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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
Step2:  <a href="https
Step3:  Step by Step Code Order
#1. How to find the order of differencing (d) in ARIMA model
p is the order of the AR term
q is the order of the MA term
d is the number of differencing required to make the time series stationary as I term
Step4:  A p-value less than 0.05 (typically ≤ 0.05) is statistically significant. It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null hypo is correct (and the results are by random). Therefore, we reject the null hypothesis, and accept the alternative hypothesis (there is  correlation).
Our timeserie is not significant, why?
Example
Step5:  For the above series, the time series reaches stationarity with two orders of differencing. But we use for the beginning 1 order as a conservative part. Let me explain that
Step6:  #3. How to find the order of the MA term (q)
Step7:  4. How to build the ARIMA Model
Step8:  Notice here the coefficient of the MA2 term is close to zero (-0.0010 ) and the P-Value in ‘P>|z|’ column is highly insignificant (0.9). It should ideally be less than 0.05 for the respective X to be significant << 0.05.
5. Plot residual errors
Let’s plot the residuals to ensure there are no patterns (that is, look for constant mean and variance).
Step9:  6. Plot Predict Actual vs Fitted
When you set dynamic=False in-sample lagged values are used for prediction.
That is, the model gets trained up until the previous values to make next prediction. This can make a fitted forecast and actuals look artificially good.
Step10:  7. Now Create Training and Test Validation
We can see that ARIMA is adequately forecasting the seasonal pattern in the series. In terms of the model performance, the RMSE (root mean squared error) and MFE (mean forecast error) and also best in terms of the lowest BIC .
Step11:  8. Some scores and performance
The 20 observations depends on the train/test set fc,se,conf = fitted.forecast(20, alpha=0.05)  # 95% conf
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Python Code:
#sign:max: MAXBOX8: 03/02/2021 18:34:41 
# optimal moving average OMA for market index signals ARIMA study- Max Kleiner
# v2 shell argument forecast days - 4 lines compare - ^GDAXI for DAX
# pip install pandas-datareader
# C:\maXbox\mX46210\DataScience\princeton\AB_NYC_2019.csv   AB_NYC_2019.csv
#https://medium.com/abzuai/the-qlattice-a-new-machine-learning-model-you-didnt-know-you-needed-c2e037878cd
#https://www.kaggle.com/dgomonov/data-exploration-on-nyc-airbnb     41
#https://www.kaggle.com/duygut/airbnb-nyc-price-prediction
#https://www.machinelearningplus.com/time-series/arima-model-time-series-forecasting-python/
 
import numpy as np
import matplotlib.pyplot as plt
import sys
import numpy as np, pandas as pd
from statsmodels.tsa.arima_model import ARIMA
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
from statsmodels.tsa.stattools import adfuller, acf
import matplotlib.pyplot as plt
plt.rcParams.update({'figure.figsize':(9,7), 'figure.dpi':120})
# Import data
wwwus = pd.read_csv('https://raw.githubusercontent.com/selva86/datasets/master/wwwusage.csv', names=['value'], header=0)
import pandas as pd
# Accuracy metrics
def forecast_accuracy(forecast, actual):
    mape = np.mean(np.abs(forecast - actual)/np.abs(actual))  # MAPE
    me = np.mean(forecast - actual)             # ME
    mae = np.mean(np.abs(forecast - actual))    # MAE
    mpe = np.mean((forecast - actual)/actual)   # MPE
    rmse = np.mean((forecast - actual)**2)**.5  # RMSE
    corr = np.corrcoef(forecast, actual)[0,1]   # corr
    mins = np.amin(np.hstack([forecast[:,None], 
                              actual[:,None]]), axis=1)
    maxs = np.amax(np.hstack([forecast[:,None], 
                              actual[:,None]]), axis=1)
    minmax = 1 - np.mean(mins/maxs)             # minmax
    acf1 = acf(fc-test)[1]                      # ACF1
    return({'mape':mape, 'me':me, 'mae': mae, 
            'mpe': mpe, 'rmse':rmse, 'acf1':acf1, 
            'corr':corr, 'minmax':minmax})
#wwwus = pd.read_csv(r'C:\maXbox\mX46210\DataScience\princeton\1022dataset.txt', \
#                                                            names=['value'], header=0)
print(wwwus.head(10).T) #Transposed for column overview
#1. How to find the order of differencing (d) in ARIMA model
result = adfuller(wwwus.value.dropna())
print('ADF Statistic: %f' % result[0])
print('p-value: %f' % result[1])
#
# Original Series
fig, axes = plt.subplots(3, 2, sharex=True)
axes[0, 0].plot(wwwus.value); axes[0, 0].set_title('Orig Series')
plot_acf(wwwus.value, ax=axes[0, 1], lags=60)
# 1st Differencing
axes[1, 0].plot(wwwus.value.diff()); axes[1, 0].set_title('1st Order Differencing')
plot_acf(wwwus.value.diff().dropna(), ax=axes[1, 1], lags=60)
# 2nd Differencing
axes[2, 0].plot(wwwus.value.diff().diff()); axes[2, 0].set_title('2nd Order Differencing')
plot_acf(wwwus.value.diff().diff().dropna(), ax=axes[2, 1], lags=60)
plt.show()
#2. How to find the order of the AR term (p)
plt.rcParams.update({'figure.figsize':(9,3), 'figure.dpi':120})
fig, axes = plt.subplots(1, 2, sharex=True)
axes[0].plot(wwwus.value.diff()); axes[0].set_title('1st Differencing')
axes[1].set(ylim=(0,5))
plot_pacf(wwwus.value.diff().dropna(), ax=axes[1], lags=100)
plt.show()
#3. How to find the order of the MA term (q)
fig, axes = plt.subplots(1, 2, sharex=True)
axes[0].plot(wwwus.value.diff()); axes[0].set_title('1st Differencing')
axes[1].set(ylim=(0,1.2))
plot_acf(wwwus.value.diff().dropna(), ax=axes[1] , lags=60)
plt.show()
#
#4. How to build the ARIMA Model
model = ARIMA(wwwus.value, order=(1,1,2))
model_fit = model.fit(disp=0)
print('first fit ',model_fit.summary())
# Plot residual errors
residuals = pd.DataFrame(model_fit.resid)
plt.rcParams.update({'figure.figsize':(9,3), 'figure.dpi':120})
fig, ax = plt.subplots(1,2)
residuals.plot(title="Residuals", ax=ax[0])
residuals.plot(kind='kde', title='Density', ax=ax[1])
plt.show()
#5. Plot Predict Actual vs Fitted
# When you set dynamic=False in-sample lagged values are used for prediction.
plt.rcParams.update({'figure.figsize':(9,3), 'figure.dpi':120})
model_fit.plot_predict(dynamic=False)
plt.show()
#That is, the model gets trained up until the previous value to make next prediction. This can make a fitted forecast and actuals look artificially good.
# Now Create Training and Test
train = wwwus.value[:80]
test = wwwus.value[80:]
#model = ARIMA(train, order=(3, 2, 1))  
model = ARIMA(train, order=(2, 2, 3))  
fitted = model.fit(disp=-1)
print('second fit ',fitted.summary())  
# Forecast
fc,se,conf = fitted.forecast(20, alpha=0.05)  # 95% conf
# Make as pandas series
fc_series = pd.Series(fc, index=test.index)
lower_series = pd.Series(conf[:,0], index=test.index)
upper_series = pd.Series(conf[:,1], index=test.index)
# Plot
plt.figure(figsize=(12,5), dpi=100)
plt.plot(train, label='training')
plt.plot(test, label='actual')
plt.plot(fc_series, label='forecast')
plt.fill_between(lower_series.index, lower_series, upper_series, 
                                            color='k', alpha=.15)
plt.title('maXbox4 Forecast vs Actuals ARIMA')
plt.legend(loc='upper left', fontsize=8)
plt.show()
print(forecast_accuracy(fc, test.values))
print('Around 5% MAPE implies a model is about 95% accurate in predicting next 20 observations.')
Explanation: <a href="https://colab.research.google.com/github/maxkleiner/maXbox4/blob/master/ARIMA_Predictor2.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a>
How to find the order of (p,d,q) in ARIMA timeseries model
A time series is a sequence where a metric is recorded over regular time intervals.
Inspired by 
https://www.machinelearningplus.com/time-series/arima-model-time-series-forecasting-python/
End of explanation
#1. How to find the order of differencing (d) in ARIMA model
result = adfuller(wwwus.value.dropna())
print('ADF Statistic: %f' % result[0])
print('p-value: %f' % result[1])
Explanation: Step by Step Code Order
#1. How to find the order of differencing (d) in ARIMA model
p is the order of the AR term
q is the order of the MA term
d is the number of differencing required to make the time series stationary as I term
End of explanation
# Original Series
fig, axes = plt.subplots(3, 2, sharex=True)
axes[0, 0].plot(wwwus.value); axes[0, 0].set_title('Orig Series')
plot_acf(wwwus.value, ax=axes[0, 1], lags=60)
# 1st Differencing
axes[1, 0].plot(wwwus.value.diff()); axes[1, 0].set_title('1st Order Differencing')
plot_acf(wwwus.value.diff().dropna(), ax=axes[1, 1], lags=60)
# 2nd Differencing
axes[2, 0].plot(wwwus.value.diff().diff()); axes[2, 0].set_title('2nd Order Differencing')
plot_acf(wwwus.value.diff().diff().dropna(), ax=axes[2, 1], lags=60)
plt.show()
Explanation: A p-value less than 0.05 (typically ≤ 0.05) is statistically significant. It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null hypo is correct (and the results are by random). Therefore, we reject the null hypothesis, and accept the alternative hypothesis (there is  correlation).
Our timeserie is not significant, why?
Example:
ADF Statistic: -2.464240
p-value: 0.124419
0-Hypothesis non stationary
0.12 > 0.05 -> not significant, therefore we can not reject the 0-hypthesis so our time series is non stationary and we had to differencing it to make it stationary.
The purpose of differencing it is to make the time series stationary.
End of explanation
plt.rcParams.update({'figure.figsize':(9,3), 'figure.dpi':120})
fig, axes = plt.subplots(1, 2, sharex=True)
axes[0].plot(wwwus.value.diff()); axes[0].set_title('1st Differencing')
axes[1].set(ylim=(0,5))
plot_pacf(wwwus.value.diff().dropna(), ax=axes[1], lags=100)
plt.show()
Explanation: For the above series, the time series reaches stationarity with two orders of differencing. But we use for the beginning 1 order as a conservative part. Let me explain that:
D>2 is not allowed in statsmodels.tsa.arima_model!
Maybe d>2 is not allowed means our best bet is to start simple, check if integrating once grants stationarity. If so, we can fit a simple ARIMA model and examine the ACF of the residual values to get a better feel about what orders of differencing to use. Also a drawback, if we integrate more than two times (d>2), we lose n observations, one for each integration. And one of the most common errors in ARIMA modeling is to "overdifference" the series and end up adding extra AR or MA terms to undo the forecast damage, so the author (I assume) decides to raise this exception.
#2. How to find the order of the AR term (p)
End of explanation
fig, axes = plt.subplots(1, 2, sharex=True)
axes[0].plot(wwwus.value.diff()); axes[0].set_title('1st Differencing')
axes[1].set(ylim=(0,1.2))
plot_acf(wwwus.value.diff().dropna(), ax=axes[1] , lags=90)
plt.show()
Explanation: #3. How to find the order of the MA term (q)
End of explanation
model = ARIMA(wwwus.value, order=(1,1,2))
model_fit = model.fit(disp=0)
print('first fit ',model_fit.summary())
Explanation: 4. How to build the ARIMA Model
End of explanation
residuals = pd.DataFrame(model_fit.resid)
plt.rcParams.update({'figure.figsize':(9,3), 'figure.dpi':120})
fig, ax = plt.subplots(1,2)
residuals.plot(title="Residuals", ax=ax[0])
residuals.plot(kind='kde', title='Density', ax=ax[1])
plt.show()
Explanation: Notice here the coefficient of the MA2 term is close to zero (-0.0010 ) and the P-Value in ‘P>|z|’ column is highly insignificant (0.9). It should ideally be less than 0.05 for the respective X to be significant << 0.05.
5. Plot residual errors
Let’s plot the residuals to ensure there are no patterns (that is, look for constant mean and variance).
End of explanation
plt.rcParams.update({'figure.figsize':(9,3), 'figure.dpi':120})
model_fit.plot_predict(dynamic=False)
plt.show()
Explanation: 6. Plot Predict Actual vs Fitted
When you set dynamic=False in-sample lagged values are used for prediction.
That is, the model gets trained up until the previous values to make next prediction. This can make a fitted forecast and actuals look artificially good.
End of explanation
train = wwwus.value[:80]
test = wwwus.value[80:]
#model = ARIMA(train, order=(3, 2, 1))  
model = ARIMA(train, order=(2, 2, 3))  
fitted = model.fit(disp=-1)
print('second fit ',fitted.summary())  
# Forecast
fc,se,conf = fitted.forecast(20, alpha=0.05)  # 95% conf
# Make as pandas series
fc_series = pd.Series(fc, index=test.index)
lower_series = pd.Series(conf[:,0], index=test.index)
upper_series = pd.Series(conf[:,1], index=test.index)
# Plot
plt.figure(figsize=(12,5), dpi=100)
plt.plot(train, label='training')
plt.plot(test, label='actual')
plt.plot(fc_series, label='forecast')
plt.fill_between(lower_series.index, lower_series, upper_series, 
                                            color='k', alpha=.15)
plt.title('maXbox4 Forecast vs Actuals ARIMA')
plt.legend(loc='upper left', fontsize=8)
plt.show()
Explanation: 7. Now Create Training and Test Validation
We can see that ARIMA is adequately forecasting the seasonal pattern in the series. In terms of the model performance, the RMSE (root mean squared error) and MFE (mean forecast error) and also best in terms of the lowest BIC .
End of explanation
print(forecast_accuracy(fc, test.values))
print('Around 5% MAPE implies a model is about 95% accurate in predicting next 20 observations.')
Explanation: 8. Some scores and performance
The 20 observations depends on the train/test set fc,se,conf = fitted.forecast(20, alpha=0.05)  # 95% conf
End of explanation
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