Outline¶
- Task of Anomaly Detection
- Gaussian Probability Distrobution
- Anamoly Detection Algorithms
Task of Anamoly Detection¶
Features of Google data center model
$\begin{cases} x_1 \text{ : Memory usage}\\ x_2 \text{ : }\frac{\# Disk access} { sec}\\ x_3 \text{ : CPU load}\\ x_4 \text{ : Network traffic} \end{cases}\xrightarrow{P(X)} Y $
Complimentary Features +¶
- Average Voltage
- Avergae Current
- Digital microphones
- Wireless modules
- Illumination conditions
- Dimentionality inspection
- Phsical condition
- Power profile
In [1]:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
def plot_3d(x, y, z, xlabel='X', ylabel='Y', zlabel='Z', title='3D Scatter Plot'):
"""
Plots a 3D scatter plot using matplotlib.
Args:
x (list or numpy array): x-axis data
y (list or numpy array): y-axis data
z (list or numpy array): z-axis data
xlabel (str): label for x-axis (default is 'X')
ylabel (str): label for y-axis (default is 'Y')
zlabel (str): label for z-axis (default is 'Z')
title (str): title of plot (default is '3D Scatter Plot')
"""
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(x, y, z)
ax.set_xlabel(xlabel)
ax.set_ylabel(ylabel)
ax.set_zlabel(zlabel)
ax.set_title(title)
plt.show()
np.random.seed(23)
In [2]:
N = 100
mu, sigma = 2, .1
ax = np.zeros(N)
samples = np.random.normal(mu, sigma, N).round(3)
plt.figure(figsize=(8,6))
plt.title('Gaussian Feature e.g. Processor Voltage')
plt.xlabel('Voltage')
plt.ylim(-1,1)
plt.scatter(samples,ax,marker='x',color='r')
plt.show()
In [3]:
plt.figure(figsize=(8, 6))
plt.title('Gaussian Feature Outlayer plot e.g. Processor Voltage')
plt.xlabel('Voltage')
plt.scatter(samples, ax, marker='x', color='r')
plt.boxplot(samples, vert=False)
plt.show()
In [4]:
def gaussian(mu,sigma,samples):
return 1/(sigma * np.sqrt(2 * np.pi)) * np.exp(- (samples - mu)**2 / (2 * sigma**2))
plt.figure(figsize=(8, 6))
plt.title('Gaussian Feature e.g. Processor Voltage')
plt.xlabel('Voltage')
plt.ylabel('# Samples')
count, bins, ignored = plt.hist(samples, N//5, density=False)
plt.scatter(samples, ax, marker='x', color='r')
plt.plot(bins, gaussian(mu,sigma,bins),
linewidth=2, color='y')
plt.show()
In [11]:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
mu, sigma = 2, .1
# Generate data for the Gaussian distribution
x = np.linspace(-5, 5, 100)
y = gaussian(mu, sigma, x)
# Create a meshgrid for the x and y values
X, Y = np.meshgrid(x, y)
# Compute the corresponding z-values using the Gaussian PDF
Z = gaussian(X, mu, sigma)
# Plot the 3D surface
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X, Y, Z, cmap='viridis')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('PDF')
plt.show()
In [17]:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
mu, sigma = 2, .1
# Generate data for the Gaussian distribution
x = np.linspace(-5, 5, 100)
y = gaussian(mu, sigma, x)
# Create a meshgrid for the x and y values
X, Y = np.meshgrid(x, y)
# Compute the corresponding z-values using the Gaussian PDF
Z = gaussian(X, mu, sigma)
# Plot the 3D surface
fig = plt.figure()
ax = fig.add_subplot(131, projection='3d')
ax.plot_surface(X, Y, Z, cmap='viridis')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('PDF')
ax = fig.add_subplot(132, projection='3d')
ax.plot_surface(X, Y, Z, cmap='viridis')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('PDF')
ax = fig.add_subplot(133, projection='3d')
ax.plot_surface(X, Y, Z**2, cmap='viridis')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('PDF')
plt.show()
In [20]:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
mu, sigma = 2, .1
# Generate data for the Gaussian distribution
x = np.linspace(-5, 5, 100)
y = gaussian(mu, sigma, x)
# Create a meshgrid for the x and y values
X, Y = np.meshgrid(x, y)
# Compute the corresponding z-values using the Gaussian PDF
Z = gaussian(X, mu, sigma)
# Plot the 3D surface
fig = plt.figure()
ax = fig.add_subplot(131, projection='3d')
ax.plot_surface(X, Y, Z, cmap='viridis')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('PDF')
ax = fig.add_subplot(132, projection='3d')
ax.plot_surface(Y, X, Z, cmap='viridis')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('PDF')
ax = fig.add_subplot(133, projection='3d')
ax.plot_surface(Y, X, Z, cmap='viridis')
ax.plot_surface(X, Y, Z, cmap='viridis')
# ax.plot_surface(X, Y, Z**2, cmap='viridis')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('PDF')
plt.show()
In [13]:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# Define the mean of the 2D Gaussian distribution
mean = [2, 1]
# Define the standard deviations (sigmas) of the 2D Gaussian distribution
sigma_x = 1
sigma_y = 2
# Create a grid of points
x, y = np.meshgrid(np.linspace(-5, 5, 100), np.linspace(-5, 5, 100))
# Compute the probability density function (PDF) of the 2D Gaussian at each point
pdf = np.exp(-((x - mean[0])**2 / (2 * sigma_x**2) + (y - mean[1])**2 / (2 * sigma_y**2))) / (2 * np.pi * sigma_x * sigma_y)
# Create a 3D figure
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# Plot the 2D Gaussian distribution in 3D
ax.plot_surface(x, y, pdf, cmap='viridis')
# Set the labels for the axes and the title of the plot
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('PDF')
ax.set_title('2 feature ')
# Show the plot
plt.show()
