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%% Cell type:markdown id: tags:
# *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:
* 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". The functions that will be implemented there will be indicated in this notebook.
### Assignment Goals:
In this assignment, we will:
* we'll implement 'Euclidean' and 'Manhattan' distance metrics
* use the validation dataset to find a good value for *k*
* Evaluate our model with respect to performance measures:
* evaluate our model with respect to performance measures:
* accuracy, generalization error and ROC curve
* try to assess if *k*-NN is suitable for the dataset you used
%% 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 |
### Bonus for 478/878
| | Tasks | 478 | 878 |
|---|----------------|-----|-----|
| 7 | Assess suitability of *k*-NN | 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 20 bonus points for 478 students and 10 bonus points for 878 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.
%% Cell type:markdown id: tags:
### Rubric:
* Euclidean +5, +5
* Manhattan +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_manhattan = distance(x, y, 'Manhattan')
print('Euclidean distance: {}, Manhattan distance: {}'.format(dist_euclidean, dist_manhattan))
```
%% 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. 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, training_features, training_labels, k, distance_f,**kwargs):
'''
Fit the model. This is pretty straightforward for k-NN.
Args:
training_features: ndarray
training_labels: ndarray
k: int
distance_f: function
kwargs: dict
Contains keyword arguments that will be passed to distance_f
'''
# TODO
# set self.train_features, self.train_labels, self.k, self.distance_f, self.distance_metric
raise NotImplementedError
return
def predict(self, test_features):
'''
Args:
test_features: ndarray
mxd array containing features for the points to be predicted
Returns:
ndarray
'''
raise NotImplementedError
pred = []
# TODO
# for each point in test_features
# 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).
# when calculating learning curve you can make use of
# self.learning_curve and self.training_proportion
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:
Preprocess the data files and partition the data.
%% Cell type:code id: tags:
``` python
# initialize the model
my_model = kNN()
# obtain features and labels from files
features, labels = preprocess(feature_file=..., label_file=...)
# partition the data set
val_indices, test_indices, train_indices = partition(size=..., t = 0.3, v = 0.1)
```
%% Cell type:markdown id: tags:
Assign a value to *k* and fit the *k*-NN model.
%% Cell type:code id: tags:
``` python
# pass the training features and labels to the fit method
kwargs_f = {'metric': 'Euclidean'}
my_model.fit(training_features=..., training_labels-..., k=10, distance_f=..., **kwargs_f)
```
%% 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.
%% Cell type:code id: tags:
``` python
# TODO
# get model predictions
pred_ratios = my_model.predict(my_model.features[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
# convert predicted ratios to predicted labels
pred_labels = None
# obtain true positive, true negative,
#false positive and false negative counts using conf_matrix
tp,tn, fp, fn = conf_matrix(...)
```
%% Cell type:markdown id: tags:
Evaluate your model on the test data and report your **accuracy**. Also, calculate and report the 95% confidence interval on the generalization **error** estimate.
%% Cell type:code id: tags:
``` python
# TODO
# Calculate and report accuracy and generalization error with confidence interval here. Show your work in this cell.
print('Accuracy: {}'.format(accuracy))
print('Confidence interval: {}-{}'.format(lower_bound, upper_bound))
```
%% 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
# train using %10, %20, %30, ..., 100% of training data
training_proportions = np.arange(0.10, 1.01, 0.10)
train_size = len(train_indices)
training_sizes = np.int(np.ceil(size*proportion))
# TODO
error_train = []
error_val = []
# For each size in training_sizes
for size in training_sizes:
# fit the model using "size" data porint
# Calculate error for training and validation sets
# 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*
%% Cell type:markdown id: tags:
### Rubric:
* 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.
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.
%% 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.
%% Cell type:code id: tags:
``` python
# TODO
# 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()
```
%% Cell type:markdown id: tags:
## TASK 7: Assess suitability of *k*-NN to your dataset
%% Cell type:markdown id: tags:
Use this cell to write about your understanding of why *k*-NN performed well if it did or why not if it didn't. What properties of the dataset affect the performance of the algorithm?
......
%% Cell type:markdown id: tags:
# *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:
* 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". The functions that will be implemented there will be indicated in this notebook.
### Assignment Goals:
In this assignment, we will:
* we'll implement 'Euclidean' and 'Manhattan' distance metrics
* use the validation dataset to find a good value for *k*
* Evaluate our model with respect to performance measures:
* evaluate our model with respect to performance measures:
* accuracy, generalization error and ROC curve
* Try to assess if *k*-NN is suitable for the dataset you used
* try to assess if *k*-NN is suitable for the dataset you used
%% 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 |
### Bonus for 478/878
| | Tasks | 478 | 878 |
|---|----------------|-----|-----|
| 7 | Assess suitability of *k*-NN | 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 20 bonus points for 478 students and 10 bonus points for 878 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_solution.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.
%% Cell type:markdown id: tags:
### Rubric:
* Euclidean +5, +5
* Manhattan +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_manhattan = distance(x, y, 'Manhattan')
print('Euclidean distance: {}, Manhattan distance: {}'.format(dist_euclidean, dist_manhattan))
```
%% Output
Euclidean distance: 1000.0, Manhattan distance: 10000
%% 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. 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, training_features, training_labels, k, distance_f, **kwargs):
'''
Fit the model. This is pretty straightforward for k-NN.
Args:
training_features: ndarray
training_labels: ndarray
k: int
distance_f: function
kwargs: dict
Contains keyword arguments that will be passed to distance_f
'''
# TODO
# set self.train_features, self.train_labels,self.k, self.distance_f, self.distance_metric
self.train_features = training_features
self.train_labels = training_labels
self.k = k
self.distance_f = distance_f
self.distance_metric = kwargs['metric']
return
def predict(self, test_features):
test_size = len(test_features)
train_size = len(self.train_labels)
pred = np.empty(len(test_features))
# TODO
# for each point in test points
for idx in range(test_size):
point = test_features[idx]
distances = []
labels = []
for tr_idx in range(train_size):
train_example = self.train_features[tr_idx]
train_label = self.train_labels[tr_idx]
dist = self.distance_f(point, train_example, metric = self.distance_metric)
distances.append(dist)
labels.append(train_label)
# get the order of distances
dist_order = np.argsort(distances)
# get the labels of k points that are closest to test point
k_labels = list(np.array(labels)[dist_order[::-1]][:self.k])
# get number of positive labels in k neighbours
b = k_labels.count(1)
pred[idx] = b/self.k
return 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:
Preprocess the data files and partition the data.
%% Cell type:code id: tags:
``` python
# initialize the model
my_model = kNN()
# obtain features and labels from files
features, labels = preprocess('../data/madelon.data', '../data/madelon.labels')
# partition the data set
val_indices, test_indices, train_indices = partition(features.shape[0], t = 0.3, v = 0.1)
```
%% Cell type:markdown id: tags:
Assign a value to *k* and fit the *k*-NN model.
%% Cell type:code id: tags:
``` python
# pass the training features and labels to the fit method
kwargs_f = {'metric': 'Euclidean'}
my_model.fit(features[train_indices], labels[train_indices], k=10, distance_f=distance, **kwargs_f)
```
%% 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.
%% Cell type:code id: tags:
``` python
# TODO
# get model predictions
pred_ratios = my_model.predict(features[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
# convert predicted ratios to predicted labels
pred_labels = [1 if x >= threshold else 0 for x in pred_ratios]
tp,tn, fp, fn = conf_matrix(labels[test_indices], pred_labels)
```
%% Cell type:markdown id: tags:
Evaluate your model on the test data and report your **accuracy**. Also, calculate and report the 95% confidence interval on the generalization **error** estimate.
%% Cell type:code id: tags:
``` python
# TODO
# Calculate and report accuracy and generalization error with confidence interval here. Show your work in this cell.
accuracy = (tp+tn)/len(test_indices)
error = 1 - accuracy
diff = 0.96 * np.sqrt((error * (1 - error)) / len(test_indices))
lower_bound = error - diff
upper_bound = error + diff
print('Accuracy: {}'.format(accuracy))
print('Confidence interval: {}-{}'.format(lower_bound, upper_bound))
```
%% Output
Accuracy: 0.475
Confidence interval: 0.49110132745961876-0.5588986725403813
%% 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_proportions = np.arange(0.10, 1.01, 0.10)
# 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.
error_train = []
error_val = []
training_sizes = []
for proportion in training_proportions:
size = len(train_indices)
size_avail = np.int(np.ceil(size*proportion))
training_sizes.append(size_avail)
idx_avail = train_indices[:size_avail]
kwargs_f = {'metric': 'Euclidean'}
my_model.fit(features[idx_avail], labels[idx_avail], k = 10, distance_f=distance, **kwargs_f)
val_pred_ratios = my_model.predict(features[val_indices])
val_pred_labels = [1 if x >= threshold else 0 for x in val_pred_ratios]
tp,tn, fp, fn = conf_matrix(labels[val_indices], val_pred_labels)
val_accuracy = (tp+tn)/len(val_indices)
val_error = 1 - val_accuracy
error_val.append(val_error)
train_pred_ratios = my_model.predict(features[train_indices])
train_pred_labels = [1 if x >= threshold else 0 for x in train_pred_ratios]
tp,tn, fp, fn = conf_matrix(labels[idx_avail], train_pred_labels)
train_accuracy = (tp+tn)/size_avail
train_error = 1 - train_accuracy
error_train.append(train_error)
plt.plot(training_sizes, error_train, 'r', label = 'training_error')
plt.plot(training_sizes, error_val, 'g', label = 'validation_error')
plt.legend()
plt.show()
print('{} is a good training size'.format(600))
```
%% Output
600 is a good training size
%% Cell type:markdown id: tags:
## TASK 5: Determining *k*
%% Cell type:markdown id: tags:
### Rubric:
* 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.
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. Hint: Try values both smaller and larger than 10.
%% Cell type:code id: tags:
``` python
# TODO
k_accuracies = []
# Change values of k.
for k in [1, 5, 10, 50, 100, 150, 200]:
# Calculate accuracies for the validation set.
my_model.fit(features[train_indices], labels[train_indices], k=k, distance_f=distance, **kwargs_f)
pred_ratios = my_model.predict(features[val_indices])
pred_labels = [1 if x >= threshold else 0 for x in pred_ratios]
tp,tn, fp, fn = conf_matrix(labels[val_indices], pred_labels)
accuracy = (tp+tn)/len(val_indices)
k_accuracies.append(accuracy)
k_accuracies
# Report a good k value.
```
%% Output
[0.49666666666666665, 0.55, 0.545, 0.505, 0.495, 0.465, 0.445]
%% Cell type:markdown id: tags:
## TASK 6: ROC curve analysis
* ROC curve has correct shape +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.
%% 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.
%% Cell type:code id: tags:
``` python
# TODO
# ROC curve
my_model.fit(features[train_indices], labels[train_indices], k=100, distance_f=distance, **kwargs_f)
pred_ratios = my_model.predict(features[test_indices])
roc_sens, roc_spec_ = ROC(labels[test_indices], pred_ratios, np.arange(0.1, 1.0, 0.1))
plt.plot(roc_sens, roc_spec_)
plt.xlabel('Sensitivity')
plt.ylabel('Specificity')
plt.show()
```
%% Output
%% Cell type:markdown id: tags:
## TASK 7: Assess suitability of *k*-NN to your dataset
%% Cell type:markdown id: tags:
Use this cell to write about your understanding of why *k*-NN performed well if it did or why not if it didn't. What properties of the dataset affect the performance of the algorithm?
......
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