How do I count thee? Let me count the ways?

Find the next number in the sequence

Between ages two and four, most children can count up to at least ten. If you ask your child, "What number comes next after 1, 2, 3, ...

Saturday, October 29, 2022

Find the next number in the sequence

Between ages two and four, most children can count up to at least ten.

If you ask your child, "What number comes next after 1, 2, 3, 4, 5?" they will probably say "6."

But to math nerds, any number can be the next number in a finite sequence. I like -14.

Given a sequence of n real numbers f(x1), f(x2), f(x3), ... , f(xn), there is always a mathematical procedure to find the next number f(x n+1) of the sequence. The resulting solution may not appear to be satisfying to students, but it is mathematically logical.

I can draw a smooth curve through the points (1,1), (2,2), (3,3), (4,4), (5,5), (6, -14). If I can find an equation for that smooth curve, then I know my answer of -14 has some logic to it. Actually many equations will work.

In my example one equation is of the form y = (x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(A/120) + x, where A is chosen so that when x is 6, the first term reduces to A, and A + 6 equals the -14 I want. So A is -20. This is called a collocation polynomial.

There is a theorem that for n+1 distinct values of xi and their corresponding yi values, there is a unique polynomial P of degree n with P(xi) = yi. One method to find P is to use polynomial regression. Another way is to use Newton's Forward Difference Formula (probably no longer taught in Numerical Analysis courses).

Higher degree polynomials than degree n is one reason why additional equations will work.

The equation does not have to be a polynomial, which then adds rational functions among others.

Of course the next number after -14 can be any number. It could be 7 :)

There are many famous sequences, and of course someone catalogued them.

Here is some R code.

xpoints <- c(1,2,3,4,5,6)
ypoints <- c(1,2,3,4,5,-14)
y <- vector()
x <- seq(from=1, to=6, by=.01)
y <- (x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(-20/120) + x
plot(xpoints, ypoints, pch=18, type="p", cex=2, col="blue", xlim=c(1,6), ylim=c(-14,6), xlab="x", ylab="y")
lines(x,y, pch = 19, cex=1.3, col = "red")
fit <- lm(ypoints ~ xpoints + I(xpoints^2) + I(xpoints^3) +I(xpoints^4) +I(xpoints^5) )
s <- summary(fit)
bo <- s$coefficient[1]
b1 <- s$coefficient[2]
b2 <- s$coefficient[3]
b3 <- s$coefficient[4]
b4 <- s$coefficient[5]
b5 <- s$coefficient[6]
x <- seq(from=1, to=6, by=.01)
z <- bo+b1*x+b2*x^2+b3*x^3+b4*x^4+b5*x^5
plot(xpoints, ypoints, pch=18, type="p", cex=2, col="blue", xlim=c(1,6), ylim=c(-14,6), xlab="x", ylab="y")
lines(x,z, pch = 19, cex=1.3, col = "red")

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Saturday, September 17, 2022

How do I count thee? Let me count the ways?

by Jerry Tuttle   

In Major League Baseball, a player who hits 50 home runs in a single season has hit a lot of home runs. Suppose I want to count the number of 50 homer seasons by team, and also the number of 50 homer seasons by New York Yankees. (I will count Maris and Mantle in 1961 as two.) Here is the data including Aaron Judge's 62 in 2022 :

You would think base R would have a count function such as count(df$Team) and count(df$Team == "NYY") but this gives the error "could not find function 'count'". Base R does not have a count function.

Base R has at last four ways to perform a count:

1. The table function will count items in a vector.    table(df$Team) presents results horizontally, and data.frame(table(df$Team)) presents results vertically.    table(df$Team == "NYY") displays results 37 false and 10 true, while table(df$Team == "NYY")[2] just displays the result 10 true.

2. The sum function can be used to count the number of rows meeting a condition.    sum(df$Team == "NYY") displays the result 10. Here df$Team == "NYY" is creating a logical vector, and sum is summing the number of true = 1.

3. Similar to sum, nrow(df[df$Team == "NYY", ]) counts the number of rows meeting the NYY condition.

4. The length function counts the number of elements in an R object.    length(which(df$Team == "NYY")) , length(df$Team[df$Team == "NYY"]) , and length(grep("NYY", df[ , "Team"])) are all ways that will count the 10 Yankees.

The more direct solution to counting uses the count function in the dplyr library. Note that dplyr's count function applies to a data frame or tibble, but not to a vector. After loading library(dplyr) ,

1. df %>% count(Team) lists the count for each team.

2. df %>% filter(Team = "NYY") lists each Yankee, and you can see there are 10.

3. df %>% count(Team == "NYY") displays 37 false and 10 true, while df %>% filter(Team == "NYY") %>% count() just displays the 10 true.

The following is a bar chart of the results by team for teams with at least 1 50 homer season:

Finally, "How do I count thee? Let me count the ways?" is of course adapted from Elizabeth Barrett Browning's poem "How do I love thee? Let me count the ways?" But in her poem, just how would we count the number of times "love" is mentioned? The tidytext library makes counting words fairly easy, and the answer is ten, the same number of 50 homer Yankee seasons. Coincidence?

The following is all the R code. Happy counting!


df <- data.frame(
   Player=c('Ruth','Ruth','Ruth','Ruth','Wilson','Foxx','Greenberg','Foxx','Kiner','Mize','Kiner','Mays','Mantle','Maris', 'Mantle','Mays','Foster','Fielder','Belle','McGwire','Anderson','McGwire','Griffey','McGwire','Sosa','Griffey', 'Vaughn','McGwire','Sosa','Sosa','Bonds','Sosa','Gonzalez','Rodriguez','Rodriguez','Thome','Jones','Howard','Ortiz', 'Rodriguez','Fielder','Bautista','Davis','Stanton','Judge','Alonso','Judge'),
   Year=c(1920,1921,1927,1928,1930,1932,1938,1938,1947,1947,1949,1955,1956,1961,1961,1965,1977,1990,1995,1996,1996,1997,1997, 1998,1998,1998,1998,1999,1999,2000,2001,2001,2001,2001,2002,2002,2005,2006,2006,2007,2007,2010,2013,2017,2017,2019,2022),
   Homers=c(54,59,60,54,56,58,58,50,51,51,54,51,52,61,54,52,52,51,50,52,50,58,56,70,66,56,50,65,63,50,73,64,57,52,57,52,51, 58,54,54,50,54,53,59,52,53,62),
   Team=c('NYY','NYY','NYY','NYY','CHC','PHA','DET','BOS','PIT','NYG','PIT','NYG','NYY','NYY','NYY','SF','CIN','DET','CLE', 'OAK','BAL','OAK/SLC','SEA','SLC','CHC','SEA','SD','SLC','CHC','CHC','SF','CHC','ARI','TEX','TEX','CLE','ATL','PHP', 'BOR','NYY','MIL','TOR','BAL','MIA','NYY','NYM','NYY'))


# base R ways to count:

table(df$Team)    # shows results horizontally
data.frame(table(df$Team))    #shows results vertically
table(df$Team == "NYY")    # displays 37 false and 10 true
table(df$Team == "NYY")[2]

sum(df$Team == "NYY")    # displays the result 10.

nrow(df[df$Team == "NYY", ])    # counts the number of rows meeting the NYY condition.

length(which(df$Team == "NYY"))     # which returns a vector of indices which are true
length(df$Team[df$Team == "NYY"])
length(grep("NYY", df[ , "Team"]))     # grep returns a vector of indices that match the pattern

# dplyr R ways to count; remember to load library(dplyr):

df %>% count(Team)    # lists the count for each team.

df %>% filter(Team == "NYY")    # lists each Yankee, and you can see there are 10.

df %>% count(Team == "NYY")    # displays 37 false and 10 true, while
df %>% filter(Team == "NYY") %>% count()    # just displays the 10 true.

# barplot of all teams with at least 1 50 homer season; remember to load library(ggplot2)

df %>%
    group_by(Team) %>%
    summarise(count = n()) %>%
    ggplot(aes(x=reorder(Team, count), y=count, fill=Team)) +
    geom_bar(stat = 'identity') +
    ggtitle("Count of 50 Homer Seasons") +
    xlab("Team") +
    scale_y_continuous(breaks=c(1,2,3,4,5,6,7,8,9,10)) +
    coord_flip() +
    theme(plot.title = element_text(face="bold", size=18)) +
    theme(axis.title.y = element_text(face="bold")) +
    theme(axis.title.x = element_blank()) +
    theme(axis.text.x = element_text(size=12, face="bold"),
    axis.text.y = element_text(size=12, face="bold")) +

# count number of times "love" is mentioned in Browning's poem; remember to load library(tidytext)

textfile <- c("How do I love thee? Let me count the ways.",
"I love thee to the depth and breadth and height",
"My soul can reach, when feeling out of sight",
"For the ends of being and ideal grace.",
"I love thee to the level of every day's",
"Most quiet need, by sun and candle-light.",
"I love thee freely, as men strive for right.",
"I love thee purely, as they turn from praise.",
"I love thee with the passion put to use",
"In my old griefs, and with my childhood's faith.",
"I love thee with a love I seemed to lose",
"With my lost saints. I love thee with the breath,",
"Smiles, tears, of all my life; and, if God choose,",
"I shall but love thee better after death.")

df<-data.frame(line=1:length(textfile), text=textfile)
df_words <- df %>% unnest_tokens(word, text)
cleaned_words <- df_words %>% anti_join(get_stopwords())
cleaned_words %>% count(word, sort = TRUE) %>% head(6)
cleaned_words %>% filter(word == "love") %>% count()

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Wednesday, June 22, 2022

What is the difference between statistics and data analysis?

What is the difference between statistics and data analysis?

Of course to answer this we need to define those terms, and definitions of such things are hardly standard. But they are nor particularly standard in other disciplines either. Can you define art? Music? How about mathematics?

Would you have defined mathematics as "including such topics as numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes," as in Wikipedia? Is this all-encompassing?

Statistics and data analysis have some overlaps. Both involve defining, exploring, cleaning, visualizing, and describing data. Data analyst students study some traditional statistics. Statistics students nowadays study some data analysis.

The father and daughter Larose team have suggested a working distinction of inferential statistics versus data mining (so neither of these is identical to the terms in the first sentence above) as follows:

Inferential statistics involves having a prior hypothesis about a population and testing that hypothesis with a sample from that population. The test may result in statistical significance, even if there is no practical significance.

Data mining does not begin with a prior hypothesis, but rather the analyst "freely trolls through the data for actionable results." (Larose, p. 161)

Larose, D.T. & Larose, C.D. (2015). Data mining and predictive analytics. Wiley. Hoboken, NJ.

Wednesday, February 3, 2021

ROC for Decision Trees – where did the data come from?

      In doing decision tree classification problems, I have often graphed the ROC (Receiver Operating Characteristic) curve. The True Positive Rate (TPR) is on the y-axis, and the False Positive Rate (FPR) is on the x-axis. True Positive is when the lab test predicts you have the disease and you actually do have it. False Positive is when the lab test predicts you have the disease but you actually do not have it.
      The following code uses the sample dataset kyphosis from the rpart package, creates a default decision tree, prints the confusion matrix, and plots the ROC curve. (Kyphosis is a type of spinal deformity.)

df <- kyphosis
mytree <- rpart(Kyphosis ~ Age + Number + Start, data = df, method="class")
fancyRpartPlot(mytree, uniform=TRUE, main=”Kyphosis Tree”)
predicted <- predict(mytree, type="class")
pred <- prediction(predict(mytree, type="prob")[, 2], df$Kyphosis)
plot(performance(pred, “tpr”, “fpr”), col=”blue”, main=”ROC Kyphosis, using library ROCR”)
abline(0, 1, lty=2)
auc <- performance(pred, "auc")

However, for a long time there has been a disconnect in my mind between the confusion matrix and the ROC curve.  The confusion matrix provides a single value of the ordered pair  (x=FPR, y=TPR) on the ROC curve, but the ROC curve has a range of values.  Where are the other values coming from?

The answer is that a default decision tree confusion matrix uses a single probability threshold value of 50%.  A decision tree ends in a set of terminal nodes.  Every observation falls in exactly one of those terminal nodes.  The predicted classification for the entire node is based on whichever classification has the greater percentage of observations, which for binary classifications requires a probability greater than 50%.  So for example if a single observation has predicted probability based on its terminal node of 58% that the disease is present, then a 50% threshold would classify this observation as disease being present.  But if the threshold were changed to 60%, then the observation would be classified as disease not being present. 

The ROC curve uses a variety of probability thresholds, reclassifies each observation, recalculates the confusion matrix, and recalculates a new value of the ordered pair (x=FPR, y=TPR) for the ROC curve.  The resulting curve shows the spread of these ordered pairs over all (including interpolated, and possibly extrapolated) probability thresholds, but the threshold values are not commonly displayed on the curve.  Plotting performance in the ROCR package does this behind the scenes, but I wanted to verify this myself.  This dataset has a small number of predictions of disease present, and at large threshold values the prediction is zero, resulting in a one column confusion matrix and zero FPR and TPR.  The following code individually applies different probability thresholds to build the ROC curve, although it does not extrapolate for large values of FPR and TPR.

dat <- data.frame()
s <- predict(mytree, type="prob")[, 2]
for (i in 1:21){
p <- .05*(i-1)
thresh <- ifelse(s>p, “present”, “absent”)
t <- table(df$Kyphosis,thresh)
fpr <- ifelse(ncol(t)==1, 0, t[1,2] / (t[1,2] + t[1,1]))
tpr <- ifelse(ncol(t)==1, 0, t[2,2] / (t[2,2] + t[2,1]))
dat[i,1] <- fpr
dat[i,2] <- tpr
colnames(dat) <- c("fpr", "tpr")
plot(x=dat$fpr, y=dat$tpr, xlab=”FPR”, ylab=”TPR”, xlim=c(0,1),
main=”ROC Kyphosis, using indiv threshold calcs”, type=”b”, col=”blue”)
abline(0, 1, lty=2)

Monday, July 6, 2020

Outliers and Domain Knowledge

      I would like to share some thoughts about outliers and domain knowledge.
      One of the common steps during the data exploration stage is the search for outliers. Some analysis methods such as regression are very sensitive to outliers. As an example of sensitivity, in the following data (10,10) is an outlier. Including the outlier produces a regression line y = .26 + .91x, while excluding the outlier produces the very different regression line y = 2.

x <- c(1,1,1,2,2,2,3,3,3,10)
y <- c(1,2,3,1,2,3,1,2,3,10)
df <- data.frame(cbind(x,y))
lm(y ~ x, df)
plot(x,y, pch=16)
abline(lm(y ~ x, df)

      Statistics books sometimes define an outlier as being outside the range of Q1 ± 1.5*IQR or Q1 ± 3*IQR, where Q1 is the lower quartile (25th percentile value), Q3 is the upper quartile (75th percentile value), and the interquartle range IQR = Q3 – Q1.
      What does one do with an outlier? It could be bad data. It is pretty unlikely that there is a graduate student who is age 9, but we don’t know whether the value should be 19 (very rare, but possible), or 29 (likely), or 39 or more (not so rare). If we have the opportunity to ask the owner of the data, perhaps we can get the value corrected. More likely is we can not ask the owner. We can delete the entire observation, or we can pretend to correct the value with a mode or median value or a judgmental value.

      Perhaps the outlier is not bad data but rather just an unusual value. In a portfolio of property or liability insurance claims, the distribution is often positively skewed (mean greater than mode, a long tail to the positive side of the mode). Most claims are small, but occasionally there is that one enormous claim. What does one do with that outlier value? Some authors consider data science to be the Venn diagram intersection among math/statistics, computer science, and domain knowledge (see for example Drew Conway, above, in ) If the data scientist is not the domain expert, he or she should consult with one. With insurance claims there are several possibilities. One is that the enormous claim is one that is unlikely to reoccur for any number of reasons. Hopefully there will never be another September 11 type destruction of two World Trade Center buildings owned by a single owner. Another example is when the insurance policy terms are revised to literally prohibit a specific kind of claim in the future. Another possibility is that the specific claim is unlikely to reoccur (the insurance company stopped insuring wheelchairs, so there won’t be another wheelchair claim), but that claim is representative of another kind of claim that is likely to occur. In this case, the outlier should not be deleted. One author has said it takes Solomon-like wisdom to discern which possibility to believe.
      An interesting example of outliers occurs with sports data. For many reasons, US major league baseball player statistics have changed over the years. There are more great home run seasons nowadays than decades ago, but there are fewer great batting average seasons. Baseball fanatics know the last .400 hitter (40% ratio of hits divided by at bats over the entire season) was Ted Williams in 1941. If we have 80 years of baseball data and we are predicting the probability of another .400 hitter, we would predict close to zero. It’s possible, but extremely unlikely, right? Actually no. Assuming there will still be a shortened season in 2020, a decision that may change, this author is willing to forecast that there will be a .400 hitter in a shortened season. This is due to the theory that batters need less time in spring training practice to be at full ability than pitchers, and it is easier to achieve .400 in a small number of at bats earlier in the season when the pitchers are not at full ability. This is another example of domain expertise as a lifetime baseball fan.

Tuesday, April 21, 2020

“Those who can, do; those who can’t – use computer simulation.”

          “Those who can, do; those who can’t – use computer simulation.” This quote was inspired by playwright George Bernard Shaw. Computer simulation is a powerful tool that attempts to reproduce the behavior of some real-world system by sampling from one or more probability distributions. It can help explain and illustrate difficult concepts such as the Monty Hall game show problem. It can also solve problems that are hard or impossible to solve directly.

          One example of a problem of this latter type is the probability distribution of the sum of n independent random variables each having probability distribution g, where n is also a random variable having probability distribution f. This is a special case of a statistical subject called convolutions, where one calculates the convolution of all possible values of the individual distributions. For certain choices of f and g, such as Poisson for f and gamma for g, the resulting convolution is an integrateable function and there is no need for simulation. But for many choices of f and g, simulation or some other numerical method is needed.

          A real-world use of this type of simulation is in modeling loss events of a business entity such as a bank or insurance company. The entity will have some random number of events n per year of random sizes that is modeled by a frequency distribution f such as Poisson or negative binomial. A number n is randomly selected from the frequency distribution. Then n random numbers are randomly selected from a severity distribution g such as lognormal or gamma to simulate the sizes of the n loss events. The n loss amounts are added to produce a total value of losses for one year. The process is repeated some large number of times, say 10,000, and the 10,000 numbers are ranked. If the entity is interested in the 99.9th percentile such loss, that value is the 9,990th largest value.

          Base R provides functions to simulate from many probability distributions. For example, rpois(n, lamda) produces n Poisson distributed samples from a distribution having population mean lambda. Poisson is a single parameter distribution with variance equal to the mean. An alternative frequency distribution with a more flexible option for the variance is the negative binomial distribution. Its R function is rnbinom(n, mu, size) which produces n negative binomial distributed samples from a distribution having population mean mu and dispersion parameter size, where size = mu^2 / (variance – mu).

          Most severity distributions are not symmetrical but rather are positively skewed with low mean and long positive tail. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Its R function is rlnorm(n, meanlog, sdlog) which produces n lognormally distributed samples from a distribution having population mean m and standard deviation s, and here meanlog = LN(m) – .5*LN((s/m)^2 + 1)) and sdlog = .5*LN((s/m)^2+1)).

          It is helpful to plot the resulting histogram of the 10,000 simulations. The real-world purpose of the exercise may be to identify the 99.9th percentile value. The R quantile function quantile(x, probs) returns the percentile value equal to probs. To display the percentile value on a plot, use the text and arrow functions. R has a text function that adds text a plot, text(x, y, label), which adds a label at coordinate (x,y). Further, R has an arrows function, arrows(x0, y0, x1, y1, code=2), which draws an arrow from (x0, y0) to (x1, y1) with code 2 drawing the arrowhead at (x1, y1).

          The following is the R code and resulting histogram plot of a negative binomial frequency and lognormal severity simulation. The user input values are based on the mean and standard deviation values of a dataset.

# Monte Carlo simulation. Negative binomial frequency, lognormal severity.
# negbinom
nb_m <- 50 # This is a user input.
nb_sd <- 10 # This is a user input.
nb_var <- nb_sd^2
nb_size <- nb_m^2/(nb_var – nb_m)
# lognorm
xbar <- 60 # This is a user input.
sd <- 40 # This is a user input.
l_mean <- log(xbar) – .5*log((sd/xbar)^2 + 1)
l_sd <- sqrt(log((sd/xbar)^2 + 1))
num_sims <- 10000 # This is a user input.
set.seed(1234) # This is a user input.
rtotal <- vector()
for (i in 1:num_sims)
nb_random <- rnbinom(n=1, mu=nb_m, size=nb_size)
l_random <- rlnorm(nb_random, meanlog=l_mean, sdlog=l_sd)
rtotal[i] <- sum(l_random)
rtotal <- sort(rtotal)
m <- round(mean(rtotal), digits=0)
percentile_999 <- round(quantile(rtotal, probs=.999), digits=0)
print(paste(“Mean = “, m, ” 99.9th percentile = “, percentile_999))
hist(rtotal, breaks=20, col=”red”, xlab=”Annual Loss”, ylab=”Frequency”,
main=”Monte Carlo Simulation”)
text(percentile_999, 100, “99.9th Percentile”)
arrows(percentile_999,75, percentile_999,0, code=2)

8 Crayola Crayons

Many of my friends know I am color-challenged. Not colorblind; I see some colors, just not as many as most people see. I don't recall always being this way. In my earliest elementary school years I had a box of 8 Crayola crayons, and I did just fine. My problems began when Crayola started adding more and more colors including combinations like green yellow and yellow green. With nothing better to do, I just wrote a little program in R, and here are the original eight colors, which I still identify just fine. There is no meaning to the heights of the bars.

title <- c("Original 8 Crayola Crayons")
subtitle <- c("According to")
temp <- c(5,7,6,4,8,5,2,5) # meaningless
colors <- c("red", "yellow", "blue", "green", "orange", "brown", "violet", "black")
hex <- c("#FF0000", "#FFFF00", "#0000FF", "#008001", "#FF6600", "#964B00", "#6A0DAD", "#000000")
barplot(temp, col=hex, main=title, sub=subtitle, names.arg=colors)