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Rating distribution in the dataset: | data.rating.value_counts().sort_index().plot.bar() | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Building our first recommender model
Preparing data
RecommenderData class provides a set of tools for manipulating the data and preparing it for experimentation.
Input parameters are: the data itself (pandas dataframe) and mapping of the data fields (column names) to internal representation: userid, itemid and feedback: | data_model = RecommenderData(data, userid='userid', itemid='movieid', feedback='rating') | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Verify correct mapping: | data.columns
data_model.fields | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
RecommenderData class has a number of parameters to control how the data is processed. Defaults are fine to start with: | data_model.get_configuration() | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Use prepare method to split the dataset into 2 parts: training data and test data. | data_model.prepare() | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
As the original data possibly contains gaps in users' and items' indices, the data preparation process will clean this up: items from the training data will be indexed starting from zero with no gaps and the result will be stored in: | data_model.index.itemid.head() | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Similarly, all userid's from both training and test set are reindexed and stored in: | data_model.index.userid.training.head()
data_model.index.userid.test.head() | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Internally only new inices are used. This ensures consistency of various methods used by the model.
The dataset is split according to test_fold and test_ratio attributes. By default it uses first 80% of users for training and last 20% of the users as test data. | data_model.training.head()
data_model.training.shape | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
The test data is further split into testset and evaluation set (evalset). Testset is used to generate recommendations, which are than evaluated against the evaluation set. | data_model.test.testset.head()
data_model.test.testset.shape
data_model.test.evalset.head()
data_model.test.evalset.shape | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
The users in the test and evaluation sets are the same (but this users are not in the training set!).
For every test user the evaluation set contains a fixed number of items which are held out from the original test data. The number of holdout items is controlled by holdout_size parameter. By default it's set to 3: | data_model.holdout_size
data_model.test.evalset.groupby('userid').movieid.count().head() | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Creating recommender model
You can create your own model by subclassing RecommenderModel class and defining two required methods: self.build() and self.get_recommendations(): | class TopMovies(RecommenderModel):
def build(self):
self._recommendations = None # this is required line in order to ensure consitency in experiments
itemid = self.data.fields.itemid # get the name of the column, that corresponds to movieid
# calculate popularity of the movies based on the number of ratings
item_scores = self.data.training[itemid].value_counts().sort_index().values
# store it for later use in some attribute
self.item_scores = item_scores
def get_recommendations(self):
userid = self.data.fields.userid #get the name of the column, that corresponds to userid
# get the number of test users
# we expect that userid doesn't have gaps in numbering (as it might be in original dataset,
# RecommenderData class takes care of that)
num_users = self.data.test.testset[userid].max() + 1
# repeat computed popularity scores in accordance with the number of test users
scores = np.repeat(self.item_scores[None, :], num_users, axis=0)
# we got the scores, but what we actually need is items (their id)
# we also need only top-k items, not all of them (for top-k recommendation task)
# here's how to get it:
top_recs = self.get_topk_items(scores)
# here leftmost items are those with the highest scores
return top_recs | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Note, that recommendations, generated by this model, do not take into account the fact, that some of the recommended items may be present in the test set and thus, should not be recommended (they are considered seen by a test user). In order to fix that you can use filter_seen parameter along with downvote_seen_items method as follows:
if self.filter_seen:
#prevent seen items from appearing in recommendations
itemid = self.data.fields.itemid
test_idx = (test_data[userid].values.astype(np.int64),
test_data[itemid].values.astype(np.int64))
self.downvote_seen_items(scores, test_idx)
With this procedure "seen" items will get the lowest scores and they will be sorted out. Place this code snippet inside the get_recommendations routine before handovering scores into get_top_k_items. This will improve the baseline.
Alternative way
Another way is to define slice_recommendations instead of get_recommendations method. With slice_recommendations defined, the model will scale better when huge datasets are used.
The method slice_recommendations takes a piece of the test data slice by slice instead of processing it as a whole. Slice if defined by start and stop parameter (which are simply a userid to start with and userid to stop at). Slicing the data avoids memory overhead and leads to a faster evaluation of models. Slicing is done automatically behind the scene and you don't have to specify anything else. Another advantage: seen items will be automatically sorted out from recommendations as long as filter_seen attribute is set to True (it is by default). So it will requires less line of code. | class TopMoviesALT(RecommenderModel):
def build(self):
# should be the same as in TopMovies
def slice_recommendations(self, test_data, shape, start, stop):
# current implementation requires handovering slice data in specific format further,
# and the easiest way to get it is via get_test_matrix method. It also returns
# test data in sparse matrix format, but as our recommender model is non-personalized
# we don't actually need it. See SVDModel implementation to see when it's useful.
test_matrix, slice_data = self.get_test_matrix(test_data, shape, (start, stop))
nusers = stop - start
scores = np.repeat(self.item_scores[None, :], nusers, axis=0)
return scores, slice_data | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Now everything is set to create an instance of the recommender model and produce recommendations.
generating recommendations: | top = TopMovies(data_model) # the model takes as input parameter the recommender data model
top.build()
recs = top.get_recommendations()
recs
recs.shape
top.topk | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
You can evaluate your model befotre submitting the results (to ensure that you have improved above baseline): | top.evaluate() | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Try to change your model to maximize the true_positive score.
submitting your model:
After you have created your perfect recsys model, firstly, save your recommendation into file. Please, use your name as the name for file (this will be used to display at leaderboard) | np.savez('your_full_name', recs=recs) | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Now you can uppload your results: | import requests
files = {'upload': open('your_full_name.npz','rb')}
url = "http://isp2017.azurewebsites.net/upload"
r = requests.post(url, files=files) | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Verify, that upload is successful: | print r.status_code, r.reason | polara_intro.ipynb | Evfro/RecSys_ISP2017 | mit |
Guia inicial de TensorFlow 2.0 para expertos
<table class="tfo-notebook-buttons" align="left">
<td>
<a target="_blank" href="https://www.tensorflow.org/tutorials/quickstart/advanced"><img src="https://www.tensorflow.org/images/tf_logo_32px.png" />Ver en TensorFlow.org</a>
</td>
<td>
<a target="_blank" href="https://colab.research.google.com/github/tensorflow/docs-l10n/blob/master/site/es-419/tutorials/quickstart/advanced.ipynb"><img src="https://www.tensorflow.org/images/colab_logo_32px.png" />Ejecutar en Google Colab</a>
</td>
<td>
<a target="_blank" href="https://github.com/tensorflow/docs-l10n/blob/master/site/es-419/tutorials/quickstart/advanced.ipynb"><img src="https://www.tensorflow.org/images/GitHub-Mark-32px.png" />Ver codigo en GitHub</a>
</td>
<td>
<a href="https://storage.googleapis.com/tensorflow_docs/docs-l10n/site/es-419/tutorials/quickstart/advanced.ipynb"><img src="https://www.tensorflow.org/images/download_logo_32px.png" />Descargar notebook</a>
</td>
</table>
Note: Nuestra comunidad de Tensorflow ha traducido estos documentos. Como las traducciones de la comunidad
son basados en el "mejor esfuerzo", no hay ninguna garantia que esta sea un reflejo preciso y actual
de la Documentacion Oficial en Ingles.
Si tienen sugerencias sobre como mejorar esta traduccion, por favor envian un "Pull request"
al siguiente repositorio tensorflow/docs.
Para ofrecerse como voluntario o hacer revision de las traducciones de la Comunidad
por favor contacten al siguiente grupo [email protected] list.
Este es un notebook de Google Colaboratory. Los programas de Python se executan directamente en tu navegador โuna gran manera de aprender y utilizar TensorFlow. Para poder seguir este tutorial, ejecuta este notebook en Google Colab presionando el boton en la parte superior de esta pagina.
En Colab, selecciona "connect to a Python runtime": En la parte superior derecha de la barra de menus selecciona: CONNECT.
Para ejecutar todas las celdas de este notebook: Selecciona Runtime > Run all.
Descarga e installa el paquete TensorFlow 2.0 version.
Importa TensorFlow en tu programa:
Import TensorFlow into your program: | import tensorflow as tf
from tensorflow.keras.layers import Dense, Flatten, Conv2D
from tensorflow.keras import Model | site/es-419/tutorials/quickstart/advanced.ipynb | tensorflow/docs-l10n | apache-2.0 |
Carga y prepara el conjunto de datos MNIST | mnist = tf.keras.datasets.mnist
(x_train, y_train), (x_test, y_test) = mnist.load_data()
x_train, x_test = x_train / 255.0, x_test / 255.0
# Agrega una dimension de canales
x_train = x_train[..., tf.newaxis]
x_test = x_test[..., tf.newaxis] | site/es-419/tutorials/quickstart/advanced.ipynb | tensorflow/docs-l10n | apache-2.0 |
Utiliza tf.data to separar por lotes y mezclar el conjunto de datos: | train_ds = tf.data.Dataset.from_tensor_slices(
(x_train, y_train)).shuffle(10000).batch(32)
test_ds = tf.data.Dataset.from_tensor_slices((x_test, y_test)).batch(32) | site/es-419/tutorials/quickstart/advanced.ipynb | tensorflow/docs-l10n | apache-2.0 |
Construye el modelo tf.keras utilizando la API de Keras model subclassing API: | class MyModel(Model):
def __init__(self):
super(MyModel, self).__init__()
self.conv1 = Conv2D(32, 3, activation='relu')
self.flatten = Flatten()
self.d1 = Dense(128, activation='relu')
self.d2 = Dense(10, activation='softmax')
def call(self, x):
x = self.conv1(x)
x = self.flatten(x)
x = self.d1(x)
return self.d2(x)
# Crea una instancia del modelo
model = MyModel() | site/es-419/tutorials/quickstart/advanced.ipynb | tensorflow/docs-l10n | apache-2.0 |
Escoge un optimizador y una funcion de perdida para el entrenamiento de tu modelo: | loss_object = tf.keras.losses.SparseCategoricalCrossentropy()
optimizer = tf.keras.optimizers.Adam() | site/es-419/tutorials/quickstart/advanced.ipynb | tensorflow/docs-l10n | apache-2.0 |
Escoge metricas para medir la perdida y exactitud del modelo.
Estas metricas acumulan los valores cada epoch y despues imprimen el resultado total. | train_loss = tf.keras.metrics.Mean(name='train_loss')
train_accuracy = tf.keras.metrics.SparseCategoricalAccuracy(name='train_accuracy')
test_loss = tf.keras.metrics.Mean(name='test_loss')
test_accuracy = tf.keras.metrics.SparseCategoricalAccuracy(name='test_accuracy') | site/es-419/tutorials/quickstart/advanced.ipynb | tensorflow/docs-l10n | apache-2.0 |
Utiliza tf.GradientTape para entrenar el modelo. | @tf.function
def train_step(images, labels):
with tf.GradientTape() as tape:
predictions = model(images)
loss = loss_object(labels, predictions)
gradients = tape.gradient(loss, model.trainable_variables)
optimizer.apply_gradients(zip(gradients, model.trainable_variables))
train_loss(loss)
train_accuracy(labels, predictions) | site/es-419/tutorials/quickstart/advanced.ipynb | tensorflow/docs-l10n | apache-2.0 |
Prueba el modelo: | @tf.function
def test_step(images, labels):
predictions = model(images)
t_loss = loss_object(labels, predictions)
test_loss(t_loss)
test_accuracy(labels, predictions)
EPOCHS = 5
for epoch in range(EPOCHS):
for images, labels in train_ds:
train_step(images, labels)
for test_images, test_labels in test_ds:
test_step(test_images, test_labels)
template = 'Epoch {}, Perdida: {}, Exactitud: {}, Perdida de prueba: {}, Exactitud de prueba: {}'
print(template.format(epoch+1,
train_loss.result(),
train_accuracy.result()*100,
test_loss.result(),
test_accuracy.result()*100))
# Reinicia las metricas para el siguiente epoch.
train_loss.reset_states()
train_accuracy.reset_states()
test_loss.reset_states()
test_accuracy.reset_states() | site/es-419/tutorials/quickstart/advanced.ipynb | tensorflow/docs-l10n | apache-2.0 |
Data: Preparing for the model
Importing the raw data | DIR = os.getcwd() + "/../data/"
t = pd.read_csv(DIR + 'raw/lending-club-loan-data/loan.csv', low_memory=False)
t.head() | notebooks/5-aa-second_model.ipynb | QuinnLee/cs109a-Project | mit |
Cleaning, imputing missing values, feature engineering (some NLP) | t2 = md.clean_data(t)
t3 = md.impute_missing(t2)
df = md.simple_dataset(t3)
# df = md.spelling_mistakes(t3) - skipping for now, so computationally expensive! | notebooks/5-aa-second_model.ipynb | QuinnLee/cs109a-Project | mit |
Train, test split: Splitting on 2015 | df['issue_d'].hist(bins = 50)
plt.title('Seasonality in lending')
plt.ylabel('Frequency')
plt.xlabel('Year')
plt.show() | notebooks/5-aa-second_model.ipynb | QuinnLee/cs109a-Project | mit |
We can use past years as predictors of future years. One challenge with this approach is that we confound time-sensitive trends (for example, global economic shocks to interest rates - such as the financial crisis of 2008, or the growth of Lending Club to broader and broader markets of debtors) with differences related to time-insensitive factors (such as a debtor's riskiness).
To account for this, we can bundle our training and test sets into the following blocks:
- Before 2015: Training set
- 2015 to current: Test set | old = df[df['issue_d'] < '2015']
new = df[df['issue_d'] >= '2015']
old.shape, new.shape | notebooks/5-aa-second_model.ipynb | QuinnLee/cs109a-Project | mit |
We'll use the pre-2015 data on interest rates (old) to fit a model and cross-validate it. We'll then use the post-2015 data as a 'wild' dataset to test against.
Fitting the model | X = old.drop(['int_rate', 'issue_d', 'earliest_cr_line', 'grade'], 1)
y = old['int_rate']
X.shape, y.shape
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=42)
X_train.shape, X_test.shape, y_train.shape, y_test.shape
rfr = RandomForestRegressor(n_estimators = 10, max_features='sqrt')
scores = cross_val_score(rfr, X, y, cv = 3)
print("Accuracy: {:.2f} (+/- {:.2f})".format(scores.mean(), scores.std() * 2))
X_new = new.drop(['int_rate', 'issue_d', 'earliest_cr_line', 'grade'], 1)
y_new = new['int_rate']
new_scores = cross_val_score(rfr, X_new, y_new, cv = 3)
print("Accuracy: {:.2f} (+/- {:.2f})".format(new_scores.mean(), new_scores.std() * 2))
# QUINN: Let's just use this - all data
X_total = df.drop(['int_rate', 'issue_d', 'earliest_cr_line', 'grade'], 1)
y_total = df['int_rate']
total_scores = cross_val_score(rfr, X_total, y_total, cv = 3)
print("Accuracy: {:.2f} (+/- {:.2f})".format(total_scores.mean(), total_scores.std() * 2)) | notebooks/5-aa-second_model.ipynb | QuinnLee/cs109a-Project | mit |
Fitting the model
We fit the model on all the data, and evaluate feature importances. | rfr.fit(X_total, y_total)
fi = [{'importance': x, 'feature': y} for (x, y) in \
sorted(zip(rfr.feature_importances_, X_total.columns))]
fi = pd.DataFrame(fi)
fi.sort_values(by = 'importance', ascending = False, inplace = True)
fi.head()
top5 = fi.head()
top5.plot(kind = 'bar')
plt.xticks(range(5), top5['feature'])
plt.title('Feature importances (top 5 features)')
plt.ylabel('Relative importance')
plt.show() | notebooks/5-aa-second_model.ipynb | QuinnLee/cs109a-Project | mit |
๋ณ๋ถ ์ถ๋ก ์ผ๋ก ์ผ๋ฐํ๋ ์ ํ ํผํฉ ํจ๊ณผ ๋ชจ๋ธ ๋ง์ถค ์กฐ์ ํ๊ธฐ
<table class="tfo-notebook-buttons" align="left">
<td> <a target="_blank" href="https://www.tensorflow.org/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference"><img src="https://www.tensorflow.org/images/tf_logo_32px.png">TensorFlow.org์์ ๋ณด๊ธฐ</a> </td>
<td><a target="_blank" href="https://colab.research.google.com/github/tensorflow/docs-l10n/blob/master/site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb"><img src="https://www.tensorflow.org/images/colab_logo_32px.png">Google Colab์์ ์คํ</a></td>
<td> <a target="_blank" href="https://github.com/tensorflow/docs-l10n/blob/master/site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb"><img src="https://www.tensorflow.org/images/GitHub-Mark-32px.png">GitHub์์ ์์ค ๋ณด๊ธฐ</a>
</td>
<td><a href="https://storage.googleapis.com/tensorflow_docs/docs-l10n/site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb"><img src="https://www.tensorflow.org/images/download_logo_32px.png">๋
ธํธ๋ถ ๋ค์ด๋ก๋</a></td>
</table> | #@title Install { display-mode: "form" }
TF_Installation = 'System' #@param ['TF Nightly', 'TF Stable', 'System']
if TF_Installation == 'TF Nightly':
!pip install -q --upgrade tf-nightly
print('Installation of `tf-nightly` complete.')
elif TF_Installation == 'TF Stable':
!pip install -q --upgrade tensorflow
print('Installation of `tensorflow` complete.')
elif TF_Installation == 'System':
pass
else:
raise ValueError('Selection Error: Please select a valid '
'installation option.')
#@title Install { display-mode: "form" }
TFP_Installation = "System" #@param ["Nightly", "Stable", "System"]
if TFP_Installation == "Nightly":
!pip install -q tfp-nightly
print("Installation of `tfp-nightly` complete.")
elif TFP_Installation == "Stable":
!pip install -q --upgrade tensorflow-probability
print("Installation of `tensorflow-probability` complete.")
elif TFP_Installation == "System":
pass
else:
raise ValueError("Selection Error: Please select a valid "
"installation option.") | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
์์ฝ
์ด colab์์๋ TensorFlow Probability์ ๋ณ๋ถ ์ถ๋ก ์ ์ฌ์ฉํ์ฌ ์ผ๋ฐํ๋ ์ ํ ํผํฉ ํจ๊ณผ ๋ชจ๋ธ์ ๋ง์ถค ์กฐ์ ํ๋ ๋ฐฉ๋ฒ์ ๋ณด์ฌ์ค๋๋ค.
๋ชจ๋ธ ํจ๋ฐ๋ฆฌ
์ผ๋ฐํ๋ ์ ํ ํผํฉ ํจ๊ณผ ๋ชจ๋ธ(GLMM)์ ์ํ๋ณ ๋
ธ์ด์ฆ๋ฅผ ์์ธก๋ ์ ํ ์๋ต์ ํตํฉํ๋ค๋ ์ ์ ์ ์ธํ๋ฉด ์ผ๋ฐํ๋ ์ ํ ๋ชจ๋ธ(GLM)๊ณผ ์ ์ฌํฉ๋๋ค. ์ด๊ฒ์ ๊ฑฐ์ ๋ณด์ด์ง ์๋ ํน์ฑ์ด ๋ ์ผ๋ฐ์ ์ผ๋ก ๋ณด์ด๋ ํน์ฑ๊ณผ ์ ๋ณด๋ฅผ ๊ณต์ ํ ์ ์๊ธฐ ๋๋ฌธ์ ๋ถ๋ถ์ ์ผ๋ก ์ ์ฉํฉ๋๋ค.
์์ฑ ํ๋ก์ธ์ค๋ก์ ์ผ๋ฐํ๋ ์ ํ ํผํฉ ํจ๊ณผ ๋ชจ๋ธ(GLMM)์ ๋ค์๊ณผ ๊ฐ์ ํน์ง์ด ์์ต๋๋ค.
$$ \begin{align} \text{for } & r = 1\ldots R: \hspace{2.45cm}\text{# for each random-effect group}\ &\begin{aligned} \text{for } &c = 1\ldots |C_r|: \hspace{1.3cm}\text{# for each category ("level") of group $r$}\ &\begin{aligned} \beta_{rc} &\sim \text{MultivariateNormal}(\text{loc}=0_{D_r}, \text{scale}=\Sigma_r^{1/2}) \end{aligned} \end{aligned}\ \text{for } & i = 1 \ldots N: \hspace{2.45cm}\text{# for each sample}\ &\begin{aligned} &\eta_i = \underbrace{\vphantom{\sum_{r=1}^R}x_i^\top\omega}\text{fixed-effects} + \underbrace{\sum{r=1}^R z_{r,i}^\top \beta_{r,C_r(i) }}\text{random-effects} \ &Y_i|x_i,\omega,{z{r,i} , \beta_r}_{r=1}^R \sim \text{Distribution}(\text{mean}= g^{-1}(\eta_i)) \end{aligned} \end{align} $$
์ฌ๊ธฐ์
$$ \begin{align} R &= \text{number of random-effect groups}\ |C_r| &= \text{number of categories for group $r$}\ N &= \text{number of training samples}\ x_i,\omega &\in \mathbb{R}^{D_0}\ D_0 &= \text{number of fixed-effects}\ C_r(i) &= \text{category (under group $r$) of the $i$th sample}\ z_{r,i} &\in \mathbb{R}^{D_r}\ D_r &= \text{number of random-effects associated with group $r$}\ \Sigma_{r} &\in {S\in\mathbb{R}^{D_r \times D_r} : S \succ 0 }\ \eta_i\mapsto g^{-1}(\eta_i) &= \mu_i, \text{inverse link function}\ \text{Distribution} &=\text{some distribution parameterizable solely by its mean} \end{align} $$
์ฆ, ๊ฐ ๊ทธ๋ฃน์ ๋ชจ๋ ์นดํ
๊ณ ๋ฆฌ๊ฐ ๋ค๋ณ๋ ์ ๊ท ๋ถํฌ์ ์ํ $\beta_{rc}$์ ์ฐ๊ฒฐ๋์ด ์์์ ์๋ฏธํฉ๋๋ค. $\beta_{rc}$ ์ถ์ถ์ ํญ์ ๋
๋ฆฝ์ ์ด์ง๋ง $r$ ๊ทธ๋ฃน์ ๋ํด์๋ง ๋์ผํ๊ฒ ๋ถํฌ๋ฉ๋๋ค. $r\in{1,\ldots,R}$๋น ์ ํํ ํ๋์ $\Sigma_r$๊ฐ ์์ต๋๋ค.
์ํ ๊ทธ๋ฃน์ ํน์ฑ($z_{r,i}$)๊ณผ ์ ์ฌํ๊ฒ ๊ฒฐํฉํ๋ฉด ๊ฒฐ๊ณผ๋ $i$๋ฒ์งธ ์์ธก ์ ํ ์๋ต(๊ทธ๋ ์ง ์์ผ๋ฉด $x_i^\top\omega$)์ ๋ํ ์ํ๋ณ ๋
ธ์ด์ฆ์
๋๋ค.
${\Sigma_r:r\in{1,\ldots,R}}$๋ฅผ ์ถ์ ํ ๋ ๋ณธ์ง์ ์ผ๋ก ์์ ํจ๊ณผ ๊ทธ๋ฃน์ด ์ ๋ฌํ๋ ๋
ธ์ด์ฆ์ ์์ ์ถ์ ํฉ๋๋ค. ๊ทธ๋ ์ง ์์ผ๋ฉด $x_i^\top\omega$์ ์๋ ์ ํธ๋ฅผ ์ถ์ถํฉ๋๋ค.
$\text{Distribution}$, ์ญ๋งํฌ ํจ์ ๋ฐ $g^{-1}$์ ๋ํ ๋ค์ํ ์ต์
์ด ์์ต๋๋ค. ์ผ๋ฐ์ ์ธ ์ต์
์ ๋ค์๊ณผ ๊ฐ์ต๋๋ค.
$Y_i\sim\text{Normal}(\text{mean}=\eta_i, \text{scale}=\sigma)$,
$Y_i\sim\text{Binomial}(\text{mean}=n_i \cdot \text{sigmoid}(\eta_i), \text{total_count}=n_i)$, ๋ฐ
$Y_i\sim\text{Poisson}(\text{mean}=\exp(\eta_i))$.
๋ ๋ง์ ๊ฐ๋ฅ์ฑ์ tfp.glm ๋ชจ๋์ ์ฐธ์กฐํ์ธ์.
๋ณ๋ถ ์ถ๋ก
๋ถํํ๋, ๋งค๊ฐ๋ณ์ $\beta,{\Sigma_r} _r^R$์ ์ต๋ ๊ฐ๋ฅ์ฑ ์ถ์ ์น๋ฅผ ์ฐพ๋ ๊ฒ์ ๋น ๋ถ์ ์ ๋ถ์ ์๋ฐํฉ๋๋ค. ์ด๋ฌํ ๋ฌธ์ ๋ฅผ ํผํ๊ณ ์ ๋์ ๋ค์์ ์ํํฉ๋๋ค.
๋ถ๋ก์ $q_{\lambda}$๋ก ํ์๋ ๋งค๊ฐ๋ณ์ํ๋ ๋ถํฌ ํจ๋ฐ๋ฆฌ('๋๋ฆฌ ๋ฐ๋')๋ฅผ ์ ์ํฉ๋๋ค.
$q_{\lambda}$๊ฐ ์ค์ ๋ชฉํ ๋ฐ๋์ ๊ฐ๊น๋๋ก ๋งค๊ฐ๋ณ์ $\lambda$๋ฅผ ์ฐพ์ต๋๋ค.
๋ถํฌ ํจ๋ฐ๋ฆฌ๋ ์ ์ ํ ์ฐจ์์ ๋
๋ฆฝ์ ์ธ ๊ฐ์ฐ์์์ด ๋ ๊ฒ์ด๋ฉฐ, '๋ชฉํ ๋ฐ๋์ ๊ฐ๊น์'์ด๋ '์ฟจ๋ฐฑ-๋ผ์ฐ๋ธ๋ฌ(Kullbakc-Leibler) ๋ฐ์ฐ ์ต์ํ'๋ฅผ ์๋ฏธํฉ๋๋ค. ์๋ฅผ ๋ค์ด, ์ ์์ฑ๋ ์ ๋ ๋ฐ ๋๊ธฐ๋ '๋ณ๋ถ ์ถ๋ก : ํต๊ณํ์๋ฅผ ์ํ ๊ฒํ '์ ์น์
2.2๋ฅผ ์ฐธ์กฐํ์ธ์. ํนํ, KL ๋ฐ์ฐ์ ์ต์ํํ๋ ๊ฒ์ ELBO(evidence lower bound)๋ฅผ ์ต์ํํ๋ ๊ฒ๊ณผ ๋์ผํจ์ ๋ณด์ฌ์ค๋๋ค.
์ฅ๋๊ฐ ๋ฌธ์
Gelman ๋ฑ(2007)์ '๋ผ๋ ๋ฐ์ดํฐ์ธํธ'๋ ํ๊ท์ ๋ํ ์ ๊ทผ ๋ฐฉ์์ ์
์ฆํ๋ ๋ฐ ์ฌ์ฉ๋๋ ๋ฐ์ดํฐ์ธํธ์
๋๋ค(์: ๋ฐ์ ํ๊ฒ ๊ด๋ จ๋ PyMC3 ๋ธ๋ก๊ทธ ๊ฒ์๋ฌผ). ๋ผ๋ ๋ฐ์ดํฐ์ธํธ์๋ ๋ฏธ๊ตญ ์ ์ญ์์ ์ธก์ ๋ ๋ผ๋์ ์ค๋ด ์ธก์ ๊ฐ์ด ํฌํจ๋์ด ์์ต๋๋ค. ๋ผ๋์ ์์ฐ์ ์ผ๋ก ๋ฐ์ํ๋ ๋ฐฉ์ฌ์ฑ ๊ฐ์ค๋ก ๊ณ ๋๋์์ ๋
์ฑ์ด ์์ต๋๋ค.
๋ฐ๋ชจ๋ฅผ ์ํด, ์งํ์ค์ด ์๋ ๊ฐ์ ์์ ๋ผ๋ ์์น๊ฐ ๋ ๋๋ค๋ ๊ฐ์ค์ ๊ฒ์ฆํ๋ ๋ฐ ๊ด์ฌ์ด ์๋ค๊ณ ๊ฐ์ ํด ๋ณด๊ฒ ์ต๋๋ค. ๋ํ ๋ผ๋ ๋๋๊ฐ ํ ์ ์ ํ, ์ฆ ์ง๋ฆฌ ๋ฌธ์ ์ ๊ด๋ จ์ด ์๋ค๊ณ ์์ฌํฉ๋๋ค.
์ด๋ฅผ ML ๋ฌธ์ ๋ก ๋ง๋ค๊ธฐ ์ํด, ํ๋
๊ฐ์ด ์ธก์ ๋ ์ธต์ ์ ํ ํจ์๋ฅผ ๊ธฐ๋ฐ์ผ๋ก ๋ก๊ทธ ๋ผ๋ ์์ค์ ์์ธกํ๋ ค๊ณ ํฉ๋๋ค. ๋ํ ์นด์ดํฐ(county)๋ฅผ ์์ ํจ๊ณผ๋ก ์ฌ์ฉํ์ฌ ์ง๋ฆฌ๋ก ์ธํ ๋ถ์ฐ์ ์ค๋ช
ํ ๊ฒ์
๋๋ค. ์ฆ, ์ผ๋ฐํ๋ ์ ํ ํผํฉ ํจ๊ณผ ๋ชจ๋ธ์ ์ฌ์ฉํฉ๋๋ค. | %matplotlib inline
%config InlineBackend.figure_format = 'retina'
import os
from six.moves import urllib
import matplotlib.pyplot as plt; plt.style.use('ggplot')
import numpy as np
import pandas as pd
import seaborn as sns; sns.set_context('notebook')
import tensorflow_datasets as tfds
import tensorflow.compat.v2 as tf
tf.enable_v2_behavior()
import tensorflow_probability as tfp
tfd = tfp.distributions
tfb = tfp.bijectors | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
๋ํ GPU์ ๊ฐ์ฉ์ฑ์ ๋น ๋ฅด๊ฒ ํ์ธํฉ๋๋ค. | if tf.test.gpu_device_name() != '/device:GPU:0':
print("We'll just use the CPU for this run.")
else:
print('Huzzah! Found GPU: {}'.format(tf.test.gpu_device_name())) | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
๋ฐ์ดํฐ์ธํธ ์ป๊ธฐ
TensorFlow ๋ฐ์ดํฐ์ธํธ์์ ๋ฐ์ดํฐ์ธํธ๋ฅผ ๋ก๋ํ๊ณ ์ฝ๊ฐ์ ๊ฐ๋ฒผ์ด ์ ์ฒ๋ฆฌ๋ฅผ ์ํํฉ๋๋ค. | def load_and_preprocess_radon_dataset(state='MN'):
"""Load the Radon dataset from TensorFlow Datasets and preprocess it.
Following the examples in "Bayesian Data Analysis" (Gelman, 2007), we filter
to Minnesota data and preprocess to obtain the following features:
- `county`: Name of county in which the measurement was taken.
- `floor`: Floor of house (0 for basement, 1 for first floor) on which the
measurement was taken.
The target variable is `log_radon`, the log of the Radon measurement in the
house.
"""
ds = tfds.load('radon', split='train')
radon_data = tfds.as_dataframe(ds)
radon_data.rename(lambda s: s[9:] if s.startswith('feat') else s, axis=1, inplace=True)
df = radon_data[radon_data.state==state.encode()].copy()
df['radon'] = df.activity.apply(lambda x: x if x > 0. else 0.1)
# Make county names look nice.
df['county'] = df.county.apply(lambda s: s.decode()).str.strip().str.title()
# Remap categories to start from 0 and end at max(category).
df['county'] = df.county.astype(pd.api.types.CategoricalDtype())
df['county_code'] = df.county.cat.codes
# Radon levels are all positive, but log levels are unconstrained
df['log_radon'] = df['radon'].apply(np.log)
# Drop columns we won't use and tidy the index
columns_to_keep = ['log_radon', 'floor', 'county', 'county_code']
df = df[columns_to_keep].reset_index(drop=True)
return df
df = load_and_preprocess_radon_dataset()
df.head() | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
GLMM ํจ๋ฐ๋ฆฌ ์ ๋ฌธํํ๊ธฐ
์ด ์น์
์์๋ GLMM ํจ๋ฐ๋ฆฌ๋ฅผ ๋ผ๋ ์์ค ์์ธก ์์
์ ์ ๋ฌธํํฉ๋๋ค. ์ด๋ฅผ ์ํด ๋จผ์ GLMM์ ๊ณ ์ ํจ๊ณผ ํน์ ์ผ์ด์ค๋ฅผ ๊ณ ๋ คํฉ๋๋ค. $$ \mathbb{E}[\log(\text{radon}_j)] = c + \text{floor_effect}_j $$
์ด ๋ชจ๋ธ์ ๊ด์ธก์น $j$์ ๋ก๊ทธ ๋ผ๋์ด $j$๋ฒ์งธ ํ๋
๊ฐ์ด ์ธก์ ๋ ์ธต๊ณผ ์ผ์ ํ ์ ํธ์ ์ํด ์์๋๋ก ๊ฒฐ์ ๋๋ค๊ณ ๊ฐ์ ํฉ๋๋ค. ์์ฌ ์ฝ๋์์๋ ๋ค์๊ณผ ๊ฐ์ด ์์ฑํ ์ ์์ต๋๋ค.
def estimate_log_radon(floor):
return intercept + floor_effect[floor]
๋ชจ๋ ์ธต์ ๋ํด ํ์ต๋ ๊ฐ์ค์น์ ๋ณดํธ์ ์ธ intercept ํญ์ด ์์ต๋๋ค. 0์ธต๊ณผ 1์ธต์ ๋ผ๋ ์ธก์ ๊ฐ์ ๋ณด๋ฉด ๋ค์๊ณผ ๊ฐ์ด ์์ํ๋ ๊ฒ์ด ์ข์ต๋๋ค. | fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(12, 4))
df.groupby('floor')['log_radon'].plot(kind='density', ax=ax1);
ax1.set_xlabel('Measured log(radon)')
ax1.legend(title='Floor')
df['floor'].value_counts().plot(kind='bar', ax=ax2)
ax2.set_xlabel('Floor where radon was measured')
ax2.set_ylabel('Count')
fig.suptitle("Distribution of log radon and floors in the dataset"); | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
์ง๋ฆฌ์ ๊ดํ ๋ด์ฉ์ ํฌํจํ์ฌ ๋ชจ๋ธ์ ์ข ๋ ์ ๊ตํ๊ฒ ๋ง๋๋ ๊ฒ์ด ์๋ง๋ ๋ ์ข์ ๊ฒ์
๋๋ค. ๋ผ๋์ ๋
์ ์กด์ฌํ ์ ์๋ ์ฐ๋ผ๋์ ๋ถ๊ดด ์ฌ์ฌ์ ์ผ๋ถ์ด๋ฏ๋ก ์ง๋ฆฌ๋ฅผ ์ค๋ช
ํ๋ ๊ฒ์ด ์ค์ํฉ๋๋ค.
$$ \mathbb{E}[\log(\text{radon}_j)] = c + \text{floor_effect}_j + \text{county_effect}_j $$
๋ค์ ํ๋ฉด, ์์ฌ ์ฝ๋์์ ๋ค์๊ณผ ๊ฐ์ต๋๋ค.
def estimate_log_radon(floor, county):
return intercept + floor_effect[floor] + county_effect[county]
์นด์ดํฐ๋ณ ๊ฐ์ค์น๋ฅผ ์ ์ธํ๊ณ ๋ ์ด์ ๊ณผ ๋์ผํฉ๋๋ค.
์ถฉ๋ถํ ํฐ ํ๋ จ ์ธํธ๊ฐ ์ฃผ์ด์ง๋ฉด ์ด๋ ํฉ๋ฆฌ์ ์ธ ๋ชจ๋ธ์
๋๋ค. ํ์ง๋ง ๋ฏธ๋ค์ํ์ ๋ฐ์ดํฐ๋ฅผ ๊ณ ๋ คํ ๋ ๊ด์ธก์น ์๊ฐ ์์ ์นด์ดํฐ๊ฐ ๋ง์์ ์ ์ ์์ต๋๋ค. ์๋ฅผ ๋ค์ด, 85๊ฐ ์นด์ดํฐ ์ค 39๊ฐ ์นด์ดํฐ์ ๊ด์ธก์น๊ฐ 5๊ฐ ๋ฏธ๋ง์
๋๋ค.
์ด๋ ์นด์ดํฐ๋น ๊ด์ธก์น ์๊ฐ ์ฆ๊ฐํจ์ ๋ฐ๋ผ ์์ ๋ชจ๋ธ๋ก ์๋ ดํ๋ ๋ฐฉ์์ผ๋ก ๋ชจ๋ ๊ด์ธก์น ๊ฐ์ ํต๊ณ์ ๊ฐ๋๋ฅผ ๊ณต์ ํ๋๋ก ๋๊ธฐ๋ฅผ ๋ถ์ฌํฉ๋๋ค. | fig, ax = plt.subplots(figsize=(22, 5));
county_freq = df['county'].value_counts()
county_freq.plot(kind='bar', ax=ax)
ax.set_xlabel('County')
ax.set_ylabel('Number of readings'); | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
์ด ๋ชจ๋ธ์ ๋ง์ถค ์กฐ์ ํ๋ฉด county_effect ๋ฒกํฐ๋ ํ๋ จ ์ํ์ด ๊ฑฐ์ ์๋ ์นด์ดํฐ์ ๋ํ ๊ฒฐ๊ณผ๋ฅผ ๊ธฐ์ตํ๊ฒ ๋ ๊ฒ์
๋๋ค. ์๋ง๋ ๊ณผ๋์ ํฉ์ด ๋ฐ์ํ์ฌ ์ผ๋ฐํ๊ฐ ๋ถ๋ํ ์ ์์ต๋๋ค.
GLMM์ ์์ ๋ GLM์ ๋ํด ์ ์ ํ ํํ์ ์ ์ ๊ณตํฉ๋๋ค. ๋ค์๊ณผ ๊ฐ์ด ๋ง์ถค ์กฐ์ ํ๋ ๊ฒ์ ๊ณ ๋ คํ ์ ์์ต๋๋ค.
$$ \log(\text{radon}_j) \sim c + \text{floor_effect}_j + \mathcal{N}(\text{county_effect}_j, \text{county_scale}) $$
์ด ๋ชจ๋ธ์ ์ฒซ ๋ฒ์งธ ๋ชจ๋ธ๊ณผ ๊ฐ์ง๋ง, ๊ฐ๋ฅ์ฑ์ด ์ ๊ท ๋ถํฌ๊ฐ ๋๋๋ก ๊ณ ์ ํ์ผ๋ฉฐ ๋จ์ผ ๋ณ์ county_scale์ ํตํด ๋ชจ๋ ์นด์ดํฐ์์ ๋ถ์ฐ์ ๊ณต์ ํฉ๋๋ค. ์์ฌ ์ฝ๋๋ ๋ค์๊ณผ ๊ฐ์ต๋๋ค.
def estimate_log_radon(floor, county):
county_mean = county_effect[county]
random_effect = np.random.normal() * county_scale + county_mean
return intercept + floor_effect[floor] + random_effect
๊ด์ธก๋ ๋ฐ์ดํฐ๋ก county_scale, county_mean ๋ฐ random_effect์ ๋ํ ๊ฒฐํฉ ๋ถํฌ๋ฅผ ์ถ๋ก ํฉ๋๋ค. ๊ธ๋ก๋ฒ county_scale์ ์ฌ์ฉํ๋ฉด ์นด์ดํฐ ๊ฐ์ ํต๊ณ์ ๊ฐ๋๋ฅผ ๊ณต์ ํ ์ ์์ต๋๋ค. ๊ด์ธก์น๊ฐ ๋ง์ ๊ฒฝ์ฐ ๊ด์ธก์น๊ฐ ๊ฑฐ์ ์๋ ์นด์ดํฐ ๋ถ์ฐ์ ๋์์ด ๋ฉ๋๋ค. ๋ํ ๋ ๋ง์ ๋ฐ์ดํฐ๋ฅผ ์์งํ๋ฉด ์ด ๋ชจ๋ธ์ scale ๋ณ์๊ฐ ํ๋งํ์ง ์๋ ๋ชจ๋ธ๋ก ์๋ ด๋ฉ๋๋ค. ์ด ๋ฐ์ดํฐ์ธํธ๋ฅผ ์ฌ์ฉํ๋๋ผ๋ ๋ ๋ชจ๋ธ ์ค ํ๋๋ฅผ ์ฌ์ฉํ์ฌ ๊ด์ธก์น๊ฐ ๊ฐ์ฅ ๋ง์ ์นด์ดํฐ์ ๋ํ ์ ์ฌํ ๊ฒฐ๋ก ์ ๋๋ฌํ๊ฒ ๋ฉ๋๋ค.
์คํ
์ด์ TensorFlow์์ ๋ณ๋ถ ์ถ๋ก ์ผ๋ก ์์ GLMM์ ๋ง์ถค ์กฐ์ ํ๋ ค๊ณ ํฉ๋๋ค. ๋จผ์ ๋ฐ์ดํฐ๋ฅผ ํน์ฑ๊ณผ ๋ ์ด๋ธ๋ก ๋ถํ ํฉ๋๋ค. | features = df[['county_code', 'floor']].astype(int)
labels = df[['log_radon']].astype(np.float32).values.flatten() | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
๋ชจ๋ธ์ ์ง์ ํฉ๋๋ค. | def make_joint_distribution_coroutine(floor, county, n_counties, n_floors):
def model():
county_scale = yield tfd.HalfNormal(scale=1., name='scale_prior')
intercept = yield tfd.Normal(loc=0., scale=1., name='intercept')
floor_weight = yield tfd.Normal(loc=0., scale=1., name='floor_weight')
county_prior = yield tfd.Normal(loc=tf.zeros(n_counties),
scale=county_scale,
name='county_prior')
random_effect = tf.gather(county_prior, county, axis=-1)
fixed_effect = intercept + floor_weight * floor
linear_response = fixed_effect + random_effect
yield tfd.Normal(loc=linear_response, scale=1., name='likelihood')
return tfd.JointDistributionCoroutineAutoBatched(model)
joint = make_joint_distribution_coroutine(
features.floor.values, features.county_code.values, df.county.nunique(),
df.floor.nunique())
# Define a closure over the joint distribution
# to condition on the observed labels.
def target_log_prob_fn(*args):
return joint.log_prob(*args, likelihood=labels) | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
์ฌํ ํ๋ฅ ๋๋ฆฌ๋ฅผ ์ง์ ํฉ๋๋ค.
๋งค๊ฐ๋ณ์ $\lambda$๊ฐ ํ๋ จ ๊ฐ๋ฅํ ๋๋ฆฌ ํจ๋ฐ๋ฆฌ $q_{\lambda}$๋ฅผ ๊ตฌ์ฑํฉ๋๋ค. ์ด ๊ฒฝ์ฐ์ ํจ๋ฐ๋ฆฌ๋ ๊ฐ ๋งค๊ฐ๋ณ์์ ๋ํด ํ๋์ ๋ถํฌ๋ฅผ ๊ฐ๋ ๋
๋ฆฝ์ ์ธ ๋ค๋ณ๋ ์ ๊ท ๋ถํฌ์ด๊ณ $\lambda = {(\mu_j, \sigma_j)}$์
๋๋ค. ์ฌ๊ธฐ์ $j$๋ 4๊ฐ์ ๋งค๊ฐ๋ณ์๋ฅผ ์ธ๋ฑ์ฑํฉ๋๋ค.
๋๋ฆฌ ํจ๋ฐ๋ฆฌ๋ฅผ ๋ง์ถค ์กฐ์ ํ๊ธฐ ์ํ ๋ฉ์๋๋ tf.Variables๋ฅผ ์ฌ์ฉํ๋ ๊ฒ์
๋๋ค. ๋ํ tfp.util.TransformedVariable์ Softplus์ ๊ฐ์ด ์ฌ์ฉํ์ฌ scale ๋งค๊ฐ๋ณ์(ํ๋ จ ๊ฐ๋ฅํจ)๋ฅผ ์์๋ก ์ ํํฉ๋๋ค. ๋ํ ์์ ๋งค๊ฐ๋ณ์์ธ ์ ์ฒด scale_prior์ Softplus๋ฅผ ์ ์ฉํฉ๋๋ค.
์ต์ ํ๋ฅผ ๋๊ธฐ ์ํด ์ฝ๊ฐ์ ์งํฐ๋ฅผ ์ฌ์ฉํ์ฌ ์ด๋ฌํ ํ๋ จ ๊ฐ๋ฅํ ๋ณ์๋ฅผ ์ด๊ธฐํํฉ๋๋ค. | # Initialize locations and scales randomly with `tf.Variable`s and
# `tfp.util.TransformedVariable`s.
_init_loc = lambda shape=(): tf.Variable(
tf.random.uniform(shape, minval=-2., maxval=2.))
_init_scale = lambda shape=(): tfp.util.TransformedVariable(
initial_value=tf.random.uniform(shape, minval=0.01, maxval=1.),
bijector=tfb.Softplus())
n_counties = df.county.nunique()
surrogate_posterior = tfd.JointDistributionSequentialAutoBatched([
tfb.Softplus()(tfd.Normal(_init_loc(), _init_scale())), # scale_prior
tfd.Normal(_init_loc(), _init_scale()), # intercept
tfd.Normal(_init_loc(), _init_scale()), # floor_weight
tfd.Normal(_init_loc([n_counties]), _init_scale([n_counties]))]) # county_prior | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
์ด ์
์ ๋ค์๊ณผ ๊ฐ์ด tfp.experimental.vi.build_factored_surrogate_posterior๋ก ๋์ฒดํ ์ ์์ต๋๋ค.
python
surrogate_posterior = tfp.experimental.vi.build_factored_surrogate_posterior(
event_shape=joint.event_shape_tensor()[:-1],
constraining_bijectors=[tfb.Softplus(), None, None, None])
๊ฒฐ๊ณผ
๋ค๋ฃจ๊ธฐ ์ฌ์ด ๋งค๊ฐ๋ณ์ํ๋ ๋ถํฌ ํจ๋ฐ๋ฆฌ๋ฅผ ์ ์ํ ๋ค์, ๋ชฉํ ๋ถํฌ์ ๊ฐ๊น์ด ๋ค๋ฃจ๊ธฐ ์ฌ์ด ๋ถํฌ๋ฅผ ๊ฐ๋๋ก ๋งค๊ฐ๋ณ์๋ฅผ ์ ํํ๋ ๊ฒ์ด ๋ชฉํ์์ ๊ธฐ์ตํ์ธ์.
์์ ๋๋ฆฌ ๋ถํฌ๋ฅผ ๋น๋ํ์ผ๋ฉฐ ์ตํฐ๋ง์ด์ ์ ์ฃผ์ด์ง ์คํ
์๋ฅผ ํ์ฉํ๋ tfp.vi.fit_surrogate_posterior๋ฅผ ์ฌ์ฉํ์ฌ ์์ฑ ELBO๋ฅผ ์ต์ํํ๋ ๋๋ฆฌ ๋ชจ๋ธ์ ๋ํ ๋งค๊ฐ๋ณ์๋ฅผ ์ฐพ์ต๋๋ค(๋๋ฆฌ์์ ๋์ ๋ถํฌ ์ฌ์ด์ ์ฟจ๋ฐฑ-๋ผ์ด๋ธ๋ฌ ๋ฐ์ฐ์ ์ต์ํํ๋ ๊ฒ๊ณผ ์ผ์นํจ).
๋ฐํ ๊ฐ์ ๊ฐ ์คํ
์์ ์์ ELBO์ด๋ฉฐ surrogate_posterior์ ๋ถํฌ๋ ์ตํฐ๋ง์ด์ ์์ ์ฐพ์ ๋งค๊ฐ๋ณ์๋ก ์
๋ฐ์ดํธ๋ฉ๋๋ค. | optimizer = tf.optimizers.Adam(learning_rate=1e-2)
losses = tfp.vi.fit_surrogate_posterior(
target_log_prob_fn,
surrogate_posterior,
optimizer=optimizer,
num_steps=3000,
seed=42,
sample_size=2)
(scale_prior_,
intercept_,
floor_weight_,
county_weights_), _ = surrogate_posterior.sample_distributions()
print(' intercept (mean): ', intercept_.mean())
print(' floor_weight (mean): ', floor_weight_.mean())
print(' scale_prior (approx. mean): ', tf.reduce_mean(scale_prior_.sample(10000)))
fig, ax = plt.subplots(figsize=(10, 3))
ax.plot(losses, 'k-')
ax.set(xlabel="Iteration",
ylabel="Loss (ELBO)",
title="Loss during training",
ylim=0); | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
์ถ์ ๋ ํ๊ท ์นด์ดํฐ(county) ํจ๊ณผ์ ํด๋น ํ๊ท ์ ๋ถํ์ค์ฑ์ ํ๋กฏํ ์ ์์ต๋๋ค. ์ด๋ฅผ ๊ด์ฐฐ ํ์๋ก ์ ๋ ฌํ์ผ๋ฉฐ ๊ฐ์ฅ ํฐ ์๋ ์ผ์ชฝ์ ์์ต๋๋ค. ๊ด์ธก์น๊ฐ ๋ง์ ์นด์ดํฐ์์๋ ๋ถํ์ค์ฑ์ด ์์ง๋ง, ๊ด์ธก์น๊ฐ ํ๋ ๊ฐ๋ง ์๋ ์นด์ดํฐ์์๋ ๋ถํ์ค์ฑ์ด ๋ ํฝ๋๋ค. | county_counts = (df.groupby(by=['county', 'county_code'], observed=True)
.agg('size')
.sort_values(ascending=False)
.reset_index(name='count'))
means = county_weights_.mean()
stds = county_weights_.stddev()
fig, ax = plt.subplots(figsize=(20, 5))
for idx, row in county_counts.iterrows():
mid = means[row.county_code]
std = stds[row.county_code]
ax.vlines(idx, mid - std, mid + std, linewidth=3)
ax.plot(idx, means[row.county_code], 'ko', mfc='w', mew=2, ms=7)
ax.set(
xticks=np.arange(len(county_counts)),
xlim=(-1, len(county_counts)),
ylabel="County effect",
title=r"Estimates of county effects on log radon levels. (mean $\pm$ 1 std. dev.)",
)
ax.set_xticklabels(county_counts.county, rotation=90); | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
์ค์ ๋ก ์ถ์ ๋ ํ์ค ํธ์ฐจ์ ๋ํ ๋ก๊ทธ ์์ ๊ด์ธก์น๋ฅผ ํ๋กฏํ์ฌ ์ด๋ฅผ ๋ ์ง์ ์ ์ผ๋ก ๋ณผ ์ ์์ผ๋ฉฐ ๊ด๊ณ๊ฐ ๊ฑฐ์ ์ ํ์์ ์ ์ ์์ต๋๋ค. | fig, ax = plt.subplots(figsize=(10, 7))
ax.plot(np.log1p(county_counts['count']), stds.numpy()[county_counts.county_code], 'o')
ax.set(
ylabel='Posterior std. deviation',
xlabel='County log-count',
title='Having more observations generally\nlowers estimation uncertainty'
); | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
R์์ lme4์ ๋น๊ตํ๊ธฐ | %%shell
exit # Trick to make this block not execute.
radon = read.csv('srrs2.dat', header = TRUE)
radon = radon[radon$state=='MN',]
radon$radon = ifelse(radon$activity==0., 0.1, radon$activity)
radon$log_radon = log(radon$radon)
# install.packages('lme4')
library(lme4)
fit <- lmer(log_radon ~ 1 + floor + (1 | county), data=radon)
fit
# Linear mixed model fit by REML ['lmerMod']
# Formula: log_radon ~ 1 + floor + (1 | county)
# Data: radon
# REML criterion at convergence: 2171.305
# Random effects:
# Groups Name Std.Dev.
# county (Intercept) 0.3282
# Residual 0.7556
# Number of obs: 919, groups: county, 85
# Fixed Effects:
# (Intercept) floor
# 1.462 -0.693 | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
๋ค์ ํ์ ๊ฒฐ๊ณผ๊ฐ ์์ฝ๋์ด ์์ต๋๋ค. | print(pd.DataFrame(data=dict(intercept=[1.462, tf.reduce_mean(intercept_.mean()).numpy()],
floor=[-0.693, tf.reduce_mean(floor_weight_.mean()).numpy()],
scale=[0.3282, tf.reduce_mean(scale_prior_.sample(10000)).numpy()]),
index=['lme4', 'vi'])) | site/ko/probability/examples/Linear_Mixed_Effects_Model_Variational_Inference.ipynb | tensorflow/docs-l10n | apache-2.0 |
Remove the smaller objects to retrieve the large galaxy using a boolean array, and then use skimage.exposure.histogram and plt.plot to show the light distribution from the galaxy.
<div style="height: 400px;"></div> | %reload_ext load_style
%load_style ../themes/tutorial.css | notebooks/3_morphological_operations.ipynb | jni/numpy-skimage-tutorial | bsd-3-clause |
If we call the keys of the pj.alignments dictionary, we can see the names of the alignments it contains: | pj.alignments.keys() | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
3.7.1 Configuring an alignment trimming process
Like the sequence alignment phase, alignment trimming has its own configuration class, the TrimalConf class. An object of this class will generate a command-line and the required input files for the program TrimAl, but will not execute the process (this is shown below). Once the process has been successfully executed, this TrimalConf object is also stored in pj.used_methods and it can be invoked as a report.
3.7.1.1 Example1, the default gappyput algorithm
With TrimalConf, instead of specifying loci names, we provide alignment names, as they appear in the keys of pj.alignments | gappyout = TrimalConf(pj, # The Project
method_name='gappyout', # Any unique string ('gappyout' is default)
program_name='trimal', # No alternatives in this ReproPhylo version
cmd='default', # the default is trimal. Change it here
# or in pj.defaults['trimal']
alns=['MT-CO1@mafftLinsi'], # 'all' by default
trimal_commands={'gappyout': True} # By default, the gappyout algorithm is used.
) | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
3.7.1.2 List comprehension to subset alignments
In this example, it is easy enough to copy and paste alignment names into a list and pass it to TrimalConf. But this is more difficult if we want to fish out a subset of alignments from a very large list of alignments. In such cases, Python's list comprehension is very useful. Below I show two uses of list comprehension, but the more you feel comfortable with this approach, the better.
Getting locus names of rRNA loci
If you read the code line that follows very carefully, you will see it quite literally says "take the name of each Locus found in pj.loci if its feature type is rRNA, and put it in a list": | rRNA_locus_names = [locus.name for locus in pj.loci if locus.feature_type == 'rRNA']
print rRNA_locus_names | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
what we get is a list of names of our rRNA loci.
Getting alignment names that have locus names of rRNA loci
The following line says: "take the key of each alignment from the pj.alignments dictionary if the first word before the '@' symbol is in the list of rRNA locus names, and put this key in a list": | rRNA_alignment_names = [key for key in pj.alignments.keys() if key.split('@')[0] in rRNA_locus_names]
print rRNA_alignment_names | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
We get a list of keys, of the rRNA loci alignments we produced on the previous section, and which are stored in the pj.alignments dictionary. We can now pass this list to a new TrimalConf instance that will only process rRNA locus alignments: | gt50 = TrimalConf(pj,
method_name='gt50',
alns = rRNA_alignment_names,
trimal_commands={'gt': 0.5} # This will keep positions with up to
# 50% gaps.
) | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
3.7.2 Executing the alignment trimming process
As for the alignment phase, this is done with a Project method, which accepts a list of TrimalConf objects. | pj.trim([gappyout, gt50]) | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
Once used, these objects are also placed in the pj.used_methods dictionary, and they can be printed out for observation: | print pj.used_methods['gappyout'] | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
3.7.3 Accessing trimmed sequence alignments
3.7.3.1 The pj.trimmed_alignments dictionary
The trimmed alignments themselves are stored in the pj.trimmed_alignments dictionary, using keys that follow this pattern: locus_name@alignment_method_name@trimming_method_name where alignment_method_name is the name you have provided to your AlnConf object and trimming_method_name is the one you provided to your TrimalConf object. | pj.trimmed_alignments | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
3.7.3.2 Accessing a MultipleSeqAlignment object
A trimmed alignment can be easily accessed and manipulated with any of Biopython's AlignIO tricks using the fta Project method: | print pj.fta('18s@muscleDefault@gt50')[:4,410:420].format('phylip-relaxed') | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
3.7.3.3 Writing trimmed sequence alignment files
Trimmed alignment text files can be dumped in any AlignIO format for usage in an external command line or GUI program. When writing to files, you can control the header of the sequence by, for example, adding the organism name of the gene name, or by replacing the feature ID with the record ID: | # record_id and source_organism are feature qualifiers in the SeqRecord object
# See section 3.4
files = pj.write_trimmed_alns(id=['record_id','source_organism'],
format='fasta')
files | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
The files will always be written to the current working directory (where this notebook file is), and can immediately be moved programmatically to avoid clutter: | # make a new directory for your trimmed alignment files:
if not os.path.exists('trimmed_alignment_files'):
os.mkdir('trimmed_alignment_files')
# move the files there
for f in files:
os.rename(f, "./trimmed_alignment_files/%s"%f) | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
3.7.3.4 Viewing trimmed alignments
Trimmed alignments can be viewed in the same way as alignments, but using this command: | pj.show_aln('MT-CO1@mafftLinsi@gappyout',id=['source_organism'])
pickle_pj(pj, 'outputs/my_project.pkpj') | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
3.7.4 Quick reference | # Make a TrimalConf object
trimconf = TrimalConf(pj, **kwargs)
# Execute alignment process
pj.trim([trimconf])
# Show AlnConf description
print pj.used_methods['method_name']
# Fetch a MultipleSeqAlignment object
trim_aln_obj = pj.fta('locus_name@aln_method_name@trim_method_name')
# Write alignment text files
pj.write_trimmed_alns(id=['some_feature_qualifier'], format='fasta')
# the default feature qualifier is 'feature_id'
# 'fasta' is the default format
# View alignment in browser
pj.show_aln('locus_name@aln_method_name@trim_method_name',id=['some_feature_qualifier']) | notebooks/Tutorials/Basic/3.7 Alignment trimming.ipynb | szitenberg/ReproPhyloVagrant | mit |
$g(x)\rightarrow 1$ for $x\rightarrow\infty$
$g(x)\rightarrow 0$ for $x\rightarrow -\infty$
$g(0) = 1/2$
Finally, to go from the regression to the classification, we can simply apply the following condition:
$$
y=\left{
\begin{array}{@{}ll@{}}
1, & \text{if}\ h_w(x)>=1/2 \
0, & \text{otherwise}
\end{array}\right.
$$
Let's clarify the notation. We have $m$ training samples and $n$ features, our training examples can be represented by a $m$-by-$n$ matrix $\underline{\underline{X}}=(x_{ij})$ ($m$-by-$n+1$, if we include the intercept term) that contains the training examples, $x^{(i)}$, in its rows.
The target values of the training set can be represented as a $m$-dimensional vector $\underline{y}$ and the parameters
of our model as
a $n$-dimensional vector $\underline{w}$ ($n+1$ if we take into account the intercept).
Now, for a given training example $x^{(i)}$, the function that we want to learn (or fit) can be written:
$$
h_\underline{w}(x^{(i)}) = \frac{1}{1+e^{-\sum_{j=0}^n w_j x_{ij}}}
$$ | # Simple example:
# we have 20 students that took an exam and we want to know if we can use
# the number of hours they studied to predict if they pass or fail the
# exam
# m = 20 training samples
# n = 1 feature (number of hours)
X = np.array([0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 1.75, 2.00, 2.25, 2.50,
2.75, 3.00, 3.25, 3.50, 4.00, 4.25, 4.50, 4.75, 5.00, 5.50])
# 1 = pass, 0 = fail
y = np.array([0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1])
print(X.shape)
print(y.shape)
p = plt.plot(X,y,'o')
tx = plt.xlabel('x [h]')
ty = plt.ylabel('y ')
| 03_Introduction_To_Supervised_Machine_Learning.ipynb | SchwaZhao/networkproject1 | mit |
Likelihood of the model
How to find the parameters, also called weights, $\underline{w}$ that best fit our training data?
We want to find the weights $\underline{w}$ that maximize the likelihood of observing the target $\underline{y}$ given the observed features $\underline{\underline{X}}$.
We need a probabilistic model that gives us the probability of observing the value $y^{(i)}$ given the features $x^{(i)}$.
The function $h_\underline{w}(x^{(i)})$ can be used precisely for that:
$$
P(y^{(i)}=1|x^{(i)};\underline{w}) = h_\underline{w}(x^{(i)})
$$
$$
P(y^{(i)}=0|x^{(i)};\underline{w}) = 1 - h_\underline{w}(x^{(i)})
$$
we can write it more compactly as:
$$
P(y^{(i)}|x^{(i)};\underline{w}) = (h_\underline{w}(x^{(i)}))^{y^{(i)}} ( 1 - h_\underline{w}(x^{(i)}))^{1-y^{(i)}}
$$
where $y^{(i)}\in{0,1}$
We see that $y^{(i)}$ is a random variable following a Bernouilli distribution with expectation $h_\underline{w}(x^{(i)})$.
The Likelihood function of a statistical model is defined as:
$$
\mathcal{L}(\underline{w}) = \mathcal{L}(\underline{w};\underline{\underline{X}},\underline{y}) = P(\underline{y}|\underline{\underline{X}};\underline{w}).
$$
The likelihood takes into account all the $m$ training samples of our training dataset and estimates the likelihood
of observing $\underline{y}$ given $\underline{\underline{X}}$ and $\underline{w}$. Assuming that the $m$ training examples were generated independently, we can write:
$$
\mathcal{L}(\underline{w}) = P(\underline{y}|\underline{\underline{X}};\underline{w}) = \prod_{i=1}^m P(y^{(i)}|x^{(i)};\underline{w}) = \prod_{i=1}^m (h_\underline{w}(x^{(i)}))^{y^{(i)}} ( 1 - h_\underline{w}(x^{(i)}))^{1-y^{(i)}}.
$$
This is the function that we want to maximize. It is usually much simpler to maximize the logarithm of this function, which is equivalent.
$$
l(\underline{w}) = \log\mathcal{L}(\underline{w}) = \sum_{i=1}^{m} \left(y^{(i)} \log h_\underline{w}(x^{(i)}) + (1- y^{(i)})\log\left(1- h_\underline{w}(x^{(i)})\right) \right)
$$
Loss function and linear models
An other way of formulating this problem is by defining a Loss function $L\left(y^{(i)}, f(x^{(i)})\right)$ such that:
$$
\sum_{i=1}^{m} L\left(y^{(i)}, f(x^{(i)})\right) = - l(\underline{w}).
$$
And now the problem consists of minimizing $\sum_{i=1}^{m} L\left(y^{(i)}, f(x^{(i)})\right)$ over all the possible values of $\underline{w}$.
Using the definition of $h_\underline{w}(x^{(i)})$ you can show that $L$ can be written as:
$$
L\left(y^{(i)}=1, f(x^{(i)})\right) = \log_2\left(1+e^{-f(x^{(i)})}\right)
$$
and
$$
L\left(y^{(i)}=0, f(x^{(i)})\right) = \log_2\left(1+e^{-f(x^{(i)})}\right) - \log_2\left(e^{-f(x^{(i)})}\right)
$$
where $f(x^{(i)}) = \sum_{j=0}^n w_j x_{ij}$ is called the decision function. | import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
fx = np.linspace(-5,5)
Ly1 = np.log2(1+np.exp(-fx))
Ly0 = np.log2(1+np.exp(-fx)) - np.log2(np.exp(-fx))
p = plt.plot(fx,Ly1,label='L(1,f(x))')
p = plt.plot(fx,Ly0,label='L(0,f(x))')
plt.xlabel('f(x)')
plt.ylabel('L')
plt.legend()
# coming back to our simple example
def Loss(x_i,y_i, w0, w1):
fx = w0 + x_i*w1
if y_i == 1:
return np.log2(1+np.exp(-fx))
if y_i == 0:
return np.log2(1+np.exp(-fx)) - np.log2(np.exp(-fx))
else:
raise Exception('y_i must be 0 or 1')
def sumLoss(x,y, w0, w1):
sumloss = 0
for x_i, y_i in zip(x,y):
sumloss += Loss(x_i,y_i, w0, w1)
return sumloss
# lets compute the loss function for several values
w0s = np.linspace(-10,20,100)
w1s = np.linspace(-10,20,100)
sumLoss_vals = np.zeros((w0s.size, w1s.size))
for k, w0 in enumerate(w0s):
for l, w1 in enumerate(w1s):
sumLoss_vals[k,l] = sumLoss(X,y,w0,w1)
# let's find the values of w0 and w1 that minimize the loss
ind0, ind1 = np.where(sumLoss_vals == sumLoss_vals.min())
print((ind0,ind1))
print((w0s[ind0], w1s[ind1]))
# plot the loss function
p = plt.pcolor(w0s, w1s, sumLoss_vals)
c = plt.colorbar()
p2 = plt.plot(w1s[ind1], w0s[ind0], 'ro')
tx = plt.xlabel('w1')
ty = plt.ylabel('w0')
| 03_Introduction_To_Supervised_Machine_Learning.ipynb | SchwaZhao/networkproject1 | mit |
Here we found the minimum of the loss function simply by computing it over a large range of values. In practice, this approach is not possible when the dimensionality of the loss function (number of weights) is very large. To find the minimum of the loss function, the gradient descent algorithm (or stochastic gradient descent) is often used. | # plot the solution
x = np.linspace(0,6,100)
def h_w(x, w0=w0s[ind0], w1=w1s[ind1]):
return 1/(1+np.exp(-(w0+x*w1)))
p1 = plt.plot(x, h_w(x))
p2 = plt.plot(X,y,'ro')
tx = plt.xlabel('x [h]')
ty = plt.ylabel('y ')
# probability of passing the exam if you worked 5 hours:
print(h_w(5)) | 03_Introduction_To_Supervised_Machine_Learning.ipynb | SchwaZhao/networkproject1 | mit |
We will use the package sci-kit learn (http://scikit-learn.org/) that provide access to many tools for machine learning, data mining and data analysis. | # The same thing using the sklearn module
from sklearn.linear_model import LogisticRegression
model = LogisticRegression(C=1e10)
# to train our model we use the "fit" method
# we have to reshape X because we have only one feature here
model.fit(X.reshape(-1,1),y)
# to see the weights
print(model.coef_)
print(model.intercept_)
# use the trained model to predict new values
print(model.predict_proba(5))
print(model.predict(5)) | 03_Introduction_To_Supervised_Machine_Learning.ipynb | SchwaZhao/networkproject1 | mit |
Note that although the loss function is not linear, the decision function is a linear function of the weights and features. This is why the Logistic regression is called a linear model.
Other linear models are defined by different loss functions. For example:
- Perceptron: $L \left(y^{(i)}, f(x^{(i)})\right) = \max(0, -y^{(i)}\cdot f(x^{(i)}))$
- Hinge-loss (soft-margin Support vector machine (SVM) classification): $L \left(y^{(i)}, f(x^{(i)})\right) = \max(0, 1-y^{(i)}\cdot f(x^{(i)}))$
See http://scikit-learn.org/stable/modules/sgd.html for more examples. | import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
fx = np.linspace(-5,5, 200)
Logit = np.log2(1+np.exp(-fx))
Percep = np.maximum(0,- fx)
Hinge = np.maximum(0, 1- fx)
ZeroOne = np.ones(fx.size)
ZeroOne[fx>=0] = 0
p = plt.plot(fx,Logit,label='Logistic Regression')
p = plt.plot(fx,Percep,label='Perceptron')
p = plt.plot(fx,Hinge,label='Hinge-loss')
p = plt.plot(fx,ZeroOne,label='Zero-One loss')
plt.xlabel('f(x)')
plt.ylabel('L')
plt.legend()
ylims = plt.ylim((0,7)) | 03_Introduction_To_Supervised_Machine_Learning.ipynb | SchwaZhao/networkproject1 | mit |
Evaluating the performance of a binary classifier
The confusion matrix allows to visualize the performance of a classifier:
| | predicted positive | predicted negative |
| --- |:---:|:---:|
| real positive | TP | FN |
| real negative | FP | TN |
For each prediction $y_p$, we put it in one of the four categories based on the true value of $y$:
- TP = True Positive
- FP = False Positive
- TN = True Negative
- FN = False Negative
We can then evalute several measures, for example:
Accuracy:
$\text{Accuracy}=\frac{TP+TN}{TP+TN+FP+FN}$
Accuracy is the proportion of true results (both true positives and true negatives) among the total number of cases examined. However, accuracy is not necessarily a good measure of the predictive power of a model. See the example below:
Accuracy paradox:
A classifier with these results:
| |Predicted Negative | Predicted Positive|
| --- |---|---|
|Negative Cases |9,700 | 150|
|Positive Cases |50 |100|
has an accuracy = 98%.
Now consider the results of a classifier that systematically predict a negative result independently of the input:
| |Predicted Negative| Predicted Positive|
|---|---|---|
|Negative Cases| 9,850 | 0|
|Positive Cases| 150 |0 |
The accuracy of this classifier is 98.5% while it is clearly useless. Here the less accurate model is more useful than the more accurate one. This is why accuracy should not be used (alone) to evaluate the performance of a classifier.
Precision and Recall are usually prefered:
Precision:
$\text{Precision}=\frac{TP}{TP+FP}$
Precision measures the fraction of correct positive or the lack of false positive.
It answers the question: "Given a positive prediction from the classifier, how likely is it to be correct ?"
Recall:
$\text{Recall}=\frac{TP}{TP+FN}$
Recall measures the proportion of positives that are correctly identified as such or the lack of false negative.
It answers the question: "Given a positive example, will the classifier detect it ?"
$F_1$ score:
In order to account for the precision and recall of a classifier, $F_1$ score is the harmonic mean of both measures:
$F_1 = 2 \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{ \mathrm{precision} + \mathrm{recall}} = 2 \frac{TP}{2TP +FP+FN}$
When evaluating the performance of a classifier it is important to test is on a different set of values than then set we used to train it. Indeed, we want to know how the classifier performs on new data not on the training data. For this purpose we separate the training set in two: a part that we use to train the model and a part that we use to test it. This method is called cross-validation. Usually, we split the training set in N parts (typically 3 or 10), train the model on N-1 parts and test it on the remaining part. We then repeat this procedure with all the combination of training and testing parts and average the performance metrics from each tests. Sci-kit learn allows to easily perform cross-validation: http://scikit-learn.org/stable/modules/cross_validation.html
Regularization and over-fitting
Overfitting happens when your model is too complicated to generalise for new data. When your model fits your data perfectly, it is unlikely to fit new data well.
<img src="https://upload.wikimedia.org/wikipedia/commons/1/19/Overfitting.svg" style="width: 250px;"/>
The model in green is over-fitted. It performs very well on the training set, but it does not generalize well to new data compared to the model in black.
To avoid over-fitting, it is important to have a large training set and to use cross-validation to evaluate the performance of a model. Additionally, regularization is used to make the model less "complex" and more general.
Regularization consists in adding a term $R(\underline{w})$, that penalizes too "complex" models, to the loss function, so that the training error that we want to minimize is:
$E(\underline{w}) = \sum_{i=1}^{m} L\left(y^{(i)}, f(x^{(i)})\right) + \lambda R(\underline{w})$,
where $\lambda$ is a parameter that controls the strength of the regularization.
Usual choices for $R(\underline{w})$ are:
- L2 norm of the weights: $R(\underline{w}) := \frac{1}{2} \sum_{i=1}^{n} w_j^2$, which forces small weights in the solution,
- L1 norm of the weights: $R(\underline{w}) := \sum_{i=1}^{n} |w_j|$, (also refered as Lasso) which leads to sparse solutions (with several zero weights).
The choice of the regularization and of the its strength are usually done by selecting the best choice during the cross-validation. | # for example
from sklearn.model_selection import cross_val_predict
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score, confusion_matrix
# logistic regression with L2 regularization, C controls the strength of the regularization
# C = 1/lambda
model = LogisticRegression(C=1, penalty='l2')
# cross validation using 10 folds
y_pred = cross_val_predict(model, X.reshape(-1,1), y=y, cv=10)
print(confusion_matrix(y,y_pred))
print('Accuracy = ' + str(accuracy_score(y, y_pred)))
print('Precision = ' + str(precision_score(y, y_pred)))
print('Recall = ' + str(precision_score(y, y_pred)))
print('F_1 = ' + str(f1_score(y, y_pred)))
# try to run it with different number of folds for the cross-validation
# and different values of the regularization strength
| 03_Introduction_To_Supervised_Machine_Learning.ipynb | SchwaZhao/networkproject1 | mit |
Vectorizer | from helpers.tokenizer import TextWrangler
from sklearn.feature_extraction.text import CountVectorizer, TfidfVectorizer
bow_stem = CountVectorizer(strip_accents="ascii", tokenizer=TextWrangler(kind="stem"))
X_bow_stem = bow_stem.fit_transform(corpus.data)
tfidf_stem = TfidfVectorizer(strip_accents="ascii", tokenizer=TextWrangler(kind="stem"))
X_tfidf_stem = tfidf_stem.fit_transform(corpus.data) | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Models | from sklearn.decomposition import LatentDirichletAllocation, TruncatedSVD, NMF
n_topics = 5
lda = LatentDirichletAllocation(n_components=n_topics,
learning_decay=0.5, learning_offset=1.,
random_state=23)
lsa = TruncatedSVD(n_components=n_topics, random_state=23)
nmf = NMF(n_components=n_topics, solver="mu", beta_loss="kullback-leibler", alpha=0.1, random_state=23)
lda_params = {"lda__learning_decay": [0.5, 0.7, 0.9],
"lda__learning_offset": [1., 5., 10.]} | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Pipelines | from sklearn.pipeline import Pipeline
lda_pipe = Pipeline([
("bow", bow_stem),
("lda", lda)
])
lsa_pipe = Pipeline([
("tfidf", tfidf_stem),
("lsa", lsa)
])
nmf_pipe = Pipeline([
("tfidf", tfidf_stem),
("nmf", nmf)
]) | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Gridsearch | from sklearn.model_selection import GridSearchCV
lda_model = GridSearchCV(lda_pipe, param_grid=lda_params, cv=5, n_jobs=-1)
#lda_model.fit(corpus.data)
#lda_model.best_params_ | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Training | lda_pipe.fit(corpus.data)
nmf_pipe.fit(corpus.data)
lsa_pipe.fit(corpus.data) | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Evaluation | print("LDA")
print("Log Likelihood:", lda_pipe.score(corpus.data)) | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Visual Inspection | def df_topic_model(vectorizer, model, n_words=20):
keywords = np.array(vectorizer.get_feature_names())
topic_keywords = []
for topic_weights in model.components_:
top_keyword_locs = (-topic_weights).argsort()[:n_words]
topic_keywords.append(keywords.take(top_keyword_locs))
df_topic_keywords = pd.DataFrame(topic_keywords)
df_topic_keywords.columns = ['Word '+str(i) for i in range(df_topic_keywords.shape[1])]
df_topic_keywords.index = ['Topic '+str(i) for i in range(df_topic_keywords.shape[0])]
return df_topic_keywords
print("LDA")
df_topic_model(vectorizer=bow_stem, model=lda_pipe.named_steps.lda, n_words=15)
print("LSA")
df_topic_model(vectorizer=tfidf_stem, model=lsa_pipe.named_steps.lsa, n_words=15)
print("NMF")
df_topic_model(vectorizer=tfidf_stem, model=nmf_pipe.named_steps.nmf, n_words=15)
import pyLDAvis
from pyLDAvis.sklearn import prepare
pyLDAvis.enable_notebook()
prepare(lda_pipe.named_steps.lda, X_bow_stem, bow_stem, mds="tsne")
prepare(nmf_pipe.named_steps.nmf, X_tfidf_stem, tfidf_stem, mds="tsne") | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Conclusion:
Topic models derived from different approaches look dissimilar. Top word distribution of NMF appears most
meaningful, mostly because its topics doesn't share same words (due to NMF algorithm). LSA topic model is
better interpretable than its LDA counterpart. Nonetheless, topics from both are hard to distinguish and
doesn't make much sense. Therefore I'll go with the NMF topic model for the assginment to novel collections
step.
Jaccard Index | df_topic_word_lda = df_topic_model(vectorizer=bow_stem, model=lda_pipe.named_steps.lda, n_words=10)
df_topic_word_lsa = df_topic_model(vectorizer=tfidf_stem, model=lsa_pipe.named_steps.lsa, n_words=10)
df_topic_word_nmf = df_topic_model(vectorizer=tfidf_stem, model=nmf_pipe.named_steps.nmf, n_words=10)
def jaccard_index(list1, list2):
s1 = set(list1)
s2 = set(list2)
jaccard_index = len(s1.intersection(s2)) / len(s1.union(s2))
return jaccard_index
sims_lda_lsa, sims_lda_nmf, sims_lsa_nmf = {}, {}, {}
assert df_topic_word_lda.shape[0] == df_topic_word_lsa.shape[0] == df_topic_word_nmf.shape[0], "n_topics mismatch"
for ix, row in df_topic_word_lda.iterrows():
l1 = df_topic_word_lda.loc[ix, :].values.tolist()
l2 = df_topic_word_lsa.loc[ix, :].values.tolist()
l3 = df_topic_word_nmf.loc[ix, :].values.tolist()
sims_lda_lsa[ix] = jaccard_index(l1, l2)
sims_lda_nmf[ix] = jaccard_index(l1, l3)
sims_lsa_nmf[ix] = jaccard_index(l2, l3)
df_jaccard_sims = pd.DataFrame([sims_lda_lsa, sims_lda_nmf, sims_lsa_nmf])
df_jaccard_sims.index = ["LDA vs LSA", "LDA vs NMF", "LSA vs NMF"]
df_jaccard_sims["mean_sim"] = df_jaccard_sims.mean(axis=1)
df_jaccard_sims | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Conclusion:
Topics derived from different topic modeling approaches are fundamentally dissimilar.
Document-topic Assignment | nmf_topic_distr = nmf_pipe.transform(corpus.data)
collections_map = {0: "His Last Bow", 1: "The Adventures of Sherlock Holmes",
2: "The Case-Book of Sherlock_Holmes", 3: "The Memoirs of Sherlock Holmes",
4: "The Return of Sherlock Holmes"}
# Titles created from dominant words in topics
novel_collections_map = {0: "The Whispering Ways Sherlock Holmes Waits to Act on Waste",
1: "Vengeful Wednesdays: Unexpected Incidences on the Tapering Train by Sherlock Holmes",
2: "A Private Journey of Sherlock Holmes: Thirteen Unfolded Veins on the Move",
3: "Sherlock Holmes Tumbling into the hanging arms of Scylla",
4: "The Shooking Jaw of Sherlock Holmes in the Villa of the Baronet"}
print("Novel Sherlock Holmes Short Stories Collections:")
for _,title in novel_collections_map.items():
print("*", title)
topics = ["Topic" + str(i) for i in range(n_topics)]
docs = [" ".join(f_name.split("/")[-1].split(".")[0].split("_"))
for f_name in corpus.filenames]
df_document_topic = pd.DataFrame(np.round(nmf_topic_distr, 3), columns=topics, index=docs)
df_document_topic["assigned_topic"] = np.argmax(df_document_topic.values, axis=1)
df_document_topic["orig_collection"] = [collections_map[item] for item in corpus.target]
df_document_topic["novel_collection"] = [novel_collections_map.get(item, item)
for item in df_document_topic.assigned_topic.values]
df_novel_assignment = df_document_topic.sort_values("assigned_topic").loc[:, ["orig_collection",
"novel_collection"]]
df_novel_assignment
from yellowbrick.text import TSNEVisualizer
tsne = TSNEVisualizer()
tsne.fit(X_tfidf_stem, df_document_topic.novel_collection)
tsne.poof() | HolmesTopicModels/holmes_topic_models/notebook/2_Modeling.ipynb | donK23/pyData-Projects | apache-2.0 |
Open a GeoTIFF with GDAL
Let's look at the SERC Canopy Height Model (CHM) to start. We can open and read this in Python using the gdal.Open function: | # Note that you will need to update the filepath below according to your local machine
chm_filename = '/Users/olearyd/Git/data/NEON_D02_SERC_DP3_368000_4306000_CHM.tif'
chm_dataset = gdal.Open(chm_filename) | tutorials/Python/Lidar/intro-lidar/classify_raster_with_threshold-2018-py/classify_raster_with_threshold-2018-py.ipynb | NEONScience/NEON-Data-Skills | agpl-3.0 |
On your own, adjust the number of bins, and range of the y-axis to get a good idea of the distribution of the canopy height values. We can see that most of the values are zero. In SERC, many of the zero CHM values correspond to bodies of water as well as regions of land without trees. Let's look at a histogram and plot the data without zero values: | chm_nonzero_array = copy.copy(chm_array)
chm_nonzero_array[chm_array==0]=np.nan
chm_nonzero_nonan_array = chm_nonzero_array[~np.isnan(chm_nonzero_array)]
# Use weighting to plot relative frequency
plt.hist(chm_nonzero_nonan_array,bins=50);
# plt.hist(chm_nonzero_nonan_array.flatten(),50)
plt.title('Distribution of SERC Non-Zero Canopy Height')
plt.xlabel('Tree Height (m)'); plt.ylabel('Relative Frequency') | tutorials/Python/Lidar/intro-lidar/classify_raster_with_threshold-2018-py/classify_raster_with_threshold-2018-py.ipynb | NEONScience/NEON-Data-Skills | agpl-3.0 |
Note that it appears that the trees don't have a smooth or normal distribution, but instead appear blocked off in chunks. This is an artifact of the Canopy Height Model algorithm, which bins the trees into 5m increments (this is done to avoid another artifact of "pits" (Khosravipour et al., 2014).
From the histogram we can see that the majority of the trees are < 30m. We can re-plot the CHM array, this time adjusting the color bar limits to better visualize the variation in canopy height. We will plot the non-zero array so that CHM=0 appears white. | plot_band_array(chm_array,
chm_ext,
(0,35),
title='SERC Canopy Height',
cmap_title='Canopy Height, m',
colormap='BuGn') | tutorials/Python/Lidar/intro-lidar/classify_raster_with_threshold-2018-py/classify_raster_with_threshold-2018-py.ipynb | NEONScience/NEON-Data-Skills | agpl-3.0 |
Threshold Based Raster Classification
Next, we will create a classified raster object. To do this, we will use the numpy.where function to create a new raster based off boolean classifications. Let's classify the canopy height into five groups:
- Class 1: CHM = 0 m
- Class 2: 0m < CHM <= 10m
- Class 3: 10m < CHM <= 20m
- Class 4: 20m < CHM <= 30m
- Class 5: CHM > 30m
We can use np.where to find the indices where a boolean criteria is met. | chm_reclass = copy.copy(chm_array)
chm_reclass[np.where(chm_array==0)] = 1 # CHM = 0 : Class 1
chm_reclass[np.where((chm_array>0) & (chm_array<=10))] = 2 # 0m < CHM <= 10m - Class 2
chm_reclass[np.where((chm_array>10) & (chm_array<=20))] = 3 # 10m < CHM <= 20m - Class 3
chm_reclass[np.where((chm_array>20) & (chm_array<=30))] = 4 # 20m < CHM <= 30m - Class 4
chm_reclass[np.where(chm_array>30)] = 5 # CHM > 30m - Class 5 | tutorials/Python/Lidar/intro-lidar/classify_raster_with_threshold-2018-py/classify_raster_with_threshold-2018-py.ipynb | NEONScience/NEON-Data-Skills | agpl-3.0 |
Use an arbitary distribution
NOTE this requires Pymc3 3.1
pymc3.distributions.DensityDist | # pymc3.distributions.DensityDist?
import matplotlib.pyplot as plt
import matplotlib as mpl
from pymc3 import Model, Normal, Slice
from pymc3 import sample
from pymc3 import traceplot
from pymc3.distributions import Interpolated
from theano import as_op
import theano.tensor as tt
import numpy as np
from scipy import stats
%matplotlib inline
%load_ext version_information
%version_information pymc3
from sklearn.neighbors.kde import KernelDensity
import numpy as np
X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
kde = KernelDensity(kernel='gaussian', bandwidth=0.2).fit(X)
kde.score_samples(X)
plt.scatter(X[:,0], X[:,1]) | updating_info/Arb_dist.ipynb | balarsen/pymc_learning | bsd-3-clause |
The class-labels are One-Hot encoded, which means that each label is a vector with 10 elements, all of which are zero except for one element. The index of this one element is the class-number, that is, the digit shown in the associated image. We also need the class-numbers as integers for the test-set, so we calculate it now. | data.test.cls = np.argmax(data.test.labels, axis=1)
feed_dict_test = {x: data.test.images,
y_true: data.test.labels,
y_true_cls: data.test.cls} | learn_stem/machine_learning/tensorflow/02_Convolutional_Neural_Network.ipynb | wgong/open_source_learning | apache-2.0 |
Function for performing a number of optimization iterations so as to gradually improve the variables of the network layers. In each iteration, a new batch of data is selected from the training-set and then TensorFlow executes the optimizer using those training samples. The progress is printed every 100 iterations. | # Counter for total number of iterations performed so far.
total_iterations = 0
def optimize(num_iterations, ndisplay_interval=100):
# Ensure we update the global variable rather than a local copy.
global total_iterations
# Start-time used for printing time-usage below.
start_time = time.time()
for i in range(total_iterations,
total_iterations + num_iterations):
# Get a batch of training examples.
# x_batch now holds a batch of images and
# y_true_batch are the true labels for those images.
x_batch, y_true_batch = data.train.next_batch(train_batch_size)
# Put the batch into a dict with the proper names
# for placeholder variables in the TensorFlow graph.
feed_dict_train = {x: x_batch,
y_true: y_true_batch}
# Run the optimizer using this batch of training data.
# TensorFlow assigns the variables in feed_dict_train
# to the placeholder variables and then runs the optimizer.
session.run(optimizer, feed_dict=feed_dict_train)
# Print status every 100 iterations.
if i % ndisplay_interval == 0:
# Calculate the accuracy on the training-set.
acc = session.run(accuracy, feed_dict=feed_dict_train)
# Message for printing.
msg = "* Optimization Iteration: {0:>6}, Training Accuracy: {1:>6.1%}"
# Print it.
print(msg.format(i + 1, acc))
# Update the total number of iterations performed.
total_iterations += num_iterations
# Ending time.
end_time = time.time()
# Difference between start and end-times.
time_dif = end_time - start_time
# Print the time-usage.
print("* Time usage: " + str(timedelta(seconds=int(round(time_dif))))) | learn_stem/machine_learning/tensorflow/02_Convolutional_Neural_Network.ipynb | wgong/open_source_learning | apache-2.0 |
helper-function to plot sample digits | def plot_sample9():
# Use TensorFlow to get a list of boolean values
# whether each test-image has been correctly classified,
# and a list for the predicted class of each image.
prediction, cls_pred = session.run([correct_prediction, y_pred_cls],
feed_dict=feed_dict_test)
num_imgs = data.test.images.shape[0]
i_start = np.random.choice(num_imgs-10, 1)[0]
# Plot the first 9 images.
plot_images(images=data.test.images[i_start:i_start+9],
cls_true=data.test.cls[i_start:i_start+9],
cls_pred=cls_pred[i_start:i_start+9]) | learn_stem/machine_learning/tensorflow/02_Convolutional_Neural_Network.ipynb | wgong/open_source_learning | apache-2.0 |
Performance after 1000 optimization iterations
After 1000 optimization iterations, the model has greatly increased its accuracy on the test-set to more than 90%. | optimize(num_iterations=900) # We performed 100 iterations above. | learn_stem/machine_learning/tensorflow/02_Convolutional_Neural_Network.ipynb | wgong/open_source_learning | apache-2.0 |
test-run on 6/12/2017
Optimization Iteration: 101, Training Accuracy: 70.3%
Optimization Iteration: 201, Training Accuracy: 81.2%
Optimization Iteration: 301, Training Accuracy: 84.4%
Optimization Iteration: 401, Training Accuracy: 89.1%
Optimization Iteration: 501, Training Accuracy: 93.8%
Optimization Iteration: 601, Training Accuracy: 87.5%
Optimization Iteration: 701, Training Accuracy: 98.4%
Optimization Iteration: 801, Training Accuracy: 93.8%
Optimization Iteration: 901, Training Accuracy: 92.2%
Time usage: 0:01:28 | plot_sample9()
print_test_accuracy(show_example_errors=True) | learn_stem/machine_learning/tensorflow/02_Convolutional_Neural_Network.ipynb | wgong/open_source_learning | apache-2.0 |
Performance after 10,000 optimization iterations
After 10,000 optimization iterations, the model has a classification accuracy on the test-set of about 99%. | optimize(num_iterations=9000, ndisplay_interval=500) # We performed 1000 iterations above. | learn_stem/machine_learning/tensorflow/02_Convolutional_Neural_Network.ipynb | wgong/open_source_learning | apache-2.0 |
Optimization Iteration: 1, Training Accuracy: 92.2%
Optimization Iteration: 501, Training Accuracy: 98.4%
Optimization Iteration: 1001, Training Accuracy: 95.3%
Optimization Iteration: 1501, Training Accuracy: 100.0%
Optimization Iteration: 2001, Training Accuracy: 96.9%
Optimization Iteration: 2501, Training Accuracy: 100.0%
Optimization Iteration: 3001, Training Accuracy: 96.9%
Optimization Iteration: 3501, Training Accuracy: 98.4%
Optimization Iteration: 4001, Training Accuracy: 96.9%
Optimization Iteration: 4501, Training Accuracy: 100.0%
Optimization Iteration: 5001, Training Accuracy: 96.9%
Optimization Iteration: 5501, Training Accuracy: 100.0%
Optimization Iteration: 6001, Training Accuracy: 98.4%
Optimization Iteration: 6501, Training Accuracy: 96.9%
Optimization Iteration: 7001, Training Accuracy: 100.0%
Optimization Iteration: 7501, Training Accuracy: 98.4%
Optimization Iteration: 8001, Training Accuracy: 100.0%
Optimization Iteration: 8501, Training Accuracy: 100.0%
Time usage: 0:14:56 | plot_sample9()
print_test_accuracy(show_example_errors=True,
show_confusion_matrix=True) | learn_stem/machine_learning/tensorflow/02_Convolutional_Neural_Network.ipynb | wgong/open_source_learning | apache-2.0 |
From these images, it looks like the second convolutional layer might detect lines and patterns in the input images, which are less sensitive to local variations in the original input images.
These images are then flattened and input to the fully-connected layer, but that is not shown here.
Close TensorFlow Session
We are now done using TensorFlow, so we close the session to release its resources. | # This has been commented out in case you want to modify and experiment
# with the Notebook without having to restart it.
session.close() | learn_stem/machine_learning/tensorflow/02_Convolutional_Neural_Network.ipynb | wgong/open_source_learning | apache-2.0 |
Flip the plot by assigning the data variable to the y axis: | sns.ecdfplot(data=penguins, y="flipper_length_mm") | doc/docstrings/ecdfplot.ipynb | arokem/seaborn | bsd-3-clause |
If neither x nor y is assigned, the dataset is treated as wide-form, and a histogram is drawn for each numeric column: | sns.ecdfplot(data=penguins.filter(like="bill_", axis="columns")) | doc/docstrings/ecdfplot.ipynb | arokem/seaborn | bsd-3-clause |
You can also draw multiple histograms from a long-form dataset with hue mapping: | sns.ecdfplot(data=penguins, x="bill_length_mm", hue="species") | doc/docstrings/ecdfplot.ipynb | arokem/seaborn | bsd-3-clause |
The default distribution statistic is normalized to show a proportion, but you can show absolute counts instead: | sns.ecdfplot(data=penguins, x="bill_length_mm", hue="species", stat="count") | doc/docstrings/ecdfplot.ipynb | arokem/seaborn | bsd-3-clause |
It's also possible to plot the empirical complementary CDF (1 - CDF): | sns.ecdfplot(data=penguins, x="bill_length_mm", hue="species", complementary=True) | doc/docstrings/ecdfplot.ipynb | arokem/seaborn | bsd-3-clause |
There are several features of $g$ to note,
- For larger values of $z$ $g(z)$ approaches 1
- For more negative values of $z$ $g(z)$ approaches 0
- The value of $g(0) = 0.5$
- For $z \ge 0$, $g(z)\ge 0.5$
- For $z \lt 0$, $g(z)\lt 0.5$
0.5 will be the cutoff for decisions. That is, if $g(z) \ge 0.5$ then the "answer" is "the positive case", 1, if $g(z) \lt 0.5$ then the answer is "the negative case", 0.
Decision Boundary
The value 0.5 mentioned above creates a boundary for classification by our model (hypothesis) $h_a(x)$
$$
\begin{align} \text{if } h_a(x) \ge 0.5 & \text{ then we say } &y=1 \ \
\text{if } h_a(x) \lt 0.5 & \text{ then } &y=0
\end{align} $$
Looking at $g(z)$ more closely gives,
$$
\begin{align} h_a(x) = g(a'x) \ge 0.5 & \text{ when} & a'x \ge 0 \ \
h_a(x) = g(a'x) \lt 0.5 & \text{ when} & a'x \le 0
\end{align} $$
Therefore,
$$ \bbox[25px,border:2px solid green]{
\begin{align} a'x \ge 0.5 & \text{ implies } & y = 1 \ \
a'x \lt 0.5 & \text{ implies} & y = 0
\end{align} }$$
The Decision Boundary is the "line" defined by $a'x$ that separates the area where $y=0$ and $y=1$. The "line" defined by $a'x$ can be non-linear since the feature variables $x_i$ can be non-linear. The decision boundary can be any shape (curve) that fits the data. We use a Cost Function derived from the logistic regression sigmoid function to helps us find the parameters $a$ that define the optimal decision boundary $a'x$. After we have found the optimal values of $a$ the model function $h_a(x$), which uses the sigmoid function, will tell us which side of the decision boundary our "question" lies on based on the values of the features $x$ that we give it.
If you understand the paragraph above then you have a good idea of what logistic regression about!
Here's some examples of what that Decision Boundary might look like; | # Generate 2 clusters of data
S = np.eye(2)
x1, y1 = np.random.multivariate_normal([1,1], S, 40).T
x2, y2 = np.random.multivariate_normal([-1,-1], S, 40).T
fig, ax = plt.subplots()
ax.plot(x1,y1, "o", label='neg data' )
ax.plot(x2,y2, "P", label='pos data')
xb = np.linspace(-3,3,100)
a = [0.55,-1.3]
ax.plot(xb, a[0] + a[1]*xb , label='b(x) = %.2f + %.2f x' %(a[0], a[1]))
plt.title("Decision Boundary", fontsize=24)
plt.legend(); | ML-Logistic-Regression-theory.ipynb | dbkinghorn/blog-jupyter-notebooks | gpl-3.0 |
The plot above shows 2 sets of training-data. The positive case is represented by green '+' and the negative case by blue 'o'. The red line is the decision boundary $b(x) = 0.55 -1.3x$. Any test cases that are above the line are negative and any below are positive. The parameters for that red line would be what we could have determined from doing a Logistic Regression run on those 2 sets of training data.
The next plot shows a case where the decision boundry is more complicated. It's represented by $b(x_1,x_2) = x_1^2 +x_2^2 - 2.5$ | fig, ax = plt.subplots()
x3, y3 = np.random.multivariate_normal([0,0], [[.5,0],[0,.5]] , 400).T
t = np.linspace(0,2*np.pi,400)
ax.plot((3+x3)*np.sin(t), (3+y3)*np.cos(t), "o")
ax.plot(x3, y3, "P")
xb1 = np.linspace(-5.0, 5.0, 100)
xb2 = np.linspace(-5.0, 5.0, 100)
Xb1, Xb2 = np.meshgrid(xb1,xb2)
b = Xb1**2 + Xb2**2 - 2.5
ax.contour(Xb1,Xb2,b,[0], colors='r')
plt.title("Decision Boundary", fontsize=24)
ax.axis('equal') | ML-Logistic-Regression-theory.ipynb | dbkinghorn/blog-jupyter-notebooks | gpl-3.0 |
In this plot the positive outcomes are in a circular region in the center of the plot. The decision boundary the red circle.
## Cost Function for Logistic Regression
A cost function's main purpose is to penalize bad choices for the parameters to be optimized and reward good ones. It should be easy to minimize by having a single global minimum and not be overly sensitive to changes in its arguments. It is also nice if it is differentiable, (without difficulty) so you can find the gradient for the minimization problem. That is, it's best if it is "convex", "well behaved" and "smooth".
The cost function for logistic regression is written with logarithmic functions. An argument for using the log form of the cost function comes from the statistical derivation of the likelihood estimation for the probabilities. With the exponential form that's is a product of probabilities and the log-likelihood is a sum. [The statistical derivations are always interesting but usually complex. We don't really need to look at that to justify the cost function we will use.] The log function is also a monotonically increasing function so the negative of the log is decreasing. The minimization of a function and minimizing the negative log of that function will give the same values for the parameters. The log form will also be convex which means it will have a single global minimum whereas a simple "least-squares" cost function using the sigmoid function can have multiple minimum and abrupt changes. The log form is just better behaved!
To see some of this lets looks at a plot of the sigmoid function and the negative log of the sigmoid function. | z = np.linspace(-10,10,100)
fig, ax = plt.subplots()
ax.plot(z, g(z))
ax.set_title('Sigmoid Function 1/(1 + exp(-z))', fontsize=24)
ax.annotate('Convex', (-7.5,0.2), fontsize=18 )
ax.annotate('Concave', (3,0.8), fontsize=18 )
z = np.linspace(-10,10,100)
plt.plot(z, -np.log(g(z)))
plt.title("Log Sigmoid Function -log(1/(1 + exp(-z)))", fontsize=24)
plt.annotate('Convex', (-2.5,3), fontsize=18 ) | ML-Logistic-Regression-theory.ipynb | dbkinghorn/blog-jupyter-notebooks | gpl-3.0 |
Recall that in the training-set $y$ are labels with a values or 0 or 1. The cost function will be broken down into two cases for each data point $(i)$, one for $y=1$ and one for $y=0$. These two cases can then be combined into a single cost function $J$
$$ \bbox[25px,border:2px solid green]{
\begin{align} J^{(i)}{y=1}(a) & = -log(h_a(x^{(i)})) \ \
J^{(i)}{y=0}(a) & = -log(1 - h_a(x^{(i)})) \ \
J(a) & = -\frac{1}{m}\sum^{m}_{i=1} y^{(i)} log(h_a(x^{(i)})) + (1-y^{(i)})log(1 - h_a(x^{(i)}))
\end{align} }$$
You can see that the factors $y$ and $(1-y)$ effectively pick out the terms for the cases $y=1$ and $y=0$.
Vectorized form of $J(a)$
$J(a)$ can be written in vector form eliminating the summation sign as,
$$ \bbox[25px,border:2px solid green]{
\begin{align} h_a(X) &= g(Xa) \
J(a) &= -\frac{1}{m} \left( y' log(h_a(X) + (1-y)'log(1 - h_a(X) \right)
\end{align} }$$
To visualize how the cost functions works look at the following plots, | x = np.linspace(-10,10,50)
plt.plot(g(x), -np.log(g(x)))
plt.title("h(x) vs J(a)=-log(h(x)) for y = 1", fontsize=24)
plt.xlabel('h(x)')
plt.ylabel('J(a)') | ML-Logistic-Regression-theory.ipynb | dbkinghorn/blog-jupyter-notebooks | gpl-3.0 |
You can see from this plot that when $y=1$ the cost $J(a)$ is large if $h(x)$ goes toward 0. That is, it favors $h(x)$ going to 1 which is what we want. | x = np.linspace(-10,10,50)
plt.plot(g(x), -np.log(1-g(x)))
plt.title("h(x) vs J(a)=-log(1-h(x)) for y = 0", fontsize=24)
plt.xlabel('h(x)')
plt.ylabel('J(a)') | ML-Logistic-Regression-theory.ipynb | dbkinghorn/blog-jupyter-notebooks | gpl-3.0 |
Data Generation
A set of periodic microstructures and their volume averaged elastic stress values $\bar{\sigma}_{xx}$ can be generated by importing the make_elastic_stress_random function from pymks.datasets. This function has several arguments. n_samples is the number of samples that will be generated, size specifies the dimensions of the microstructures, grain_size controls the effective microstructure feature size, elastic_modulus and poissons_ratio are used to indicate the material property for each of the
phases, macro_strain is the value of the applied uniaxixial strain, and the seed can be used to change the the random number generator seed.
Let's go ahead and create 6 different types of microstructures each with 200 samples with dimensions 21 x 21. Each of the 6 samples will have a different microstructure feature size. The function will return and the microstructures and their associated volume averaged stress values. | from pymks.datasets import make_elastic_stress_random
sample_size = 200
grain_size = [(15, 2), (2, 15), (7, 7), (8, 3), (3, 9), (2, 2)]
n_samples = [sample_size] * 6
elastic_modulus = (410, 200)
poissons_ratio = (0.28, 0.3)
macro_strain = 0.001
size = (21, 21)
X, y = make_elastic_stress_random(n_samples=n_samples, size=size, grain_size=grain_size,
elastic_modulus=elastic_modulus, poissons_ratio=poissons_ratio,
macro_strain=macro_strain, seed=0)
| notebooks/stress_homogenization_2D.ipynb | XinyiGong/pymks | mit |
These default parameters may not be the best model for a given problem, we will now show one method that can be used to optimize them.
Optimizing the Number of Components and Polynomial Order
To start with, we can look at how the variance changes as a function of the number of components.
In general for SVD as well as PCA, the amount of variance captured in each component decreases
as the component number increases.
This means that as the number of components used in the dimensionality reduction increases, the percentage of the variance will asymptotically approach 100%. Let's see if this is true for our dataset.
In order to do this we will change the number of components to 40 and then
fit the data we have using the fit function. This function performs the dimensionality reduction and
also fits the regression model. Because our microstructures are periodic, we need to
use the periodic_axes argument when we fit the data. | model.n_components = 40
model.fit(X, y, periodic_axes=[0, 1])
| notebooks/stress_homogenization_2D.ipynb | XinyiGong/pymks | mit |
Roughly 90 percent of the variance is captured with the first 5 components. This means our model may only need a few components to predict the average stress.
Next we need to optimize the number of components and the polynomial order. To do this we are going to split the data into testing and training sets. This can be done using the train_test_spilt function from sklearn. | from sklearn.cross_validation import train_test_split
flat_shape = (X.shape[0],) + (np.prod(X.shape[1:]),)
X_train, X_test, y_train, y_test = train_test_split(X.reshape(flat_shape), y,
test_size=0.2, random_state=3)
print(X_train.shape)
print(X_test.shape)
| notebooks/stress_homogenization_2D.ipynb | XinyiGong/pymks | mit |
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