> ## 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.

# Optimized Theta Model

> Step-by-step guide on using the `OptimizedTheta 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: 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).

## Table of Contents

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

## Introduction <a class="anchor" id="introduction" />

The optimized Theta model is a time series forecasting method that is
based on the decomposition of the time series into three components:
trend, seasonality and noise. The model then forecasts the long-term
trend and seasonality, and uses the noise to adjust the short-term
forecasts. The optimized Theta model has been shown to be more accurate
than other time series forecasting methods, especially for time series
with complex trends and seasonality.

The optimized Theta model was developed by Athanasios N. Antoniadis and
Nikolaos D. Tsonis in 2013. The model is based on the Theta forecasting
method, which was developed by George E. P. Box and Gwilym M. Jenkins in
1976\. Theta method is a time series forecasting method that is based on
the decomposition of the time series into three components: trend,
seasonality, and noise. The Theta model then forecasts the long-term
trend and seasonality, and uses the noise to adjust the short-term
forecasts.

The Theta Optimized model improves on the Theta method by using an
optimization algorithm to find the best parameters for the model. The
optimization algorithm is based on the Akaike loss function (AIC), which
is a measure of the goodness of fit of a model to the data. The
optimization algorithm looks for the parameters that minimize the AIC
function.

The optimized Theta model has been shown to be more accurate than other
time series forecasting methods, especially for time series with complex
trends and seasonality. The model has been used to forecast a variety of
time series, including sales, production, prices, and weather.

Below are some of the benefits of the optimized Theta model:

* It is more accurate than other time series forecasting methods.
* It’s easy to use.
* Can be used to forecast a variety of time series.
* It is flexible and can be adapted to different scenarios.

If you are looking for an easy-to-use and accurate time series
forecasting method, the Optimized Theta model is a good choice.

The optimized Theta model can be applied in a variety of areas,
including:

* **Sales:** The optimized Theta model can be used to forecast sales
  of products or services. This can help companies make decisions
  about production, inventory, and marketing.
* **Production:** The optimized Theta model can be used to forecast
  the production of goods or services. This can help companies ensure
  they have the capacity to meet demand and avoid overproduction.
* **Prices:** The optimized Theta model can be used to forecast the
  prices of goods or services. This can help companies make decisions
  about pricing and marketing strategy.
* **Weather:** The optimized Theta model can be used to forecast the
  weather. This can help companies make decisions about agricultural
  production, travel planning and risk management.
* **Other:** The optimized Theta model can also be used to forecast
  other types of time series, including traffic, energy demand, and
  population.

The Optimized Theta model is a powerful tool that can be used to improve
the accuracy of time series forecasts. It is easy to use and can be
applied to a variety of areas. If you are looking for a tool to improve
your time series forecasts, the Optimized Theta model is a good choice.

## Optimized Theta Model (OTM) <a class="anchor" id="model" />

Assume that either the time series $Y_1, \cdots Y_n$ is non-seasonal or
it has been seasonally adjusted using the multiplicative classical
decomposition approach.

Let $X_t$ be the linear combination of two theta lines,

$$
\begin{equation}
X_t=\omega \text{Z}_t (\theta_1) +(1-\omega) \text{Z}_t (\theta_2) \tag 1
\end{equation}
$$

where $\omega \in [0,1]$ is the weight parameter. Assuming that
$\theta_1 <1$ and $\theta_2 \geq 1$, the weight $\omega$ can be derived
as

$$
\begin{equation}
\omega:=\omega(\theta_1, \theta_2)=\frac{\theta_2 -1}{\theta_2 -\theta_1} \tag 2
\end{equation}
$$

It is straightforward to see from Eqs. (1), (2) that
$X_t=Y_t, \ t=1, \cdots n$ i.e., the weights are calculated properly in
such a way that Eq. (1) reproduces the original series.

**Theorem 1:** Let $\theta_1 <1$ and $\theta_2 \geq 1$. We will prove
that

1. the linear system given by $X_t=Y_t$ for all $t=1, \cdots, n$, where
   $X_t$ is given by Eq.(4), has the single solution

$\omega= (\theta_2 -1)/(\theta_2 - \theta_1)$

1. the error of choosing a non-optimal weight
   $\omega_{\delta} =\omega + \delta$ is proportional to the error for
   a simple linear regression model.

In Theorem 1 , we prove that the solution is unique and that the error
from not choosing the optimal weights ($\omega$ and $1-\omega$) s
proportional to the error of a linear regression model. As a
consequence, the STheta method is given simply by setting $\theta_1=0$
and $\theta_2=2$ while from Eq. (2) we get $\omega=0.5$. Thus, Eqs. (1),
(2) allow us to construct a generalisation of the Theta model that
maintains the re-composition propriety of the original time series for
any theta lines $\text{Z}_t (\theta_1)$ and $\text{Z}_t (\theta_2)$.

In order to maintain the modelling of the long-term component and retain
a fair comparison with the STheta method, in this work we fix
$\theta_1=0$ and focus on the optimisation of the short-term component,
$\theta_2=0$ with $\theta \geq 1$. Thus, $\theta$ is the only parameter
that requires estimation so far. The theta decomposition is now given by

$Y_t=(1-\frac{1}{\theta}) (\text{A}_n+\text{B}_n t)+ \frac{1}{\theta} \text{Z}_t (\theta), \ t=1, \cdots , n$

The $h$ -step-ahead forecasts calculated at origin are given by

$$
\begin{equation}
\hat Y_{n+h|n} = (1-\frac{1}{\theta}) [\text{A}_n+\text{B}_n (n+h)]+ \frac{1}{\theta} \tilde {\text{Z}}_{n+h|n} (\theta) \tag 3
\end{equation}
$$

where
$\tilde {\text{Z}}_{n+h|n} (\theta)=\tilde {\text{Z}}_{n+1|n} (\theta)=\alpha \sum_{i=0}^{n-1}(1-\alpha)^i \text{Z}_{n-i}(\theta)+(1-\alpha)^n \ell_{0}^{*}$
is the extrapolation of $\text{Z}_t(\theta)$ by an SES model with
$\ell_{0}^{*} \in \mathbb{R}$ as the initial level parameter and
$\alpha \in (0,1)$ as the smoothing parameter. Note that for $\theta=2$
Eq. (3) corresponds to Step 4 of the STheta algorithm. After some
algebra, we can write

$$
\begin{equation}
\tilde {\text{Z}}_{n+1|n} (\theta)=\theta \ell{n}+(1-\theta) \{ \text{A}_n [1-(1-\alpha)^n] + \text{B}_n [n+(1-\frac{1}{\alpha}) [1-(1-\alpha)^n] ]  \}   \tag 4
\end{equation}
$$

where $\ell_{t}=\alpha Y_t +(1-\alpha) \ell_{t-1}$ for $t=1, \cdots, n$
and $\ell_{0}=\ell_{0}^{*}/\theta$.

In the light of Eqs. (3), (4), we suggest four stochastic approaches.
These approaches differ due to the parameter $\theta$ which may be
either fixed at two or optimised, and the coefficients $\text{A}_n$ and
$\text{B}_n$, which can be either fixed or dynamic functions. To
formulate the state space models, it is helpful to adopt $\mu_{t}$ as
the one-step-ahead forecast at origin $t-1$ and $\varepsilon_{t}$ as the
respective additive error, i.e., $\varepsilon_{t}=Y_t - \mu_{t}$ if
$\mu_{t}= \hat Y_{t|t-1}$. We assume $\{ \varepsilon_{t} \}$ to be a
Gaussian white noise process with mean zero and variance $\sigma^2$.

### More on Optimised Theta models

Let $\text{A}_n$ and $\text{B}_n$ be fixed coefficients for all
$t=1, \cdots, n$ so that Eqs. (3), (4) configure the state space model
given by

$$
\begin{equation}
Y_t=\mu_{t}+\varepsilon_{t} \tag 5
\end{equation}
$$

$$
\begin{equation}
\mu_{t}=\ell_{t-1}+(1-\frac{1}{\theta}) \{(1-\alpha)^{t-1} \text{A}_n +[\frac{1-(1-\alpha)^t}{\alpha} \text{B}_n]  \tag 6
\end{equation}
$$

$$
\begin{equation}
\ell_{t}=\alpha Y_t +(1-\alpha)\ell_{t-1} \tag 7
\end{equation}
$$

with parameters $\ell_{0} \in \mathbb{R}$, $\alpha \in (0,1)$ and
$\theta \in [1,\infty)$ . The parameter $\theta$ is to be estimated
along with $\alpha$ and $\ell_{0}$ We call this the optimised Theta
model (OTM).

The $h$-step-ahead forecast at origin $n$ is given by

$\hat Y_{n+h|n}=E[Y_{n+h}|Y_1,\cdots, Y_n]=\ell_{n}+(1-\frac{1}{\theta}) \{(1-\alpha)^n \text{A}_n +[(h-1) + \frac{1-(1-\alpha)^{n+1}}{\alpha}] \text{B}_n \}$

which is equivalent to Eq. (3). The conditional variance
$\text{Var}[Y_{n+h}|Y_1, \cdots, Y_n]=[1+(h-1)\alpha^2]\sigma^2$ can be
computed easily from the state space model. Thus, the $(1-\alpha)\%$
prediction interval for $Y_{n+h}$ is given by
$\hat Y_{n+h|n} \ \pm  \ q_{1-\alpha/2} \sqrt{[1+(h-1)\alpha^2 ]\sigma^2 }$

For $\theta=2$ OTM reproduces the forecasts of the STheta method;
hereafter, we will refer to this particular case as the standard Theta
model (STM).

**Theorem 2:** The SES-d $(\ell_{0}^{**}, \alpha, b)$ model, where
$\ell_{0}^{**} \in \mathbb{R}, \alpha \in (0,1)$ and $b \in \mathbb{R}$
is equivalent to $\text{OTM} (\ell_{0}, \alpha, \theta )$ where
$\ell_{0} \in \mathbb{R}$ and $\theta \geq 1$, if

$\ell_{0}^{**} = \ell_{0} + (1- \frac{1}{\theta} )A_n \ \ and \ \ b=(1-\frac{1}{\theta} )B_n$

In Theorem 2, we show that OTM is mathematically equivalent to the SES-d
model. As a corollary of Theorem 2, STM is mathematically equivalent to
SES-d with $b=\frac{1}{2} \text{B}_n$. Therefore, for $\theta=2$ the
corollary also re-confirms the H\&B result on the relationship between
STheta and the SES-d model.

## 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/cWAHyBcGjSWwLJ1B/statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-8-output-1.png?fit=max&auto=format&n=cWAHyBcGjSWwLJ1B&q=85&s=aec456cc260c86ef0368d4ef6f0b7eb0" alt="" width="1710" height="361" data-path="statsforecast/docs/models/OptimizedTheta_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/cWAHyBcGjSWwLJ1B/statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-9-output-1.png?fit=max&auto=format&n=cWAHyBcGjSWwLJ1B&q=85&s=83e7cd152664875c395d822fe3f59ca8" alt="" width="1475" height="610" data-path="statsforecast/docs/models/OptimizedTheta_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/CTL3io3Vj231Jv4v/statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-10-output-1.png?fit=max&auto=format&n=CTL3io3Vj231Jv4v&q=85&s=38b2fc9f27900d87045b2fba0432275f" alt="" width="1784" height="684" data-path="statsforecast/docs/models/OptimizedTheta_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/cWAHyBcGjSWwLJ1B/statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-11-output-1.png?fit=max&auto=format&n=cWAHyBcGjSWwLJ1B&q=85&s=49a9619bfb07acee7edcbe0a96d13782" alt="" width="1784" height="684" data-path="statsforecast/docs/models/OptimizedTheta_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
`Optimized 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))
```

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

### Load libraries

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

### 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 = [OptimizedTheta(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 the Model

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

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

Let’s see the results of our `Optimized Theta Model (OTM)`. 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([-83.14191626,   0.73681394,  12.45013763]), fn=10.448217519858636, nit=47, simplex=array([[-58.73988124,   0.7441127 ,  11.69842922],
       [-49.97233449,   0.73580297,  11.41787513],
       [-83.14191626,   0.73681394,  12.45013763],
       [-77.04867427,   0.73498431,  11.99254037]]))
```

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   | -271.899414    |
| 1   | -114.671692    |
| 2   | 4.768066       |
| ... | ...            |
| 153 | -60.233887     |
| 154 | -92.472839     |
| 155 | -44.143982     |

```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/cWAHyBcGjSWwLJ1B/statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-20-output-1.png?fit=max&auto=format&n=cWAHyBcGjSWwLJ1B&q=85&s=783768143e51c6699642d17f10f50b30" alt="" width="1494" height="633" data-path="statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-20-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. (If you want to speed things up to a couple of seconds,
remove the AutoModels like `ARIMA` and `Theta`)

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

|     | unique\_id | ds         | OptimizedTheta |
| --- | ---------- | ---------- | -------------- |
| 0   | 1          | 1975-01-01 | 839.682800     |
| 1   | 1          | 1975-02-01 | 802.071838     |
| 2   | 1          | 1975-03-01 | 896.117126     |
| ... | ...        | ...        | ...            |
| 9   | 1          | 1975-10-01 | 824.135498     |
| 10  | 1          | 1975-11-01 | 795.691223     |
| 11  | 1          | 1975-12-01 | 833.316345     |

Let’s visualize the fitted values

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

|   | unique\_id | ds         | y     | OptimizedTheta |
| - | ---------- | ---------- | ----- | -------------- |
| 0 | 1          | 1962-01-01 | 589.0 | 860.899414     |
| 1 | 1          | 1962-02-01 | 561.0 | 675.671692     |
| 2 | 1          | 1962-03-01 | 640.0 | 635.231934     |
| 3 | 1          | 1962-04-01 | 656.0 | 614.731323     |
| 4 | 1          | 1962-05-01 | 727.0 | 609.770752     |

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

<img src="https://mintcdn.com/nixtla/cWAHyBcGjSWwLJ1B/statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-23-output-1.png?fit=max&auto=format&n=cWAHyBcGjSWwLJ1B&q=85&s=20d999a80801b4cdcf5edae8451fffa1" alt="" width="1710" height="361" data-path="statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-23-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         | OptimizedTheta | OptimizedTheta-lo-95 | OptimizedTheta-hi-95 |
| --- | ---------- | ---------- | -------------- | -------------------- | -------------------- |
| 0   | 1          | 1975-01-01 | 839.682800     | 742.509583           | 955.414307           |
| 1   | 1          | 1975-02-01 | 802.071838     | 643.581360           | 945.119202           |
| 2   | 1          | 1975-03-01 | 896.117126     | 710.785095           | 1065.057495          |
| ... | ...        | ...        | ...            | ...                  | ...                  |
| 9   | 1          | 1975-10-01 | 824.135498     | 555.948669           | 1084.320190          |
| 10  | 1          | 1975-11-01 | 795.691223     | 503.147858           | 1036.519531          |
| 11  | 1          | 1975-12-01 | 833.316345     | 530.259705           | 1106.636597          |

```python theme={null}
sf.plot(train, Y_hat)
```

<img src="https://mintcdn.com/nixtla/cWAHyBcGjSWwLJ1B/statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-25-output-1.png?fit=max&auto=format&n=cWAHyBcGjSWwLJ1B&q=85&s=f5956118521b9c93cb0040698489c086" alt="" width="1832" height="361" data-path="statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-25-output-1.png" />

### 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         | OptimizedTheta |
| --- | ---------- | ---------- | -------------- |
| 0   | 1          | 1975-01-01 | 839.682800     |
| 1   | 1          | 1975-02-01 | 802.071838     |
| 2   | 1          | 1975-03-01 | 896.117126     |
| ... | ...        | ...        | ...            |
| 9   | 1          | 1975-10-01 | 824.135498     |
| 10  | 1          | 1975-11-01 | 795.691223     |
| 11  | 1          | 1975-12-01 | 833.316345     |

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

|     | unique\_id | ds         | OptimizedTheta | OptimizedTheta-lo-80 | OptimizedTheta-hi-80 | OptimizedTheta-lo-95 | OptimizedTheta-hi-95 |
| --- | ---------- | ---------- | -------------- | -------------------- | -------------------- | -------------------- | -------------------- |
| 0   | 1          | 1975-01-01 | 839.682800     | 766.665955           | 928.326172           | 742.509583           | 955.414307           |
| 1   | 1          | 1975-02-01 | 802.071838     | 704.290039           | 899.335815           | 643.581360           | 945.119202           |
| 2   | 1          | 1975-03-01 | 896.117126     | 761.334778           | 1007.408447          | 710.785095           | 1065.057495          |
| ... | ...        | ...        | ...            | ...                  | ...                  | ...                  | ...                  |
| 9   | 1          | 1975-10-01 | 824.135498     | 623.903992           | 996.567200           | 555.948669           | 1084.320190          |
| 10  | 1          | 1975-11-01 | 795.691223     | 576.546570           | 975.490784           | 503.147858           | 1036.519531          |
| 11  | 1          | 1975-12-01 | 833.316345     | 606.713623           | 1033.885742          | 530.259705           | 1106.636597          |

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

<img src="https://mintcdn.com/nixtla/cWAHyBcGjSWwLJ1B/statsforecast/docs/models/OptimizedTheta_files/figure-markdown_strict/cell-28-output-1.png?fit=max&auto=format&n=cWAHyBcGjSWwLJ1B&q=85&s=ed4ebe4366b29ccf6579c72d531d59dc" alt="" width="1910" height="361" data-path="statsforecast/docs/models/OptimizedTheta_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     | OptimizedTheta |
| --- | ---------- | ---------- | ---------- | ----- | -------------- |
| 0   | 1          | 1972-01-01 | 1971-12-01 | 826.0 | 828.836365     |
| 1   | 1          | 1972-02-01 | 1971-12-01 | 799.0 | 792.592346     |
| 2   | 1          | 1972-03-01 | 1971-12-01 | 890.0 | 883.269592     |
| ... | ...        | ...        | ...        | ...   | ...            |
| 33  | 1          | 1974-10-01 | 1973-12-01 | 812.0 | 812.183838     |
| 34  | 1          | 1974-11-01 | 1973-12-01 | 773.0 | 783.898376     |
| 35  | 1          | 1974-12-01 | 1973-12-01 | 813.0 | 821.124329     |

## 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 | OptimizedTheta |
| - | ---------- | ------ | -------------- |
| 0 | 1          | mae    | 6.740204       |
| 1 | 1          | mape   | 0.007828       |
| 2 | 1          | mase   | 0.303120       |
| 3 | 1          | rmse   | 8.701501       |
| 4 | 1          | smape  | 0.003893       |

## 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 OptimizedTheta
   API](../../src/core/models.html#optimizedtheta)
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 (3rd
   ed)”](https://otexts.com/fpp3/tscv.html).
6. [Seasonal periods- Rob J
   Hyndman](https://robjhyndman.com/hyndsight/seasonal-periods/).