> ## Documentation Index
> Fetch the complete documentation index at: https://nixtlaverse.nixtla.io/llms.txt
> Use this file to discover all available pages before exploring further.

# Dynamic Standard Theta Model

> Step-by-step guide on using the `DynamicStandardTheta Model` with
> `Statsforecast`.

During this walkthrough, we will become familiar with the main
`StatsForecast` class and some relevant methods such as
`StatsForecast.plot`, `StatsForecast.forecast` and
`StatsForecast.cross_validation` in other.

The text in this article is largely taken from [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](https://www.sciencedirect.com/science/article/pii/S0169207016300243).

## Table of Contents

* [Dynamic Standard Theta Model (DOTM)](#model)
* [Loading libraries and data](#loading)
* [Explore data with the plot method](#plotting)
* [Split the data into training and testing](#splitting)
* [Implementation of DynamicStandardTheta with
  StatsForecast](#implementation)
* [Cross-validation](#cross_validate)
* [Model evaluation](#evaluate)
* [References](#references)

## Dynamic Standard Theta Models (DSTM) <a class="anchor" id="model" />

The Dynamic Standard Theta Model is a case-specific variation of the
[Optimized Dynamic Theta Model](./dynamicoptimizedtheta.html).

Also, for $\theta=2$, we have a stochastic approach of Theta, which is
referred to hereafter as the dynamic standard Theta model (DSTM).

## Loading libraries and data <a class="anchor" id="loading" />

> **Tip**
>
> Statsforecast will be needed. To install, see
> [instructions](../getting-started/installation.html).

Next, we import plotting libraries and configure the plotting style.

```python theme={null}
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

```python theme={null}
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.

```python theme={null}
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          |

```python theme={null}
print(df.dtypes)
```

```text theme={null}
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

```python theme={null}
df["ds"] = pd.to_datetime(df["ds"])
```

## Explore Data with the plot method <a class="anchor" id="plotting" />

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.

```python theme={null}
from statsforecast import StatsForecast

StatsForecast.plot(df)
```

<img src="https://mintcdn.com/nixtla/A8PLgF1BjgyAtVfE/statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-8-output-1.png?fit=max&auto=format&n=A8PLgF1BjgyAtVfE&q=85&s=d51d832e072bc0893403aba3587a6193" alt="" width="1710" height="361" data-path="statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-8-output-1.png" />

### Autocorrelation plots

```python theme={null}
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();
```

<img src="https://mintcdn.com/nixtla/A8PLgF1BjgyAtVfE/statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-9-output-1.png?fit=max&auto=format&n=A8PLgF1BjgyAtVfE&q=85&s=918e10eaefd52eaf648bd1c52d795fd2" alt="" width="1475" height="610" data-path="statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-9-output-1.png" />

### 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

```python theme={null}
from statsmodels.tsa.seasonal import seasonal_decompose
a = seasonal_decompose(df["y"], model = "additive", period=12)
a.plot();
```

<img src="https://mintcdn.com/nixtla/GiexohgQnWrBlEOB/statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-10-output-1.png?fit=max&auto=format&n=GiexohgQnWrBlEOB&q=85&s=141be3805faa2e2bfb1291d0cef0dc72" alt="" width="1784" height="684" data-path="statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-10-output-1.png" />

### Multiplicative

```python theme={null}
from statsmodels.tsa.seasonal import seasonal_decompose
a = seasonal_decompose(df["y"], model = "Multiplicative", period=12)
a.plot();
```

<img src="https://mintcdn.com/nixtla/A8PLgF1BjgyAtVfE/statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-11-output-1.png?fit=max&auto=format&n=A8PLgF1BjgyAtVfE&q=85&s=25074df0de3799b0b505a47b4b6c6dc1" alt="" width="1784" height="684" data-path="statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-11-output-1.png" />

## Split the data into training and testing<a class="anchor" id="splitting" />

Let’s divide our data into sets 1. Data to train our
`Dynamic Standard 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.

```python theme={null}
train = df[df.ds<='1974-12-01']
test = df[df.ds>'1974-12-01']
```

```python theme={null}
train.shape, test.shape
```

```text theme={null}
((156, 3), (12, 3))
```

Now let’s plot the training data and the test data.

```python theme={null}
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()
```

<img src="https://mintcdn.com/nixtla/A8PLgF1BjgyAtVfE/statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-14-output-1.png?fit=max&auto=format&n=A8PLgF1BjgyAtVfE&q=85&s=5e453e29e48a4e5fbcbac503bb97f6d3" alt="" width="1479" height="633" data-path="statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-14-output-1.png" />

## Implementation of DynamicStandardTheta with StatsForecast <a class="anchor" id="implementation" />

### Load libraries

```python theme={null}
from statsforecast import StatsForecast
from statsforecast.models import DynamicTheta
```

### Instantiating Model

Import and instantiate the models. Setting the argument is sometimes
tricky. This article on [Seasonal
periods](https://robjhyndman.com/hyndsight/seasonal-periods/) by the
master, Rob Hyndmann, can be useful for `season_length`.

```python theme={null}
season_length = 12 # Monthly data
horizon = len(test) # number of predictions

models = [DynamicTheta(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](https://pandas.pydata.org/pandas-docs/stable/user_guide/timeseries.html#offset-aliases).)

* `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.

```python theme={null}
sf = StatsForecast(models=models, freq='MS')
```

### Fit Model

```python theme={null}
sf.fit(df=train)
```

```text theme={null}
StatsForecast(models=[DynamicTheta])
```

Let’s see the results of our `Dynamic Standard Theta model`. We can
observe it with the following instruction:

```python theme={null}
result=sf.fitted_[0,0].model_
print(result.keys())
print(result['fit'])
```

```text theme={null}
dict_keys(['mse', 'amse', 'fit', 'residuals', 'm', 'states', 'par', 'n', 'modeltype', 'mean_y', 'decompose', 'decomposition_type', 'seas_forecast', 'fitted'])
results(x=array([393.28739991,   0.76875   ]), fn=10.787112115489622, nit=20, simplex=array([[399.92916541,   0.771875  ],
       [393.28739991,   0.76875   ],
       [384.74798713,   0.771875  ]]))
```

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()`.

```python theme={null}
residual=pd.DataFrame(result.get("residuals"), columns=["residual Model"])
residual
```

|     | residual Model |
| --- | -------------- |
| 0   | -18.247131     |
| 1   | -46.247131     |
| 2   | 17.140198      |
| ... | ...            |
| 153 | -58.941711     |
| 154 | -91.055420     |
| 155 | -42.624939     |

```python theme={null}
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();
```

<img src="https://mintcdn.com/nixtla/A8PLgF1BjgyAtVfE/statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-21-output-1.png?fit=max&auto=format&n=A8PLgF1BjgyAtVfE&q=85&s=355c5c3b1633c038c5f5a707115823c1" alt="" width="1502" height="633" data-path="statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-21-output-1.png" />

### 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.

```python theme={null}
Y_hat = sf.forecast(df=train, h=horizon, fitted=True)
Y_hat
```

|     | unique\_id | ds         | DynamicTheta |
| --- | ---------- | ---------- | ------------ |
| 0   | 1          | 1975-01-01 | 838.531555   |
| 1   | 1          | 1975-02-01 | 800.154968   |
| 2   | 1          | 1975-03-01 | 893.430786   |
| ... | ...        | ...        | ...          |
| 9   | 1          | 1975-10-01 | 815.959351   |
| 10  | 1          | 1975-11-01 | 786.716431   |
| 11  | 1          | 1975-12-01 | 823.539368   |

```python theme={null}
values=sf.forecast_fitted_values()
values.head()
```

|   | unique\_id | ds         | y     | DynamicTheta |
| - | ---------- | ---------- | ----- | ------------ |
| 0 | 1          | 1962-01-01 | 589.0 | 607.247131   |
| 1 | 1          | 1962-02-01 | 561.0 | 607.247131   |
| 2 | 1          | 1962-03-01 | 640.0 | 622.859802   |
| 3 | 1          | 1962-04-01 | 656.0 | 606.987793   |
| 4 | 1          | 1962-05-01 | 727.0 | 605.021179   |

```python theme={null}
StatsForecast.plot(values)
```

<img src="https://mintcdn.com/nixtla/A8PLgF1BjgyAtVfE/statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-24-output-1.png?fit=max&auto=format&n=A8PLgF1BjgyAtVfE&q=85&s=11b5adafe55bad74c40e949437ad6bd4" alt="" width="1710" height="361" data-path="statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-24-output-1.png" />

Adding 95% confidence interval with the forecast method

```python theme={null}
sf.forecast(df=train, h=horizon, level=[95])
```

|     | unique\_id | ds         | DynamicTheta | DynamicTheta-lo-95 | DynamicTheta-hi-95 |
| --- | ---------- | ---------- | ------------ | ------------------ | ------------------ |
| 0   | 1          | 1975-01-01 | 838.531555   | 741.237366         | 954.407166         |
| 1   | 1          | 1975-02-01 | 800.154968   | 640.697205         | 945.673096         |
| 2   | 1          | 1975-03-01 | 893.430786   | 703.900635         | 1065.418701        |
| ... | ...        | ...        | ...          | ...                | ...                |
| 9   | 1          | 1975-10-01 | 815.959351   | 536.422791         | 1086.643433        |
| 10  | 1          | 1975-11-01 | 786.716431   | 484.476593         | 1033.687134        |
| 11  | 1          | 1975-12-01 | 823.539368   | 509.187256         | 1104.107788        |

### 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.

```python theme={null}
sf.predict(h=horizon)
```

|     | unique\_id | ds         | DynamicTheta |
| --- | ---------- | ---------- | ------------ |
| 0   | 1          | 1975-01-01 | 838.531555   |
| 1   | 1          | 1975-02-01 | 800.154968   |
| 2   | 1          | 1975-03-01 | 893.430786   |
| ... | ...        | ...        | ...          |
| 9   | 1          | 1975-10-01 | 815.959351   |
| 10  | 1          | 1975-11-01 | 786.716431   |
| 11  | 1          | 1975-12-01 | 823.539368   |

```python theme={null}
forecast_df = sf.predict(h=horizon, level=[80,95])
forecast_df
```

|     | unique\_id | ds         | DynamicTheta | DynamicTheta-lo-80 | DynamicTheta-hi-80 | DynamicTheta-lo-95 | DynamicTheta-hi-95 |
| --- | ---------- | ---------- | ------------ | ------------------ | ------------------ | ------------------ | ------------------ |
| 0   | 1          | 1975-01-01 | 838.531555   | 765.423828         | 927.285339         | 741.237366         | 954.407166         |
| 1   | 1          | 1975-02-01 | 800.154968   | 701.099854         | 899.316162         | 640.697205         | 945.673096         |
| 2   | 1          | 1975-03-01 | 893.430786   | 758.326416         | 1007.631165        | 703.900635         | 1065.418701        |
| ... | ...        | ...        | ...          | ...                | ...                | ...                | ...                |
| 9   | 1          | 1975-10-01 | 815.959351   | 608.699463         | 992.552673         | 536.422791         | 1086.643433        |
| 10  | 1          | 1975-11-01 | 786.716431   | 558.429810         | 970.648376         | 484.476593         | 1033.687134        |
| 11  | 1          | 1975-12-01 | 823.539368   | 588.706787         | 1031.564941        | 509.187256         | 1104.107788        |

```python theme={null}
sf.plot(train, test.merge(forecast_df), level=[80, 95])
```

<img src="https://mintcdn.com/nixtla/A8PLgF1BjgyAtVfE/statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-28-output-1.png?fit=max&auto=format&n=A8PLgF1BjgyAtVfE&q=85&s=e6aa6654756bcbe8d61226b156aa03e6" alt="" width="1898" height="361" data-path="statsforecast/docs/models/DynamicStandardTheta_files/figure-markdown_strict/cell-28-output-1.png" />

## Cross-validation <a class="anchor" id="cross_validate" />

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:

![](https://raw.githubusercontent.com/Nixtla/statsforecast/main/nbs/imgs/ChainedWindows.gif)

### 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.

```python theme={null}
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 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.

```python theme={null}
crossvalidation_df
```

|     | unique\_id | ds         | cutoff     | y     | DynamicTheta |
| --- | ---------- | ---------- | ---------- | ----- | ------------ |
| 0   | 1          | 1972-01-01 | 1971-12-01 | 826.0 | 827.107239   |
| 1   | 1          | 1972-02-01 | 1971-12-01 | 799.0 | 789.924194   |
| 2   | 1          | 1972-03-01 | 1971-12-01 | 890.0 | 879.664429   |
| ... | ...        | ...        | ...        | ...   | ...          |
| 33  | 1          | 1974-10-01 | 1973-12-01 | 812.0 | 804.398560   |
| 34  | 1          | 1974-11-01 | 1973-12-01 | 773.0 | 775.329285   |
| 35  | 1          | 1974-12-01 | 1973-12-01 | 813.0 | 811.767639   |

## Model Evaluation <a class="anchor" id="evaluate" />

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.

```python theme={null}
from functools import partial

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

```python theme={null}
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 | DynamicTheta |
| - | ---------- | ------ | ------------ |
| 0 | 1          | mae    | 8.182119     |
| 1 | 1          | mape   | 0.009736     |
| 2 | 1          | mase   | 0.367965     |
| 3 | 1          | rmse   | 9.817624     |
| 4 | 1          | smape  | 0.004874     |

## References <a class="anchor" id="references" />

1. [Kostas I. Nikolopoulos, Dimitrios D. Thomakos. Forecasting with the
   Theta Method-Theory and Applications. 2019 John Wiley & Sons
   Ltd.](https://onlinelibrary.wiley.com/doi/book/10.1002/9781119320784)
2. [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](https://www.sciencedirect.com/science/article/pii/S0169207016300243).
3. [Nixtla DynamicTheta API](../../src/core/models.html#dynamictheta)
4. [Pandas available
   frequencies](https://pandas.pydata.org/pandas-docs/stable/user_guide/timeseries.html#offset-aliases).
5. [Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting
   principles and practice, Time series
   cross-validation”.](https://otexts.com/fpp3/tscv.html).
6. [Seasonal periods- Rob J
   Hyndman](https://robjhyndman.com/hyndsight/seasonal-periods/).