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In our previous video, we
found the distance matrix,

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which computes the pairwise
distances between all

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the intensity values
in the flower vector.

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Now we can cluster
the intensity values

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using hierarchical clustering.

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So we're going to type
"cluster intensity."

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And then we're going to use
the hclust function, which

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is the hierarchical clustering
function in R, which

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takes as an input
the distance matrix.

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And then we're going to
specify the clustering method

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to be "word."

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As a reminder,
the "words" method

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is a minimum variants
method, which

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tries to find compact
and spherical clusters.

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We can think about it as
trying to minimize the variance

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within each cluster and the
distance among clusters.

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Now we can plot the
cluster dendrogram.

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So-- plot(clusterIntensity).

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And now we obtain the
cluster dendrogram.

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Let's have here a little
aside or a quick reminder

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about how to read a dendrogram
and make sense of it.

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Let us first consider this
toy dendrogram example.

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The lowest row of
nodes represent

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the data or the
individual observations,

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and the remaining nodes
represent the clusters.

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The vertical lines
depict the distance

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between two nodes or clusters.

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The taller the line, the more
dissimilar the clusters are.

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For instance, cluster D-E-F
is closer to cluster B-C-D-E-F

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than cluster B-C is.

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And this is well depicted by the
height of the lines connecting

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each of clusters B-C and
D-E-F to their parent node.

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Now cutting the dendrogram
at a given level

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yields a certain
partitioning of the data.

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For instance, if we cut the tree
between levels two and three,

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we obtain four clusters,
A, B-C, D-E, and F.

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If we cut the dendrogram
between levels three and four,

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then we obtain three
clusters, A, B-C, and D-E-F.

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And if we were to cut the
dendrogram between levels four

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and five, then we obtain two
clusters, A and B-C-D-E-F.

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What to choose, two,
three, or four clusters?

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Well, the smaller the number
of clusters, the coarser

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the clustering is.

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But at the same time,
having many clusters

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may be too much of a stretch.

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We should always have
this trade-off in mind.

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Now the distance
information between clusters

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can guide our choice of
the number of clusters.

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A good partition belongs to a
cut that has a good enough room

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to move up and down.

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For instance, the cut between
levels two and three can go up

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until it reaches cluster D-E-F.
The cut between levels three

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and four has more room to move
until it reaches the cluster

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B-C-D-E-F. And the cut between
levels four and five has

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the least room.

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So it seems like
choosing three clusters

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is reasonable in this case.

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Going back to our
dendrogram, it seems

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that having two clusters
or three clusters

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is reasonable in our case.

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We can actually
visualize the cuts

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by plotting rectangles around
the clusters on this tree.

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To do so, we can use the
rect.hclust function,

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which takes as an input
clusterIntensiy, which

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is our tree.

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And then we can specify
the number of clusters

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that we want.

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So let's set k=3.

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And we can color the
borders of the rectangles.

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And let's color them,
for instance, in red.

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Now going back to
our dendrogram,

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now we can see
the three clusters

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in these red rectangles.

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Now let us split the data
into these three clusters.

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We're going to
call our clusters,

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for instance, flowerClusters.

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And then we're going to
use the function cut tree.

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And literally, this
function cuts the dendrogram

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into however many
clusters we want.

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The input would be
clusterIntensity.

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And then we have to specify k=3,
because we would like to have

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three clusters.

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Now let us output
the flower clusters

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variable to see how it looks.

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So flowerClusters.

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And we see that the
flower cluster is actually

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a vector that assigns each
intensity value in the flower

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vector to a cluster.

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It actually has the same
length, which is 2,005,

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and has values 1,
2, and 3, which

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correspond to each cluster.

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To find the mean intensity
value of each of our clusters,

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we can use the tapply
function and ask R to group

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the values in the flower
vector according to the flower

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clusters, and then
apply the mean statistic

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to each of the groups.

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What we obtain is that the first
cluster has a mean intensity

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value of 0.08, which
is closest to zero,

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and this means
that it corresponds

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to the darkest
shape in our image.

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And then the third cluster
here, which is closest to 1,

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corresponds to
the fairest shade.

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And now the fun part.

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Let us see how the
image was segmented.

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To output an image, we
can use the image function

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in R, which takes a
matrix as an input.

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But the variable flowerClusters,
as we just saw, is a vector.

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So we need to convert
it into a matrix.

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We can do this by setting the
dimension of this variable

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by using the dimension function.

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So, let's use the
dimension function, or dim,

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which takes as input
flowerClusters.

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And then we're going to
use the combined function,

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or the c function.

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And its first argument
will be the number of rows

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that we want for the matrix,
and that would be 50.

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And the second argument would
be the number of columns.

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Why did we use 50?

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Simply because we have a 50
by 50 resolution picture.

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Now pressing Enter,
and flowerClusters

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looks like a matrix.

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Now we can use the
function image,

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which takes as an input the
"flower cl clusters" matrix.

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And let's turn off the axes
by writing axes="false."

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And now, going back to
our graphics window,

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we can see our
segmented image here.

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The darkest shade corresponds
to the background,

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and this is actually associated
with the first cluster.

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The one in the middle is
the core of the flower,

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and this is cluster 2.

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And then the petals
correspond to cluster 3,

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which has the fairest
shade in our image.

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Let us now check how the
original image looked.

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Let's go back to the console
and then maximize it here.

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So let's go back to
our image function,

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but now this time the
input is the flower matrix.

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And then let's keep
the axis as false.

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But now, how about we add an
additional argument regarding

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the color scheme?

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Let's make it grayscale.

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So we're going to
take the color,

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and it's going to take
the function gray.

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And the input to this
function is a sequence

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of values that goes
from 0 to 1, which

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actually is from black to white.

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And then we have to
also specify its length,

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and that's specified as 256,
because this corresponds

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to the convention for grayscale.

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And now, going
back to our image,

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now we can see our original
grayscale image here.

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You can see that it has a
very, very low resolution.

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But in our next video,
we will try to segment

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an MRI image of
the brain that has

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a much, much higher resolution.