import matplotlib.pyplot as plt
import numpy as np
import torch
import torch.nn as nn
import pandas as pd
# Retina mode
%matplotlib inline
%config InlineBackend.figure_format = 'retina'PDF of some common continuous random variables and applications
ML
PDF of Normal Distribution
Let \(X\) be a random variable that follows a normal distribution with mean \(\mu\) and variance \(\sigma^2\). The probability density function (PDF) of \(X\) is given by:
\[ f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right). \]
Let \(X \sim \mathcal{N}(\mu, \sigma^2)\) denote that \(X\) is drawn from a normal distribution with mean \(\mu\) and variance \(\sigma^2\).
dist = torch.distributions.Normal(0, 1)dist.supportReal()dist.meantensor(0.)dist.variancetensor(1.)dist.modetensor(0.)## Plotting the PDF
def plot_pdf_normal(mu, sigma):
dist = torch.distributions.Normal(mu, sigma)
x = torch.linspace(-10, 10, 1000)
y = dist.log_prob(x).exp()
plt.plot(x, y, label=f"PDF N({mu}, {sigma})")plot_pdf_normal(0, 1)
plot_pdf_normal(0, 2)
plot_pdf_normal(1, 2)
plt.legend()
# Simulating data with normal distributed noise
x_true = torch.linspace(-5, 5, 1000)
y_true = 2 * x_true + 1
eps = torch.distributions.Normal(0, 1).sample(y_true.shape)
y_obs = y_true + eps
plt.scatter(x_true, y_obs,
label="Observed data",
marker='o', s=2,
alpha = 0.5, color='red')
plt.plot(x_true, y_true, label="True data")
plt.legend()
Heights and weights data
The dataset contains 25,000 rows and 3 columns. Each row represents a person and the columns represent the person’s index, height, and weight.
df = pd.read_html("http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_Data_Dinov_020108_HeightsWeights.html")store_df = df[0]
store_df.columns = store_df.iloc[0]
store_df = store_df.iloc[1:]
store_df = store_df.astype(float)
store_df = store_df.drop(columns=["Index"])
store_df = store_df.dropna()store_df.describe()| Height(Inches) | Weight(Pounds) | |
|---|---|---|
| count | 25000.000000 | 25000.000000 |
| mean | 67.993114 | 127.079421 |
| std | 1.901679 | 11.660898 |
| min | 60.278360 | 78.014760 |
| 25% | 66.704397 | 119.308675 |
| 50% | 67.995700 | 127.157750 |
| 75% | 69.272958 | 134.892850 |
| max | 75.152800 | 170.924000 |
store_df.head()| Height(Inches) | Weight(Pounds) | |
|---|---|---|
| 1 | 65.78331 | 112.9925 |
| 2 | 71.51521 | 136.4873 |
| 3 | 69.39874 | 153.0269 |
| 4 | 68.21660 | 142.3354 |
| 5 | 67.78781 | 144.2971 |
fig, ax = plt.subplots(nrows=2, sharex=True)
store_df["Height(Inches)"].plot(kind='density', ax=ax[0])
store_df["Height(Inches)"].plot(kind='hist', bins=30, ax=ax[1])
# Fit a normal distribution to the data
mu = store_df["Height(Inches)"].mean().item()
sigma = store_df["Height(Inches)"].std().item()
dist = torch.distributions.Normal(mu, sigma)
x = torch.linspace(50, 80, 1000)
y = dist.log_prob(x).exp()
plt.plot(x, y, label="Fitted PDF")
store_df["Height(Inches)"].plot(kind='hist', label="Histogram", density=True, bins=30)
plt.legend()
store_df["Weight(Pounds)"].plot(kind='density')
Grading
Note: I DO NOT FOLLOW or endorse using a normal distribution to model grades in a class. This is just an exercise to practice the PDF of a normal distribution and show how to use percentiles.
marks = torch.distributions.Normal(70, 8).sample((400,))_ = plt.hist(marks, bins=20, density=True)
mu_marks, sigma_marks = marks.mean().item(), marks.std().item()
dist = torch.distributions.Normal(mu_marks, sigma_marks)
x = torch.linspace(30, 110, 1000)
y = dist.log_prob(x).exp()
plt.plot(x, y, label="Fitted PDF", color='gray', lw=2)
# 99% percentile and above get A+
marks_99_per = dist.icdf(torch.tensor(0.99))
num_students_getting_A_plus = marks[marks>marks_99_per].shape[0]
plt.fill_between(x, y, where=x>marks_99_per, alpha=0.5, label=f"A+ ({num_students_getting_A_plus})")
# 90th percntile to 99th percentile get A
marks_90_per = dist.icdf(torch.tensor(0.90))
num_students_getting_A = marks[(marks>marks_90_per) & (marks<marks_99_per)].shape[0]
plt.fill_between(x, y, where=(x>marks_90_per) & (x<marks_99_per), alpha=0.5, label=f"A ({num_students_getting_A})")
# 75th percentile to 90th percentile get A-
marks_75_per = dist.icdf(torch.tensor(0.75))
num_students_getting_B = marks[(marks>marks_75_per) & (marks<marks_90_per)].shape[0]
plt.fill_between(x, y, where=(x>marks_75_per) & (x<marks_90_per), alpha=0.5, label=f"B ({num_students_getting_B})")
# 60th percentile to 75th percentile get B
marks_60_per = dist.icdf(torch.tensor(0.60))
num_students_getting_B = marks[(marks>marks_60_per) & (marks<marks_75_per)].shape[0]
plt.fill_between(x, y, where=(x>marks_60_per) & (x<marks_75_per), alpha=0.5, label=f"B- ({num_students_getting_B})")
# 45th percentile to 60th percentile get C
marks_45_per = dist.icdf(torch.tensor(0.45))
num_students_getting_B_minus = marks[(marks>marks_45_per) & (marks<marks_60_per)].shape[0]
plt.fill_between(x, y, where=(x>marks_45_per) & (x<marks_60_per), alpha=0.5, label=f"C ({num_students_getting_B_minus})")
#35th percentile to 45th percentile get C-
marks_35_per = dist.icdf(torch.tensor(0.35))
num_students_getting_C = marks[(marks>marks_35_per) & (marks<marks_45_per)].shape[0]
plt.fill_between(x, y, where=(x>marks_35_per) & (x<marks_45_per), alpha=0.5, label=f"C- ({num_students_getting_C})")
# 20th percentile to 35th percentile get D
marks_20_per = dist.icdf(torch.tensor(0.20))
num_students_getting_C_minus = marks[(marks>marks_20_per) & (marks<marks_35_per)].shape[0]
plt.fill_between(x, y, where=(x>marks_20_per) & (x<marks_35_per), alpha=0.5, label=f"D ({num_students_getting_C_minus})")
# 3rd percentile to 20th percentile get E
marks_3_per = dist.icdf(torch.tensor(0.03))
num_students_getting_D = marks[(marks>marks_3_per) & (marks<marks_20_per)].shape[0]
plt.fill_between(x, y, where=(x>marks_3_per) & (x<marks_20_per), alpha=0.5, label=f"E ({num_students_getting_D})")
# 3rd percentile and below get F
num_students_getting_F = marks[marks<marks_3_per].shape[0]
plt.fill_between(x, y, where=x<marks_3_per, alpha=0.5, label=f"F ({num_students_getting_F})")
plt.legend()
Laplace Distribution
Let \(X\) be a random variable that follows a Laplace distribution with mean \(\mu\) and scale \(\lambda\). The probability density function (PDF) of \(X\) is given by:
\[ f_X(x) = \frac{1}{2\lambda} \exp\left(-\frac{|x-\mu|}{\lambda}\right). \]
unit_normal = torch.distributions.Normal(0, 1)
unit_laplace = torch.distributions.Laplace(0, 1)
x = torch.linspace(-10, 10, 1000)
y_normal = unit_normal.log_prob(x).exp()
y_laplace = unit_laplace.log_prob(x).exp()
plt.plot(x, y_normal, label="Normal")
plt.plot(x, y_laplace, label="Laplace")
plt.legend()
Half Normal Distribution
Let \(Y\) follow the normal distribution with mean \(0\) and variance \(\sigma^2\). The half normal distribution is obtained by taking the absolute value of \(Y\). \(X = |Y|\) follows a half normal distribution with mean \(\sqrt{\frac{2}{\pi}}\sigma\) and variance \(\sigma^2(1-\frac{2}{\pi})\).
The probability density function (PDF) of \(X\) is given by:
$$ f_X(x) = (-).
$$
hn = torch.distributions.HalfNormal(1)
x = torch.linspace(-10, 10, 1000)
try:
y = hn.log_prob(x).exp()
plt.plot(x, y, label="HalfNormal")
except Exception as e:
print(e)Expected value argument (Tensor of shape (1000,)) to be within the support (GreaterThanEq(lower_bound=0.0)) of the distribution HalfNormal(), but found invalid values:
tensor([-10.0000, -9.9800, -9.9600, -9.9399, -9.9199, -9.8999, -9.8799,
-9.8599, -9.8398, -9.8198, -9.7998, -9.7798, -9.7598, -9.7397,
-9.7197, -9.6997, -9.6797, -9.6597, -9.6396, -9.6196, -9.5996,
-9.5796, -9.5596, -9.5395, -9.5195, -9.4995, -9.4795, -9.4595,
-9.4394, -9.4194, -9.3994, -9.3794, -9.3594, -9.3393, -9.3193,
-9.2993, -9.2793, -9.2593, -9.2392, -9.2192, -9.1992, -9.1792,
-9.1592, -9.1391, -9.1191, -9.0991, -9.0791, -9.0591, -9.0390,
-9.0190, -8.9990, -8.9790, -8.9590, -8.9389, -8.9189, -8.8989,
-8.8789, -8.8589, -8.8388, -8.8188, -8.7988, -8.7788, -8.7588,
-8.7387, -8.7187, -8.6987, -8.6787, -8.6587, -8.6386, -8.6186,
-8.5986, -8.5786, -8.5586, -8.5385, -8.5185, -8.4985, -8.4785,
-8.4585, -8.4384, -8.4184, -8.3984, -8.3784, -8.3584, -8.3383,
-8.3183, -8.2983, -8.2783, -8.2583, -8.2382, -8.2182, -8.1982,
-8.1782, -8.1582, -8.1381, -8.1181, -8.0981, -8.0781, -8.0581,
-8.0380, -8.0180, -7.9980, -7.9780, -7.9580, -7.9379, -7.9179,
-7.8979, -7.8779, -7.8579, -7.8378, -7.8178, -7.7978, -7.7778,
-7.7578, -7.7377, -7.7177, -7.6977, -7.6777, -7.6577, -7.6376,
-7.6176, -7.5976, -7.5776, -7.5576, -7.5375, -7.5175, -7.4975,
-7.4775, -7.4575, -7.4374, -7.4174, -7.3974, -7.3774, -7.3574,
-7.3373, -7.3173, -7.2973, -7.2773, -7.2573, -7.2372, -7.2172,
-7.1972, -7.1772, -7.1572, -7.1371, -7.1171, -7.0971, -7.0771,
-7.0571, -7.0370, -7.0170, -6.9970, -6.9770, -6.9570, -6.9369,
-6.9169, -6.8969, -6.8769, -6.8569, -6.8368, -6.8168, -6.7968,
-6.7768, -6.7568, -6.7367, -6.7167, -6.6967, -6.6767, -6.6567,
-6.6366, -6.6166, -6.5966, -6.5766, -6.5566, -6.5365, -6.5165,
-6.4965, -6.4765, -6.4565, -6.4364, -6.4164, -6.3964, -6.3764,
-6.3564, -6.3363, -6.3163, -6.2963, -6.2763, -6.2563, -6.2362,
-6.2162, -6.1962, -6.1762, -6.1562, -6.1361, -6.1161, -6.0961,
-6.0761, -6.0561, -6.0360, -6.0160, -5.9960, -5.9760, -5.9560,
-5.9359, -5.9159, -5.8959, -5.8759, -5.8559, -5.8358, -5.8158,
-5.7958, -5.7758, -5.7558, -5.7357, -5.7157, -5.6957, -5.6757,
-5.6557, -5.6356, -5.6156, -5.5956, -5.5756, -5.5556, -5.5355,
-5.5155, -5.4955, -5.4755, -5.4555, -5.4354, -5.4154, -5.3954,
-5.3754, -5.3554, -5.3353, -5.3153, -5.2953, -5.2753, -5.2553,
-5.2352, -5.2152, -5.1952, -5.1752, -5.1552, -5.1351, -5.1151,
-5.0951, -5.0751, -5.0551, -5.0350, -5.0150, -4.9950, -4.9750,
-4.9550, -4.9349, -4.9149, -4.8949, -4.8749, -4.8549, -4.8348,
-4.8148, -4.7948, -4.7748, -4.7548, -4.7347, -4.7147, -4.6947,
-4.6747, -4.6547, -4.6346, -4.6146, -4.5946, -4.5746, -4.5546,
-4.5345, -4.5145, -4.4945, -4.4745, -4.4545, -4.4344, -4.4144,
-4.3944, -4.3744, -4.3544, -4.3343, -4.3143, -4.2943, -4.2743,
-4.2543, -4.2342, -4.2142, -4.1942, -4.1742, -4.1542, -4.1341,
-4.1141, -4.0941, -4.0741, -4.0541, -4.0340, -4.0140, -3.9940,
-3.9740, -3.9540, -3.9339, -3.9139, -3.8939, -3.8739, -3.8539,
-3.8338, -3.8138, -3.7938, -3.7738, -3.7538, -3.7337, -3.7137,
-3.6937, -3.6737, -3.6537, -3.6336, -3.6136, -3.5936, -3.5736,
-3.5536, -3.5335, -3.5135, -3.4935, -3.4735, -3.4535, -3.4334,
-3.4134, -3.3934, -3.3734, -3.3534, -3.3333, -3.3133, -3.2933,
-3.2733, -3.2533, -3.2332, -3.2132, -3.1932, -3.1732, -3.1532,
-3.1331, -3.1131, -3.0931, -3.0731, -3.0531, -3.0330, -3.0130,
-2.9930, -2.9730, -2.9530, -2.9329, -2.9129, -2.8929, -2.8729,
-2.8529, -2.8328, -2.8128, -2.7928, -2.7728, -2.7528, -2.7327,
-2.7127, -2.6927, -2.6727, -2.6527, -2.6326, -2.6126, -2.5926,
-2.5726, -2.5526, -2.5325, -2.5125, -2.4925, -2.4725, -2.4525,
-2.4324, -2.4124, -2.3924, -2.3724, -2.3524, -2.3323, -2.3123,
-2.2923, -2.2723, -2.2523, -2.2322, -2.2122, -2.1922, -2.1722,
-2.1522, -2.1321, -2.1121, -2.0921, -2.0721, -2.0521, -2.0320,
-2.0120, -1.9920, -1.9720, -1.9520, -1.9319, -1.9119, -1.8919,
-1.8719, -1.8519, -1.8318, -1.8118, -1.7918, -1.7718, -1.7518,
-1.7317, -1.7117, -1.6917, -1.6717, -1.6517, -1.6316, -1.6116,
-1.5916, -1.5716, -1.5516, -1.5315, -1.5115, -1.4915, -1.4715,
-1.4515, -1.4314, -1.4114, -1.3914, -1.3714, -1.3514, -1.3313,
-1.3113, -1.2913, -1.2713, -1.2513, -1.2312, -1.2112, -1.1912,
-1.1712, -1.1512, -1.1311, -1.1111, -1.0911, -1.0711, -1.0511,
-1.0310, -1.0110, -0.9910, -0.9710, -0.9510, -0.9309, -0.9109,
-0.8909, -0.8709, -0.8509, -0.8308, -0.8108, -0.7908, -0.7708,
-0.7508, -0.7307, -0.7107, -0.6907, -0.6707, -0.6507, -0.6306,
-0.6106, -0.5906, -0.5706, -0.5506, -0.5305, -0.5105, -0.4905,
-0.4705, -0.4505, -0.4304, -0.4104, -0.3904, -0.3704, -0.3504,
-0.3303, -0.3103, -0.2903, -0.2703, -0.2503, -0.2302, -0.2102,
-0.1902, -0.1702, -0.1502, -0.1301, -0.1101, -0.0901, -0.0701,
-0.0501, -0.0300, -0.0100, 0.0100, 0.0300, 0.0500, 0.0701,
0.0901, 0.1101, 0.1301, 0.1502, 0.1702, 0.1902, 0.2102,
0.2302, 0.2503, 0.2703, 0.2903, 0.3103, 0.3303, 0.3504,
0.3704, 0.3904, 0.4104, 0.4304, 0.4505, 0.4705, 0.4905,
0.5105, 0.5305, 0.5506, 0.5706, 0.5906, 0.6106, 0.6306,
0.6507, 0.6707, 0.6907, 0.7107, 0.7307, 0.7508, 0.7708,
0.7908, 0.8108, 0.8308, 0.8509, 0.8709, 0.8909, 0.9109,
0.9309, 0.9510, 0.9710, 0.9910, 1.0110, 1.0310, 1.0511,
1.0711, 1.0911, 1.1111, 1.1311, 1.1512, 1.1712, 1.1912,
1.2112, 1.2312, 1.2513, 1.2713, 1.2913, 1.3113, 1.3313,
1.3514, 1.3714, 1.3914, 1.4114, 1.4314, 1.4515, 1.4715,
1.4915, 1.5115, 1.5315, 1.5516, 1.5716, 1.5916, 1.6116,
1.6316, 1.6517, 1.6717, 1.6917, 1.7117, 1.7317, 1.7518,
1.7718, 1.7918, 1.8118, 1.8318, 1.8519, 1.8719, 1.8919,
1.9119, 1.9319, 1.9520, 1.9720, 1.9920, 2.0120, 2.0320,
2.0521, 2.0721, 2.0921, 2.1121, 2.1321, 2.1522, 2.1722,
2.1922, 2.2122, 2.2322, 2.2523, 2.2723, 2.2923, 2.3123,
2.3323, 2.3524, 2.3724, 2.3924, 2.4124, 2.4324, 2.4525,
2.4725, 2.4925, 2.5125, 2.5325, 2.5526, 2.5726, 2.5926,
2.6126, 2.6326, 2.6527, 2.6727, 2.6927, 2.7127, 2.7327,
2.7528, 2.7728, 2.7928, 2.8128, 2.8328, 2.8529, 2.8729,
2.8929, 2.9129, 2.9329, 2.9530, 2.9730, 2.9930, 3.0130,
3.0330, 3.0531, 3.0731, 3.0931, 3.1131, 3.1331, 3.1532,
3.1732, 3.1932, 3.2132, 3.2332, 3.2533, 3.2733, 3.2933,
3.3133, 3.3333, 3.3534, 3.3734, 3.3934, 3.4134, 3.4334,
3.4535, 3.4735, 3.4935, 3.5135, 3.5335, 3.5536, 3.5736,
3.5936, 3.6136, 3.6336, 3.6537, 3.6737, 3.6937, 3.7137,
3.7337, 3.7538, 3.7738, 3.7938, 3.8138, 3.8338, 3.8539,
3.8739, 3.8939, 3.9139, 3.9339, 3.9540, 3.9740, 3.9940,
4.0140, 4.0340, 4.0541, 4.0741, 4.0941, 4.1141, 4.1341,
4.1542, 4.1742, 4.1942, 4.2142, 4.2342, 4.2543, 4.2743,
4.2943, 4.3143, 4.3343, 4.3544, 4.3744, 4.3944, 4.4144,
4.4344, 4.4545, 4.4745, 4.4945, 4.5145, 4.5345, 4.5546,
4.5746, 4.5946, 4.6146, 4.6346, 4.6547, 4.6747, 4.6947,
4.7147, 4.7347, 4.7548, 4.7748, 4.7948, 4.8148, 4.8348,
4.8549, 4.8749, 4.8949, 4.9149, 4.9349, 4.9550, 4.9750,
4.9950, 5.0150, 5.0350, 5.0551, 5.0751, 5.0951, 5.1151,
5.1351, 5.1552, 5.1752, 5.1952, 5.2152, 5.2352, 5.2553,
5.2753, 5.2953, 5.3153, 5.3353, 5.3554, 5.3754, 5.3954,
5.4154, 5.4354, 5.4555, 5.4755, 5.4955, 5.5155, 5.5355,
5.5556, 5.5756, 5.5956, 5.6156, 5.6356, 5.6557, 5.6757,
5.6957, 5.7157, 5.7357, 5.7558, 5.7758, 5.7958, 5.8158,
5.8358, 5.8559, 5.8759, 5.8959, 5.9159, 5.9359, 5.9560,
5.9760, 5.9960, 6.0160, 6.0360, 6.0561, 6.0761, 6.0961,
6.1161, 6.1361, 6.1562, 6.1762, 6.1962, 6.2162, 6.2362,
6.2563, 6.2763, 6.2963, 6.3163, 6.3363, 6.3564, 6.3764,
6.3964, 6.4164, 6.4364, 6.4565, 6.4765, 6.4965, 6.5165,
6.5365, 6.5566, 6.5766, 6.5966, 6.6166, 6.6366, 6.6567,
6.6767, 6.6967, 6.7167, 6.7367, 6.7568, 6.7768, 6.7968,
6.8168, 6.8368, 6.8569, 6.8769, 6.8969, 6.9169, 6.9369,
6.9570, 6.9770, 6.9970, 7.0170, 7.0370, 7.0571, 7.0771,
7.0971, 7.1171, 7.1371, 7.1572, 7.1772, 7.1972, 7.2172,
7.2372, 7.2573, 7.2773, 7.2973, 7.3173, 7.3373, 7.3574,
7.3774, 7.3974, 7.4174, 7.4374, 7.4575, 7.4775, 7.4975,
7.5175, 7.5375, 7.5576, 7.5776, 7.5976, 7.6176, 7.6376,
7.6577, 7.6777, 7.6977, 7.7177, 7.7377, 7.7578, 7.7778,
7.7978, 7.8178, 7.8378, 7.8579, 7.8779, 7.8979, 7.9179,
7.9379, 7.9580, 7.9780, 7.9980, 8.0180, 8.0380, 8.0581,
8.0781, 8.0981, 8.1181, 8.1381, 8.1582, 8.1782, 8.1982,
8.2182, 8.2382, 8.2583, 8.2783, 8.2983, 8.3183, 8.3383,
8.3584, 8.3784, 8.3984, 8.4184, 8.4384, 8.4585, 8.4785,
8.4985, 8.5185, 8.5385, 8.5586, 8.5786, 8.5986, 8.6186,
8.6386, 8.6587, 8.6787, 8.6987, 8.7187, 8.7387, 8.7588,
8.7788, 8.7988, 8.8188, 8.8388, 8.8589, 8.8789, 8.8989,
8.9189, 8.9389, 8.9590, 8.9790, 8.9990, 9.0190, 9.0390,
9.0591, 9.0791, 9.0991, 9.1191, 9.1391, 9.1592, 9.1792,
9.1992, 9.2192, 9.2392, 9.2593, 9.2793, 9.2993, 9.3193,
9.3393, 9.3594, 9.3794, 9.3994, 9.4194, 9.4394, 9.4595,
9.4795, 9.4995, 9.5195, 9.5395, 9.5596, 9.5796, 9.5996,
9.6196, 9.6396, 9.6597, 9.6797, 9.6997, 9.7197, 9.7397,
9.7598, 9.7798, 9.7998, 9.8198, 9.8398, 9.8599, 9.8799,
9.8999, 9.9199, 9.9399, 9.9600, 9.9800, 10.0000])hn = torch.distributions.HalfNormal(1)
x = torch.linspace(-10, 10, 1000)
x_mask = x>0
y = torch.zeros_like(x)
y[x_mask] = hn.log_prob(x[x_mask]).exp()
plt.plot(x, y, label="Half Normal")
normal = torch.distributions.Normal(0, 1)
y_norm = normal.log_prob(x).exp()
plt.plot(x, y_norm, label="Normal")
plt.legend()
dist = torch.distributions.Normal(0, 1)
x_lin = torch.linspace(-17, 17, 1000)
log_probs = dist.log_prob(x_lin)
probs = log_probs.exp()
fig, ax = plt.subplots(nrows=2, sharex=True)
ax[0].plot(x_lin, log_probs)
ax[0].set_title("Log Prob")
ax[1].plot(x_lin, probs)
ax[1].set_title("Prob")Text(0.5, 1.0, 'Prob')
print(probs[0], probs[1])
print(log_probs[0], log_probs[1])tensor(0.) tensor(0.)
tensor(-145.4189) tensor(-144.8409)Log Normal Distribution
Let \(Y \sim \mathcal{N}(\mu, \sigma^2)\) be a normally distributed random variable.
Let us define a new random variable \(X = \exp(Y)\).
We can say that log of \(X\) is normally distributed, i.e., \(\log(X) \sim \mathcal{N}(\mu, \sigma^2)\).
We can also say that \(X\) is log-normally distributed.
The probability density function (PDF) of \(X\) is given by:
\[ f_X(x) = \frac{1}{x\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(\log(x)-\mu)^2}{2\sigma^2}\right). \]
We can derive the PDF of \(X\) using the change of variables formula. (will be covered later in the course)
mu = 1.0
sigma = 1.0
log_normal = torch.distributions.LogNormal(mu, sigma)log_normal.supportGreaterThan(lower_bound=0.0)log_normal.meantensor(4.4817)log_normal.modetensor(1.)log_normal.variancetensor(34.5126)
x = torch.linspace(-10, 10, 1000)
x_non_neg_mask = x > 0.001
y = torch.zeros_like(x)
y[x_non_neg_mask] = log_normal.log_prob(x[x_non_neg_mask]).exp()
plt.plot(x, y, label="PDF LogNormal(1, 1)")
normal = torch.distributions.Normal(mu, sigma)
plt.plot(x, normal.log_prob(x).exp(), label="PDF Normal(1, 1)")
plt.legend()
Applications
See: https://en.wikipedia.org/wiki/Log-normal_distribution
See https://chess.stackexchange.com/questions/2506/what-is-the-average-length-of-a-game-of-chess/4899#4899
import kagglehub
# Download latest version
path = kagglehub.dataset_download("datasnaek/chess")
print("Path to dataset files:", path)Path to dataset files: /Users/nipun/.cache/kagglehub/datasets/datasnaek/chess/versions/1import os
df = pd.read_csv(os.path.join(path, "games.csv"))df.head()| id | rated | created_at | last_move_at | turns | victory_status | winner | increment_code | white_id | white_rating | black_id | black_rating | moves | opening_eco | opening_name | opening_ply | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | TZJHLljE | False | 1.504210e+12 | 1.504210e+12 | 13 | outoftime | white | 15+2 | bourgris | 1500 | a-00 | 1191 | d4 d5 c4 c6 cxd5 e6 dxe6 fxe6 Nf3 Bb4+ Nc3 Ba5... | D10 | Slav Defense: Exchange Variation | 5 |
| 1 | l1NXvwaE | True | 1.504130e+12 | 1.504130e+12 | 16 | resign | black | 5+10 | a-00 | 1322 | skinnerua | 1261 | d4 Nc6 e4 e5 f4 f6 dxe5 fxe5 fxe5 Nxe5 Qd4 Nc6... | B00 | Nimzowitsch Defense: Kennedy Variation | 4 |
| 2 | mIICvQHh | True | 1.504130e+12 | 1.504130e+12 | 61 | mate | white | 5+10 | ischia | 1496 | a-00 | 1500 | e4 e5 d3 d6 Be3 c6 Be2 b5 Nd2 a5 a4 c5 axb5 Nc... | C20 | King's Pawn Game: Leonardis Variation | 3 |
| 3 | kWKvrqYL | True | 1.504110e+12 | 1.504110e+12 | 61 | mate | white | 20+0 | daniamurashov | 1439 | adivanov2009 | 1454 | d4 d5 Nf3 Bf5 Nc3 Nf6 Bf4 Ng4 e3 Nc6 Be2 Qd7 O... | D02 | Queen's Pawn Game: Zukertort Variation | 3 |
| 4 | 9tXo1AUZ | True | 1.504030e+12 | 1.504030e+12 | 95 | mate | white | 30+3 | nik221107 | 1523 | adivanov2009 | 1469 | e4 e5 Nf3 d6 d4 Nc6 d5 Nb4 a3 Na6 Nc3 Be7 b4 N... | C41 | Philidor Defense | 5 |
# Distribution of the number of turns in the games
df["turns"].plot(kind='hist', bins=50)
# Logarithm of the number of turns
df["turns"].apply(np.log).plot(kind='hist', bins=50)
# Log of turns seems to be normally distributed
mu, sigma = df["turns"].apply(np.log).mean(), df["turns"].apply(np.log1p).std()
print(mu, sigma)3.9070571274448245 0.6822030192719669# Plot PDF of the fitted log-normal distribution
x = torch.linspace(0.001, 300, 1000)
with torch.no_grad():
log_normal = torch.distributions.LogNormal(mu, sigma)
y = log_normal.log_prob(x).exp()
plt.plot(x, y, label="Fitted PDF")
plt.hist(df["turns"], bins=50, density=True, alpha=0.5, label="KDE")
plt.legend()
Gamma distribution
Let \(X\) be a random variable that follows a gamma distribution with shape parameter \(k\) and scale parameter \(\theta\). The probability density function (PDF) of \(X\) is given by:
\[ f_X(x) = \frac{1}{\Gamma(k)\theta^k} x^{k-1} \exp\left(-\frac{x}{\theta}\right). \]
where \(\Gamma(k)\) is the gamma function defined as:
\[ \Gamma(k) = \int_0^\infty x^{k-1} e^{-x} dx. \]
gamma_dist = torch.distributions.Gamma(2, 1)
x = torch.linspace(0.001, 10, 1000)
y = gamma_dist.log_prob(x).exp()
plt.plot(x, y, label="PDF Gamma(2, 1)")
# Fit a gamma distribution to the data
alpha, beta = torch.tensor([1.0], requires_grad=True), torch.tensor([1.0], requires_grad=True)
gamma_dist = torch.distributions.Gamma(alpha, beta)
optimizer = torch.optim.Adam([alpha, beta], lr=0.01)
x = torch.tensor(df["turns"].values, dtype=torch.float32)
for i in range(1000):
optimizer.zero_grad()
loss = -gamma_dist.log_prob(x).mean()
loss.backward()
optimizer.step()
print(alpha.item(), beta.item())2.315873384475708 0.03829348832368851learnt_gamma_dist = torch.distributions.Gamma(alpha.detach(), beta.detach())
x = torch.linspace(0.001, 300, 1000)
y = learnt_gamma_dist.log_prob(x).exp()
plt.plot(x, y, label="Fitted PDF")
plt.hist(df["turns"], bins=50, density=True, alpha=0.5, label="KDE")
plt.legend()
Uniform Distribution
Let \(X\) be a random variable that follows a uniform distribution on the interval \([a, b]\). The probability density function (PDF) of \(X\) is given by:
$$ f_X(x) = \[\begin{cases} \frac{1}{b-a} & \text{if } x \in [a, b], \\ 0 & \text{otherwise}. \end{cases}\]$$
We can say that \(X \sim \text{Uniform}(a, b)\).
a = 0.0
b = 2.0
dist = torch.distributions.Uniform(a, b)dist.supportInterval(lower_bound=0.0, upper_bound=2.0)dist.high, dist.low(tensor(2.), tensor(0.))dist.meantensor(1.)dist.variancetensor(0.3333)x_range = torch.linspace(-1, 3, 1000)
try:
y = dist.log_prob(x_range).exp()
except Exception as e:
print(e)Expected value argument (Tensor of shape (1000,)) to be within the support (Interval(lower_bound=0.0, upper_bound=2.0)) of the distribution Uniform(low: 0.0, high: 2.0), but found invalid values:
tensor([-1.0000e+00, -9.9600e-01, -9.9199e-01, -9.8799e-01, -9.8398e-01,
-9.7998e-01, -9.7598e-01, -9.7197e-01, -9.6797e-01, -9.6396e-01,
-9.5996e-01, -9.5596e-01, -9.5195e-01, -9.4795e-01, -9.4394e-01,
-9.3994e-01, -9.3594e-01, -9.3193e-01, -9.2793e-01, -9.2392e-01,
-9.1992e-01, -9.1592e-01, -9.1191e-01, -9.0791e-01, -9.0390e-01,
-8.9990e-01, -8.9590e-01, -8.9189e-01, -8.8789e-01, -8.8388e-01,
-8.7988e-01, -8.7588e-01, -8.7187e-01, -8.6787e-01, -8.6386e-01,
-8.5986e-01, -8.5586e-01, -8.5185e-01, -8.4785e-01, -8.4384e-01,
-8.3984e-01, -8.3584e-01, -8.3183e-01, -8.2783e-01, -8.2382e-01,
-8.1982e-01, -8.1582e-01, -8.1181e-01, -8.0781e-01, -8.0380e-01,
-7.9980e-01, -7.9580e-01, -7.9179e-01, -7.8779e-01, -7.8378e-01,
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-7.5976e-01, -7.5576e-01, -7.5175e-01, -7.4775e-01, -7.4374e-01,
-7.3974e-01, -7.3574e-01, -7.3173e-01, -7.2773e-01, -7.2372e-01,
-7.1972e-01, -7.1572e-01, -7.1171e-01, -7.0771e-01, -7.0370e-01,
-6.9970e-01, -6.9570e-01, -6.9169e-01, -6.8769e-01, -6.8368e-01,
-6.7968e-01, -6.7568e-01, -6.7167e-01, -6.6767e-01, -6.6366e-01,
-6.5966e-01, -6.5566e-01, -6.5165e-01, -6.4765e-01, -6.4364e-01,
-6.3964e-01, -6.3564e-01, -6.3163e-01, -6.2763e-01, -6.2362e-01,
-6.1962e-01, -6.1562e-01, -6.1161e-01, -6.0761e-01, -6.0360e-01,
-5.9960e-01, -5.9560e-01, -5.9159e-01, -5.8759e-01, -5.8358e-01,
-5.7958e-01, -5.7558e-01, -5.7157e-01, -5.6757e-01, -5.6356e-01,
-5.5956e-01, -5.5556e-01, -5.5155e-01, -5.4755e-01, -5.4354e-01,
-5.3954e-01, -5.3554e-01, -5.3153e-01, -5.2753e-01, -5.2352e-01,
-5.1952e-01, -5.1552e-01, -5.1151e-01, -5.0751e-01, -5.0350e-01,
-4.9950e-01, -4.9550e-01, -4.9149e-01, -4.8749e-01, -4.8348e-01,
-4.7948e-01, -4.7548e-01, -4.7147e-01, -4.6747e-01, -4.6346e-01,
-4.5946e-01, -4.5546e-01, -4.5145e-01, -4.4745e-01, -4.4344e-01,
-4.3944e-01, -4.3544e-01, -4.3143e-01, -4.2743e-01, -4.2342e-01,
-4.1942e-01, -4.1542e-01, -4.1141e-01, -4.0741e-01, -4.0340e-01,
-3.9940e-01, -3.9540e-01, -3.9139e-01, -3.8739e-01, -3.8338e-01,
-3.7938e-01, -3.7538e-01, -3.7137e-01, -3.6737e-01, -3.6336e-01,
-3.5936e-01, -3.5536e-01, -3.5135e-01, -3.4735e-01, -3.4334e-01,
-3.3934e-01, -3.3534e-01, -3.3133e-01, -3.2733e-01, -3.2332e-01,
-3.1932e-01, -3.1532e-01, -3.1131e-01, -3.0731e-01, -3.0330e-01,
-2.9930e-01, -2.9530e-01, -2.9129e-01, -2.8729e-01, -2.8328e-01,
-2.7928e-01, -2.7528e-01, -2.7127e-01, -2.6727e-01, -2.6326e-01,
-2.5926e-01, -2.5526e-01, -2.5125e-01, -2.4725e-01, -2.4324e-01,
-2.3924e-01, -2.3524e-01, -2.3123e-01, -2.2723e-01, -2.2322e-01,
-2.1922e-01, -2.1522e-01, -2.1121e-01, -2.0721e-01, -2.0320e-01,
-1.9920e-01, -1.9520e-01, -1.9119e-01, -1.8719e-01, -1.8318e-01,
-1.7918e-01, -1.7518e-01, -1.7117e-01, -1.6717e-01, -1.6316e-01,
-1.5916e-01, -1.5516e-01, -1.5115e-01, -1.4715e-01, -1.4314e-01,
-1.3914e-01, -1.3514e-01, -1.3113e-01, -1.2713e-01, -1.2312e-01,
-1.1912e-01, -1.1512e-01, -1.1111e-01, -1.0711e-01, -1.0310e-01,
-9.9099e-02, -9.5095e-02, -9.1091e-02, -8.7087e-02, -8.3083e-02,
-7.9079e-02, -7.5075e-02, -7.1071e-02, -6.7067e-02, -6.3063e-02,
-5.9059e-02, -5.5055e-02, -5.1051e-02, -4.7047e-02, -4.3043e-02,
-3.9039e-02, -3.5035e-02, -3.1031e-02, -2.7027e-02, -2.3023e-02,
-1.9019e-02, -1.5015e-02, -1.1011e-02, -7.0070e-03, -3.0030e-03,
1.0010e-03, 5.0050e-03, 9.0090e-03, 1.3013e-02, 1.7017e-02,
2.1021e-02, 2.5025e-02, 2.9029e-02, 3.3033e-02, 3.7037e-02,
4.1041e-02, 4.5045e-02, 4.9049e-02, 5.3053e-02, 5.7057e-02,
6.1061e-02, 6.5065e-02, 6.9069e-02, 7.3073e-02, 7.7077e-02,
8.1081e-02, 8.5085e-02, 8.9089e-02, 9.3093e-02, 9.7097e-02,
1.0110e-01, 1.0511e-01, 1.0911e-01, 1.1311e-01, 1.1712e-01,
1.2112e-01, 1.2513e-01, 1.2913e-01, 1.3313e-01, 1.3714e-01,
1.4114e-01, 1.4515e-01, 1.4915e-01, 1.5315e-01, 1.5716e-01,
1.6116e-01, 1.6517e-01, 1.6917e-01, 1.7317e-01, 1.7718e-01,
1.8118e-01, 1.8519e-01, 1.8919e-01, 1.9319e-01, 1.9720e-01,
2.0120e-01, 2.0521e-01, 2.0921e-01, 2.1321e-01, 2.1722e-01,
2.2122e-01, 2.2523e-01, 2.2923e-01, 2.3323e-01, 2.3724e-01,
2.4124e-01, 2.4525e-01, 2.4925e-01, 2.5325e-01, 2.5726e-01,
2.6126e-01, 2.6527e-01, 2.6927e-01, 2.7327e-01, 2.7728e-01,
2.8128e-01, 2.8529e-01, 2.8929e-01, 2.9329e-01, 2.9730e-01,
3.0130e-01, 3.0531e-01, 3.0931e-01, 3.1331e-01, 3.1732e-01,
3.2132e-01, 3.2533e-01, 3.2933e-01, 3.3333e-01, 3.3734e-01,
3.4134e-01, 3.4535e-01, 3.4935e-01, 3.5335e-01, 3.5736e-01,
3.6136e-01, 3.6537e-01, 3.6937e-01, 3.7337e-01, 3.7738e-01,
3.8138e-01, 3.8539e-01, 3.8939e-01, 3.9339e-01, 3.9740e-01,
4.0140e-01, 4.0541e-01, 4.0941e-01, 4.1341e-01, 4.1742e-01,
4.2142e-01, 4.2543e-01, 4.2943e-01, 4.3343e-01, 4.3744e-01,
4.4144e-01, 4.4545e-01, 4.4945e-01, 4.5345e-01, 4.5746e-01,
4.6146e-01, 4.6547e-01, 4.6947e-01, 4.7347e-01, 4.7748e-01,
4.8148e-01, 4.8549e-01, 4.8949e-01, 4.9349e-01, 4.9750e-01,
5.0150e-01, 5.0551e-01, 5.0951e-01, 5.1351e-01, 5.1752e-01,
5.2152e-01, 5.2553e-01, 5.2953e-01, 5.3353e-01, 5.3754e-01,
5.4154e-01, 5.4555e-01, 5.4955e-01, 5.5355e-01, 5.5756e-01,
5.6156e-01, 5.6557e-01, 5.6957e-01, 5.7357e-01, 5.7758e-01,
5.8158e-01, 5.8559e-01, 5.8959e-01, 5.9359e-01, 5.9760e-01,
6.0160e-01, 6.0561e-01, 6.0961e-01, 6.1361e-01, 6.1762e-01,
6.2162e-01, 6.2563e-01, 6.2963e-01, 6.3363e-01, 6.3764e-01,
6.4164e-01, 6.4565e-01, 6.4965e-01, 6.5365e-01, 6.5766e-01,
6.6166e-01, 6.6567e-01, 6.6967e-01, 6.7367e-01, 6.7768e-01,
6.8168e-01, 6.8569e-01, 6.8969e-01, 6.9369e-01, 6.9770e-01,
7.0170e-01, 7.0571e-01, 7.0971e-01, 7.1371e-01, 7.1772e-01,
7.2172e-01, 7.2573e-01, 7.2973e-01, 7.3373e-01, 7.3774e-01,
7.4174e-01, 7.4575e-01, 7.4975e-01, 7.5375e-01, 7.5776e-01,
7.6176e-01, 7.6577e-01, 7.6977e-01, 7.7377e-01, 7.7778e-01,
7.8178e-01, 7.8579e-01, 7.8979e-01, 7.9379e-01, 7.9780e-01,
8.0180e-01, 8.0581e-01, 8.0981e-01, 8.1381e-01, 8.1782e-01,
8.2182e-01, 8.2583e-01, 8.2983e-01, 8.3383e-01, 8.3784e-01,
8.4184e-01, 8.4585e-01, 8.4985e-01, 8.5385e-01, 8.5786e-01,
8.6186e-01, 8.6587e-01, 8.6987e-01, 8.7387e-01, 8.7788e-01,
8.8188e-01, 8.8589e-01, 8.8989e-01, 8.9389e-01, 8.9790e-01,
9.0190e-01, 9.0591e-01, 9.0991e-01, 9.1391e-01, 9.1792e-01,
9.2192e-01, 9.2593e-01, 9.2993e-01, 9.3393e-01, 9.3794e-01,
9.4194e-01, 9.4595e-01, 9.4995e-01, 9.5395e-01, 9.5796e-01,
9.6196e-01, 9.6597e-01, 9.6997e-01, 9.7397e-01, 9.7798e-01,
9.8198e-01, 9.8599e-01, 9.8999e-01, 9.9399e-01, 9.9800e-01,
1.0020e+00, 1.0060e+00, 1.0100e+00, 1.0140e+00, 1.0180e+00,
1.0220e+00, 1.0260e+00, 1.0300e+00, 1.0340e+00, 1.0380e+00,
1.0420e+00, 1.0460e+00, 1.0501e+00, 1.0541e+00, 1.0581e+00,
1.0621e+00, 1.0661e+00, 1.0701e+00, 1.0741e+00, 1.0781e+00,
1.0821e+00, 1.0861e+00, 1.0901e+00, 1.0941e+00, 1.0981e+00,
1.1021e+00, 1.1061e+00, 1.1101e+00, 1.1141e+00, 1.1181e+00,
1.1221e+00, 1.1261e+00, 1.1301e+00, 1.1341e+00, 1.1381e+00,
1.1421e+00, 1.1461e+00, 1.1502e+00, 1.1542e+00, 1.1582e+00,
1.1622e+00, 1.1662e+00, 1.1702e+00, 1.1742e+00, 1.1782e+00,
1.1822e+00, 1.1862e+00, 1.1902e+00, 1.1942e+00, 1.1982e+00,
1.2022e+00, 1.2062e+00, 1.2102e+00, 1.2142e+00, 1.2182e+00,
1.2222e+00, 1.2262e+00, 1.2302e+00, 1.2342e+00, 1.2382e+00,
1.2422e+00, 1.2462e+00, 1.2503e+00, 1.2543e+00, 1.2583e+00,
1.2623e+00, 1.2663e+00, 1.2703e+00, 1.2743e+00, 1.2783e+00,
1.2823e+00, 1.2863e+00, 1.2903e+00, 1.2943e+00, 1.2983e+00,
1.3023e+00, 1.3063e+00, 1.3103e+00, 1.3143e+00, 1.3183e+00,
1.3223e+00, 1.3263e+00, 1.3303e+00, 1.3343e+00, 1.3383e+00,
1.3423e+00, 1.3463e+00, 1.3504e+00, 1.3544e+00, 1.3584e+00,
1.3624e+00, 1.3664e+00, 1.3704e+00, 1.3744e+00, 1.3784e+00,
1.3824e+00, 1.3864e+00, 1.3904e+00, 1.3944e+00, 1.3984e+00,
1.4024e+00, 1.4064e+00, 1.4104e+00, 1.4144e+00, 1.4184e+00,
1.4224e+00, 1.4264e+00, 1.4304e+00, 1.4344e+00, 1.4384e+00,
1.4424e+00, 1.4464e+00, 1.4505e+00, 1.4545e+00, 1.4585e+00,
1.4625e+00, 1.4665e+00, 1.4705e+00, 1.4745e+00, 1.4785e+00,
1.4825e+00, 1.4865e+00, 1.4905e+00, 1.4945e+00, 1.4985e+00,
1.5025e+00, 1.5065e+00, 1.5105e+00, 1.5145e+00, 1.5185e+00,
1.5225e+00, 1.5265e+00, 1.5305e+00, 1.5345e+00, 1.5385e+00,
1.5425e+00, 1.5465e+00, 1.5506e+00, 1.5546e+00, 1.5586e+00,
1.5626e+00, 1.5666e+00, 1.5706e+00, 1.5746e+00, 1.5786e+00,
1.5826e+00, 1.5866e+00, 1.5906e+00, 1.5946e+00, 1.5986e+00,
1.6026e+00, 1.6066e+00, 1.6106e+00, 1.6146e+00, 1.6186e+00,
1.6226e+00, 1.6266e+00, 1.6306e+00, 1.6346e+00, 1.6386e+00,
1.6426e+00, 1.6466e+00, 1.6507e+00, 1.6547e+00, 1.6587e+00,
1.6627e+00, 1.6667e+00, 1.6707e+00, 1.6747e+00, 1.6787e+00,
1.6827e+00, 1.6867e+00, 1.6907e+00, 1.6947e+00, 1.6987e+00,
1.7027e+00, 1.7067e+00, 1.7107e+00, 1.7147e+00, 1.7187e+00,
1.7227e+00, 1.7267e+00, 1.7307e+00, 1.7347e+00, 1.7387e+00,
1.7427e+00, 1.7467e+00, 1.7508e+00, 1.7548e+00, 1.7588e+00,
1.7628e+00, 1.7668e+00, 1.7708e+00, 1.7748e+00, 1.7788e+00,
1.7828e+00, 1.7868e+00, 1.7908e+00, 1.7948e+00, 1.7988e+00,
1.8028e+00, 1.8068e+00, 1.8108e+00, 1.8148e+00, 1.8188e+00,
1.8228e+00, 1.8268e+00, 1.8308e+00, 1.8348e+00, 1.8388e+00,
1.8428e+00, 1.8468e+00, 1.8509e+00, 1.8549e+00, 1.8589e+00,
1.8629e+00, 1.8669e+00, 1.8709e+00, 1.8749e+00, 1.8789e+00,
1.8829e+00, 1.8869e+00, 1.8909e+00, 1.8949e+00, 1.8989e+00,
1.9029e+00, 1.9069e+00, 1.9109e+00, 1.9149e+00, 1.9189e+00,
1.9229e+00, 1.9269e+00, 1.9309e+00, 1.9349e+00, 1.9389e+00,
1.9429e+00, 1.9469e+00, 1.9510e+00, 1.9550e+00, 1.9590e+00,
1.9630e+00, 1.9670e+00, 1.9710e+00, 1.9750e+00, 1.9790e+00,
1.9830e+00, 1.9870e+00, 1.9910e+00, 1.9950e+00, 1.9990e+00,
2.0030e+00, 2.0070e+00, 2.0110e+00, 2.0150e+00, 2.0190e+00,
2.0230e+00, 2.0270e+00, 2.0310e+00, 2.0350e+00, 2.0390e+00,
2.0430e+00, 2.0470e+00, 2.0511e+00, 2.0551e+00, 2.0591e+00,
2.0631e+00, 2.0671e+00, 2.0711e+00, 2.0751e+00, 2.0791e+00,
2.0831e+00, 2.0871e+00, 2.0911e+00, 2.0951e+00, 2.0991e+00,
2.1031e+00, 2.1071e+00, 2.1111e+00, 2.1151e+00, 2.1191e+00,
2.1231e+00, 2.1271e+00, 2.1311e+00, 2.1351e+00, 2.1391e+00,
2.1431e+00, 2.1471e+00, 2.1512e+00, 2.1552e+00, 2.1592e+00,
2.1632e+00, 2.1672e+00, 2.1712e+00, 2.1752e+00, 2.1792e+00,
2.1832e+00, 2.1872e+00, 2.1912e+00, 2.1952e+00, 2.1992e+00,
2.2032e+00, 2.2072e+00, 2.2112e+00, 2.2152e+00, 2.2192e+00,
2.2232e+00, 2.2272e+00, 2.2312e+00, 2.2352e+00, 2.2392e+00,
2.2432e+00, 2.2472e+00, 2.2513e+00, 2.2553e+00, 2.2593e+00,
2.2633e+00, 2.2673e+00, 2.2713e+00, 2.2753e+00, 2.2793e+00,
2.2833e+00, 2.2873e+00, 2.2913e+00, 2.2953e+00, 2.2993e+00,
2.3033e+00, 2.3073e+00, 2.3113e+00, 2.3153e+00, 2.3193e+00,
2.3233e+00, 2.3273e+00, 2.3313e+00, 2.3353e+00, 2.3393e+00,
2.3433e+00, 2.3473e+00, 2.3514e+00, 2.3554e+00, 2.3594e+00,
2.3634e+00, 2.3674e+00, 2.3714e+00, 2.3754e+00, 2.3794e+00,
2.3834e+00, 2.3874e+00, 2.3914e+00, 2.3954e+00, 2.3994e+00,
2.4034e+00, 2.4074e+00, 2.4114e+00, 2.4154e+00, 2.4194e+00,
2.4234e+00, 2.4274e+00, 2.4314e+00, 2.4354e+00, 2.4394e+00,
2.4434e+00, 2.4474e+00, 2.4515e+00, 2.4555e+00, 2.4595e+00,
2.4635e+00, 2.4675e+00, 2.4715e+00, 2.4755e+00, 2.4795e+00,
2.4835e+00, 2.4875e+00, 2.4915e+00, 2.4955e+00, 2.4995e+00,
2.5035e+00, 2.5075e+00, 2.5115e+00, 2.5155e+00, 2.5195e+00,
2.5235e+00, 2.5275e+00, 2.5315e+00, 2.5355e+00, 2.5395e+00,
2.5435e+00, 2.5475e+00, 2.5516e+00, 2.5556e+00, 2.5596e+00,
2.5636e+00, 2.5676e+00, 2.5716e+00, 2.5756e+00, 2.5796e+00,
2.5836e+00, 2.5876e+00, 2.5916e+00, 2.5956e+00, 2.5996e+00,
2.6036e+00, 2.6076e+00, 2.6116e+00, 2.6156e+00, 2.6196e+00,
2.6236e+00, 2.6276e+00, 2.6316e+00, 2.6356e+00, 2.6396e+00,
2.6436e+00, 2.6476e+00, 2.6517e+00, 2.6557e+00, 2.6597e+00,
2.6637e+00, 2.6677e+00, 2.6717e+00, 2.6757e+00, 2.6797e+00,
2.6837e+00, 2.6877e+00, 2.6917e+00, 2.6957e+00, 2.6997e+00,
2.7037e+00, 2.7077e+00, 2.7117e+00, 2.7157e+00, 2.7197e+00,
2.7237e+00, 2.7277e+00, 2.7317e+00, 2.7357e+00, 2.7397e+00,
2.7437e+00, 2.7477e+00, 2.7518e+00, 2.7558e+00, 2.7598e+00,
2.7638e+00, 2.7678e+00, 2.7718e+00, 2.7758e+00, 2.7798e+00,
2.7838e+00, 2.7878e+00, 2.7918e+00, 2.7958e+00, 2.7998e+00,
2.8038e+00, 2.8078e+00, 2.8118e+00, 2.8158e+00, 2.8198e+00,
2.8238e+00, 2.8278e+00, 2.8318e+00, 2.8358e+00, 2.8398e+00,
2.8438e+00, 2.8478e+00, 2.8519e+00, 2.8559e+00, 2.8599e+00,
2.8639e+00, 2.8679e+00, 2.8719e+00, 2.8759e+00, 2.8799e+00,
2.8839e+00, 2.8879e+00, 2.8919e+00, 2.8959e+00, 2.8999e+00,
2.9039e+00, 2.9079e+00, 2.9119e+00, 2.9159e+00, 2.9199e+00,
2.9239e+00, 2.9279e+00, 2.9319e+00, 2.9359e+00, 2.9399e+00,
2.9439e+00, 2.9479e+00, 2.9520e+00, 2.9560e+00, 2.9600e+00,
2.9640e+00, 2.9680e+00, 2.9720e+00, 2.9760e+00, 2.9800e+00,
2.9840e+00, 2.9880e+00, 2.9920e+00, 2.9960e+00, 3.0000e+00])x_range_mask = (x_range >= a) & (x_range <= b)
y = torch.zeros_like(x_range)
y[x_range_mask] = dist.log_prob(x_range[x_range_mask]).exp()
plt.plot(x_range, y)
Modeling quantisation error using uniform distribution
Quantization error is the error that arises when representing continuous signals with discrete signals.
NOTE: I am using a simplified convention here. Study DSP for a rigorous treatment (with N and T used in the equations).
Let original signal represented in computer be \(y(t)\). We will quantize the signal to \(x(t)\). The quantization error is given by:
\[ e(t) = y(t) - x(t). \]
We will quantize the signal to \(x(t)\) such that \(x(t)\) can take on \(N\) discrete values. Let \(\Delta\) be the quantization step size. The quantization error is given by:
If the quantization error is uniformly distributed between \(-\Delta/2\) and \(\Delta/2\), then we can model the quantization error as a uniform random variable.
\[ e(t) \sim \text{Uniform}(-\Delta/2, \Delta/2). \]
x_t = torch.linspace(-2, 8, 5000)
y_t = torch.sin(x_t)
plt.plot(x_t, y_t, label="y_t")
max_y_t = y_t.max()
min_y_t = y_t.min()
# Divide the range of y_t into N (=10) equal parts
N = 10
y_bins = torch.linspace(min_y_t, max_y_t, N+1)
# Draw the N levels as horizontal lines
for y_level in y_bins:
plt.axhline(y_level, color='gray', linestyle='--')
plt.text(3, y_level+.01, f"{y_level:.2f}")
delta = (max_y_t - min_y_t)/N
print(delta)
tensor(0.2000)
# For x = 3, find the bin in which y_t falls
y_t_x_3 = torch.sin(torch.tensor(3))
print(y_t_x_3)
plt.plot(x_t, y_t, label="y_t")
plt.axvline(3, color='red', linestyle='--')
plt.axhline(y_t_x_3, color='red', linestyle='--')
# Draw the N levels as horizontal lines
for y_level in y_bins:
plt.axhline(y_level, color='gray', linestyle='--', alpha=0.2)
plt.text(3, y_level+.01, f"{y_level:.2f}")
# Find the bin in which y_t falls
bin_idx = torch.searchsorted(y_bins, y_t_x_3)
plt.axhline(y_bins[bin_idx], color='green', linestyle='--', label="Closest level")
plt.legend()tensor(0.1411)
y_t.shape, y_bins.shape(torch.Size([5000]), torch.Size([11]))# Find closest bin for each y_t
bins = (y_t - y_bins.reshape(-1, 1)).abs().argmin(dim=0)
y_binned = y_bins[bins]
plt.plot(x_t, y_t, label="y_t")
plt.plot(x_t, y_binned, label="y_binned")
plt.legend()
plt.plot(y_t - y_binned)
_ = plt.hist(y_t - y_binned, density=True)
plt.xlabel("Error in binning")
plt.ylabel("Density")
theoretical_uniform = torch.distributions.Uniform(-delta/2, delta/2)
x = torch.linspace(-0.1, 0.1, 1000)
x_mask = (x >= -delta/2) & (x <= delta/2)
y = torch.zeros_like(x)
y[x_mask] = theoretical_uniform.log_prob(x[x_mask]).exp()
plt.plot(x, y, label="Theoretical PDF")
plt.legend()
Amount of bits saved
Originally, each sample of the signal was represented using \(B=32\) bits. After quantization, each sample is represented using \(B_q = \log_2(10)\) bits. The amount of bits saved is given by:
\[ (B - B_q) \times \text{number of samples}. \]
from pydub import AudioSegment
import numpy as np
# Load MP3
audio = AudioSegment.from_mp3("vlog-music.mp3")
# Convert to NumPy array
samples = np.array(audio.get_array_of_samples(), dtype=np.float32)
sample_rate = audio.frame_rate
print(f"Samples shape: {samples.shape}")
print(f"Sample rate: {sample_rate}")Samples shape: (5430528,)
Sample rate: 44100from IPython.display import Audio
Audio("vlog-music.mp3")plt.plot(samples)
# Plot 2nd second to 5th second
filtered_audio = samples[sample_rate*2:sample_rate*5]
fig, ax = plt.subplots(figsize=(20, 5))
ax.plot(filtered_audio)
# Quantize to 10 levels
min_audio = filtered_audio.min()
max_audio = filtered_audio.max()
N = 10
audio_bins = torch.linspace(min_audio, max_audio, N+1)
# Plotting audio bins
for audio_bin in audio_bins:
plt.axhline(audio_bin, color='gray', linestyle='--', alpha=0.5)
# Quantize the audio
audio_bins = np.linspace(min_audio, max_audio, N+1)
# Find closest bin for each audio sample
bins = np.abs(filtered_audio - audio_bins.reshape(-1, 1)).argmin(0)
audio_binned = audio_bins[bins]
fig, ax = plt.subplots(figsize=(20, 5))
ax.plot(filtered_audio, label="Original audio", alpha=0.2)
ax.plot(audio_binned, label="Quantized audio")
ax.legend()
Audio(filtered_audio, rate=sample_rate)# Play the quantized audio
Audio(audio_binned, rate=sample_rate)Beta Distribution
Let \(X\) be a random variable that follows a beta distribution with parameters \(\alpha\) and \(\beta\). The probability density function (PDF) of \(X\) is given by:
\[ f_X(x) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1}, \]
where \(\Gamma(\cdot)\) is the gamma function given as:
\[ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt. \]
We can say that \(X \sim \text{Beta}(\alpha, \beta)\).
beta_dist = torch.distributions.Beta(2, 2)beta_dist.supportInterval(lower_bound=0.0, upper_bound=1.0)beta_dist.meantensor(0.5000)Beta distribution is used to model random variables that are constrained to lie within a fixed interval. For example, the probability of success in a Bernoulli trial is a random variable that lies in the interval \([0, 1]\). We can model this probability using a beta distribution.
x_lin = torch.linspace(0.001, 0.999, 1000)
for alpha in range(1, 3):
for beta in range(1, 3):
beta_dist = torch.distributions.Beta(alpha, beta)
y = beta_dist.log_prob(x_lin).exp()
plt.plot(x_lin, y, label=f"PDF Beta({alpha}, {beta})")
plt.legend()
Exponential Distribution
Let \(X\) be a random variable that follows an exponential distribution with rate parameter \(\lambda\). The probability density function (PDF) of \(X\) is given by:
$$ f_X(x) = \[\begin{cases} \lambda \exp(-\lambda x) & \text{if } x \geq 0, \\ 0 & \text{otherwise}. \end{cases}\]$$
We can say that \(X \sim \text{Exponential}(\lambda)\).
The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.
l = 5.0
dist = torch.distributions.Exponential(l)dist.supportGreaterThanEq(lower_bound=0.0)dist.ratetensor(5.)dist.meantensor(0.2000)dist.modetensor(0.)# Plotting the PDF
x_range = torch.linspace(0.001, 10, 1000)
y = dist.log_prob(x_range).exp()
plt.plot(x_range, y)