Positional Encoding

Basic Imports

import numpy as np
import matplotlib.pyplot as plt
import torch
import seaborn as sns
import pandas as pd

dist =torch.distributions

sns.reset_defaults()
sns.set_context(context="talk", font_scale=1)
%matplotlib inline
%config InlineBackend.figure_format='retina'

# Generate complex data
x = np.linspace(-2, 2, 100)
freq = lambda x: x**2
f = lambda x: np.sin(2 * np.pi * x * freq(x))
y = f(x) + np.random.randn(x.shape[0]) * 0.1

# Plot
plt.figure(figsize=(8, 5))
plt.plot(x, y, 'o', label='data')
plt.plot(x, f(x), label='true function')
plt.legend()

<matplotlib.legend.Legend at 0x7f58b9eb0c70>

# Learn linear model on data
from sklearn.linear_model import LinearRegression, Ridge

lr1 = Ridge()
lr1.fit(x.reshape(-1, 1), y.reshape(-1, 1))

# Predict on linspace and plot
y_pred = lr1.predict(x.reshape(-1, 1))
plt.figure(figsize=(8, 5))
plt.plot(x, y, 'o', label='data')
plt.plot(x, f(x), label='true function')
plt.plot(x, y_pred, label='linear model')

# Add position encoding
# Gamma(x) = [sin(2^0*pi*x), cos(2^0*pi*x), sin(2^1*pi*x), cos(2^1*pi*x), ..., sin(2^k*pi*x), cos(2^k*pi*x)]

def gamma(x, k):
    """
    x: (N, 1)
    k: int
    Output: (N, 2k)
    """
    x = x.reshape(-1, 1)
    x = np.repeat(x, k, axis=1)
    x = x * (2 ** np.arange(k) * np.pi)
    z = np.concatenate([np.sin(x), np.cos(x)], axis=1)
    # Concatenate x and z
    z = np.concatenate([x, z], axis=1)
    return z


plt.plot(x, gamma(x, 2), alpha=0.2, color='k')

# Fit linear model with position encoding for k

def fit_plot(x, k):
    lr = Ridge()
    X_new = gamma(x, k)
    lr.fit(X_new, y.reshape(-1, 1))
    y_pred = lr.predict(X_new)
    plt.figure(figsize=(8, 5))
    plt.plot(x, y, 'o', label='data')
    plt.plot(x, f(x), label='true function')
    plt.plot(x, y_pred, label='linear model with position encoding of order {}'.format(k), lw=3)
    # Legend outside plot
    plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)

    # Title is the score
    plt.title('Score: {:.2f}'.format(lr.score(X_new, y.reshape(-1, 1))))

    # Show extrapolation also
    # extrapolation points are set difference between linspace and data points
    
    x_extra = np.linspace(-3, 3, 100)
    X_extra = gamma(x_extra, k)
    y_extra = lr.predict(X_extra)
    plt.plot(x_extra, y_extra, '--', label='extrapolation', alpha=0.2)
    plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)
    
    

fit_plot(x, 1)

ValueError: Found input variables with inconsistent numbers of samples: [100, 768]

fit_plot(x, 2)

fit_plot(x, 3)

fit_plot(x, 7)

fit_plot(x, 10)

fit_plot(x, 70)

!wget https://gml.noaa.gov/webdata/ccgg/trends/co2/co2_mm_mlo.csv

--2023-06-09 15:39:49--  https://gml.noaa.gov/webdata/ccgg/trends/co2/co2_mm_mlo.csv
Resolving gml.noaa.gov (gml.noaa.gov)... 140.172.200.41, 2610:20:8800:6101::29
Connecting to gml.noaa.gov (gml.noaa.gov)|140.172.200.41|:443... connected.
HTTP request sent, awaiting response... 200 OK
Length: 38018 (37K) [text/csv]
Saving to: ‘co2_mm_mlo.csv’

co2_mm_mlo.csv      100%[===================>]  37.13K   146KB/s    in 0.3s    

2023-06-09 15:39:51 (146 KB/s) - ‘co2_mm_mlo.csv’ saved [38018/38018]

df = pd.read_csv('co2_mm_mlo.csv', header=None, skiprows=72)

X = df.index.values
y = df[3].values.reshape(-1, 1)

plt.plot(X, y)