scikit-笔记06:非监督学习之特征转换

A very basic example is the rescaling of our data, which is a requirement for many machine learning algorithms as they are not scale-invariant – rescaling falls into the category of data pre-processing and can barely be called learning. There exist many different rescaling techniques, and in the following example, we will take a look at a particular method that is commonly called "standardization."

STANDARDIZATION:

---

we will recale the data so that each feature is centered at zero (mean = 0) with unit variance (standard deviation = 1).

For example, if we have a 1D dataset with the values [1, 2, 3, 4, 5], the standardized values are

1 -> -1.41 2 -> -0.71 3 -> 0.0 4 -> 0.71 5 -> 1.41

computed via the equation \(x_{standardized} = \frac{x - \mu_x}{\sigma_x}\) , where μ is the sample mean, and σ the standard deviation, respectively.

ary = np.array([1, 2, 3, 4, 5])
ary_standardized = (ary - ary.mean()) / ary.std()
ary_standardized, ary_standardized.mean(), ary_standardized.std()

(array([-1.41421356, -0.70710678,  0.        ,  0.70710678,  1.41421356]),
0.0,
0.99999999999999989)

Applying such a preprocessing has a very similar interface to the supervised learning algorithms we saw so far. To get some more practice with scikit-learn's "Transformer" interface, let's start by loading the iris dataset and rescale it:

from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split

iris = load_iris() #<- return a Bunch obj, essentially a dict
X_train, X_test, y_train, y_test = train_test_split(iris.data, iris.target, random_state=0)
print(X_train.shape)

The iris dataset is not "centered" that is it has non-zero mean and the standard deviation is different for each component:

print("mean : %s " % X_train.mean(axis=0))
print("standard deviation : %s " % X_train.std(axis=0))

To use a preprocessing method, we first import the estimator, here StandardScaler and instantiate it:

from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()

scaler.fit(X_train)

StandardScaler(copy=True, with_mean=True, with_std=True)

Now we can rescale our data by applying the transform (not predict) method:

X_train_scaled = scaler.transform(X_train) #<- get a ndarray

X_train_scaled has the same number of samples and features, but the mean was subtracted and all features were scaled to have unit standard deviation:

print(X_train_scaled.shape)

print("mean : %s " % X_train_scaled.mean(axis=0)) # <- axis = 0 means accumulating vertically
print("standard deviation : %s " % X_train_scaled.std(axis=0))

To summarize:

Via the fit method, the estimator is fitted to the data we provide. In this step, the estimator estimates the parameters from the data (here: mean and standard deviation).

Then, if we transform data, these parameters are used to transform a dataset. (Please note that the transform method does not update these parameters).

It's important to note that the same transformation is applied to the training and the test set. That has the consequence that usually the mean of the test data is not zero after scaling:

X_test_scaled = scaler.transform(X_test)
print("mean test data: %s" % X_test_scaled.mean(axis=0))

It is important for the training and test data to be transformed in exactly the same way, for the following processing steps to make sense of the data, as is illustrated in the figure below:

from figures import plot_relative_scaling
plot_relative_scaling()

• StandardScaler
• MinMaxScaler
• etc.

There are several common ways to scale the data. The most common one is the StandardScaler we just introduced, but rescaling the data to a fix minimum an maximum value with MinMaxScaler (usually between 0 and 1), or using more robust statistics like median and quantile, instead of mean and standard deviation (with RobustScaler), are also useful.

from figures import plot_scaling
plot_scaling()

As with the classification and regression algorithms, we call fit to learn the model from the data. As this is an unsupervised model, we only pass X, not y. This simply estimates mean and standard deviation.

X: training dataset; y: training datset labels; new_X: testing dataset or new data

For prediction model:

• create: estimator = [estimator](), eg: estimator = LinearRegression, LogisticRegression, etc
• fitting: supervised model : [estimator].fit(X,y) unsupervised model : [estimator].fit(X)
• predict: [estimator].predict(new_X)

For standardization model:

• create: scaler = StandardScaler()
• fitting: scaler.fit(X)
• transformation: scaler.transform(X)

An unsupervised transformation that is somewhat more interesting is Principal Component Analysis (PCA). It is a technique to reduce the dimensionality of the data, by creating a:

linear projection.

That is, we find new features to represent the data that are a linear combination of the old data (i.e. we rotate it).

Thus, we can think of PCA as a projection of our data onto a new feature space.

The way PCA finds these new directions is by looking for the directions of maximum variance. Usually only few components that explain most of the variance in the data are kept.

Here, the premise is to reduce the size (dimensionality) of a dataset while capturing most of its information. There are many reason why dimensionality reduction can be useful:

• It can reduce the computational cost when running learning algorithms,
• decrease the storage space,
• may help with the so-called "curse of dimensionality".

To illustrate how a rotation might look like, we first show it on two-dimensional data and keep both principal components. Here is an illustration:

from figures import plot_pca_illustration
plot_pca_illustration()

1. from sklearn.datasets import make_blobs
2. from sklearn.datasets import load_iris
3. from sklearn.model_selection import train_test_split
4. from sklearn.linear_model import LogisticRegression
5. from sklearn.linear_model import LinearRegression
6. from sklearn.neighbors import KNeighborsClassifier
7. from sklearn.neighbors import KNeighborsRegressor
8. from sklearn.preprocessing import StandardScaler
9. from sklearn.decomposition import PCA

Give different colors for data points who satisfy different conditions.

import matplotlib.pyplot as plt
import numpy as np
X = np.array([1,2,3,4,5,6])
y = np.array([4,5,6,7,8,9])
plt.scatter(X,y,c=y>5) # <- here pass an array of boolean to 'c' --- color parameter
plt.show()

import matplotlib.pyplot as plt
import numpy as np
X = np.array([1,2,3,4,5,6])
y = np.array([4,5,6,7,8,9])
plt.scatter(X,y,c=y%2) # <- here pass an array of '01' to 'c' --- color parameter
plt.show()