Holt-Winters time series forecasting with statsmodels

Question

I tried forecasting with holt-winters model as shown below but I keep getting a prediction that is not consistent with what I expect. I also showed a visualization of the plot

Train = Airline[:130]
Test = Airline[129:]

from statsmodels.tsa.holtwinters import Holt

y_hat_avg = Test.copy()
fit1 = Holt(np.asarray(Train['Passengers'])).fit()
y_hat_avg['Holt_Winter'] = fit1.predict(start=1,end=15)
plt.figure(figsize=(16,8))
plt.plot(Train.index, Train['Passengers'], label='Train')
plt.plot(Test.index,Test['Passengers'], label='Test')
plt.plot(y_hat_avg.index,y_hat_avg['Holt_Winter'], label='Holt_Winter')
plt.legend(loc='best')
plt.savefig('Holt_Winters.jpg')

I am unsure of what I'm missing here.

The prediction seems to be fitted to the earlier part of the Training data

Gibber

Can you post the time series data here? – Generative

The data can be found here datamarket.com/data/set/22u3/… Click on export. I did some preprocessing on the data and converted the months column to index. – Gibber

My guess is your indices start=1,end=15 are wrong. In the plot the prediction looks like it's for the first few observations. Try to predict with start=129 or start=130. – Katowice

Answer 1

The main reason for the mistake is your start and end values. It forecasts the value for the first observation until the fifteenth. However, even if you correct that, Holt only includes the trend component and your forecasts will not carry the seasonal effects. Instead, use ExponentialSmoothing with seasonal parameters.

Here's a working example for your dataset:

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.holtwinters import ExponentialSmoothing

df = pd.read_csv('/home/ayhan/international-airline-passengers.csv', 
                 parse_dates=['Month'], 
                 index_col='Month'
)
df.index.freq = 'MS'
train, test = df.iloc[:130, 0], df.iloc[130:, 0]
model = ExponentialSmoothing(train, seasonal='mul', seasonal_periods=12).fit()
pred = model.predict(start=test.index[0], end=test.index[-1])

plt.plot(train.index, train, label='Train')
plt.plot(test.index, test, label='Test')
plt.plot(pred.index, pred, label='Holt-Winters')
plt.legend(loc='best')

which yields the following plot:

Generative

Hey, could you please tell me the role of df.index.freq='MS' in your code? – Brackish

In order to build a smoothing model statsmodels needs to know the frequency of your data (whether it is daily, monthly or so on). MS means start of the month so we are saying that it is monthly data that we observe at the start of each month. – Generative

Thanks for the reply. My data points are at a time lag of 5 mins. So, what should be my data's frequency? Any ideas? – Brackish

I think it should be '5T' – Generative

@ayhan If I have one month data and the frequency is '10T' and there's a trend for each day i.e (24 hours) what should be my seasonal and seasonal_periods parameters be? Should I use seasonal be 'mul' and seasonal_periods=1 for one month. – Vestigial

Answer 2

This is an improvisation of the above answer https://stackoverflow.com/users/2285236/ayhan

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.holtwinters import ExponentialSmoothing
from sklearn.metrics import mean_squared_error
from math import sqrt

from matplotlib.pylab import rcParams
rcParams['figure.figsize'] = 15, 7

df = pd.read_csv('D:/WORK/international-airline-passengers.csv', 
                 parse_dates=['Month'], 
                 index_col='Month'
)
df.index.freq = 'MS'
train, test = df.iloc[:132, 0], df.iloc[132:, 0]
# model = ExponentialSmoothing(train, seasonal='mul', seasonal_periods=12).fit()
model = ExponentialSmoothing(train, trend='add', seasonal='add', seasonal_periods=12, damped=True)
hw_model = model.fit(optimized=True, use_boxcox=False, remove_bias=False)
pred = hw_model.predict(start=test.index[0], end=test.index[-1])

plt.plot(train.index, train, label='Train')
plt.plot(test.index, test, label='Test')
plt.plot(pred.index, pred, label='Holt-Winters')
plt.legend(loc='best');

Here is how I got the best parameters

def exp_smoothing_configs(seasonal=[None]):
    models = list()
    # define config lists
    t_params = ['add', 'mul', None]
    d_params = [True, False]
    s_params = ['add', 'mul', None]
    p_params = seasonal
    b_params = [True, False]
    r_params = [True, False]
    # create config instances
    for t in t_params:
        for d in d_params:
            for s in s_params:
                for p in p_params:
                    for b in b_params:
                        for r in r_params:
                            cfg = [t,d,s,p,b,r]
                            models.append(cfg)
    return models

cfg_list = exp_smoothing_configs(seasonal=[12]) #[0,6,12]

---

edf = df['Passengers']
ts = edf[:'1959-12-01'].copy()
ts_v = edf['1960-01-01':].copy()
ind = edf.index[-12:]  # this will select last 12 months' indexes

print("Holt's Winter Model")
best_RMSE = np.inf
best_config = []
t1 = d1 = s1 = p1 = b1 = r1 = ''
for j in range(len(cfg_list)):
    print(j)
    try:
        cg = cfg_list[j]
        print(cg)
        t,d,s,p,b,r = cg
        train = edf[:'1959'].copy()
        test = edf['1960-01-01':'1960-12-01'].copy()
        # define model
        if (t == None):
            model = ExponentialSmoothing(ts, trend=t, seasonal=s, seasonal_periods=p)
        else:
            model = ExponentialSmoothing(ts, trend=t, damped=d, seasonal=s, seasonal_periods=p)
        # fit model
        model_fit = model.fit(optimized=True, use_boxcox=b, remove_bias=r)
        # make one step forecast
        y_forecast = model_fit.forecast(12)
        rmse = np.sqrt(mean_squared_error(ts_v,y_forecast))
        print(rmse)
        if rmse < best_RMSE:
            best_RMSE = rmse
            best_config = cfg_list[j]
    except:
       continue

---

Function to evaluate model

def model_eval(y, predictions):
    
    # Import library for metrics
    from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error

    # Mean absolute error (MAE)
    mae = mean_absolute_error(y, predictions)

    # Mean squared error (MSE)
    mse = mean_squared_error(y, predictions)

    
    # SMAPE is an alternative for MAPE when there are zeros in the testing data. It
    # scales the absolute percentage by the sum of forecast and observed values
    SMAPE = np.mean(np.abs((y - predictions) / ((y + predictions)/2))) * 100


    # Calculate the Root Mean Squared Error
    rmse = np.sqrt(mean_squared_error(y, predictions))
    
    # Calculate the Mean Absolute Percentage Error
    # y, predictions = check_array(y, predictions)
    MAPE = np.mean(np.abs((y - predictions) / y)) * 100
    
    # mean_forecast_error
    mfe = np.mean(y - predictions)
    
    # NMSE normalizes the obtained MSE after dividing it by the test variance. It
    # is a balanced error measure and is very effective in judging forecast
    # accuracy of a model.

    # normalised_mean_squared_error
    NMSE = mse / (np.sum((y - np.mean(y)) ** 2)/(len(y)-1))

    
    # theil_u_statistic
    # It is a normalized measure of total forecast error.
    error = y - predictions
    mfe = np.sqrt(np.mean(predictions**2))
    mse = np.sqrt(np.mean(y**2))
    rmse = np.sqrt(np.mean(error**2))
    theil_u_statistic =  rmse / (mfe*mse)


    # mean_absolute_scaled_error
    # This evaluation metric is used to over come some of the problems of MAPE and
    # is used to measure if the forecasting model is better than the naive model or
    # not.
    

    # Print metrics
    print('Mean Absolute Error:', round(mae, 3))
    print('Mean Squared Error:', round(mse, 3))
    print('Root Mean Squared Error:', round(rmse, 3))
    print('Mean absolute percentage error:', round(MAPE, 3))
    print('Scaled Mean absolute percentage error:', round(SMAPE, 3))
    print('Mean forecast error:', round(mfe, 3))
    print('Normalised mean squared error:', round(NMSE, 3))
    print('Theil_u_statistic:', round(theil_u_statistic, 3))

---

print(best_RMSE, best_config)

t1,d1,s1,p1,b1,r1 = best_config

if t1 == None:
    hw_model1 = ExponentialSmoothing(ts, trend=t1, seasonal=s1, seasonal_periods=p1)
else:
    hw_model1 = ExponentialSmoothing(ts, trend=t1, seasonal=s1, seasonal_periods=p1, damped=d1)
    
fit2 = hw_model1.fit(optimized=True, use_boxcox=b1, remove_bias=r1)

pred_HW = fit2.predict(start=pd.to_datetime('1960-01-01'), end = pd.to_datetime('1960-12-01'))
# pred_HW = fit2.forecast(12)

pred_HW = pd.Series(data=pred_HW, index=ind)
df_pass_pred = pd.concat([df, pred_HW.rename('pred_HW')], axis=1)

print(model_eval(ts_v, pred_HW))
print('-*-'*20)

# 15.570830579664698 ['add', True, 'add', 12, False, False]
# Mean Absolute Error: 10.456
# Mean Squared Error: 481.948
# Root Mean Squared Error: 15.571
# Mean absolute percentage error: 2.317
# Scaled Mean absolute percentage error: 2.273
# Mean forecast error: 483.689
# Normalised mean squared error: 0.04
# Theil_u_statistic: 0.0
# None
# -*--*--*--*--*--*--*--*--*--*--*--*--*--*--*--*--*--*--*--*-

Summary:

New Model results:

Mean Absolute Error: 10.456
Mean Squared Error: 481.948
Root Mean Squared Error: 15.571
Mean absolute percentage error: 2.317
Scaled Mean absolute percentage error: 2.273
Mean forecast error: 483.689
Normalised mean squared error: 0.04
Theil_u_statistic: 0.0

Old Model Results:

Mean Absolute Error: 20.682
Mean Squared Error: 481.948
Root Mean Squared Error: 23.719
Mean absolute percentage error: 4.468
Scaled Mean absolute percentage error: 4.56
Mean forecast error: 466.704
Normalised mean squared error: 0.093
Theil_u_statistic: 0.0

Bonus:

You will get this nice dataframe where you can compare the original values with predicted values.

df_pass_pred['1960':]

output

            Passengers     pred_HW
Month                             
1960-01-01         417  417.826543
1960-02-01         391  400.452916
1960-03-01         419  461.804259
1960-04-01         461  450.787208
1960-05-01         472  472.695903
1960-06-01         535  528.560823
1960-07-01         622  601.265794
1960-08-01         606  608.370401
1960-09-01         508  508.869849
1960-10-01         461  452.958727
1960-11-01         390  407.634391
1960-12-01         432  437.385058

Aspersorium

Consider itertools.product for that deeply nested loop. – Bellarmine