Standard Theta Model
Stepbystep guide on using the Standard Theta Model
with Statsforecast
.
Table of Contents
 Introduction
 Standard Theta
 Loading libraries and data
 Explore data with the plot method
 Split the data into training and testing
 Implementation of StandardTheta with StatsForecast
 Crossvalidation
 Model evaluation
 References
Introduction
The Theta method (Assimakopoulos & Nikolopoulos, 2000, hereafter A&N) is applied to nonseasonal or deseasonalised time series, where the deseasonalisation is usually performed via the multiplicative classical decomposition. The method decomposes the original time series into two new lines through the socalled theta coefficients, denoted by ${\theta}_1$ and ${\theta}_2$ for ${\theta}_1, {\theta}_2 \in \mathbb{R}$, which are applied to the second difference of the data. The second differences are reduced when ${\theta}<1$, resulting in a better approximation of the longterm behaviour of the series (Assimakopoulos, 1995). If ${\theta}$ is equal to zero, the new line is a straight line. When ${\theta}>1$ the local curvatures are increased, magnifying the shortterm movements of the time series (A&N). The new lines produced are called theta lines, denoted here by $\text{Z}(\theta_1)$ and $\text{Z}(\theta_2)$. These lines have the same mean value and slope as the original data, but the local curvatures are either filtered out or enhanced, depending on the value of the $\theta$ coefficient.
In other words, the decomposition process has the advantage of exploiting information in the data that usually cannot be captured and modelled completely through the extrapolation of the original time series. The theta lines can be regarded as new time series and are extrapolated separately using an appropriate forecasting method. Once the extrapolation of each theta line has been completed, recomposition takes place through a combination scheme in order to calculate the point forecasts of the original time series. Combining has long been considered as a useful practice in the forecasting literature (for example, Clemen, 1989, Makridakis and Winkler, 1983, Petropoulos et al., 2014), and therefore its application to the Theta method is expected to result in more accurate and robust forecasts.
The Theta method is quite versatile in terms of choosing the number of theta lines, the theta coefficients and the extrapolation methods, and combining these to obtain robust forecasts. However, A&N proposed a simplified version involving the use of only two theta lines with prefixed $\theta$ coefficients that are extrapolated over time using a linear regression (LR) model for the theta line with ${\theta}_1 =0$ and simple exponential smoothing (SES) for the theta line with ${\theta}_2 =2$. The final forecasts are produced by combining the forecasts of the two theta lines with equal weights.
The performance of the Theta method has also been confirmed by other empirical studies (for example Nikolopoulos et al., 2012, Petropoulos and Nikolopoulos, 2013). Moreover, Hyndman and Billah (2003), hereafter H&B, showed that the simple exponential smoothing with drift model (SESd) is a statistical model for the simplified version of the Theta method. More recently, Thomakos and Nikolopoulos (2014) provided additional theoretical insights, while Thomakos and Nikolopoulos (2015) derived new theoretical formulations for the application of the method to multivariate time series, and investigated the conditions under which the bivariate Theta method is expected to forecast better than the univariate one. Despite these advances, we believe that the Theta method deserves more attention from the forecasting community, given its simplicity and superior forecasting performance.
One key aspect of the Theta method is that, by definition, it is dynamic. One can choose different theta lines and combine the produced forecasts using either equal or unequal weights. However, AN limit this important property by fixing the theta coefficients to have predefined values.
Standard Theta Model
Assimakopoulos and Nikolopoulo for standard theta model proposed the Theta line as the solution of the equation
where $Y_1, \cdots , Y_T$ represent the original time series data and $DX_t = (X_t − X_{t−1})$. The initial values $\zeta_1$ and $\zeta_2$ are obtained by minimizing $\sum_{i=1}^{T} [Y_t  \zeta_t (\theta) ]^2$. However, the analytical solution of (1) is given by
where $A_T$ and $B_T$ are the minimum square coefficients of a simple linear regression over $Y_1, \cdots,Y_T$ against $1, \cdots , T$ which are only dependent on the original data and given as follow
Theta lines can be understood as functions of the linear regression model directly applied to the data from this perspective. Indeed, the Theta method’s projections for h steps ahead are an ad hoc combination (50 percent  50 percent) of the linear extrapolations of $\zeta(0)$ and $\zeta(2)$.

When $\theta < 1$ is applied to the second differences of the data, the decomposition process is defined by a theta coefficient, which reduces the second differences and improves the approximation of series behavior.

If $\theta = 0$, the deconstructed line is turned into a constant straight line. (see Fig)

If $\theta > 1$ then the short term movements of the analyzed series show more local curvatures (see fig)
We will refer to the above setup as the standard Theta method. The steps for building the theta method are as follows:
 Deseasonalisation: Firstly, the time series data is tested for statistically significant seasonal behaviour. A time series is seasonal if
$\rho_m > q_{1 \frac{\alpha}{2} } \sqrt{\frac{1+2 \sum_{i=1}^{m1} \rho_{i}^{2} }{T} }$
where ρk denotes the lag $k$ autocorrelation function, $m$ is the number of the periods within a seasonal cycle (for example, 12 for monthly data), $T$ is the sample size, $q$ is the quantile function of the standard normal distribution, and $(1 − a)\%$ is the confidence level. Assimakopoulos and Nikolopoulo [Standar Theta model] opted for a 90% confidence level. If the time series is identified as seasonal, then it is deseasonalised via the classical decomposition method, assuming the seasonal component to have a multiplicative relationship.

Decomposition: The second step consits for the decomposition of the seasonally adjusted time series into two Theta lines, the
linear regression
line $\zeta(0)$ and the theta line $\zeta(2)$. 
Extrapolation: $\zeta(2)$ is extrapolated using
simple exponential smoothing (SES)
, while $\zeta(0)$ is extrapolated as a normallinear regression
line. 
Combination: the final forecast is a combination of the forecasts of the two $\theta$ lines using equal weights.

Reseasonalisation: In the presence of seasonality in first step, then the final forecasts are multiplied by the respective seasonal indices.
Loading libraries and data
Tip
Statsforecast will be needed. To install, see instructions.
Next, we import plotting libraries and configure the plotting style.
import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
plt.style.use('grayscale') # fivethirtyeight grayscale classic
plt.rcParams['lines.linewidth'] = 1.5
dark_style = {
'figure.facecolor': '#008080', # #212946
'axes.facecolor': '#008080',
'savefig.facecolor': '#008080',
'axes.grid': True,
'axes.grid.which': 'both',
'axes.spines.left': False,
'axes.spines.right': False,
'axes.spines.top': False,
'axes.spines.bottom': False,
'grid.color': '#000000', #2A3459
'grid.linewidth': '1',
'text.color': '0.9',
'axes.labelcolor': '0.9',
'xtick.color': '0.9',
'ytick.color': '0.9',
'font.size': 12 }
plt.rcParams.update(dark_style)
from pylab import rcParams
rcParams['figure.figsize'] = (18,7)
Read Data
import pandas as pd
df = pd.read_csv("https://raw.githubusercontent.com/Naren8520/SeriedetiempoconMachineLearning/main/Data/milk_production.csv", usecols=[1,2])
df.head()
month  production  

0  19620101  589 
1  19620201  561 
2  19620301  640 
3  19620401  656 
4  19620501  727 
The input to StatsForecast is always a data frame in long format with three columns: unique_id, ds and y:

The
unique_id
(string, int or category) represents an identifier for the series. 
The
ds
(datestamp) column should be of a format expected by Pandas, ideally YYYYMMDD for a date or YYYYMMDD HH:MM:SS for a timestamp. 
The
y
(numeric) represents the measurement we wish to forecast.
df["unique_id"]="1"
df.columns=["ds", "y", "unique_id"]
df.head()
ds  y  unique_id  

0  19620101  589  1 
1  19620201  561  1 
2  19620301  640  1 
3  19620401  656  1 
4  19620501  727  1 
print(df.dtypes)
ds object
y int64
unique_id object
dtype: object
We can see that our time variable (ds)
is in an object format, we need
to convert to a date format
df["ds"] = pd.to_datetime(df["ds"])
Explore Data with the plot method
Plot some series using the plot method from the StatsForecast class. This method prints a random series from the dataset and is useful for basic EDA.
from statsforecast import StatsForecast
StatsForecast.plot(df, engine="matplotlib")
Autocorrelation plots
fig, axs = plt.subplots(nrows=1, ncols=2)
plot_acf(df["y"], lags=30, ax=axs[0],color="fuchsia")
axs[0].set_title("Autocorrelation");
plot_pacf(df["y"], lags=30, ax=axs[1],color="lime")
axs[1].set_title('Partial Autocorrelation')
plt.show();
Decomposition of the time series
How to decompose a time series and why?
In time series analysis to forecast new values, it is very important to know past data. More formally, we can say that it is very important to know the patterns that values follow over time. There can be many reasons that cause our forecast values to fall in the wrong direction. Basically, a time series consists of four components. The variation of those components causes the change in the pattern of the time series. These components are:
 Level: This is the primary value that averages over time.
 Trend: The trend is the value that causes increasing or decreasing patterns in a time series.
 Seasonality: This is a cyclical event that occurs in a time series for a short time and causes shortterm increasing or decreasing patterns in a time series.
 Residual/Noise: These are the random variations in the time series.
Combining these components over time leads to the formation of a time series. Most time series consist of level and noise/residual and trend or seasonality are optional values.
If seasonality and trend are part of the time series, then there will be effects on the forecast value. As the pattern of the forecasted time series may be different from the previous time series.
The combination of the components in time series can be of two types: * Additive * Multiplicative
Additive time series
If the components of the time series are added to make the time series. Then the time series is called the additive time series. By visualization, we can say that the time series is additive if the increasing or decreasing pattern of the time series is similar throughout the series. The mathematical function of any additive time series can be represented by: $y(t) = level + Trend + seasonality + noise$
Multiplicative time series
If the components of the time series are multiplicative together, then the time series is called a multiplicative time series. For visualization, if the time series is having exponential growth or decline with time, then the time series can be considered as the multiplicative time series. The mathematical function of the multiplicative time series can be represented as.
$y(t) = Level * Trend * seasonality * Noise$
Additive
from statsmodels.tsa.seasonal import seasonal_decompose
a = seasonal_decompose(df["y"], model = "additive", period=12)
a.plot();
Multiplicative
from statsmodels.tsa.seasonal import seasonal_decompose
a = seasonal_decompose(df["y"], model = "Multiplicative", period=12)
a.plot();
Split the data into training and testing
Let’s divide our data into sets 1. Data to train our
Theta
model 2. Data to test our model
For the test data we will use the last 12 months to test and evaluate the performance of our model.
train = df[df.ds<='19741201']
test = df[df.ds>'19741201']
train.shape, test.shape
((156, 3), (12, 3))
Now let’s plot the training data and the test data.
sns.lineplot(train,x="ds", y="y", label="Train", linestyle="")
sns.lineplot(test, x="ds", y="y", label="Test")
plt.title("Monthly Milk Production");
plt.show()
Implementation of StandardTheta with StatsForecast
To also know more about the parameters of the functions of the
StandardTheta Model
, they are listed below. For more information,
visit the
documentation.
season_length : int
Number of observations per unit of time. Ex: 24 Hourly data.
decomposition_type : str
Sesonal decomposition type, 'multiplicative' (default) or 'additive'.
alias : str
Custom name of the model.
prediction_intervals : Optional[ConformalIntervals]
Information to compute conformal prediction intervals.
By default, the model will compute the native prediction
intervals.
Load libraries
from statsforecast import StatsForecast
from statsforecast.models import Theta
Instantiating Model
Import and instantiate the models. Setting the argument is sometimes
tricky. This article on Seasonal
periods by the
master, Rob Hyndmann, can be useful for season_length
.
season_length = 12 # Monthly data
horizon = len(test) # number of predictions
models = [Theta(season_length=season_length,
decomposition_type="additive")] # multiplicative additive
We fit the models by instantiating a new StatsForecast object with the following parameters:
models: a list of models. Select the models you want from models and import them.

freq:
a string indicating the frequency of the data. (See panda’s available frequencies.) 
n_jobs:
n_jobs: int, number of jobs used in the parallel processing, use 1 for all cores. 
fallback_model:
a model to be used if a model fails.
Any settings are passed into the constructor. Then you call its fit method and pass in the historical data frame.
sf = StatsForecast(df=train,
models=models,
freq='MS',
n_jobs=1)
Fit Model
sf.fit()
StatsForecast(models=[Theta])
Let’s see the results of our Theta model. We can observe it with the following instruction:
result=sf.fitted_[0,0].model_
print(result.keys())
print(result['fit'])
dict_keys(['mse', 'amse', 'fit', 'residuals', 'm', 'states', 'par', 'n', 'modeltype', 'mean_y', 'decompose', 'decomposition_type', 'seas_forecast', 'fitted'])
results(x=array([225.82002697, 0.76015625]), fn=10.638733596938769, nit=19, simplex=array([[241.83142594, 0.76274414],
[225.82002697, 0.76015625],
[212.41789302, 0.76391602]]))
Let us now visualize the residuals of our models.
As we can see, the result obtained above has an output in a dictionary,
to extract each element from the dictionary we are going to use the
.get()
function to extract the element and then we are going to save
it in a pd.DataFrame()
.
residual=pd.DataFrame(result.get("residuals"), columns=["residual Model"])
residual
residual Model  

0  17.596375 
1  46.997192 
2  23.093933 
…  … 
153  59.003235 
154  91.150085 
155  42.749451 
import scipy.stats as stats
fig, axs = plt.subplots(nrows=2, ncols=2)
residual.plot(ax=axs[0,0])
axs[0,0].set_title("Residuals");
sns.distplot(residual, ax=axs[0,1]);
axs[0,1].set_title("Density plot  Residual");
stats.probplot(residual["residual Model"], dist="norm", plot=axs[1,0])
axs[1,0].set_title('Plot QQ')
plot_acf(residual, lags=35, ax=axs[1,1],color="fuchsia")
axs[1,1].set_title("Autocorrelation");
plt.show();
Forecast Method
If you want to gain speed in productive settings where you have multiple
series or models we recommend using the
StatsForecast.forecast
method instead of .fit
and .predict
.
The main difference is that the .forecast
doest not store the fitted
values and is highly scalable in distributed environments.
The forecast method takes two arguments: forecasts next h
(horizon)
and level
.

h (int):
represents the forecast h steps into the future. In this case, 12 months ahead. 
level (list of floats):
this optional parameter is used for probabilistic forecasting. Set the level (or confidence percentile) of your prediction interval. For example,level=[90]
means that the model expects the real value to be inside that interval 90% of the times.
The forecast object here is a new data frame that includes a column with the name of the model and the y hat values, as well as columns for the uncertainty intervals. Depending on your computer, this step should take around 1min.
# Prediction
Y_hat = sf.forecast(horizon, fitted=True)
Y_hat
ds  Theta  

unique_id  
1  19750101  838.559814 
1  19750201  800.188232 
1  19750301  893.472900 
…  …  … 
1  19751001  816.166931 
1  19751101  786.962036 
1  19751201  823.826538 
values=sf.forecast_fitted_values()
values.head()
ds  y  Theta  

unique_id  
1  19620101  589.0  606.596375 
1  19620201  561.0  607.997192 
1  19620301  640.0  616.906067 
1  19620401  656.0  608.873047 
1  19620501  727.0  607.395142 
StatsForecast.plot(values)
Adding 95% confidence interval with the forecast method
sf.forecast(h=horizon, level=[95])
ds  Theta  Thetalo95  Thetahi95  

unique_id  
1  19750101  838.559814  741.324280  954.365540 
1  19750201  800.188232  640.785645  944.996887 
1  19750301  893.472900  705.123901  1064.757324 
…  …  …  …  … 
1  19751001  816.166931  539.706848  1083.791870 
1  19751101  786.962036  487.946075  1032.028931 
1  19751201  823.826538  512.674866  1101.965942 
Y_hat=Y_hat.reset_index()
Y_hat
unique_id  ds  Theta  

0  1  19750101  838.559814 
1  1  19750201  800.188232 
2  1  19750301  893.472900 
…  …  …  … 
9  1  19751001  816.166931 
10  1  19751101  786.962036 
11  1  19751201  823.826538 
# Merge the forecasts with the true values
test['unique_id'] = test['unique_id'].astype(int)
Y_hat1 = test.merge(Y_hat, how='left', on=['unique_id', 'ds'])
Y_hat1
ds  y  unique_id  Theta  

0  19750101  834  1  838.559814 
1  19750201  782  1  800.188232 
2  19750301  892  1  893.472900 
…  …  …  …  … 
9  19751001  827  1  816.166931 
10  19751101  797  1  786.962036 
11  19751201  843  1  823.826538 
fig, ax = plt.subplots(1, 1)
plot_df = pd.concat([train, Y_hat1]).set_index('ds')
plot_df[['y', "Theta"]].plot(ax=ax, linewidth=2)
ax.set_title(' Forecast', fontsize=22)
ax.set_ylabel('Monthly Milk Production ', fontsize=20)
ax.set_xlabel('Monthly [t]', fontsize=20)
ax.legend(prop={'size': 15})
ax.grid(True)
Predict method with confidence interval
To generate forecasts use the predict method.
The predict method takes two arguments: forecasts the next h
(for
horizon) and level
.

h (int):
represents the forecast h steps into the future. In this case, 12 months ahead. 
level (list of floats):
this optional parameter is used for probabilistic forecasting. Set the level (or confidence percentile) of your prediction interval. For example,level=[95]
means that the model expects the real value to be inside that interval 95% of the times.
The forecast object here is a new data frame that includes a column with the name of the model and the y hat values, as well as columns for the uncertainty intervals.
This step should take less than 1 second.
sf.predict(h=horizon)
ds  Theta  

unique_id  
1  19750101  838.559814 
1  19750201  800.188232 
1  19750301  893.472900 
…  …  … 
1  19751001  816.166931 
1  19751101  786.962036 
1  19751201  823.826538 
forecast_df = sf.predict(h=horizon, level=[80,95])
forecast_df
ds  Theta  Thetalo80  Thetahi80  Thetalo95  Thetahi95  

unique_id  
1  19750101  838.559814  765.496155  927.260071  741.324280  954.365540 
1  19750201  800.188232  701.729797  898.807434  640.785645  944.996887 
1  19750301  893.472900  758.481018  1006.847656  705.123901  1064.757324 
…  …  …  …  …  …  … 
1  19751001  816.166931  611.404541  991.667419  539.706848  1083.791870 
1  19751101  786.962036  561.990845  969.637634  487.946075  1032.028931 
1  19751201  823.826538  591.283691  1029.491577  512.674866  1101.965942 
We can join the forecast result with the historical data using the
pandas function pd.concat()
, and then be able to use this result for
graphing.
pd.concat([df, forecast_df]).set_index('ds')
y  unique_id  Theta  Thetalo80  Thetahi80  Thetalo95  Thetahi95  

ds  
19620101  589.0  1  NaN  NaN  NaN  NaN  NaN 
19620201  561.0  1  NaN  NaN  NaN  NaN  NaN 
19620301  640.0  1  NaN  NaN  NaN  NaN  NaN 
…  …  …  …  …  …  …  … 
19751001  NaN  NaN  816.166931  611.404541  991.667419  539.706848  1083.791870 
19751101  NaN  NaN  786.962036  561.990845  969.637634  487.946075  1032.028931 
19751201  NaN  NaN  823.826538  591.283691  1029.491577  512.674866  1101.965942 
Now let’s visualize the result of our forecast and the historical data of our time series, also let’s draw the confidence interval that we have obtained when making the prediction with 95% confidence.
def plot_forecasts(y_hist, y_true, y_pred, models):
_, ax = plt.subplots(1, 1, figsize = (20, 7))
y_true = y_true.merge(y_pred, how='left', on=['unique_id', 'ds'])
df_plot = pd.concat([y_hist, y_true]).set_index('ds').tail(12*10)
df_plot[['y'] + models].plot(ax=ax, linewidth=2 )
colors = ['green']
ax.fill_between(df_plot.index,
df_plot['Thetalo80'],
df_plot['Thetahi80'],
alpha=.20,
color='lime',
label='Theta_level_80')
ax.fill_between(df_plot.index,
df_plot['Thetalo95'],
df_plot['Thetahi95'],
alpha=.2,
color='white',
label='Theta_level_95')
ax.set_title('', fontsize=22)
ax.set_ylabel("Return", fontsize=20)
ax.set_xlabel('MonthDays', fontsize=20)
ax.legend(prop={'size': 15})
ax.grid(True)
plt.show()
plot_forecasts(train, test, forecast_df, models=['Theta'])
Let’s plot the same graph using the plot function that comes in
Statsforecast
, as shown below.
sf.plot(df, forecast_df, level=[95])
Crossvalidation
In previous steps, we’ve taken our historical data to predict the future. However, to asses its accuracy we would also like to know how the model would have performed in the past. To assess the accuracy and robustness of your models on your data perform CrossValidation.
With time series data, Cross Validation is done by defining a sliding window across the historical data and predicting the period following it. This form of crossvalidation allows us to arrive at a better estimation of our model’s predictive abilities across a wider range of temporal instances while also keeping the data in the training set contiguous as is required by our models.
The following graph depicts such a Cross Validation Strategy:
Perform time series crossvalidation
Crossvalidation of time series models is considered a best practice but most implementations are very slow. The statsforecast library implements crossvalidation as a distributed operation, making the process less timeconsuming to perform. If you have big datasets you can also perform Cross Validation in a distributed cluster using Ray, Dask or Spark.
In this case, we want to evaluate the performance of each model for the
last 5 months (n_windows=5)
, forecasting every second months
(step_size=12)
. Depending on your computer, this step should take
around 1 min.
The cross_validation method from the StatsForecast class takes the following arguments.

df:
training data frame 
h (int):
represents h steps into the future that are being forecasted. In this case, 12 months ahead. 
step_size (int):
step size between each window. In other words: how often do you want to run the forecasting processes. 
n_windows(int):
number of windows used for cross validation. In other words: what number of forecasting processes in the past do you want to evaluate.
crossvalidation_df = sf.cross_validation(df=train,
h=horizon,
step_size=12,
n_windows=3)
The crossvaldation_df object is a new data frame that includes the following columns:
unique_id:
index. If you dont like working with index just run crossvalidation_df.resetindex()ds:
datestamp or temporal indexcutoff:
the last datestamp or temporal index for the n_windows.y:
true value"model":
columns with the model’s name and fitted value.
Evaluate Model
We can now compute the accuracy of the forecast using an appropiate
accuracy metric. Here we’ll use the Root Mean Squared Error (RMSE). To
do this, we first need to install datasetsforecast
, a Python library
developed by Nixtla that includes a function to compute the RMSE.
!pip install datasetsforecast
from datasetsforecast.losses import rmse
The function to compute the RMSE takes two arguments:
 The actual values.
 The forecasts, in this case, Theta.
rmse = rmse(crossvalidation_df['y'], crossvalidation_df["Theta"])
print("RMSE using crossvalidation: ", rmse)
RMSE using crossvalidation: 12.643162
As you have noticed, we have used the cross validation results to perform the evaluation of our model.
Now we are going to evaluate our model with the results of the
predictions, we will use different types of metrics
MAE, MAPE, MASE, RMSE, SMAPE
to evaluate the accuracy
.
from datasetsforecast.losses import mae, mape, mase, rmse, smape
def evaluate_performace(y_hist, y_true, y_pred, model):
y_true = y_true.merge(y_pred, how='left', on=['unique_id', 'ds'])
evaluation = {}
evaluation[model] = {}
for metric in [mase, mae, mape, rmse, smape]:
metric_name = metric.__name__
if metric_name == 'mase':
evaluation[model][metric_name] = metric(y_true['y'].values,
y_true[model].values,
y_hist['y'].values, seasonality=12)
else:
evaluation[model][metric_name] = metric(y_true['y'].values, y_true[model].values)
return pd.DataFrame(evaluation).T
evaluate_performace(train, test, Y_hat, model="Theta")
mae  mape  mase  rmse  smape  

Theta  8.111287  0.964855  0.36478  9.730347  0.965874 
Acknowledgements
We would like to thank Naren Castellon for writing this tutorial.
References
 Jose A. Fiorucci, Tiago R. Pellegrini, Francisco Louzada, Fotios Petropoulos, Anne B. Koehler (2016). “Models for optimising the theta method and their relationship to state space models”. International Journal of Forecasting.
 Nixtla Parameters.
 Pandas available frequencies.
 Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting principles and practice, Time series crossvalidation”..
 Seasonal periods Rob J Hyndman.