Dynamic Optimized Theta Model

Introduction
Dynamic Optimized Theta Model (DOTM)
Loading libraries and data
Explore data with the plot method
Split the data into training and testing
Implementation of DynamicOptimizedTheta with StatsForecast
Cross-validation
Model evaluation
References

Introduction

The Dynamic Optimized Theta Model (DOTM) in StatsForecast is a variation of the classic Theta model. It combines key features of two other extensions: the Optimized Theta Model (OTM) and the Dynamic Standard Theta Model (DSTM). DOTM introduces two main improvements over the standard Theta model: optimization of the theta parameters and dynamic updating of model components over time.

Optimization: Like OTM, this version automatically searches for the best theta values based on the data, rather than relying on fixed parameters. This flexibility allows the model to better adapt to series with complex seasonal or trend patterns.
Dynamic updating: Like DSTM, DOTM continuously updates its internal components as new data becomes available. This makes it well-suited for non-stationary series, where the underlying data structure evolves over time.

DOTM also supports seasonal decomposition, controlled by the decomposition_type parameter. You can choose between: - 'multiplicative' (default), which assumes that seasonal effects scale with the level of the series, or - 'additive', which assumes that seasonal effects remain constant in absolute magnitude. The Dynamic Optimized Theta Model is the most flexible of the Theta family and is particularly effective when forecasting series with changing trends and seasonalities.

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/Serie-de-tiempo-con-Machine-Learning/main/Data/milk_production.csv", usecols=[1,2])
df.head()

	month	production
0	1962-01-01	589
1	1962-02-01	561
2	1962-03-01	640
3	1962-04-01	656
4	1962-05-01	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 YYYY-MM-DD for a date or YYYY-MM-DD 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	1962-01-01	589	1
1	1962-02-01	561	1
2	1962-03-01	640	1
3	1962-04-01	656	1
4	1962-05-01	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)

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 short-term 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

Data to train our Dynamic Optimized Theta Model(DOTM).
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<='1974-12-01'] 
test = df[df.ds>'1974-12-01']

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 DynamicOptimizedTheta with StatsForecast

Load libraries

from statsforecast import StatsForecast
from statsforecast.models import DynamicOptimizedTheta

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

# We call the model that we are going to use
models = [DynamicOptimizedTheta(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 pandas’ 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(models=models, freq='MS')

Fit the Model

sf.fit(df=train)

StatsForecast(models=[DynamicOptimizedTheta])

Let’s see the results of our Dynamic Optimized 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([250.83206219,   0.75624902,   4.67964777]), fn=10.697554045462667, nit=55, simplex=array([[237.42074763,   0.75306547,   4.46023813],
       [250.83206219,   0.75624902,   4.67964777],
       [257.16444246,   0.75229688,   4.42377059],
       [256.90853867,   0.75757957,   4.43171897]]))

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	-18.247106
1	-75.757706
2	6.001494
…	…
153	-59.747044
154	-91.901521
155	-43.503294

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 Q-Q')

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(df=train, h=horizon, fitted=True)
Y_hat

	unique_id	ds	DynamicOptimizedTheta
0	1	1975-01-01	839.259705
1	1	1975-02-01	801.399170
2	1	1975-03-01	895.189148
…	…	…	…
9	1	1975-10-01	821.271240
10	1	1975-11-01	792.530518
11	1	1975-12-01	829.854553

values=sf.forecast_fitted_values()
values.head()

	unique_id	ds	y	DynamicOptimizedTheta
0	1	1962-01-01	589.0	607.247131
1	1	1962-02-01	561.0	636.757690
2	1	1962-03-01	640.0	633.998535
3	1	1962-04-01	656.0	608.461243
4	1	1962-05-01	727.0	604.808899

StatsForecast.plot(values)

Adding 95% confidence interval with the forecast method

sf.forecast(df=train, h=horizon, level=[95])

	unique_id	ds	DynamicOptimizedTheta	DynamicOptimizedTheta-lo-95	DynamicOptimizedTheta-hi-95
0	1	1975-01-01	839.259705	741.952332	955.151001
1	1	1975-02-01	801.399170	641.867920	946.045776
2	1	1975-03-01	895.189148	707.189087	1066.356812
…	…	…	…	…	…
9	1	1975-10-01	821.271240	546.081726	1088.193481
10	1	1975-11-01	792.530518	494.623718	1037.459839
11	1	1975-12-01	829.854553	519.661133	1108.213867

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)

	unique_id	ds	DynamicOptimizedTheta
0	1	1975-01-01	839.259705
1	1	1975-02-01	801.399170
2	1	1975-03-01	895.189148
…	…	…	…
9	1	1975-10-01	821.271240
10	1	1975-11-01	792.530518
11	1	1975-12-01	829.854553

forecast_df = sf.predict(h=horizon, level=[80,95]) 
forecast_df

	unique_id	ds	DynamicOptimizedTheta	DynamicOptimizedTheta-lo-80	DynamicOptimizedTheta-hi-80	DynamicOptimizedTheta-lo-95	DynamicOptimizedTheta-hi-95
0	1	1975-01-01	839.259705	766.142090	928.025513	741.952332	955.151001
1	1	1975-02-01	801.399170	702.981262	899.884216	641.867920	946.045776
2	1	1975-03-01	895.189148	760.125916	1008.335022	707.189087	1066.356812
…	…	…	…	…	…	…	…
9	1	1975-10-01	821.271240	617.391724	996.698364	546.081726	1088.193481
10	1	1975-11-01	792.530518	568.303162	975.070312	494.623718	1037.459839
11	1	1975-12-01	829.854553	598.098267	1035.476196	519.661133	1108.213867

sf.plot(train, test.merge(forecast_df), level=[80, 95])

Cross-validation

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 Cross-Validation. 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 cross-validation 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 cross-validation

Cross-validation of time series models is considered a best practice but most implementations are very slow. The statsforecast library implements cross-validation as a distributed operation, making the process less time-consuming 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: series identifier
ds: datestamp or temporal index
cutoff: the last datestamp or temporal index for the n_windows.
y: true value
"model": columns with the model’s name and fitted value.

crossvalidation_df

	unique_id	ds	cutoff	y	DynamicOptimizedTheta
0	1	1972-01-01	1971-12-01	826.0	828.692017
1	1	1972-02-01	1971-12-01	799.0	792.444092
2	1	1972-03-01	1971-12-01	890.0	883.122620
…	…	…	…	…	…
33	1	1974-10-01	1973-12-01	812.0	810.304688
34	1	1974-11-01	1973-12-01	773.0	781.804688
35	1	1974-12-01	1973-12-01	813.0	818.811096

Model Evaluation

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 functools import partial

import utilsforecast.losses as ufl
from utilsforecast.evaluation import evaluate

evaluate(
    test.merge(Y_hat),
    metrics=[ufl.mae, ufl.mape, partial(ufl.mase, seasonality=season_length), ufl.rmse, ufl.smape],
    train_df=train,
)

	unique_id	metric	DynamicOptimizedTheta
0	1	mae	6.861949
1	1	mape	0.008045
2	1	mase	0.308595
3	1	rmse	8.647459
4	1	smape	0.004010

Getting Started

Tutorials

How to Guides

Distributed

Experiments

Model References

API Reference

Contributing

Dynamic Optimized Theta Model

Table of Contents

Introduction

Loading libraries and data

Read Data

Explore Data with the plot method

Autocorrelation plots

Decomposition of the time series

Additive time series

Multiplicative time series

Additive

Multiplicative

Split the data into training and testing

Implementation of DynamicOptimizedTheta with StatsForecast

Load libraries

Instantiating Model

Fit the Model

Forecast Method

Predict method with confidence interval

Cross-validation

Perform time series cross-validation

Model Evaluation

References

Getting Started

Tutorials

How to Guides

Distributed

Experiments

Model References

API Reference

Contributing

​Table of Contents

​Introduction

​Loading libraries and data

​Read Data

​Explore Data with the plot method

​Autocorrelation plots

​Decomposition of the time series

​Additive time series

​Multiplicative time series

​Additive

​Multiplicative

​Split the data into training and testing

​Implementation of DynamicOptimizedTheta with StatsForecast

​Load libraries

​Instantiating Model

​Fit the Model

​Forecast Method

​Predict method with confidence interval

​Cross-validation

​Perform time series cross-validation

​Model Evaluation

​References

Table of Contents

Introduction

Loading libraries and data

Read Data

Explore Data with the plot method

Autocorrelation plots

Decomposition of the time series

Additive time series

Multiplicative time series

Additive

Multiplicative

Split the data into training and testing

Implementation of DynamicOptimizedTheta with StatsForecast

Load libraries

Instantiating Model

Fit the Model

Forecast Method

Predict method with confidence interval

Cross-validation

Perform time series cross-validation

Model Evaluation

References