Project: Smoothing#

Section Title: Smoothing

Smoothing is used to reduce the impact of fluctuations such as spikes or errors.

  • Smoothed values deviate from actual data points, yet they encapsulate the overall behavioral trends within the dataset.

import matplotlib.pyplot as plt
import numpy as np

Data#

In this project, we will use the following Apple daily stock price data for a particular period of time.

# the opening price at the beginning of each day
open = [182.60232902746063,
 182.31272011600538,
 184.89999389648438,
 185.44000244140625,
 187.50999450683594,
 187.91000366210938,
 190.47000122070312,
 189.50999450683594,
 189.3300018310547,
 191.08999633789062,
 192.27000427246094,
 190.97999572753906,
 188.82000732421875,
 191.50999450683594,
 189.61000061035156,
 190.75999450683594,
 191.44000244140625,
 192.89999389648438,
 194.63999938964844,
 195.39999389648438,
 195.69000244140625,
 194.64999389648438]

# the closing price at the end of each day
close = [182.4924774169922,
 184.32000732421875,
 183.0500030517578,
 186.27999877929688,
 187.42999267578125,
 189.72000122070312,
 189.83999633789062,
 189.8699951171875,
 191.0399932861328,
 192.35000610351562,
 190.89999389648438,
 186.8800048828125,
 189.97999572753906,
 189.99000549316406,
 190.2899932861328,
 191.2899932861328,
 192.25,
 194.02999877929688,
 194.35000610351562,
 195.8699951171875,
 194.47999572753906,
 196.88999938964844]

# The lowest price reached during each day.
low = [181.20421623621485,
 181.86333268844692,
 182.1300048828125,
 184.6199951171875,
 186.2899932861328,
 187.3699951171875,
 189.66000366210938,
 189.17999267578125,
 189.00999450683594,
 190.9199981689453,
 190.27000427246094,
 186.6300048828125,
 188.0399932861328,
 189.10000610351562,
 189.50999450683594,
 190.6300048828125,
 189.91000366210938,
 192.52000427246094,
 193.02999877929688,
 194.8699951171875,
 194.1699981689453,
 194.13999938964844]

# The highest price reached during each day.
high = [182.82203224839753,
 184.4098817621062,
 185.08999633789062,
 187.10000610351562,
 188.3000030517578,
 190.64999389648438,
 191.10000610351562,
 190.80999755859375,
 191.9199981689453,
 192.72999572753906,
 192.82000732421875,
 191.0,
 190.5800018310547,
 193.0,
 192.25,
 192.17999267578125,
 192.57000732421875,
 194.99000549316406,
 195.32000732421875,
 196.89999389648438,
 196.5,
 196.94000244140625]

# number of days
N = len(high)
N

22

# the list of days
days = ['day_'+str(i) for i in range(1, N+1)]
days

['day_1',
 'day_2',
 'day_3',
 'day_4',
 'day_5',
 'day_6',
 'day_7',
 'day_8',
 'day_9',
 'day_10',
 'day_11',
 'day_12',
 'day_13',
 'day_14',
 'day_15',
 'day_16',
 'day_17',
 'day_18',
 'day_19',
 'day_20',
 'day_21',
 'day_22']

Stock Behavior#

The list indicating the behavior of stock prices as Increasing, Decreasing, or Flat.

change = []
for i in range(N):
    if close[i] > open[i]:
        change.append('Inc')
    elif close[i] < open[i]:
        change.append('Dec')
    else:
        change.append('Flat')

change

['Dec',
 'Inc',
 'Dec',
 'Inc',
 'Dec',
 'Inc',
 'Dec',
 'Inc',
 'Inc',
 'Inc',
 'Dec',
 'Dec',
 'Inc',
 'Dec',
 'Inc',
 'Inc',
 'Inc',
 'Inc',
 'Dec',
 'Inc',
 'Dec',
 'Inc']

Candlestick#

Candlesticks are used to identify patterns in stock data.

  • Red color indicates a decrease, while green color indicates an increase.

  • The lower and upper parts of the rectangle represent the opening and closing values.

  • The lower and upper parts of the line represent the lowest and highest values.

plt.figure(figsize=(20,5))

for i in range(N):
    
    if change[i] == 'Inc':
        col = 'green'
    elif change[i] == 'Dec':
        col = 'red'
    else:
        col = 'blue'
        
    plt.plot([days[i],days[i]],[open[i], close[i]], c=col, linewidth=20)
    plt.plot([days[i],days[i]],[low[i], high[i]], c=col, linewidth=3)
  
plt.grid()
_images/59d41f7d5b46056be9bc190ea084517543ba48f0ee8c60d282326700ab30fee8.png

SMA#

  • Simple Moving Average

  • SMA calculates the average value within a chosen time frame.

  • All values are weighted equally, without distinction between old and new data points.

  • \(SMA_k\) is represents the sequence of the average of k consecutive values of the original data

  • Formula: \(\displaystyle SMA_k = \frac{x_n+x_{n+1}+x_{n+2}+...+x_{n+k-1}}{k}\)

  • For the first k−1 values, there is no average value because there are fewer than k values available.

  • The first k−1 values of \(SMA_k\) are NaN (Not a Number).

def sma(seq, k):
    sma_list = [np.nan]*(k-1)
    for n in range(len(seq)-k+1):
        sma_list.append(sum(seq[n:n+k])/k)
    return sma_list

len(sma(close, 3))

22

plt.figure(figsize=(20,5))
plt.plot(days, close, label='close', linestyle='dotted')
plt.plot(sma(close, 5), label='sma_5')
plt.plot(sma(close, 10), label='sma_10')
plt.legend();
_images/aeebadc65c411baf9cc028d433a471ff9dc6e7f97842793e1a253c8af9c4b36b.png

Moving Average Signals#

  • A buy signal happens when the shorter-term moving average (MA) crosses above the longer-term MA, indicating an upward trend shift known as a “golden cross”.

  • A sell signal occurs when the shorter-term moving average (MA) crosses below the longer-term MA, signaling a downward trend shift known as a “death cross”.

Reference: www.investopedia.com

  • There are two intersections between the \(SMA_2\) and \(SMA_4\) graphs.

plt.figure(figsize=(20,5))
plt.plot(days, close, label='close')
plt.plot(sma(close, 2), label='sma_2', linestyle='dotted')
plt.plot(sma(close, 4), label='sma_4', linestyle='dotted')
plt.xticks(range(len(days)), days)
plt.grid()
plt.legend();
_images/7f3b9e3a762f0b52439b747bfc7ce150a85cb98d6c0e60f9716f52b407850e4b.png 
for i in range(len(close)-1):
    if (sma(close, 2)[i] > sma(close, 4)[i]) & (sma(close, 2)[i+1] < sma(close, 4)[i+1]):
        print(f'Death cross  : day_{i+1} -- day_{i+2}')
    if (sma(close, 2)[i] < sma(close, 4)[i]) & (sma(close, 2)[i+1] > sma(close, 4)[i+1]):
        print(f'Golden Crsoss: day_{i+1} -- day_{i+2}')

Death cross  : day_11 -- day_12
Golden Crsoss: day_13 -- day_14

WMA#

  • Weighted Moving Average

  • WMA computes the weighted average value over a specified time period.

  • Unlike SMA, the weights assigned to values may vary, with newer data points potentially carrying more significance than older ones.

  • The influence of new points surpasses that of old points.

\(\displaystyle WMA_k = \frac{x_n + 2x_{n+1} + 3x_{n+2} + ... + kx_{n+k-1}}{1 + 2 + ... + k}\)

def wma(seq, k):
    wma_list = [np.nan]*(k-1)
    for n in range(len(seq)-k+1):
        
        num = 0
        for i in range(k):
            num += (i+1)*seq[n+i]
            
        den = sum(range(1,k+1))
        wma_list.append(num/den)
    return wma_list

plt.figure(figsize=(20,5))
plt.plot(days, close, label='close')
plt.plot(sma(close, 5), label='sma_5', linestyle='dotted')
plt.plot(wma(close, 5), label='wma_5', linestyle='dotted')
plt.legend();
_images/526695c9c140894fcccabfcf8ce1c8e10d39d1439d1b11cbc515f383168cbb7a.png

EWA#

  • Exponential Weighted Average

\( s_0 = x_0 \)

\( s_t = \alpha x_t +(1-\alpha) s_{t-1}\) for \(t>0\)

where \(\alpha\) is between 0 and 1.

\( s_0 = x_0 \)
\( s_1 = \alpha x_1 + (1-\alpha)s_0 = \alpha x_1 + (1-\alpha)x_0\)
\( s_2 = \alpha x_2 + (1-\alpha)s_1 = \alpha x_2 + (1-\alpha)(\alpha x_1 + (1-\alpha)x_0) = \alpha x_2 + (1-\alpha)\alpha x_1 + (1-\alpha)^2x_0\)

def ewa(seq, alpha):
    s = [seq[0]]
    for n in range(1,len(seq)):
        s.append(alpha*seq[n]+(1-alpha)*s[-1])
    return s

plt.figure(figsize=(20,5))
plt.plot(days, close, label='close')
plt.plot(sma(close, 5), label='sma_5', linestyle='dotted')
plt.plot(wma(close, 5), label='wma_5', linestyle='dotted')
plt.plot(ewa(close, 0.2), label='ewa_0_2', linestyle='dotted')
plt.plot(ewa(close, 0.9), label='ewa_0_9', linestyle='dotted')
plt.legend();
_images/2098a14741a2a1856dfed5e11d5c4f28926ecb250b66fc3fdee1377c2efc9e3f.png

AEWA#

Adjusted Exponential Weighted Average

  • Similar to WMA, but with different weights.

  • It is calculated using weights: \(1, (1-\alpha), (1-\alpha)^2, (1-\alpha)^3, ...\)

  • \(\displaystyle AEWA_k = \frac{(1-\alpha)^{k-1}x_n + (1-\alpha)^{k-2}x_{n+1} + ... +(1-\alpha)x_{n+k-2} + x_{n+k-1}}{ (1-\alpha)^{k-1}+(1-\alpha)^{k-2}+...+(1-\alpha)+1}\)

Example If k = 5,

\(\displaystyle AEWA_5 = \frac{(1-\alpha)^{4}x_n +(1-\alpha)^{3}x_{n+1}+(1-\alpha)^{2}x_{n+2}+(1-\alpha)x_{n+3}+x_{n+4}} { (1-\alpha)^{4}+(1-\alpha)^{3}+(1-\alpha)^{2}+(1-\alpha)+1}\)

def aewa(seq, k, alpha):
    aewa_list = [np.nan]*(k-1)
    for n in range(len(seq)-k+1):
        
        num = 0
        for i in range(k):
            num += seq[n+i]*(1-alpha)**(k-i-1)
            
        den = sum([(1-alpha)**(k-i) for i in range(1,k+1)])
        aewa_list.append(num/den)
    return aewa_list

plt.figure(figsize=(20,5))
plt.plot(days, close, label='close')
plt.plot(aewa(close, 5, 0.2), label='aewa_0_2', linestyle='dotted')
plt.plot(aewa(close, 5, 0.7), label='aewa_0_9', linestyle='dotted')
plt.legend();
_images/1eb1beb310c2f2ad78262d4ffddf4b07a91637f6689753c06dea343fbd3c42d7.png

Future work#

Similar analyses can be conducted with additional data. To import stock data, you can use the yfinance package.

  • Install it using !pip3 install yfinance.

  • Use the stock symbol to import historical price data.

  • For example, the symbol for Apple is AAPL, which is used in the following code to import data from ‘1/1/2010’ to ‘12/31/2023’.

import yfinance as yf
apple_data = yf.Ticker('AAPL').history(start='2010-1-1', end='2023-12-31')
apple = apple_data['Close'].tolist()
len(apple)

3522

  • The list apple has a length of 3,522.

  • You can use 50 and 200 periods for SMA to identify potential golden and death crosses.