Lab 6: KNNs & ID Trees

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Identification Trees

In this section, we will start by using an ID tree to classify points, then we will construct ID trees from raw data.

Using an ID Tree to classify unknown points

In this lab, we will represent an ID tree recursively as a tree of IdentificationTreeNode objects. For more information, see the API below.

For the ID trees section, we will represent a data point as a dictionary mapping attributes to values. For example:

point = {"X": 1, "Y": 2, "Angel": True}   ##(jmn todo: use a better example that matches the test data)

To begin, let's use an ID tree to classify a point! Write a function id_tree_classify_point that takes in a point (represented as a dictionary) and an ID tree (represented as a IdentificationTreeNode) and uses the ID tree to classify the point (even if the point already has a classification defined).

def id_tree_classify_point(point, id_tree):

This function can be written cleanly in 3-4 lines, either iteratively or recursively. If you're not sure how, check out the available methods in the IdentificationTreeNode API. In particular, the methods .apply_classifier(point) and .get_node_classification() may be useful.

Calculating Disorder

Now that we're able to classify points, it's time to actually construct an ID tree! We'll start by coding the information-disorder equations to calculate the disorder of a branch or test. These equations are explained in more detail on page 429 of the reading.

(todo provide equations here & explain disorder)

branch_disorder should take in a branch (of a decision stump), represented as a list of classifications (such as ["Oak", "Oak", "Maple"]), and return the disorder of the branch (a number):

def branch_disorder(branch):

The following Python tricks and shortcuts may be useful here and/or later in the lab:

log2: We've defined a function log2 that takes in a single number and returns log₂ of the number.
INF: As in previous labs, we've defined the constant INF, which is a float representing infinity
list.count: .count is a built-in list method that counts how many times an item occurs in a list. For example, [10, 20, 20, 30].count(20) -> 2.
float: Recall that if you divide two ints in Python2, it rounds down to the nearest int, so you'll need to cast one of them to a float if you want to perform normal division.
set: Recall from Lab 0 that a set is handy way to count or enumerate the unique items in a list.

average_test_disorder should take in a decision stump (a.k.a. a "test"), represented as a list of branches (such as [["Oak", "Oak", "Maple"], ["Maple", "Maple"]]), and return the disorder of the entire stump (a number):

def average_test_disorder(stump):

Constructing an ID Tree

In order to actually construct an ID tree, we'll need to work directly with classifiers, also known as tests. (todo explain further)

We will represent each classifier as a Classifier object, described in the API below.

Using the disorder functions you defined above, implement a function to select the best classifier. The function should take in some data (as a list of point dictionaries), a list of Classifiers, and the attribute you want your ID tree to ultimately classifying by (e.g. "tree_type"). It should return the Classifier that has the lowest disorder.

Edge cases:

If two Classifiers are tied for lowest disorder, break ties by preferring classifiers that occur earlier in the list.
If the Classifier with lowest disorder is no good (that is, it doesn't separate the data at all), raise the exception NoGoodClassifiersError instead of returning a Classifier.

def find_best_classifier(data, possible_classifiers, classification_key):

Next, we will incorporate the ID tree. Use find_best_classifier to implement the next function, which should take in an incomplete tree (as an IdentificationTreeNode object) and a list of possible classifiers. The function should perform one of three actions:

If the node is homogeneous (you can use the function is_homogeneous(data, classification_key) to check this), then it should be a leaf node, so add the classification to the node.
If the node is not homogeneous and the data can be divided further, add the best classifier to the node.
If the node is not homogeneous but there are no good classifiers left, raise a NoGoodClassifiersError.

The function should either modify and return the original IdentificationTreeNode, or raise an exception. Implement add_best_classifier_to_tree:

def add_best_classifier_to_tree(id_tree_node, possible_classifiers):

Now, you can implement the function finish_greedy_subtree, which takes in an incomplete tree (represented as an IndentificationTreeNode) and a list of possible classifiers, and adds classifiers to the tree until either perfect classification has been achieved, or there are no good classifiers left. If a branch cannot become homogeneous, leave its classification unassigned (which defaults to None). We recommend implementing this function recursively.

def finish_greedy_subtree(id_tree_node, possible_classifiers):

Finally, to put it all together, we've provided a function that calls your functions to construct a complete ID tree:

def construct_greedy_identification_tree(data, possible_classifiers, classification_key):
    id_tree = IdentificationTreeNode(data, classification_key)
    return finish_greedy_subtree(id_tree, possible_classifiers)

We've also provided some datasets from past quizzes, so you can now use your ID tree builder to solve problems! For example, if you run:

print construct_greedy_identification_tree(tree_data, tree_classifiers, "tree_type")

...it should compute and print the solution to the tree-identification problem from 2014 Q2.

You can also try:

print construct_greedy_identification_tree(angel_data, angel_classifiers, "Classification")  #from 2012 Q2
print construct_greedy_identification_tree(numeric_data, numeric_classifiers, "class")  #from 2013 Q2

You can also change the classification_key attribute to, for example, use tree_type to predict what type of bark_texture a tree has:

print construct_greedy_identification_tree(tree_data, tree_classifiers_reverse, "bark_texture")  #build an ID tree to predict bark_texture

Conceptual questions

todo

k-Nearest Neighbors

Multiple Choice

todo: something about drawing 1NN boundaries

Python warm-up: Distance metrics

k-nearest neighbors can use many different distance metrics. In 6.034, we cover four of them:

Euclidean distance: the straight-line distance between two points (todo add formula)
Manhattan distance: todo explain (todo add formula)
Hamming distance: todo explain (todo add formula)
Cosine distance: todo explain (todo add formula) (todo note about cosine_distance being negated, b/c it's a similarity metric where 1 = most similar, 0 = not similar)

It's also possible to transform data instead of using a different metric -- for instance, transforming to polar coordinates -- but in this lab we will only work with distance metrics in Cartesian coordinates.

We'll start with some basic functions for manipulating vectors, which may be useful for cosine_distance. For these, we will represent an n-dimensional vector as a list or tuple of n coordinates. dot_product should compute the dot product of two vectors, while norm computes the length of a vector. (There is a simple implementation of norm that uses dot_product.) Implement both functions:

def dot_product(u, v):

def norm(v):

Next, you'll implement each of the four distance metrics. For the rest of the k-nearest neighbors portion of this lab, we will represent data points as Point objects (described below in the API). Each distance function should take in two Points and return the distance between them as a float or int. Implement each distance function:

def euclidean_distance(point1, point2):

def manhattan_distance(point1, point2):

def hamming_distance(point1, point2):

def cosine_distance(point1, point2):

Classifying Points with k-Nearest Neighbors

We've provided a function: (todo explain)

def get_k_closest_points(point, data, k, distance_metric):
    "return list of k points closest to input point, based on metric"
    if k >= len(data):
        return data
    sorted_points = sorted(data, key=lambda p2: distance_metric(point, p2))
    return sorted_points[:k]

Your task is to use that to write a function that classifies a point, given a set of data (as a list of Points), a value of k, and a distance metric (a function):

def knn_classify_point(point, data, k, distance_metric):

todo hint: to get the mode of a list, do: max(set(my_list), key=my_list.count)

Cross-validation: Choosing the best k and distance metric

todo explain functions

todo clarify different definitions of cross-validation; we're using leave-one-out

def cross_validate(data, k, distance_metric):

def find_best_k_and_metric(data):

More multiple choice

todo something about overfitting, underfitting, running classify and find_best_k... on some data to see what happens

Survey

Please answer these questions at the bottom of your lab5.py file:

NAME: What is your name? (string)

COLLABORATORS: Other than 6.034 staff, with whom did you work on this lab? (string, or empty string if you worked alone)

HOW_MANY_HOURS_THIS_LAB_TOOK: Approximately how many hours did you spend on this lab? (number or string)

WHAT_I_FOUND_INTERESTING: Which parts of this lab, if any, did you find interesting? (string)

WHAT_I_FOUND_BORING: Which parts of this lab, if any, did you find boring or tedious? (string)

(optional) SUGGESTIONS: What specific changes would you recommend, if any, to improve this lab for future years? (string)

(We'd ask which parts you find confusing, but if you're confused you should really ask a TA.)

When you're done, run the online tester to submit your code.

API Reference Documentation

The file api.py defines the Classifier, IdentificationTreeNode, and Point classes, as well as some helper functions for ID trees, all described below. (todo: describe helper functions)

Classifier

Classifier objects are used for constructing and manipulating ID trees.

A Classifier has the following attributes:

classifier.name, the name of the classifier.
classifier.classify, a function that takes in a point and returns a value.

In our ID trees, a point is represented as a dict mapping attribute names to their values.

IdentificationTreeNode

In this lab, an ID tree is represented recursively as a tree of IdentificationTreeNode objects. In a completed ID tree, each node has either a classification (such as "Oak" or "Maple"), or a classifier (such as "has leaves") with branches leading to child nodes.

An IdentificationTreeNode has the following attributes and methods:

id_tree_node.data, the training points at this node in the tree.
id_tree_node.classification_key, the attribute by which the tree classifies points (e.g. "tree_type" for the tree data, or "Classification" for the angel data).
id_tree_node.get_parent_branch_name(): returns the name of the branch leading to this node, or "ID Tree" if it is the root node
id_tree_node.is_leaf(): returns True if the node is a leaf (has a classification), otherwise False
id_tree_node.set_classifier(classifier): Uses the specified classifier to add branches below the current node. Modifies and returns the original id_tree_node. May print warnings if the specified classifier is inadvisable.
id_tree_node.apply_classifier(point): Applies classifier to point by following the appropriate branch of the tree, then returns a child node. If node is leaf node (and thus doesn't have a classifier), raises error.
id_tree_node.get_classifier(): returns the classifier, if any
id_tree_node.set_node_classification(classification): Sets the node's classification, which defines the node as a leaf node. Modifies and returns original id_tree_node. May print warnings if the node already has branches defined.
id_tree_node.get_node_classification(): returns the node's classification, if any
id_tree_node.get_branches(): returns a dict mapping node's child branches to the nodes at the end of them
id_tree_node.describe_type(): returns a string describing the node type (leaf, subtree, etc)

Point (for kNN)

Point objects are used in the k-nearest neighbors section only.

A Point has the following attributes:

point.name, the name of the point (a string), if defined.
point.coords, the coordinates of the point, represented as a vector (a tuple or list of numbers).
point.classification, the classification of the point, if known.

@@ Line 11: / Line 11: @@
 * View the files individually: http://web.mit.edu/6.034/www/labs/lab5/
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-Your answers for this lab belong in the main file <tt>lab5.py</tt>.  This lab covers k-nearest neighbors and identification trees.
+Your answers for this lab belong in the main file <tt>lab5.py</tt>. This lab will cover identification trees as well as k-nearest neighbors.