Probability theory: application of convolution in calculating moving average

Application of convolution in calculating moving average

Convolution may often be encountered in the field of artificial intelligence , So what exactly is convolution ？ How to understand ？

1. Introduction of convolution

For science and engineering students , Contact convolution is mostly in 《 Probability theory and mathematical statistics 》 Encountered in the course of , I'm learning “ Random variable function distribution ” Medium Z=X+Y in ：
set up $(X,Y)$ Is a two-dimensional continuous random variable , It has a probability density $f(x,y)$, be $Z=X+Y$ It is still a continuous random variable , The probability density is ：
$$f_{X+Y}(z)={\int }_{-\infty }^{+\infty }f(x,z-x)dx$$
perhaps
$$f_{X+Y}(z)={\int }_{-\infty }^{+\infty }f(z-y,y)dy$$
You Ruo $XandY$ Are independent of each other , set up $(X,Y)$ About $X,Y$ The edge densities are $f_X(x),f_Y(y)$, Then you can get it again ：
$$f_{X+Y}(z)={\int }_{-\infty }^{+\infty }{f}_X(z-y)f_Y(y)dy(1)$$
and
$$f_{X+Y}(z)={\int }_{-\infty }^{+\infty }{f}_X(x)f_Y(z-x)dx(2)$$
The formula above （1） and （2） be called $f_{X}and{f}_Y$ Convolution formula , Write it down as $f_X\ast f_Y$, namely
$$f_X\ast f_Y={\int }_{-\infty }^{+\infty }{f}_X(x)f_Y(z-x)dx={\int }_{-\infty }^{+\infty }{f}_X(z-y)f_Y(y)dy$$

2. Exploration of convolution

Convolution is an important operation in analytical mathematics , There are convolution operations in both continuous and discrete cases . Here's the definition ：
Definition of continuous variables ： set up $f(x),g(x)$ yes R Two integrable functions on , To integrate ：
$$h(x)={\int }_{-\infty }^{+\infty }f(\tau )g(x-\tau )d\tau$$
This definite integral defines a function $h(x)$, Called a function f and g Convolution of , Write it down as $h(x)=(f\ast g)(x)$.
Definition of discrete variables ： If the variable is a sequence $x(n)andy(n)$, be
$$h(n)=\sum _{i=-\infty }^{+\infty }x(i)y(n-i)$$
that ,$h(n)$ be called $x(n)Andy(n)$ Convolution of .

3. Application of convolution

Convolution has many applications in engineering and Mathematics ： Weighted moving average in statistics is a kind of convolution ; Two independent statistics in probability theory X and Y The probability density of the sum of is X And Y Convolution of probability density function of .
Now let's calculate the share price of apple for a period 5 Daily moving average price , And show the smoothing effect of convolution by drawing .

# step 1: load data
import numpy as np
from matplotlib.pyplot import plot

# 1.1  Here I read Apple 30 Day closing price 
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)

# 1.2  Calculate the five-day average price , So construct an equivalent weight list （ You can also use unequal weights ）
N = 5
weights = np.ones(N)/N

# step 2： calulate
# 2.1  Calculate the convolution of closing price and weight 
sma = np.convolve(weithts, c)
# sma Not what we asked 5 Average daily price , Because there is only continuity 5 Day's closing price , Will produce an average price , So be right sma section ,sma[N-1:-N+1] That's what we need 5 Average daily price 
sma = sma[N-1:-N+1]

# step 3： Plot
t = np.arange(N-1, len(c))
plot(t, c[N-1:], lw=1.0)
plot(t, sma, lw=2.0)

stay jupyter notebook The lower output is as follows ：

As shown in the figure above , The relatively smooth thick line is the five-day moving average , The jagged thin lines depict the daily closing price .

4. summary

We calculate stocks by using convolution 5 Average daily price , It is demonstrated that the result of convolution calculation can produce a smoother output for the input . In fact, convolution plays a very important role in signal processing .