# Percentile Calculations in Water Quality Regulations

Percentile calculations can be more tricky than at first meets the eye. A percentile indicates the value below which a percentage of observations fall. Some percentiles have special names, such as the quartile or the decile, both of which are quantiles. This deceivingly simple definition hides the various ways to determine this number. Unfortunately, there is no standard definition for percentiles, so which method do you use?

The quantile function in R generates sample percentiles corresponding to the given probabilities. By default, the quantile function provides the quartiles and the minimum and maximum values. The code snippet below generates semi-random data, plots the histogram and visualises the third quartile.

set.seed(1969)
test.data <- rnorm(n = 10000, mean = 100, sd = 15)
library(ggplot2)
ggplot(as.data.frame(test.data), aes(test.data)) +
geom_histogram(binwidth = 1, aes(y = ..density..), fill = "dodgerblue") +
geom_line(stat = "function", fun = dnorm, args = list(mean = 100, sd = 15), colour = "red", size = 1) +
geom_area(stat = "function", fun = dnorm, args = list(mean = 100, sd = 15),
colour = "red", fill="red", alpha = 0.5, xlim = quantile(test.data, c(0.5, 0.75))) +
theme(text = element_text(size = 16))


The quantile default function and the 95th percentile give the following results:

> quantile(test.data)
0%       25%       50%       75%      100%
39.91964  89.68041 100.16437 110.01910 153.50195

> quantile(test.data, probs=0.95)
95%
124.7775


## Methods of percentile calculations

The quantile function in R provides for nine different ways to calculate percentiles. Each of these options uses a different method to interpolate between observed values. I will not discuss the mathematical nuances between these methods. Hyndman and Fan (1996) provide a detailed overview of these methods.

The differences between the nine available methods only matter in skewed distributions, such as water quality data. For the normal distribution simulated above the outcome for all methods is exactly the same, as illustrated by the following code.

> sapply(1:9, function(m) quantile(test.data, 0.95, type = m))

95%      95%      95%      95%      95%      95%      95%      95%      95%
124.7775 124.7775 124.7775 124.7775 124.7775 124.7775 124.7775 124.7775 124.7775


## Percentile calculations in water quality

The Australian Drinking Water Quality Guidelines (November 2016) specify that: “based on aesthetic considerations, the turbidity should not exceed 5 NTU at the consumer’s tap”. The Victorian Safe Drinking Water Regulations (2015) relax this requirement and require that:

“The 95th percentile of results for samples in any 12 month period must be less than or equal to 5.0 NTU.”

The Victorian regulators also specify that the percentile should be calculated with the Weibull Method. This requirement raises two questions: What is the Weibull method? How do you implement this requirement in R?

The term Weibull Method is a bit confusing as this is not a name used by statisticians. In Hyndman & Fan (1996), this method has the less poetic name $\hat{Q}_8(p)$. Waloddi Weibull, a Swedish engineer famous for his distribution, was one of the first to describe this method. Only the regulator in Victoria uses that name, which is based on McBride (2005). This theoretical interlude aside, how can we practically apply this to water quality data?

In case you are interested in how the Weibull method works, the weibull.quantile function shown below calculates a quantile p for a vector x using this method. This function gives the same result as quantile(x, p, type=6).

weibull.quantile <- function(x, p) {
# Order Samples from large to small
x <- x[order(x, decreasing = FALSE)]
# Determine ranking of percentile according to Weibull (1939)
r <- p * (length(x) + 1)
# Linear interpolation
rfrac <- (r - floor(r))
return((1 - rfrac) * x[floor(r)] + rfrac * x[floor(r) + 1])
}


## Turbidity Data Example

Turbidity data is not normally distributed as it is always larger than zero. In this example, the turbidity results for the year 2016 for the water system in Tarnagulla are used to illustrate the percentile calculations. The range of weekly turbidity measurements is between 0.,05 NTU and 0.8 NTU, well below the aesthetic limits.

Turbidity at customer tap for each zone in the Tarnagulla system in 2016 (n=53).

When we calculate the percentiles for all nine methods available in the base-R function we see that the so-called Weibull method generally provides the most conservative result.

Zone R1 R2 R3 R4 R5 R6 R7 R8 R9
Bealiba 0.300 0.300 0.200 0.240 0.290 0.300 0.245 0.300 0.300
Dunolly 0.40000 0.40000 0.30000 0.34000 0.39000 0.43500 0.34500 0.40500 0.40125
Laanecoorie 0.50000 0.50000 0.40000 0.44000 0.49000 0.53500 0.44500 0.50500 0.50125
Tarnagulla 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

The graph and the table were created with the following code snippet:

ggplot(turbidity, aes(Result)) +
geom_histogram(binwidth=.05, fill="dodgerblue", aes(y=..density..)) +
facet_wrap(~Zone) +
theme(text=element_text(size=16))

tapply(turbidity$Result, turbidity$Zone,
function(x) sapply(1:9, function(m) quantile(x, 0.95, type=m)))


# SCADA spikes in Water Treatment Data

SCADA spikes are events in the data stream of water treatment plants or similar installations. These SCADA spikes can indicate problems with the process and could result in an increased risk to public health.

The WSAA Health Based Targets Manual specifies a series of decision rules to assess the performance of filtration processes. For example, this rule assesses the performance of conventional filtration:

“Individual filter turbidity ≤ 0.2 NTU for 95% of month and not > 0.5 NTU for ≥ 15 consecutive minutes.”

Turbidity is a measure for the cloudiness of a fluid because of large numbers of individual particles otherwise invisible to the naked eye. Turbidity is an important parameter in water treatment because a high level of cloudiness strongly correlates with the presence of microbes. This article shows how to implement this specific decision rule using the R language.

## Simulation

To create a minimum working example, I first create a simulated SCADA feed for turbidity. The turbidity data frame contains 24 hours of data. The seq.POSIXt function creates 24 hours of timestamps at a one-minute spacing. In addition, the rnorm function creates 1440 turbidity readings with an average of 0.1 NTU and a standard deviation of 0.01 NTU. The image below visualises the simulated data. The next step is to assess this data in accordance with the decision rule.

# Simulate data
set.seed(1234)
turbidity <- data.frame(DateTime = seq.POSIXt(as.POSIXct("2017-01-01 00:00:00"), by = "min", length.out=24*60),
Turbidity = rnorm(n = 24*60, mean = 0.1, sd = 0.01)
)


The second section simulates five spikes in the data. The first line picks a random start time for the spike. The second line in the for-loop picks a duration between 10 and 30 minutes. In addition, the third line simulates the value of the spike. The mean value of the spike is determined by the rbinom function to create either a low or a high spike. The remainder of the spike simulation inserts the new data into the turbidity data frame.

# Simulate spikes
for (i in 1:5) {
time <- sample(turbidity$DateTime, 1) duration <- sample(10:30, 1) value <- rnorm(1, 0.5 * rbinom(1, 1, 0.5) + 0.3, 0.05) start <- which(turbidity$DateTime == time)
turbidity$Turbidity[start:(start+duration - 1)] <- rnorm(duration, value, value/10) }  The image below visualises the simulated data using the mighty ggplot. Only four spikes are visible because two of them overlap. The next step is to assess this data in accordance with the decision rule. library(ggplot2) ggplot(turbidity, aes(x = DateTime, y = Turbidity)) + geom_line(size = 0.2) + geom_hline(yintercept = 0.5, col = "red") + ylim(0,max(turbidity$Turbidity)) +
ggtitle("Simulated SCADA data")


## SCADA Spikes Detection

The following code searches for all spikes over 0.50 NTU using the run length function. This function transforms a vector into a vector of values and lengths. For example, the run length of the vector c(1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 6) is:

• lengths: int [1:5] 2 3 4 2 1
• values : num [1:5] 1 2 3 5 6

The value 1 has a length of 1, the value 2 has a length of 3 and so on. The spike detection code creates the run length for turbidity levels greater than 0.5, which results in a boolean vector. The cumsum function calculates the starting point of each spike which allows us to calculate their duration.

The code results in a data frame with all spikes higher than 0.50 NTU and longer than 15 minutes. The spike that occurred at 11:29 was higher than 0.50 NTU and lasted for 24 minutes. The other three spikes are either lower than 0.50 NTU. The first high spike lasted less than 15 minutes.

# Spike Detection
spike.detect <- function(DateTime, Value, Height, Duration) {
runlength <- rle(Value > Height)
spikes <- data.frame(Spike = runlength$values, times <- cumsum(runlength$lengths))
spikes$Times <- DateTime[spikes$times]
spikes$Event <- c(0,spikes$Times[-1] - spikes$Times[-nrow(spikes)]) spikes <- subset(spikes, Spike == TRUE & Event > Duration) return(spikes) } spike.detect(turbidity$DateTime, turbidity\$Turbidity, 0.5, 15)


This approach was used to prototype a software package to assess water treatment plant data in accordance with the Health-Based Targets Manual. The finished product has been written in SQL and is available under an Open Source sharing license.