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%% Cell type:markdown id: tags:
# $k$-Nearest Neighbor
# *k*-Nearest Neighbor
We'll implement $k$-Nearest Neighbor ($k$-NN) algorithm for this assignment. We recommend using [Madelon](https://archive.ics.uci.edu/ml/datasets/Madelon) dataset, although it is not mandatory. If you choose to use a different dataset, it should meet the following criteria:
We'll implement *k*-Nearest Neighbor (*k*-NN) algorithm for this assignment. We recommend using [Madelon](https://archive.ics.uci.edu/ml/datasets/Madelon) dataset, although it is not mandatory. If you choose to use a different dataset, it should meet the following criteria:
* dependent variable should be binary (suited for binary classification)
* number of features (attributes) should be at least 50
* number of examples (instances) should be between 1,000 - 5,000
A skeleton of a general supervised learning model is provided in "model.ipynb". Please look through it and complete the "preprocess" and "partition" methods.
### Assignment Goals:
In this assignment, we will:
* learn to split a dataset into training/validation/test partitions
* use the validation dataset to find a good value for $k$
* Having found the "best" $k$, we'll obtain final performance measures:
* use the validation dataset to find a good value for *k*
* Having found the "best" *k*, we'll obtain final performance measures:
* accuracy, generalization error and ROC curve
%% Cell type:markdown id: tags:
# GRADING
You will be graded on parts that are marked with **\#TODO** comments. Read the comments in the code to make sure you don't miss any.
### Mandatory for 478 & 878:
| | Tasks | 478 | 878 |
|---|----------------------------|-----|-----|
| 1 | Implement `distance` | 10 | 10 |
| 2 | Implement `k-NN` methods | 25 | 20 |
| 3 | Model evaluation | 25 | 20 |
| 4 | Learning curve | 20 | 20 |
| 6 | ROC curve analysis | 20 | 20 |
### Mandatory for 878, bonus for 478
| | Tasks | 478 | 878 |
|---|----------------|-----|-----|
| 5 | Optimizing $k$ | 10 | 10 |
| 5 | Optimizing *k* | 10 | 10 |
Points are broken down further below in Rubric sections. The **first** score is for 478, the **second** is for 878 students. There a total of 100 points in this assignment and extra 10 bonus points for 478 students.
%% Cell type:markdown id: tags:
You can use numpy for array operations and matplotlib for plotting for this assignment. Please do not add other libraries.
%% Cell type:code id: tags:
``` python
import numpy as np
import matplotlib.pyplot as plt
```
%% Cell type:markdown id: tags:
Following code makes the Model class and relevant functions available from model.ipynb.
%% Cell type:code id: tags:
``` python
%run 'model.ipynb'
```
%% Cell type:markdown id: tags:
## TASK 1: Implement `distance` function
%% Cell type:markdown id: tags:
Choice of distance metric plays an important role in the performance of $k$-NN. Let's start with implementing a distance method in the "distance" function in **model.ipynb**. It should take two data points and the name of the metric and return a scalar value.
Choice of distance metric plays an important role in the performance of *k*-NN. Let's start with implementing a distance method in the "distance" function in **model.ipynb**. It should take two data points and the name of the metric and return a scalar value.
%% Cell type:markdown id: tags:
### Rubric:
* Euclidean +5, +5
* Hamming +5, +5
%% Cell type:markdown id: tags:
### Test `distance`
%% Cell type:code id: tags:
``` python
x = np.array(range(100))
y = np.array(range(100, 200))
dist_euclidean = distance(x, y, 'Euclidean')
dist_hamming = distance(x, y, 'Hamming')
```
%% Cell type:markdown id: tags:
## TASK 2: Implement $k$-NN Class Methods
%% Cell type:markdown id: tags:
We can start implementing our $k$-NN classifier. $k$-NN class inherits Model class. You'll need to implement "fit" and "predict" methods. Use the "distance" function you defined above. "fit" method takes $k$ as an argument. "predict" takes as input an $mxd$ array containing $d$-dimensional $m$ feature vectors for examples and outputs the predicted class and the ratio of positive examples in $k$ nearest neighbors.
We can start implementing our *k*-NN classifier. *k*-NN class inherits Model class. You'll need to implement "fit" and "predict" methods. Use the "distance" function you defined above. "fit" method takes *k* as an argument. "predict" takes as input an *mxd* array containing *d*-dimensional *m* feature vectors for examples and outputs the predicted class and the ratio of positive examples in *k* nearest neighbors.
%% Cell type:markdown id: tags:
### Rubric:
* correct implementation of fit method +5, +5
* correct implementation of predict method +20, +15
%% Cell type:code id: tags:
``` python
class kNN(Model):
'''
Inherits Model class. Implements the k-NN algorithm for classification.
'''
def fit(self, k, distance_f, **kwargs):
'''
Fit the model. This is pretty straightforward for k-NN.
'''
# TODO
# set self.k, self.distance_f, self.distance_metric
raise NotImplementedError
return
def predict(self, test_indices):
raise NotImplementedError
pred = []
# TODO
# for each point in test points
# use your implementation of distance function
# distance_f(..., distance_metric)
# to find the labels of k-nearest neighbors.
# Find the ratio of the positive labels
# and append to pred with pred.append(ratio).
return np.array(pred)
```
%% Cell type:markdown id: tags:
## TASK 3: Build and Evaluate the Model
%% Cell type:markdown id: tags:
### Rubric:
* Reasonable accuracy values +10, +5
* Reasonable confidence intervals on the error estimate +10, +10
* Reasonable confusion matrix +5, +5
%% Cell type:markdown id: tags:
Remember you need to provide values to *t*, *v* parameters for "partition" function and to *feature_file* and *label_file* for "preprocess" function.
%% Cell type:code id: tags:
``` python
# populate the keyword arguments dictionary kwargs
kwargs = {'t': 0.3, 'v': 0.1, 'feature_file': ..., 'label_file': ...}
# initialize the model
my_model = kNN(preprocessor_f=preprocess, partition_f=partition, **kwargs)
```
%% Cell type:markdown id: tags:
Assign a value to $k$ and fit the kNN model.
Assign a value to *k* and fit the *k*-NN model.
%% Cell type:code id: tags:
``` python
kwargs_f = {'metric': 'Euclidean'}
my_model.fit(k = 10, distance_f=distance, **kwargs_f)
```
%% Cell type:markdown id: tags:
Evaluate your model on the test data and report your **accuracy**. Also, calculate and report the confidence interval on the generalization **error** estimate.
%% Cell type:code id: tags:
``` python
final_labels = my_model.predict(my_model.test_indices)
# For now, we will consider a data point as predicted in the positive class if more than 0.5
# of its k-neighbors are positive.
threshold = 0.5
# TODO
# Calculate and report accuracy and generalization error with confidence interval here. Show your work in this cell.
```
%% Cell type:markdown id: tags:
### Computing the confusion matrix for $k = 10$
Now that we have the true labels and the predicted ones from our model, we can build a confusion matrix and see how accurate our model is. Implement the "conf_matrix" function (in model.ipynb) that takes as input an array of true labels ($true$) and an array of predicted labels ($pred$). It should output a numpy.ndarray. You do not need to change the value of the threshold parameter yet.
### Computing the confusion matrix for *k* = 10
Now that we have the true labels and the predicted ones from our model, we can build a confusion matrix and see how accurate our model is. Implement the "conf_matrix" function (in model.ipynb) that takes as input an array of true labels (*true*) and an array of predicted labels (*pred*). It should output a numpy.ndarray. You do not need to change the value of the threshold parameter yet.
%% Cell type:code id: tags:
``` python
# TODO
conf_matrix(my_model.labels[my_model.test_indices], final_labels, threshold = 0.5)
```
%% Cell type:markdown id: tags:
## TASK 4: Plotting a learning curve
A learning curve shows how error changes as the training set size increases. For more information, see [learning curves](https://www.dataquest.io/blog/learning-curves-machine-learning/).
We'll plot the error values for training and validation data while varying the size of the training set. Report a good size for training set for which there is a good balance between bias and variance.
%% Cell type:markdown id: tags:
### Rubric:
* Correct training error calculation for different training set sizes +8, +8
* Correct validation error calculation for different training set sizes +8, +8
* Reasonable learning curve +4, +4
%% Cell type:code id: tags:
``` python
# try sizes 50, 100, 150, 200, ..., up to the largest multiple of 50 >= train_size
training_sizes = np.arange(50, my_model.train_size + 1, 50)
# TODO
# Calculate error for each entry in training_sizes
# for training and validation sets and populate
# error_train and error_val arrays. Each entry in these arrays
# should correspond to each entry in training_sizes.
plt.plot(training_sizes, error_train, 'r', label = 'training_error')
plt.plot(training_sizes, error_val, 'g', label = 'validation_error')
plt.legend()
plt.show()
```
%% Cell type:markdown id: tags:
## TASK 5: Determining $k$
## TASK 5: Determining *k*
%% Cell type:markdown id: tags:
### Rubric:
* Increased accuracy with new $k$ +5, +5
* Increased accuracy with new *k* +5, +5
* Improved confusion matrix +5, +5
%% Cell type:markdown id: tags:
We can use the validation set to come up with a $k$ value that results in better performance in terms of accuracy.
We can use the validation set to come up with a *k* value that results in better performance in terms of accuracy.
Below calculate the accuracies for different values of $k$ using the validation set. Report a good $k$ value and use it in the analyses that follow this section. Report confusion matrix for the new value of $k$.
Below calculate the accuracies for different values of *k* using the validation set. Report a good *k* value and use it in the analyses that follow this section. Report confusion matrix for the new value of *k*.
%% Cell type:code id: tags:
``` python
# TODO
# Change values of k.
# Calculate accuracies for the validation set.
# Report a good k value.
# Calculate the confusion matrix for new k.
```
%% Cell type:markdown id: tags:
## TASK 6: ROC curve analysis
* Correct implementation +20, +20
%% Cell type:markdown id: tags:
### ROC curve and confusion matrix for the final model
ROC curves are a good way to visualize sensitivity vs. 1-specificity for varying cut off points. Now, implement, in "model.ipynb", a "ROC" function that predicts the labels of the test set examples using different $threshold$ values in "predict" and plot the ROC curve. "ROC" takes a list containing different $threshold$ parameter values to try and returns two arrays; one where each entry is the sensitivity at a given threshold and the other where entries are 1-specificities.
ROC curves are a good way to visualize sensitivity vs. 1-specificity for varying cut off points. Now, implement, in "model.ipynb", a "ROC" function that predicts the labels of the test set examples using different *threshold* values in "predict" and plot the ROC curve. "ROC" takes a list containing different *threshold* parameter values to try and returns two arrays; one where each entry is the sensitivity at a given threshold and the other where entries are 1-specificities.
%% Cell type:markdown id: tags:
We can finally create the confusion matrix and plot the ROC curve for our optimal $k$-NN classifier. Use the $k$ value you found above, if you completed TASK 5, else use $k$ = 10. We'll plot the ROC curve for values between 0.1 and 1.0.
We can finally create the confusion matrix and plot the ROC curve for our optimal *k*-NN classifier. Use the *k* value you found above, if you completed TASK 5, else use *k* = 10. We'll plot the ROC curve for values between 0.1 and 1.0.
%% Cell type:code id: tags:
``` python
# ROC curve
roc_sens, roc_spec_ = ROC(my_model, my_model.test_indices, np.arange(0.1, 1.0, 0.1))
plt.plot(roc_sens, roc_spec_)
plt.show()
```
......
model.ipynb
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%% Cell type:markdown id: tags:
# JUPYTER NOTEBOOK TIPS
Each rectangular box is called a cell.
* ctrl+ENTER evaluates the current cell; if it contains Python code, it runs the code, if it contains Markdown, it returns rendered text.
* alt+ENTER evaluates the current cell and adds a new cell below it.
* If you click to the left of a cell, you'll notice the frame changes color to blue. You can erase a cell by hitting 'dd' (that's two "d"s in a row) when the frame is blue.
%% Cell type:markdown id: tags:
# Supervised Learning Model Skeleton
We'll use this skeleton for implementing different supervised learning algorithms. Please complete "preprocess" and "partition" methods below.
%% Cell type:markdown id: tags:
This step is for reading the dataset and for extracting features and labels. The "preprocess" function should return an $n \times d$ "features" array, and an $n \times 1$ "labels" array, where $n$ is the number of examples and $d$ is the number of features in the dataset. In cases where there is a big difference between the scales of features, we want to normalize the features to have values in the same range [0,1]. Since this is not the case with this dataset, we will not do normalization.
This step is for reading the dataset and for extracting features and labels. The "preprocess" function should return an *n x d* "features" array, and an *n x 1* "labels" array, where *n* is the number of examples and *d* is the number of features in the dataset. In cases where there is a big difference between the scales of features, we want to normalize the features to have values in the same range [0,1]. Since this is not the case with this dataset, we will not do normalization.
%% Cell type:code id: tags:
``` python
#TODO: GIVE!
def preprocess(feature_file, label_file):
'''
Args:
feature_file: str
file containing features
label_file: str
file containing labels
Returns:
features: ndarray
nxd features
labels: ndarray
nx1 labels
'''
# You might find np.genfromtxt useful for reading in the file. Be careful with the file delimiter,
# e.g. for comma-separated files use delimiter=',' argument.
# TODO
raise NotImplementedError
return features, labels
```
%% Cell type:markdown id: tags:
Next, you'll need to split your dataset into training, validation and test sets. The "partition" function should take as input the size of the whole dataset and randomly sample a proportion $t$ of the dataset as test partition and a proportion of $v$ as validation partition. The remaining will be used as training data. For example, to keep 30% of the examples as test and %10 as validation, set $t=0.3$ and $v=0.1$. You should choose these values according to the size of the data available to you. The "split" function should return indices of the training, validation and test sets. These will be used to index into the whole training set.
Next, you'll need to split your dataset into training, validation and test sets. The "partition" function should take as input the size of the whole dataset and randomly sample a proportion *t* of the dataset as test partition and a proportion of *v* as validation partition. The remaining will be used as training data. For example, to keep 30% of the examples as test and %10 as validation, set *t* = 0.3 and *v* = 0.1. You should choose these values according to the size of the data available to you. The "split" function should return indices of the training, validation and test sets. These will be used to index into the whole training set.
%% Cell type:code id: tags:
``` python
#TODO: GIVE!
def partition(size, t, v = 0):
'''
Args:
size: int
number of examples in the whole dataset
t: float
proportion kept for test
v: float
proportion kept for validation
Returns:
test_indices: ndarray
1D array containing test set indices
val_indices: ndarray
1D array containing validation set indices
'''
# np.random.permutation might come in handy. Do not sample with replacement!
# Be sure not to use the same indices in test and validation sets!
# use the first np.ceil(size*t) for test,
# the following np.ceil(size*v) for validation set.
# TODO
raise NotImplementedError
return test_indices, val_indices
```
%% Cell type:markdown id: tags:
In cases, where data is not abundantly available, we resort to getting an error estimate from average of error on different splits of error. In this case, every fold of data is used for testing and for training in turns, i.e. assuming we split our data into 3 folds, we'd
* train our model on fold-1+fold-2 and test on fold-3
* train our model on fold-1+fold-3 and test on fold-2
* train our model on fold-2+fold-3 and test on fold-1.
We'd use the average of the error we obtained in three runs as our error estimate. Implement function "kfold" below.
%% Cell type:code id: tags:
``` python
# TODO: Programming Assignment 2
def kfold(indices, k):
'''
Args:
indices: ndarray
1D array with integer entries containing indices
k: int
Number of desired splits in data.(Assume test set is already separated.)
Returns:
fold_dict: dict
A dictionary with integer keys corresponding to folds. Values are (training_indices, val_indices).
val_indices: ndarray
1/k of training indices randomly chosen and separates them as validation partition.
train_indices: ndarray
Remaining 1-(1/k) of the indices.
e.g. fold_dict = {0: (train_0_indices, val_0_indices),
1: (train_0_indices, val_0_indices), 2: (train_0_indices, val_0_indices)} for k = 3
'''
return fold_dict
```
%% Cell type:markdown id: tags:
Choice of distance metric plays an important role in the performance of $k$-NN. Let's start with implementing a distance method in the "distance" function below. It should take two data points and the name of the metric and return a scalar value.
Choice of distance metric plays an important role in the performance of *k*-NN. Let's start with implementing a distance method in the "distance" function below. It should take two data points and the name of the metric and return a scalar value.
%% Cell type:code id: tags:
``` python
#TODO: Programming Assignment 1
def distance(x, y, metric):
'''
Args:
x: ndarray
1D array containing coordinates for a point
y: ndarray
1D array containing coordinates for a point
metric: str
Euclidean, Hamming
Returns:
'''
if metric == 'Euclidean':
raise NotImplementedError
elif metric == 'Hammming':
raise NotImplementedError
else:
raise ValueError('{} is not a valid metric.'.format(metric))
return dist # scalar distance btw x and y
```
%% Cell type:markdown id: tags:
We will extend "Model" class while implementing supervised learning algorithms.
%% Cell type:code id: tags:
``` python
class Model:
# preprocess_f and partition_f expect functions
# use kwargs to pass arguments to preprocessor_f and partition_f
# kwargs is a dictionary and should contain t, v, feature_file, label_file
# e.g. {'t': 0.3, 'v': 0.1, 'feature_file': 'some_file_name', 'label_file': 'some_file_name'}
def __init__(self, preprocessor_f, partition_f, **kwargs):
self.features, self.labels = preprocessor_f(kwargs['feature_file'], kwargs['label_file'])
self.size = len(self.labels) # number of examples in dataset
self.feat_dim = self.features.shape[1] # number of features
self.val_indices, self.test_indices = partition_f(self.size, kwargs['t'], kwargs['v'])
self.val_size = len(self.val_indices)
self.test_size = len(self.test_indices)
self.train_indices = np.delete(np.arange(self.size), np.append(self.test_indices, self.val_indices), 0)
self.train_size = len(self.train_indices)
def fit(self):
raise NotImplementedError
def predict(self, indices):
raise NotImplementedError
```
%% Cell type:markdown id: tags:
## General supervised learning performance related functions
### (To be implemented later when it is indicated in other notebooks)
%% Cell type:markdown id: tags:
Implement the "conf_matrix" function that takes as input an array of true labels ($true$) and an array of predicted labels ($pred$). It should output a numpy.ndarray.
Implement the "conf_matrix" function that takes as input an array of true labels (*true*) and an array of predicted labels (*pred*). It should output a numpy.ndarray.
%% Cell type:code id: tags:
``` python
# TODO: Programming Assignment 1
def conf_matrix(true, pred):
'''
Args:
true: ndarray
nx1 array of true labels for test set
pred: ndarray
nx1 array of predicted labels for test set
Returns:
ndarray
'''
raise NotImplementedError
tp = tn = fp = fn = 0
# calculate true positives (tp), true negatives(tn)
# false positives (fp) and false negatives (fn)
# returns the confusion matrix as numpy.ndarray
return np.array([tp,tn, fp, fn])
```
%% Cell type:markdown id: tags:
ROC curves are a good way to visualize sensitivity vs. 1-specificity for varying cut off points. Now, implement a "ROC" function that predicts the labels of the test set examples using different $threshold$ values in "predict" and plot the ROC curve. "ROC" takes a list containing different $threshold$ parameter values to try and returns two arrays; one where each entry is the sensitivity at a given threshold and the other where entries are 1-specificities.
ROC curves are a good way to visualize sensitivity vs. 1-specificity for varying cut off points. Now, implement a "ROC" function that predicts the labels of the test set examples using different *threshold* values in "predict" and plot the ROC curve. "ROC" takes a list containing different *threshold* parameter values to try and returns two arrays; one where each entry is the sensitivity at a given threshold and the other where entries are 1-specificities.
%% Cell type:code id: tags:
``` python
# TODO: Programming Assignment 1
def ROC(model, indices, value_list):
'''
Args:
model: a fitted supervised learning model
indices: ndarray
1D array containing indices
value_list: ndarray
1D array containing different threshold values
Returns:
sens: ndarray
1D array containing sensitivities
spec_: ndarray
1D array containing 1-specifities
'''
# use predict method to obtain predicted labels at different threshold values
# use conf_matrix to calculate tp, tn, fp, fn
# calculate sensitivity, 1-specificity
# return two arrays
raise NotImplementedError
return sens, spec_
```
......
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