Introduction to R for Excel Users

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As the saying goes, when all you have is a hammer, everything looks like a nail. Excel was designed to do simple financial analyses and to craft financial statements. Though its capabilities have been expanded over the years, it was never designed to perform the sort of data analysis that industry scientists, engineers and Six Sigma belts need to perform on a daily basis.

Most data analyses performed in Excel look more like simple financial spreadsheets rather than actual data analysis, and this quality of work translates into bad—or at least sub-optimal—business decisions. There are alternatives to Excel, and the free, open-source data analysis platform R is one of them.

Unfortunately, R has a steep learning curve. I’m offering, for free, a short primer on R [PDF] where I’ve sought to make that learning curve a little less painful for engineers and scientists who normally work in Excel.

Background

A couple of years ago, I was developing a short course to teach R to scientists and engineers in industry who normally used Excel. The goal was to help them transition to a more capable tool. My course design notes morphed into a handout, and when plans for the course fell through, that handout grew into a self-study guide, which I later adapted into this seventy-page, stand-alone introduction for Excel users.

Organization

The primer walks the reader through the basics of R, starting with a brief overview of capabilities, then diving into installation, basic operations, graphical analysis and basic statistics. I believe that a picture is worth a thousand words, so it’s light on text and heavy on examples and visuals.

The end of the book rounds out with a look at some of the most useful add-ons, the briefest of introductions to writing your own, custom functions in R, and a cross-reference of common Excel functions with their equivalents in R.

The text is broken up into chapters and fully indexed so that it can be used either as a walk-through tutorial or as a quick reference.

Can We do Better than R-squared?

If you're anything like me, you've used Excel to plot data, then used the built-in “add fitted line” feature to overlay a fitted line to show the trend, and displayed the “goodness of fit,” the r-squared (R2) value, on the chart by checking the provided box in the chart dialog.

The R2 calculated in Excel is often used as a measure of how well a model explains a response variable, so that “R2 = 0.8” is interpreted as “80% of the variation in the 'y' variable is explained by my model.” I think that the ease with which the R2 value can be calculated and added to a plot is one of the reasons for its popularity.

There's a hidden trap, though. R2 will increase as you add terms to a model, even if those terms offer no real explanatory power. By using the R2 that Excel so helpfully provides, we can fool ourselves into believing that a model is better than it is.

Below I'll demonstrate this and show an alternative that can be implemented easily in R.

Some data to work with

First, let's create a simple, random data set, with factors a, b, c and response variable y.

head(my.df)
##       y a       b      c
## 1 2.189 1 -1.2935 -0.126
## 2 3.912 2 -0.4662  1.623
## 3 4.886 3  0.1338  2.865
## 4 5.121 4  1.2945  4.692
## 5 4.917 5  0.1178  5.102
## 6 4.745 6  0.4045  5.936

Here is what this data looks like:

Plot of the response and factors in a linear model.

Calculating R-squared

What Excel does when it displays the R2 is create a linear least-squares model, which in R looks something like:

my.lm <- lm(y ~ a + b + c, data = my.df)

Excel also does this when we call RSQ() in a worksheet. In fact, we can do this explicitly in Excel using the Regression analysis option in the Analysis Pack add-on, but I don't know many people who use this, and Excel isn't known for its reliability in producing good output from the Analysis Pack.

In R, we can obtain R2 via the summary() function on a linear model.

summary(my.lm)
## 
## Call:
## lm(formula = y ~ a + b + c, data = my.df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.2790 -0.6006  0.0473  0.5177  1.5299 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(&gt;|t|)  
## (Intercept)    2.080      0.763    2.72    0.034 *
## a             -0.337      0.776   -0.43    0.679  
## b             -0.489      0.707   -0.69    0.515  
## c              1.038      0.817    1.27    0.250  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.1 on 6 degrees of freedom
## Multiple R-squared:  0.833,  Adjusted R-squared:  0.75 
## F-statistic:   10 on 3 and 6 DF,  p-value: 0.00948

Since summary() produces a list object as output, we can grab just the R2 value.

summary(my.lm)$r.squared
## [1] 0.8333

Normally, we would (somewhat loosely) interpret this as telling us that about 83% of the variation in the response y is explained by the model.

Notice that there is also an "adjusted r-squared” value given by summary(). This tells us that only 75% of the variation is explained by the model. Which is right?

The problem with R-squared

Models that have many terms will always give higher R2 values, just because more terms will slightly improve the model fit to the given data. The unadjusted R2 is wrong. The calculation for adjusted R2 is intended to partially compensate for that “overfit,” so it's better.

It's nice that R shows us both values, and a pity that Excel won't show the adjusted value. The only way to get an adjusted R2 in Excel is to run the Regression analysis; otherwise, we have to calculate adjusted R2 manually.

Both R2 and adjusted R2 are measures of how well the model explains the given data. However, in industry we usually want to know something a little different. We don't build regression models to explain only the data we have; we build them to think about future results. We want R2 to tell us how well the model predicts the future. That is, we want a predictive R2. Minitab has added the ability to calculate predictive R2 in Minitab 17, and has a nice blog post explaining this statistic.

Calcuting predictive R-squared

Neither R nor Excel provide a means of calculating the predictive R2 within the default functions. While some free R add-on packages provide this ability (DAAG, at least), we can easily do it ourselves. We'll need a linear model, created with lm(), for the residuals so we can calculate the “PRESS” statistic, and then we need the sum of squares of the terms so we can calculate a predictive R2.

Since the predictive R2 depends entirely on the PRESS statistic, we could skip the added work of calculating predictive R2 and just use PRESS, as some authors advocate. The lower the PRESS, the better the model is at fitting future data from the same process, so we can use PRESS to compare different models. Personally, I'm used to thinking in terms of R2, and I like having the ability to compare to the old R2 statistic that I'm familiar with.

To calculate PRESS, first we calculate the predictive residuals, then take the sum of squares (thanks to (Walker’s helpful blog post) for this). This is pretty easy if we already have a linear model. It would take a little more work in Excel.

pr <- residuals(my.lm)/(1 - lm.influence(my.lm)$hat)
PRESS <- sum(pr^2)
PRESS
## [1] 19.9

The predictive R2 is then (from a helpful comment by Ibanescu on LikedIn) the PRESS divided by the total sum of squares, subtracted from one. The total sum of squares can be calculated directly as the sum of the squared residuals, or obtained by summing over Sum Sq from an anova() on our linear model. I prefer using the anova function, as any statistical subtleties are more likely to be properly accounted for there than in my simple code.

# anova to calculate residual sum of squares
my.anova <- anova(my.lm)
tss <- sum(my.anova$"Sum Sq")
# predictive R^2
pred.r.squared <- 1 - PRESS/(tss)
pred.r.squared
## [1] 0.5401

You'll notice that this is smaller than the residual R2, which is itself smaller than the basic R2. This is the point of the exercise. We don't want to fool ourselves into thinking we have a better model than we actually do. One way to think of this is that 29% (83% – 54%) of the model is explained by too many factors and random correlations, which we would have attributed to our model if we were just using Excel's built-in function.

When the model is good and has few terms, the differences are small. For example, working through the examples in Mitsa's two posts, we see that for her model 3, R2 = 0.96 and the predictive R2 = 0.94, so calculating the predictive R2 wasn't really worth the extra effort for that model. Unfortunately, we can't know, in advance, which models are “good.” For Mitsa's model 1 we have R2 = 0.95 and predictive R2 = 0.32. Even the adjusted R2 looks pretty good for model 1, at 0.94, but we see from the predictive R2 that our model is not very useful. This is the sort of thing we need to know to make correct decisions.

Automating

In R, we can easily wrap these in functions that we can source() and call directly, reducing the typing. Just create a linear model with lm() (or an equivalent) and pass that to either function. Note that pred_r_squared() calls PRESS(), so both functions have to be sourced.

pred_r_squared <- function(linear.model) {
    lm.anova <- anova(linear.model)
    tss <- sum(lm.anova$"Sum Sq")
    # predictive R^2
    pred.r.squared <- 1 - PRESS(linear.model)/(tss)
    return(pred.r.squared)
}
PRESS <- function(linear.model) {
    pr <- residuals(linear.model)/(1 - lm.influence(linear.model)$hat)
    PRESS <- sum(pr^2)
    return(PRESS)
}

Then we just call the function to get the result:

pred.r.squared <- pred_r_squared(my.lm)
pred.r.squared
## [1] 0.5401

I've posted these as Gists on GitHub, with extra comments, so you can copy and paste from here or go branch or copy them there.

References and further reading

Normality and Testing for Normality

Many of our statistical tests make assumptions about the distribution of the underlying population. Many of the most common—ImR (XmR) and XbarR control charts, ANOVA, t-tests—assume normal distributions in the underlying population (or normal distributions in the residuals, in the case of ANOVA), and we’re often told that we must carefully check the assumptions.

At the same time, there’s a lot of conflicting advice about how to test for normality. There are the statistical tests for normality, such as Shapiro-Wilk or Anderson-Darling. There’s the “fat pencil” test, where we just eye-ball the distribution and use our best judgement. We could even use control charts, as they’re designed to detect deviations from the expected distribution. We are discouraged from using the “fat pencil” because it will result in a lot of variation from person to person. We’re often told not to rely too heavily on the statistical tests because they are not sensitive with small sample sizes and too sensitive to the tails. In industrial settings, our data is often messy, and the tails are likely to be the least reliable portion of our data.

I’d like to explore what the above objections really look like. I’ll use R to generate some fake data based on the normal distribution and the t distribution, and compare the frequency of p-values obtained from the Shapiro-Wilk test for normality.

A Function to test normality many times

First, we need to load our libraries

library(ggplot2)
library(reshape2)

To make this easy to run, I’ll create a function to perform a large number of normality tests (Shapiro-Wilk) for sample sizes n = 5, 10 and 1000, all drawn from the same data:

#' @name assign_vector
#' @param data A vector of data to perform the t-test on.
#' @param n An integer indicating the number of t-tests to perform. Default is 1000
#' @return A data frame in "tall" format
assign_vector <- function(data, n = 1000) {
  # replicate the call to shapiro.test n times to build up a vector of p-values
  p.5 <- replicate(n=n, expr=shapiro.test(sample(my.data, 5, replace=TRUE))$p.value)
  p.10 <- replicate(n=n, expr=shapiro.test(sample(my.data, 10, replace=TRUE))$p.value)
  p.1000 <- replicate(n=n, expr=shapiro.test(sample(my.data, 1000, replace=TRUE))$p.value)
  #' Combine the data into a data frame, 
  #' one column for each number of samples tested.
  p.df <- cbind(p.5, p.10, p.1000)
  p.df <- as.data.frame(p.df)
  colnames(p.df) <- c("5 samples","10 samples","1000 samples")
  #' Put the data in "tall" format, one column for number of samples
  #' and one column for the p-value.
  p.df.m <- melt(p.df)
  #' Make sure the levels are sorted correctly.
  p.df.m <- transform(p.df.m, variable = factor(variable, levels = c("5 samples","10 samples","1000 samples")))
  return(p.df.m)  
}

Clean, random data

I want to simulate real-word conditions, where we have an underlying population from which we sample a limited number of times. To start, I’ll generate 100000 values from a normal distribution. To keep runtimes low I’ll have assign_vector() sample from that distribution when performing the test for normality.

n.rand <- 100000
n.test <- 10000
my.data <- rnorm(n.rand)
p.df.m <- assign_vector(my.data, n = n.test)

We would expect that normally distributed random data will have an equal probability of any given p-value. i.e. 5% of the time we’ll see p-value ≤ 0.05, 5% of the time we’ll see p-value > 0.05 and ≤ 0.10, and so on through > 0.95 and ≤ 1.00. Let’s graph that and see what we get for each sample size:

ggplot(p.df.m, aes(x = value)) + 
  geom_histogram(binwidth = 1/10) + 
  facet_grid(facets=variable ~ ., scales="free_y") + 
  xlim(0,1) +
  ylab("Count of p-values") +
  xlab("p-values") +
  theme(text = element_text(size = 16))
Histogram of p-values for the normal distribution, for sample sizes 5, 10 and 1000.

Histogram of p-values for the normal distribution, for sample sizes 5, 10 and 1000.

This is, indeed, what we expected.

Now let’s compare the normal distribution to a t distribution. The t distribution would pass the “fat pencil” test—it looks normal to the eye:

ggplot(NULL, aes(x=x, colour = distribution)) + 
  stat_function(fun=dnorm, data = data.frame(x = c(-6,6), distribution = factor(1)), size = 1) + 
  stat_function(fun=dt, args = list( df = 20), data = data.frame(x = c(-6,6), distribution = factor(2)), linetype = "dashed", size = 1) + 
  scale_colour_manual(values = c("blue","red"), labels = c("Normal","T-Distribution")) +
  theme(text = element_text(size = 12),
        legend.position = c(0.85, 0.75)) +
  xlim(-4, 4) +
  xlab(NULL) +
  ylab(NULL)

Density plot of normal and t distributions

Starting with random data generated from the t-distribution:

my.data <- rt(n.rand, df = 20)
Histogram of p-values for the t distribution, for sample sizes 5, 10 and 1000.

Histogram of p-values for the t distribution, for sample sizes 5, 10 and 1000.

The tests for normality are not very sensitive for small sample sizes, and are much more sensitive for large sample sizes. Even with a sample size of 1000, the data from a t distribution only fails the test for normality about 50% of the time (add up the frequencies for p-value > 0.05 to see this).

Testing the tails

Since the t distribution is narrower in the middle range and has longer tails than the normal distribution, the normality test might be failing because the entire distribution doesn’t look quite normal; we haven’t learned anything specifically about the tails.

To test the tails, we can construct a data set that uses the t distribution for the middle 99% of the data, and the normal distribution for the tails.

my.data <- rt(n.rand, df = 20)
my.data.2 <- rnorm(n.rand)
# Trim off the tails
my.data <- my.data[which(my.data < 3 & my.data > -3)]
# Add in tails from the other distribution
my.data <- c(my.data, my.data.2[which(my.data.2 < -3 | my.data.2 > 3)])
Histogram of p-values for sample sizes 5, 10 and 1000, from a data set constructed from the t distribution in the range -3 to +3 sigmas, with tails from the normal distribution below -3 and above +3.

Histogram of p-values for sample sizes 5, 10 and 1000, from a data set constructed from the t distribution in the range -3 to +3 sigmas, with tails from the normal distribution below -3 and above +3.

Despite 99% of the data being from the t distribution, this is almost identical to our test with data from just the normal distribution. It looks like the tails may be having a larger impact on the normality test than rest of the data

Now let’s flip this around: data that is 99% normally-distributed, but using the t distribution in the extreme tails.

my.data <- rnorm(n.rand)
my.data.2 <- rt(n.rand, df = 20)
# Trim off the tails
my.data <- my.data[which(my.data < 3 & my.data > -3)]
# Add in tails from the other distribution
my.data <- c(my.data, my.data.2[which(my.data.2 < -3 | my.data.2 > 3)])
Histogram of p-values for sample sizes 5, 10 and 1000, from a data set constructed from the normal distribution in the range -3 to +3 sigmas, with tails from the t-distribution below -3 and above +3.

Histogram of p-values for sample sizes 5, 10 and 1000, from a data set constructed from the normal distribution in the range -3 to +3 sigmas, with tails from the t-distribution below -3 and above +3.

Here, 99% of the data is from the normal distribution, yet the normality test looks almost the same as the normality test for just the t-distribution. If you check the y-axis scales carefully, you’ll see that the chance of getting p-value ≤ 0.05 is a bit lower here than for the t distribution.

To make the point further, suppose we have highly skewed data:

my.data <- rlnorm(n.rand, 0, 0.4)

This looks like:
Histogram of log-normal data

For small sample sizes, even this is likely to pass a test for normality:
Histogram of p-values for a log-normal distribution

What have we learned?

  • With small sample sizes, everything looks normal.
  • The normality tests are, indeed, very sensitive to what goes on in the extreme tails.

In other words, if we have enough data to fail a normality test, we always will because our real-world data won’t be clean enough. If we don’t have enough data to reliably fail a normality test, then there’s no point in performing the test, and we have to rely on the fat pencil test or our own understanding of the underlying processes.

Don’t get too hung up on whether your data is normally distributed or not. When evaluating and summarizing data, rely mainly on your brain and use the statistics only to catch really big errors in judgement. When attempting to make predictions about future performance, e.g. calculating Cpk or simulating a process, recognize the opportunities for errors in judgment and explicitly state you assumptions.

Scientific Consensus

I have seen a number of comments lately that “consensus” has no place in science, and that claims that there is a “scientific consensus” are just thinly veiled political double-speak. I have to take issue with such criticisms. In fact, consensus is one of the cornerstones of science.

This is not quite the same consensus used in politics or everyday life. Consensus means “general agreement” and is usually achieved through some form of discussion and negotiation. Consensus is therefore agreement over opinions, and is often agreement over a course of action despite differing opinions.

In science, consensus is derived from data and independent replication of experiments. It is the consensus that an idea—specifically, a testable hypothesis—is correct. It is the expression of scientists that a hypothesis is (a) scientifically testable and falsifiable, (b) that it has not been falsified and (c) that it explains the universe better than competing hypotheses. It is a consensus derived from the replication of observations or tests by other researchers.

Part of the problem, here, is that science has a habit of taking everyday words and developing very specific meanings around them. This happens because scientists need to communicate clearly and exactly at times, and language is messy and full of fuzzy concepts. The same thing happens in a lot of occupations. For example, accountants also develop specific meanings for everyday words.

Scientists no longer argue over the validity of Newton’s hypothesis on the gravitational force because there is broad consensus that the hypothesis is correct (as far as it goes). Objects attract each other according to their mass and the distance between them, and there have been plenty of independent experiments confirming the specific relationship, F = -G M1m2 / r2. The consensus is so strong that it’s referred to as Newton’s law of gravity. Likewise, scientists no longer argue over the geocentric model of the universe because there is broad consensus—derived from data collected over centuries by many independent researchers—that the Earth is not at the center of the solar system, let alone the universe.

Conversely, there is very little consensus when it comes to the accelerating expansion of the universe. Cosmologists agree that the universe is expanding faster than can be explained by our current understanding of the universe, but there are many, conflicting hypotheses about the causes. There is consensus over the fact of the accelerating expansion but the data does not yet support consensus on the underlying physical processes or mechanism causing it, and so there is no consensus about the physical process.

In one sense, scientific consensus is stronger, or more robust, than we are used to thinking about in politics and everyday life, precisely because it is based on observation and careful analysis by independent groups. It’s not just consensus based on what we think might be true, or what we want to be true, but based on what careful observation tells us must be true. Scientific consensus is not subject to whim.

From another perspective, though, scientific consensus is much weaker than we are used to. In science, there is no downside to abandoning or overturning a consensus when the data points in a different direction. In fact, there is significant benefit to being the person who can overturn a previous consensus; we remember Galileo, Newton, Darwin, Einstein and others precisely because their work, collecting and analyzing data, was so pivotal in altering the scientific consensus. In everyday life, if you back out of a consensus agreement, it’s likely that others party to the agreement will feel betrayed. There can be a significant social cost to pay for backing out of a consensus, even when you are convinced that you are right. Scientists may sometimes feel this same social pressure, but the scientific method provides clear guidance for adopting or abandoning consensus, and it doesn’t focus on the people involved but rather on external, objective observations of how the universe works. Scientific consensus is, perhaps, more readily changed than conventional consensus.

So consensus does exist is science and it plays an important role in science. We should be careful to distinguish, though, between consensus based on independent replication of results and consensus based on preconceptions and social negotiation.

And You Thought Physics Was *YAWN*

Part of my day job involves monitoring the renewable energy market, and particularly keeping abreast of storage technologies. It’s like combining my hobby with my job.

A new wind turbine design was recently announced, the SeaTwirl. It’s an off-shore turbine design using vertical blades. The key technological advance here is that it includes a method to store energy, so that it can continue to produce electricity when the wind stops blowing. This ability to deliver a constant output is important, because, as you may have heard, wind energy is intermittent; you only get electricity when the wind blows, and only to the degree that it’s blowing. Demand, unfortunately, doesn’t follow wind’s intermittancy—nobody stops to check that the wind is blowing before they turn on their lights—and the utilities, transmission system operators and distribution system operators all have to supply electricity to meet demand.

The SeaTwirl stores energy by integrating an unusual wind turbine design with a pumped hydro system. It has the turbine blades on a large circular ring, which rotates parallel to the water’s surface kind of like a hula-hoop. This ring is hollow (more or less), and is filled up with water when the wind blows. When the wind stops blowing, the momentum of the water keeps the tube spinning, generating electricity, and the water is allowed to drain back out, spinning a hydro turbine to generate electricity.

The nice thing about this is that the storage can always be “recharged” and it’s “free.” Or at least it seems to be. If you’ve ever held a bucket while spinning around, you know that spinning with an empty bucket takes a lot less effort than spinning around with a full bucket. In part, this is because of a property known as the moment of inertia. The heavier or larger a spinning object gets, the more it resists changes to its rate of rotation (or rpm).

If the SeaTwirl is filling up this horizontal “hula-hoop” with water, then the weight of the tube is increasing and so is the moment of inertia. As the moment of inertia increases, the energy needed to reach a given rpm increases. Wind turbines normally generate electricity in proportion to the wind speed, because the rpm of the blades is proportional to the wind speed. Increase the moment of inertia and you decrease the rpm, which means you generate less electricity for a given wind speed.

Now comes the physics. For something shaped like a hula-hoop, the moment of inertia, I, is calculated from the mass, M, and the radius of the hoop, R, according to:

I = M R^{2}

The energy, E, in a spinning object is equal to the moment of inertia times the speed of rotation, ω, according to

E=\frac{1}{2} I \omega ^{2}

If we know the energy (because we know the wind speed), then we can calculate the speed of rotation, ω, by rearranging that equation to get ω on the left-hand side:

\omega = \sqrt{\frac{2E}{I}}

We can then replace I with mass and radius from the first equation to get

\omega = \sqrt{\frac{2E}{MR^{2}}}

So we can see that, if we don’t change the energy E (or don’t change the wind speed), and don’t change the radius R of the spinning hoop, then increasing the mass M results in a slower rate of rotation.

From SeaTwirl’s website and press releases, we can estimate how big the SeaTwirl is, which will let us estimate how much slower a full SeaTwirl will spin than an empty one, and therefore how much less electricity must be generated. We can calculate this by taking the ratio of ω full to ω empty, so that the parts that we don’t have to know E and R.

\frac{\omega_{full}}{\omega_{empty}} = \frac{\sqrt{\frac{2E}{M_{full}R^{2}}}}{\sqrt{\frac{2E}{M_{empty}R^{2}}}} = \sqrt{\frac{\frac{2E}{R^{2}}}{\frac{2E}{R^{2}}}}\sqrt{\frac{M_{empty}}{M_{full}}}=\sqrt{\frac{M_{empty}}{M_{full}}}

For the SeaTwirl, we now have to find out what R is, and estimate M for both filled and empty

The whole turbine assembly is made of composite materials, which probably have a density, \rho_{c}, of around 2500 kilograms per cubic meter (similar to fiberglass). Water has a density, \rho_{w}, of near 1000 kilograms per cubit meter (depending on temperature). The diameter of the turbine will be near 180 meters, so the radius, R, of our “hula-hoop” is half that, or 90 meters. From the pictures, it looks like the thickness of that hula-hoop is a few percent of the total diameter of the turbine, so we can figure an outside diameter of the “hula-hoop” of about 2 meters, for a radius, r, of 1 meter. Figure that at least ten percent of this is composite, and the rest is the hollow, water-filled portion.

To estimate the weight of the water in the “hula-hoop,” we can approximate the water as being a cylinder of radius r_{w} = 0.9r and length equal to the circumference of the “hula-hoop,” l = 2\pi R. The volume of such a cylinder is equal to the cross-sectional area of the water column, A_{w}=\pi r_{w}^{2} times the length of the column, l. The total mass of the water, m_{w} is the density times this volume.

m_{w} = \rho_{w}\pi r_{w}^{2}2\pi R = \rho_{w}3\pi (0.9r)^{2} R

Plugging in our estimates for the above values gives us

m_{w} = 1000 \cdot 3\pi (0.9)^{2} 90 = 687000 kg

That’s a lot of water.

Now for the empty “hula-hoop.” We can treat it in the same way: a cylinder of material of radius r, length l = 2\pi R. However, we don’t want to calculate for a solid cylinder of composite; we have to subtract out the hollow part with radius r_{w}. So the mass of the composite is

m_{c} = \rho_{c} 2 \pi R ( \pi r^{2} - \pi r^{2}_{w} )

m_{c} = 2500 \cdot 3 \pi 90 (1^{2} - 0.9^{2} ) = 403000 kg

So the water more than doubles the weight of the hoop.

From the picture, you can see that there’s another hoop at the top, and the two hoops are connected by the turbines, which combined are probably worth at least another hoop in weight, so we can further assume that the mass of this bottom hoop, empty, is roughly one-third of the total mass of the movable parts of turbine.

The mass of the turbine, empty, is therefore about 1200000 kg, or 1200 tons. Filled with water, this goes up to about 1900000 kg, or 1900 tons. Empty, that’s maybe twice the largest off-shore turbine currently in existence, but this thing is easily twice as big as any current turbine, so our estimate appears to be in the right neighborhood.

Now we go back to our equation for the ratio of the rotational velocities, ω, and plug in these weights:

\frac{\omega_{full}}{\omega_{empty}} = \sqrt{\frac{M_{empty}}{M_{full}}} =\sqrt{\frac{1200}{1900}} = 0.8

So we get about 80% as much electricity from a water-filled turbine as from an empty one, when the wind blows. This is a direct efficiency loss due to the storage of energy in the spinning-water-hoop.

In addition, there’s the efficiency losses in loading water into the hoop, or “charging” the hoop, and the efficiency losses of “discharging” the hoop, running the water back out through a hydro turbine. Pumped hydro is usually about 72% efficient, or less, in each direction, so the total round-trip efficiency of storage + discharge is about 50% efficient. There are a lot of other storage technologies that do at least this good, if not better.

These two figures, the 80% efficiency loss of just operating the turbine and the 50% storage-discharge efficiency, can be used to directly compare SeaTwist with other wind turbine + storage technology solutions. Any storage technology that has at least a 50% round-trip efficiency and increases the total system cost by less than 20% over the system’s operating lifetime will outperform SeaTwist in terms of return on investment.

R Function Reference

Updated below

The R Function Reference is a mind map that I created as a guide for novice and intermediate users of the R statistics language. When you first open it, I suggest that you collapse all the nodes by clicking on the “Expand/Collapse all nodes” button in the bottom left of the screen to make the map easier to navigate. You can also adjust the zoom level with the slider next to that button.

R Function Reference screenshot

The top-level nodes of the R Function Reference

The mind map is arranged in eight sections, or main branches, arranged by task. What do you want to do? Each branch covers a general set tasks, such as learning to use R, running R, working with data, statistical analysis or plotting data. The end of each string of nodes is generally a function and example. The Reference provides code fragments, rather than details of the function or complete reproducible code blocks. Once you’ve followed the Reference and have an idea of how to accomplish something, you can look up the details in R’s help system (e.g. “?read.csv” to learn more about using the read.csv() function), or search Google or the online R-Help mailing list archives for answers using the function name.

There are a lot of useful nodes and examples, especially in the “Graphs” section, but the mind map is not complete; some trails end before you get to a useful function reference. I am sorry for that, but it’s a work in progress, and will be slowly updated over time.

Comments and suggestions are welcome.

Update 1

In comments, several users reported problems opening the mind map. With a little investigation, it appears that the size of the mind map is the problem. To try to fix the problems , I have split the mind map out into several small mind maps, all linked together.

The new main mind map is the R Function Reference, Main. The larger branches on this main map no longer expand to their own content, but contain a link to a “child” mind map. The link looks like a sheet of paper with an arrow pointing to the right, click on it and little cartoon speech bubble will pop up with a link that you have to click on to go to the child mind map. Likewise, the central nodes on the child mind maps contain a link back to the main mind map.

Due to load times and the required extra clicks, this may slightly reduce usability for users who didn’t have a problem with the all-in-one version, but will hopefully make the mind map accessible to a broader audience.

I have to offer praise to the developers of Mind42. Though I couldn’t directly split branches off into their own mind maps or duplicate the mind map, it was very easy to export the mind map as a native Mind42 file and then import it multiple times, editing the copies without any loss of data or links. The ability to link directly between mind maps within Mind42 was also a key enabling feature. Considering that this is a free web app, its capabilities are most impressive. They were also quick to respond when I posted a call for help on the Mind42 forum.

Please let me know how the new, “improved” version works.

The old mind map, containing everything, is still available, but I will not update it.

Graphing Highly Skewed Data

Recently Chandoo.org posted a question about how to graph data when you have a lot of small values and a few larger values. It’s not the first time that I’ve come across this question, and I’ve seen a lot of answers, many of them really bad. While all solutions involve trade-offs for understanding and interpreting graphs, some solutions are better than others.

Data graphs tell stories by revealing patterns in complex data. Good data graphs let the data tell the story by revealing the patterns, rather than trying to impose patterns on the data.

As William Cleveland discusses in The Elements of Graphing Data and his 1993 paper A Model for Studying Display Methods of Statistical Graphics, there are two basic visual operations that people employ when looking at and interpreting graphs: pattern perception and table look-up. Pattern perception is where we see the geometric patterns in a graph: groupings; relative differences (larger/smaller); or trends (straight/curved or increasing/decreasing). Table look-up is where we explore details of values and names on a graph. These two operations are distinct and complimentary, and it is through these two operations that the data’s story is told.

month sales
1 Feb 09 200
2 Mar 09 300
3 Apr 09 200
4 May 09 300
5 Jun 09 200
6 Jul 09 300
7 Aug 09 350
8 Sep 09 400
9 Oct 09 450
10 Nov 09 1200
11 Dec 09 100000
12 Jan 10 85000
13 Feb 10 450

So suppose that we have some data like that at right, where we are interested in the patterns of smaller, individual values, but there are also a few extremely large values, or outliers. We describe such data as being skewed. How do we plot this data? First, for such a small data set, a simple table is the best approach. People can see the numbers and interpret them, there aren’t too many numbers to make sense of and the table is very compact. For more complicated data sets, though, a graph is needed. There’s a few basic options:

  • Graph as-is;
  • Graph with a second axis;
  • Graph the logarithm of the data;
  • Use a scale break.
  • Plot the data multiple times.

Graph As-Is

Bar chart with all data plotted

A bar chart with all data, including outliers, plotted on the same scale.

This is the simplest solution, and if you’re only interested in knowing about the outliers (Dec ’09 and Jan ’10) then it will do. However, it completely hides whatever is happening in the rest of the months. Pattern recognition tells us that two months near the end of the series have the big numbers. Table-lookup tells us the approximate values and that these months are around December ’09 and February ’10, but the way the labels string together and overlap the tick marks, it’s not clear exactly what the labels are, let alone which label applies to which bar (which months are those, precisely? Is that “09 Dec” and “09 Feb?” Do the numbers even go with the text, or are they separate labels?).

For all but the simplest of messages, this rendition defeats both pattern recognition and table look-up. We definitely need a better solution.

Use a Secondary Axis

Excel gives us an easy solution: break the data into two columns (“small” numbers in one and “large” numbers in the other) and plot them on separate axes. Now we can see all the data, including the patterns in all the months.

Bar Chart with Outliers on Secondary Axis

Bar chart, with outliers plotted using a secondary axis.

Unfortunately, pattern recognition tells us that the big-sales months are about the same as all the other months. It’s only the table look-up that tells us how big of a difference there is between the two blue columns and the rest of the data. This is why I’ve added data labels to the two columns: to aid table look-up.

Even if we tweaked around with the axes to set the outliers off from the rest of the data, we’d still have the same basic problem: pattern recognition would tell us that there is a much smaller difference than there actually is. By using a secondary axis, we’ve set up a basic conflict between pattern recognition and table look-up. Worse, it’s easy to confuse the axes; which bars go with which axis? Reproduction in black and white or grayscale would make it impossible to correctly connect bars to the correct axis. Some types of color blindness would similarly make it difficult to interpret the graph. Table look-up is easily defeated with secondary axes.

The secondary axis presents so many problems that I always advise against using it. Stephen Few, author of Show Me The Numbers and Information Dashboard Design, calls graphs with secondary axes “dual-scaled graphs.” In his 2008 article Dual-Scaled Axes in Graphs, he concludes that there is always a better way to display data than by using secondary axes. Excel makes it easy to create graphs like this, but it’s always a bad idea.

Take the Logarithm

In scientific applications, skewed data is common, and the usual solution is to plot the logarithm of the values.

Bar Chart with Logarithmic Axis

Bar chart plotting skewed with logarithmic axis.

With the logarithm, it is easy to plot, and see, all of the data. Trends in small values are not hidden. Pattern perception immediately tells us the overall story of the data. Table look-up is easier than with secondary axes, and immediately tells us the scale of the differences. Plotting the logarithm allows pattern perception and table look-up to compliment each other.

Below, I’ve created the same graph using a dot plot instead of a bar chart. Dot plots have many advantages over bar charts: most obviously, dot plots provide a better arrangement for category labels (e.g. the months); also, dot plots provide a clearer view of the data by plotting the data points rather than filling in the space between the axis and the data point. There are some nice introductions to dot plots, including William Cleveland’s works and a short introduction by Naomi Robbins. The message is clear: any data that you might present with a bar chart (or pie chart) will be better presented using dot plots.

Dot plot with logarithmic scale

Skewed data plotted on a dot plot using a logarithmic scale.

Use a Scale Break

Another approach, which might be better for audiences unfamiliar with logarithmic scales, is to use a scale break, or broken axis. With some work, we can create a scale break in Excel or OpenOffice.org.

Bar chart with a subtle scale break on the Y axis.

Bar chart with outliers plotted by introducing a subtle scale break on the y-axis.

There are plenty of tutorials for how to accomplish this in Excel. For this example, I created the graph in OpenOffice.org Spreadsheet, using the same graph with the secondary axis, above. I adjusted the two scales, turned off the labels for both y-axes and turned off the tick marks for the secondary y-axis. Then I copied the graph over to the OpenOffice.org Draw application and added y-axis labels and the break marks as drawing objects.

That pretty much highlights the first problem with this approach: it takes a lot of work. The second problem is that those break marks are just too subtle; people will miss them.

The bigger problem is with interpretation. As with the secondary axis, this subtle scale break sets up a basic conflict between the two basic operations of graph interpretation. Pattern recognition tells us that the numbers are comparable; it’s only table look-up that tells us what a large difference there is.

Cleveland’s recommendation, when the logarithm won’t work, is to use a full-panel scale break. In this way, pattern recognition tells that there are two distinct groups of data, and table look-up tells us what they are.

Dot plot with full scale break

Dot plot with a full scale break to show outliers.

The potential disadvantage of this approach is that pattern perception might be fooled. While the scale break visually groups the “large” values from the “small” ones, the scale also changes, so that the broader panel on the left actually represents a much narrower range of values (about 1100 dollars range) than the narrower panel on the right (about 17000 dollars range). Our audience might have difficulties interpreting this correctly.

Small Multiples

Edward Tufte has popularized the idea of small multiples, the emphasis of differences by repeating a graph or image with small changes from one frame to the next. In this case, we could show the full data set, losing fidelity in the smaller values, and then repeat the graph while progressively zooming in on a narrower and narrower slice with each repetition.

Dot Plot showing full data (including outliers) side-by-side with zoomed view.

The full data, with outliers, is plotted on the left. On the right, a zoomed view showing detail in the smaller values.

This shares many similarities to Cleveland’s full scale break, but provides greater flexibility. With this data, there are two natural ranges: 0 – 100000 and 0 – 1200. If there were more data between 1200 and 85000, we might repeat the graph several times, zooming in more with each repetition to show lower levels of detail.

I think there are two potential pitfalls. As with the full scale break, the audience might fail to appreciate the effect of the changes to scale. Worse, the audience might be fooled into thinking that each graph represented a different set of data, rather than just a different slice of the same data. Some care  in preparing such graphs will be needed for successful communication.

Summary

When presenting data that is, like the data above, arranged by category, use a dot plot instead of bar charts. When your data is heavily skewed, the best solution is to graph the logarithm of the data. However, if your audience will be unable to correctly interpret the logarithm, try a full scale break or small multiples.