pendigits data - first analysis

Data files

Download the two following files :

• pendigits.tra
• pendigits.tes

Data description can be found here: each line represent a draw of a digit between 0 and 9, with 8 points (\((x,y)\) coordinates for each point) and the number drawn.

Importation

The data files are here easy to read:

• the names of the attributes are not present
• the values are separated by a comma ","

pen.tra = read.table("donnees/pendigits.tra", sep = ",")
pen.tes = read.table("donnees/pendigits.tes", sep = ",")

Now, to collapse them together, we use the following code

pen = rbind(pen.tra, pen.tes)

At the end of the import process, we name the attributes according to what we know: the two first attributes are \((x_1, y_1)\), the third and the fourth \((x_2, y_2)\), and so on. The last attribute is the written digit. Then, we create the vector "X1" "Y1" "X2" "Y2" ... "X8" "Y8" "digit".

names(pen) = c(paste0(c("X", "Y"), rep(1:8, each = 2)), "digit")
class(pen$digit)

## [1] "integer"

digit variable is here a numeric attribute (integer more precisely). But, it is a nominal one in reality. So, we transform it into factor.

pen$digit = factor(pen$digit)
class(pen$digit)

## [1] "factor"

We now have the following value (only 6 first are presented here).

Description

table(pen$digit)

## 
##    0    1    2    3    4    5    6    7    8    9 
## 1143 1143 1144 1055 1144 1055 1056 1142 1055 1055

ggplot(pen, aes(digit, fill = digit)) + 
    geom_bar() +
    guides(fill = FALSE)

tab = apply(pen[-17], 2, function(x) {
    return (c(Mean = mean(x),
              Median = median(x),
              StdDev = sd(x)))
})
round(t(tab), 2)

##     Mean Median StdDev
## X1 38.81   32.0  34.26
## Y1 85.12   89.0  16.22
## X2 40.61   40.0  26.34
## Y2 83.77   91.0  19.16
## X3 49.77   53.0  34.10
## Y3 65.57   71.0  27.00
## X4 51.22   53.5  30.58
## Y4 44.50   43.0  29.91
## X5 56.87   60.0  34.14
## Y5 33.70   33.0  27.25
## X6 60.52   73.0  37.29
## Y6 34.83   30.0  27.12
## X7 55.02   53.0  22.34
## Y7 34.94   27.0  33.16
## X8 47.29   40.0  41.76
## Y8 28.85    9.0  35.78

for (j in 1:16)
    boxplot(pen[,j])

pen.X = pen[seq(1, 15, 2)]
names(pen.X)

## [1] "X1" "X2" "X3" "X4" "X5" "X6" "X7" "X8"

pen.X.melt = melt(pen.X)

## No id variables; using all as measure variables

knitr::kable(head(pen.X.melt))

ggplot(pen.X.melt, aes(variable, value)) +
    geom_boxplot()

ggplot(melt(pen[seq(2, 16, 2)]), aes(variable, value)) + geom_boxplot()

## No id variables; using all as measure variables

apply(pen[-17], 2, tapply, pen$digit, mean)

agg = aggregate(. ~ digit, pen, mean)

knitr::kable(agg, digits = 2)

pen.melt = melt(pen, id.vars = 17)
head(pen.melt)

##   digit variable value
## 1     8       X1    47
## 2     2       X1     0
## 3     1       X1     0
## 4     4       X1     0
## 5     1       X1     0
## 6     6       X1   100

ggplot(pen.melt, aes(digit, value, color = digit)) + 
    geom_boxplot() +
    facet_wrap(~ variable)

Representation of a digit

We want to represent the first "0" drawn. This step is interesting for further analysis. We select the first 0 digit, and extract \(x\) and \(y\) coordinates. After we plot these with a line, and add the number of the point.

first.0 = subset(pen, digit == 0)[1,]
x = unlist(first.0[seq(1,15, by = 2)])
y = unlist(first.0[seq(2,16, by = 2)])
par(mar = c(0, 0, 0, 0) + .1)
plot(x, y, type = "b", pch = " ", axes = FALSE, col = "gray70")
text(x, y, 1:8)

Function

Now, we have the basis of a function that can reproduc each digit. We want the following constraints for this one :

• Parameters:
  • v: a vector with the 16 values of each point (if it is a data.frame, transform it as simple vector)
  • n: a numeric value (with NULL value by default), denoting the number to drawn (possibly not pass)
  • point: a boolean vector (with FALSE value by default), indicating if the point number has to be added to the graph
  • add: a boolean vector (with FALSE value by default), indicating if the plot is to added to the previous one
  • color: a specific color (with "black" value by default)
• Output:
  • line representing the drawn
  • number drawn as title (if known)
  • point number (if wanted)

Here is the final function.

drawn <- function(v, n = NULL, point = FALSE, add = FALSE, color = "black") {
    # Transformation of the data.frame if needed
    if (is.data.frame(v))
        v = unlist(v)
    # extract x and y coordinates
    x = v[seq(1, 15, by = 2)]
    y = v[seq(2, 16, by = 2)]
    # optimize space into graphics in reducing margin (sse ?par for more information)
    opar = par(mar = c(0, 0, 2, 0) + .1)
    if (!add) { # Create a graphic
        plot(x, y, 
             # Specify limits is a way to have always the same frame for plotting lines
             xlim = c(-5, 105), ylim = c(-5, 105),
             # Do not show axes
             axes = FALSE,
             # If point is TRUE, we add a space (with pch = " ") at each point
             # If not, draw a line
             type = ifelse(point, "b", "l"), 
             pch = " ", 
             # Specify color (black by default)
             col = color,
             # Add a title (NULL by default)
             main = n)
        if (point) text(x, y, 1:8)
    } else { # Add line to the plot
        # lines() add lines to an existing plot (the last produce)
        lines(x, y, 
              # same comment than before
              xlim = c(-5, 105), ylim = c(-5, 105),
              type = ifelse(point, "b", "l"), 
              pch = " ", 
              col = color)
    }
    par(opar)
}

Now, here some use of this function

drawn(first.0)

drawn(first.0, n = 0)

drawn(first.0, point = TRUE)

drawn(first.0)
first.1 = subset(pen, digit == 1)[1,]
drawn(first.1, add = TRUE, col = "red")

First drawn of each digit

For our curiosity, we can now drawn each first digit of the data. The mfrow parameter of the par() function permits us to separate the graphical window into 10 cells (with 2 rows and 5 columns).

par(mfrow = c(2, 5))
for (i in 0:9) {
    s = subset(pen, digit == i)[1,]
    drawn(s[-16], n = i)
}

Average digit

From the previous section, we now want to represent the average digit, i.e. a line produce by the average value for each coordinate. To see if there are some area that are not use for a particular digit, we represent also all the drawn in a very light gray.

par(mfrow = c(2, 5))
for (i in 0:9) {
    s = subset(pen, digit == i)
    # Computing the average values
    mean = apply(s[-17], 2, mean)
    # Drawn the first drawn
    drawn(s[1,1:16], col = "gray90", n = i)
    # Add all other drawn
    for (j in 2:nrow(s)) 
        drawn(s[j,1:16], col = "gray90", add = TRUE)
    # Last, add the average line
    drawn(mean[-17], add = TRUE)
}

PCA

A good way to represent many variables in one plot is the Principal Component Analysis. Then, we perform PCA on the data, and look at the quality of the projection on the first components.

res = PCA(pen, quali.sup = 17, graph = FALSE)
eig = data.frame(comp = 1:16,
                 setNames(res$eig, c("eigenvalue", "percent", "cum.percent")))
ggplot(eig) +
    geom_bar(aes(comp, percent), stat = "identity") +
    stat_summary(aes(comp, cum.percent, group = 1), 
                 fun.y = sum, geom = "line")

Now, we plot the first plan with a specific color for each digit. For that, we have to add the digit variable to the coordinates data.frame produce by PCA().

res2 = data.frame(res$ind$coord, digit = pen$digit)
ggplot(res2, aes(Dim.1, Dim.2, color = digit)) + 
    geom_point()

For each digit

The above plot is hard to understand. So, we choose to represent each digit separately, to have a better idea on how digit are spread (or not) in the PCA projection.

ggplot(res2, aes(Dim.1, Dim.2, color = digit)) +
    geom_point() +
    facet_wrap(~ digit) +
    guides(color = FALSE)

We conclude here that we surely need to clusters data separatly for each digit, to detect if there really are different ways to write a digit (and how).

Point representation with ggplot()

We can also represent digit drawn with the ggplot() function. We have to use the melt() function, and to be careful on how we do this.

f0 = data.frame(x = unlist(first.0[seq(1, 15, 2)]),
                y = unlist(first.0[seq(2, 16, 2)]))
ggplot(f0, aes(x, y)) + 
    geom_path() +
    theme_void()

pen$id = 1:nrow(pen)
pen.melt = melt(pen, id.vars = c("id", "digit"))
pen.melt$coord = substr(pen.melt$variable, 1, 1)
pen.melt$point = substr(pen.melt$variable, 2, 2)
pen.melt.X = subset(pen.melt, coord == "X", -c(variable, coord))
names(pen.melt.X) = c("id", "digit", "x", "point")
pen.melt.Y = subset(pen.melt, coord == "Y", -c(variable, coord))
names(pen.melt.Y) = c("id", "digit", "y", "point")
pen.merge = merge(pen.melt.X, pen.melt.Y)

agg.melt = melt(agg, id.vars = "digit")
agg.melt$coord = substr(agg.melt$variable, 1, 1)
agg.melt$point = substr(agg.melt$variable, 2, 2)
agg.melt.X = subset(agg.melt, coord == "X", -c(variable, coord))
names(agg.melt.X) = c("digit", "x", "point")
agg.melt.Y = subset(agg.melt, coord == "Y", -c(variable, coord))
names(agg.melt.Y) = c("digit", "y", "point")
agg.merge = merge(agg.melt.X, agg.melt.Y)

ggplot(pen.merge, aes(x, y)) +
    geom_path(aes(group = id), color = "gray90") +
    facet_wrap(~ digit) +
    geom_path(data = agg.merge) +
    theme_void()