: local data frame [20 x 4] ## ## winter_year COLLEGE OBSY FAIRBANKS INTL AP UNIVERSITY EXP STN ## 1 1994 -43.96 -47.92 -47.92 ## 2 1995 -45.04 -45.04 -47.92 ## 3 1996 -50.98 -50.98 -54.04 ## 4 1997 -43.96 -47.92 -47.92 ## 5 1998 -52.06 -54.94 -54.04 ## 6 1999 -50.08 -52.96 -50.98 ## 7 2000 -27.94 -36.04 -27.04 ## 8 2001 -40.00 -43.06 -36.04 ## 9 2002 -34.96 -38.92 -34.06 ## 10 2003 -45.94 -45.94 NA ## 11 2004 NA -47.02 -49.00 ## 12 2005 -47.92 -50.98 -49.00 ## 13 2006 NA -43.96 -41.98 ## 14 2007 -38.92 -47.92 -45.94 ## 15 2008 -47.02 -47.02 -49.00 ## 16 2009 -32.98 -41.08 -41.08 ## 17 2010 -36.94 -43.96 -38.02 ## 18 2011 -47.92 -50.98 -52.06 ## 19 2012 -43.96 -47.92 -45.04 ## 20 2013 -36.94 -40.90 NA
See if you can follow the code above. The pipe operator makes is easy to see each operation performed along the way.
There are a couple new functions here, group_by and summarize. group_by indicates at what level we want to group the data, and summarize uses those groupings to perform summary calculations using aggregate functions. We group by station and winter year, then we use the minimum and n functions to get the minimum temperature and number of days in each year where temperature data was available. You can see we are using n to remove winter years where more than two weeks of data are missing.
Also notice that we’re using spread again in order to make a single column for each station containing the minimum temperature data.
Here’s how we can write out the table data as a restructuredText document, which can be converted into many document formats (PDF, ODF, HTML, etc.):
sink("last_twenty.rst") print(kable(last_twenty, format="rst")) sink()
|winter_year||COLLEGE OBSY||FAIRBANKS INTL AP||UNIVERSITY EXP STN|
Finally, let’s plot the minimum temperatures for all three stations.
q <- fai_winter_year_minimum %>% gather(station_name, tmin, -winter_year) %>% arrange(winter_year) %>% ggplot(aes(x=winter_year, y=tmin, colour=station_name)) + geom_point(size=1.5, position=position_jitter(w=0.5,h=0.0)) + geom_smooth(method="lm", se=FALSE) + scale_x_continuous(name="Winter Year", breaks=pretty_breaks(n=20)) + scale_y_continuous(name="Minimum temperature (degrees F)", breaks=pretty_breaks(n=10)) + scale_color_manual(name="Station", labels=c("College Observatory", "Fairbanks Airport", "University Exp. Station"), values=c("darkorange", "blue", "darkcyan")) + theme_bw() + # theme(legend.position = c(0.150, 0.850)) + theme(axis.text.x = element_text(angle=45, hjust=1)) print(q)
To plot the data, we need the data in a slightly different format with each row containing winter year, station name and the minimum temperature. We’re plotting minimum temperature against winter year, coloring the points and trendlines using the station name. That means all three of those variables need to be on the same row.
To do that we use gather. The first parameter is the name of variable the columns will be moved into (the station names, which are currently columns, will become values in a row named station_name). The second is the name of the column that stores the observations (tmin) and the parameters after that are the list of columns to gather together. In our case, rather than specifying the names of the columns, we're specifying the inverse: all the columns except winter_year.
The result of the gather looks like this:
fai_winter_year_minimum %>% gather(station_name, tmin, -winter_year)
## Source: local data frame [321 x 3] ## ## winter_year station_name tmin ## 1 1905 COLLEGE OBSY NA ## 2 1907 COLLEGE OBSY NA ## 3 1908 COLLEGE OBSY NA ## 4 1909 COLLEGE OBSY NA ## 5 1910 COLLEGE OBSY NA ## 6 1911 COLLEGE OBSY NA ## 7 1912 COLLEGE OBSY NA ## 8 1913 COLLEGE OBSY NA ## 9 1915 COLLEGE OBSY NA ## 10 1916 COLLEGE OBSY NA ## .. ... ... ...
The plot is produced using ggplot2. A full introduction would be a seminar by itself, but the basics of our plot can be summarized as follows.
ggplot(aes(x=winter_year, y=tmin, colour=station_name)) +
aes defines variables and grouping.
geom_point(size=1.5, position=position_jitter(w=0.5,h=0.0)) + geom_smooth(method="lm", se=FALSE) +
geom_point draws points, geom_smooth draws fitted lines.
scale_x_continuous(name="Winter Year", breaks=pretty_breaks(n=20)) + scale_y_continuous(name="Minimum temperature (degrees F)", breaks=pretty_breaks(n=10)) + scale_color_manual(name="Station", labels=c("College Observatory", "Fairbanks Airport", "University Exp. Station"), values=c("darkorange", "blue", "darkcyan")) +
Scale functions define how the data is scaled into a plot and controls labelling.
theme_bw() + theme(axis.text.x = element_text(angle=45, hjust=1))
Theme functions controls the style.
For more information:
Linear regression, winter year and minimum temperature
Finally let’s look at the significance of those regression lines:
summary(lm(data=fai_winter_year_minimum, `COLLEGE OBSY` ~ winter_year))
## ## Call: ## lm(formula = `COLLEGE OBSY` ~ winter_year, data = fai_winter_year_minimum) ## ## Residuals: ## Min 1Q Median 3Q Max ## -19.0748 -5.8204 0.1907 3.8042 17.1599 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -275.01062 105.20884 -2.614 0.0114 * ## winter_year 0.11635 0.05311 2.191 0.0325 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 7.599 on 58 degrees of freedom ## (47 observations deleted due to missingness) ## Multiple R-squared: 0.07643, Adjusted R-squared: 0.06051 ## F-statistic: 4.8 on 1 and 58 DF, p-value: 0.03249
summary(lm(data=fai_winter_year_minimum, `FAIRBANKS INTL AP` ~ winter_year))
## ## Call: ## lm(formula = `FAIRBANKS INTL AP` ~ winter_year, data = fai_winter_year_minimum) ## ## Residuals: ## Min 1Q Median 3Q Max ## -15.529 -4.605 -1.025 4.007 19.764 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -171.19553 43.55177 -3.931 0.000153 *** ## winter_year 0.06250 0.02221 2.813 0.005861 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 7.037 on 104 degrees of freedom ## (1 observation deleted due to missingness) ## Multiple R-squared: 0.07073, Adjusted R-squared: 0.06179 ## F-statistic: 7.916 on 1 and 104 DF, p-value: 0.005861
summary(lm(data=fai_winter_year_minimum, `UNIVERSITY EXP STN` ~ winter_year))
## ## Call: ## lm(formula = `UNIVERSITY EXP STN` ~ winter_year, data = fai_winter_year_minimum) ## ## Residuals: ## Min 1Q Median 3Q Max ## -15.579 -5.818 -1.283 6.029 19.977 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -158.41837 51.03809 -3.104 0.00248 ** ## winter_year 0.05638 0.02605 2.164 0.03283 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 8.119 on 100 degrees of freedom ## (5 observations deleted due to missingness) ## Multiple R-squared: 0.04474, Adjusted R-squared: 0.03519 ## F-statistic: 4.684 on 1 and 100 DF, p-value: 0.03283
Essentially, all the models show a significant increase in minimum temperature over time, but none of them explain very much of the variation in minimum temperature.
Last week I gave a presentation at work about the National Climate Data Center’s GHCND climate database and methods to import and manipulate the data using the dplyr and tidyr R packages (a report-style version of it is here). Along the way, I used this function to calculate the average daily temperature from the minimum and maximum daily temperatures:
One of the people in the audience asked why the Weather Service would calculate average daily temperature this way, rather than by averaging the continuous or hourly temperatures at each station. The answer is that many, perhaps most, of the official stations in the GHCND data set are COOP stations which only report minimum and maximum temperature, and the original instrument provided to COOP observers was likely a mercury minimum / maximum thermometer. Now that these instruments are digital, they could conceivably calculate average temperature internally, and observers could report minimum, maximum and average as calculated from the device. But that’s not how it’s done.
In this analysis, I look at the difference between calculating average daily temperature using the mean of all daily temperature observations, and using the average of the minimum and maximum reported temperature each day. I’ll use five years of data collected at our house using our Arduino-based weather station.
Our weather station records temperature every few seconds, averages this data every five minutes and stores these five minute observations in a database. For our analysis, I’ll group the data by day and calculate the average daily temperature using the mean of all the five minute observations, and using the average of the minimum and maximum daily temperature. I’ll use R to perform the analysis.
Load the libraries we need:
library(dplyr) library(lubridate) library(ggplot2) library(scales) library(readr)
Retrieve the data
Connect to the database and retrieve the data. We’re using build_sql because the data table we’re interested in is a view (sort of like a stored SQL query), not a table, and dplyr::tbl can’t currently read from a view:
dw1454 <- src_postgres(dbname="goldstream_creek_wx", user="readonly") raw_data <- tbl(dw1454, build_sql("SELECT * FROM arduino_west"))
The raw data contains the timestamp for each five minute observation, and the temperature, in degrees Fahrenheit for that observation. The following series of functions aggregates the data to daily data and calculates the average daily temperature using the two methods.
daily_average <- raw_data %>% filter(obs_dt>'2009-12-31 23:59:59') %>% mutate(date=date(obs_dt)) %>% select(date, wtemp) %>% group_by(date) %>% summarize(mm_avg=(min(wtemp)+max(wtemp))/2.0, h_avg=mean(wtemp), n=n()) %>% filter(n==24*60/5) %>% # 5 minute obs collect()
All these steps are joined together using the “pipe” or “then” operator %>% as follows:
- daily_average <-: assign the result of all the operations to daily_average.
- raw_data %>%: start with the data from our database query (all the temperature observations).
- filter(obs_dt>'2009-12-31 23:59:59') %>%: use data from 2010 and after.
- mutate(date=date(obs_dt)) %>%: calculate the data from the timestamp.
- select(date, wtemp) %>%: reduce the columns to our newly calculated date variable and the temperatures.
- group_by(date) %>%: group the data by date.
- summarize(mm_avg=(min(wtemp)+max(wtemp))/2.0) %>%: summarize the data grouped by date, calculate daily average from the average of the minimum and maximum temperature.
- summarize(h_avg=mean(wtemp), n=n()) %>%: calculate another daily average from the mean of the temperaures. Also calculate the number of observations on each date.
- filter(n==24*60/5) %>%: Only include dates where we have a complete set of five minute observations. We don’t want data with too few or too many observations because those would skew the averages.
- collect(): This function retrieves the data from the database. Without collect(), the query is run on the database server, producing a subset of the full results. This allows us to tweak the query until it’s exactly what we want without having to wait to retrieve everything at each iteration.
Now we’ve got a table with one row for each date in the database where we had exactly 288 observations on that date. Columns include the average temperature calculated using the two methods and the number of observations on each date.
Save the data so we don’t have to do these calculations again:
write_csv(daily_average, "daily_average.csv") save(daily_average, file="daily_average.rdata", compress=TRUE)
How does the min/max method of calculating average daily temperature compare against the true mean of all observed temperatures in a day? We calculate the difference between the methods, the anomaly, as the mean temperature subtracted from the average of minimum and maximum. When this anomaly is positive, the min/max method is higher than the actual mean, and when it’s negative, it’s lower.
anomaly <- daily_average %>% mutate(month=month(date), anomaly=mm_avg-h_avg) %>% ungroup() %>% arrange(date)
We also populate a column with the month of each date so we can look at the seasonality of the anomalies.
This is what the results look like:
## Min. 1st Qu. Median Mean 3rd Qu. Max. ## -6.8600 -1.5110 -0.1711 -0.1341 1.0740 9.3570
The average anomaly is very close to zero (-0.13), and I suspect it would be even closer to zero as more data is included. Half the data is between -1.5 and 1.1 degrees and the full range is -6.86 to +9.36°F.
Let’s take a look at some plots of the anomalies.
Raw anomaly data
The first plot shows the raw anomaly data, with positive anomalies (min/max calculate average is higher than the mean daily average) colored red and negative anomalies in blue.
# All anomalies q <- ggplot(data=anomaly, aes(x=date, ymin=0, ymax=anomaly, colour=anomaly<0)) + geom_linerange(alpha=0.5) + theme_bw() + scale_colour_manual(values=c("red", "blue"), guide=FALSE) + scale_x_date(name="") + scale_y_continuous(name="Difference between min/max and hourly aggregation") print(q)
I don't see much in the way of trends in this data, but there are short periods where all the anomalies are in one direction or another. If there is a seasonal pattern, it's hard to see it when the data is presented this way.
To examine the seasonality of the anomalies, let’s look at some boxplots, grouped by the “month” variable we calculated when calculating the anomalies.
mean_anomaly <- mean(anomaly$anomaly) # seasonal pattern of anomaly q <- ggplot(data=anomaly, aes(x=as.factor(month), y=anomaly)) + geom_hline(data=NULL, aes(yintercept=mean_anomaly), colour="darkorange") + geom_boxplot() + scale_x_discrete(name="", labels=c("Jan", "Feb", "Mar", "Apr", "May", "Jun", "Jul", "Aug", "Sep", "Oct", "Nov", "Dec")) + scale_y_continuous(name="Difference between min/max and hourly aggregation") + theme_bw() print(q)
There does seem to be a slight seasonal pattern to the anomalies, with spring and summer daily average underestimated when using the min/max calculation (the actual daily average temperature is warmer than was calculated using minimum and maximum temperatures) and slightly overestimated in fall and late winter. The boxes in a boxplot show the range where half the observations fall, and in all months but April and May these ranges include zero, so there's a good chance that the pattern isn't statistically significant. The orange line under the boxplots show the overall average anomaly, close to zero.
Cumulative frequency distribution
Finally, we plot the cumulative frequency distribution of the absolute value of the anomalies. These plots have the variable of interest on the x-axis and the cumulative frequency of all values to the left on the y-axis. It’s a good way of seeing how much of the data falls into certain ranges.
# distribution of anomalies q <- ggplot(data=anomaly, aes(x=abs(anomaly))) + stat_ecdf() + scale_x_discrete(name="Absolute value of anomaly (+/- degrees F)", breaks=0:11, labels=0:11, expand=c(0, 0)) + scale_y_continuous(name="Cumulative frequency", labels=percent, breaks=pretty_breaks(n=10), limits=c(0,1)) + annotate("rect", xmin=-1, xmax=1, ymin=0, ymax=0.4, alpha=0.1, fill="darkcyan") + annotate("rect", xmin=-1, xmax=2, ymin=0, ymax=0.67, alpha=0.1, fill="darkcyan") + annotate("rect", xmin=-1, xmax=3, ymin=0, ymax=0.85, alpha=0.1, fill="darkcyan") + annotate("rect", xmin=-1, xmax=4, ymin=0, ymax=0.94, alpha=0.1, fill="darkcyan") + annotate("rect", xmin=-1, xmax=5, ymin=0, ymax=0.975, alpha=0.1, fill="darkcyan") + theme_bw() print(q)
The overlapping rectangles on the plot show what percentages of anomalies fall in certain ranges. Starting from the innermost and darkest rectangle, 40% of the temperatures calculated using minimum and maximum are within a degree of the actual temperature. Sixty-seven percent are within two degrees, 85% within three degrees, 94% are within four degrees, and more than 97% are within five degrees of the actual value. There's probably a way to get R to calculate these intersections along the curve for you, but I looked at the plot and manually added the annotations.
We looked at more than five years of data from our weather station in the Goldstream Valley, comparing daily average temperature calculated from the mean of all five minute temperature observations and those calculated using the average minimum and maximum daily temperature, which is the method the National Weather Service uses for it’s daily data. The results show that the difference between these methods average to zero, which means that on an annual (or greater) basis, there doesn't appear to be any bias in the method.
Two thirds of all daily average temperatures are within two degrees of the actual daily average, and with a few exceptions, the error is always below five degrees.
There is some evidence that there’s a seasonal pattern to the error, however, with April and May daily averages particularly low. If those seasonal patterns are real, this would indicate an important bias in this method of calculating average daily temperature.