> ## 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.
# Seasonal Exponential Smoothing Optimized Model
> Step-by-step guide on using the
> `SeasonalExponentialSmoothingOptimized 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. [Changquan Huang • Alla Petukhina. Springer series (2022). Applied
Time Series Analysis and Forecasting with
Python.](https://link.springer.com/book/10.1007/978-3-031-13584-2)
2. Ivan Svetunkov. [Forecasting and Analytics with the Augmented
Dynamic Adaptive Model (ADAM)](https://openforecast.org/adam/)
3. [James D. Hamilton. Time Series Analysis Princeton University Press,
Princeton, New Jersey, 1st Edition,
1994.](https://press.princeton.edu/books/hardcover/9780691042893/time-series-analysis)
4. [Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting
Principles and Practice (3rd
ed)”](https://otexts.com/fpp3/tscv.html).
## Table of Contents
* [Introduction](#introduction)
* [Seasonal Exponential Smoothing Optimized Model](#model)
* [Loading libraries and data](#loading)
* [Explore data with the plot method](#plotting)
* [Split the data into training and testing](#splitting)
* [Implementation of SeasonalExponentialSmoothingOptimized with
StatsForecast](#implementation)
* [Cross-validation](#cross_validate)
* [Model evaluation](#evaluate)
* [References](#references)
## Introduction
The Seasonal Exponential Smoothing Optimized (SESO) model is a
forecasting technique used to predict future values of a time series
that exhibits seasonal patterns. It is a variant of the exponential
smoothing method, which uses a combination of past and predicted values
to generate a prediction.
The SESO algorithm uses an optimization approach to find the optimal
values of the seasonal exponential smoothing parameters. These
parameters include the smoothing coefficients for the levels, trends,
and seasonal components of the time series.
The SESO model is particularly useful for forecasting time series with
pronounced seasonal patterns, such as seasonal product sales or seasonal
temperatures, and many other areas. By using SESO, accurate and useful
forecasts can be generated for business planning and decision making.
## Seasonal Exponential Smoothing Model
The SESO model is based on the exponential smoothing method, which uses
a combination of past and predicted values to generate a prediction. The
mathematical formula for the SESO model is as follows:
$\hat{y}{t+1,s} = \alpha y_t + (1-\alpha) \hat{y}{t-1,s}$
Where: - $\hat{y}{t+1,s}$ is the forecast for the next period of the
season $s$. - $\alpha$ is the smoothing parameter that is optimized by
minimizing the squared error. - $y_t$ is the current observation of
station $s$ in period $t$. - $\hat{y}{t-1,s}$ is the forecast for the
previous period of the season $s$.
The equation indicates that the forecast value for the next season
period $s$ is calculated as a weighted combination of the current
observation and the previous forecast for the same station. The
smoothing parameter $\alpha$ controls the relative influence of these
two terms on the final prediction. A high value of α gives more weight
to the current observation and less weight to the previous forecast,
making the model more sensitive to recent changes in the time series. A
low value of $\alpha$, on the other hand, gives more weight to the
previous forecast and less weight to the current observation, making the
model more stable and smooth.
The optimal value of the smoothing parameter $\alpha$ is determined by
minimizing the squared error between the forecasts generated by the
model and the actual values of the time series.
### Model selection
Model selection in the context of the SESO model refers to the process
of choosing the optimal values of the smoothing parameters and the
seasonal component for the model. The optimal values of these parameters
are the ones that result in the best forecast performance for the given
data set.
A great advantage of the ETS statistical framework is that information
criteria can be used for model selection. The $AIC, AIC_c$ and $BIC$,
that also can be used here to determine which of the ETS models is most
appropriate for a given time series.
For ETS models, Akaike’s Information Criterion (AIC) is defined as
$\text{AIC} = -2\log(L) + 2k,$
where $L$ is the likelihood of the model and $k$ is the total number of
parameters and initial states that have been estimated (including the
residual variance).
The AIC corrected for small sample bias ($AIC_c$) is defined as
$\text{AIC}_{\text{c}} = \text{AIC} + \frac{2k(k+1)}{T-k-1},$
and the Bayesian Information Criterion (BIC) is
$\text{BIC} = \text{AIC} + k[\log(T)-2].$
These criteria balance the goodness of fit with the complexity of the
model and provide a way to choose the model that maximizes the
likelihood of the data while minimizing the number of parameters.
In addition to these techniques, expert judgment and domain knowledge
can also be used to select the optimal SESO model. This involves
considering the underlying dynamics of the time series, the patterns of
seasonality, and any other relevant factors that may influence the
choice of the model.
Overall, the process of model selection for the SESO model involves a
combination of statistical techniques, information criteria, and expert
judgment to identify the optimal values of the smoothing parameters and
the seasonal component that result in the best forecast performance for
the given data set.
## Loading libraries and data
> **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/ads.csv")
df.head()
```
| | Time | Ads |
| - | ------------------- | ------ |
| 0 | 2017-09-13T00:00:00 | 80115 |
| 1 | 2017-09-13T01:00:00 | 79885 |
| 2 | 2017-09-13T02:00:00 | 89325 |
| 3 | 2017-09-13T03:00:00 | 101930 |
| 4 | 2017-09-13T04:00:00 | 121630 |
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 | 2017-09-13T00:00:00 | 80115 | 1 |
| 1 | 2017-09-13T01:00:00 | 79885 | 1 |
| 2 | 2017-09-13T02:00:00 | 89325 | 1 |
| 3 | 2017-09-13T03:00:00 | 101930 | 1 |
| 4 | 2017-09-13T04:00:00 | 121630 | 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
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)
```
### The Augmented Dickey-Fuller Test
An Augmented Dickey-Fuller (ADF) test is a type of statistical test that
determines whether a unit root is present in time series data. Unit
roots can cause unpredictable results in time series analysis. A null
hypothesis is formed in the unit root test to determine how strongly
time series data is affected by a trend. By accepting the null
hypothesis, we accept the evidence that the time series data is not
stationary. By rejecting the null hypothesis or accepting the
alternative hypothesis, we accept the evidence that the time series data
is generated by a stationary process. This process is also known as
stationary trend. The values of the ADF test statistic are negative.
Lower ADF values indicate a stronger rejection of the null hypothesis.
Augmented Dickey-Fuller Test is a common statistical test used to test
whether a given time series is stationary or not. We can achieve this by
defining the null and alternate hypothesis.
Null Hypothesis: Time Series is non-stationary. It gives a
time-dependent trend. Alternate Hypothesis: Time Series is stationary.
In another term, the series doesn’t depend on time.
ADF or t Statistic \< critical values: Reject the null hypothesis, time
series is stationary. ADF or t Statistic > critical values: Failed to
reject the null hypothesis, time series is non-stationary.
```python theme={null}
from statsmodels.tsa.stattools import adfuller
def Augmented_Dickey_Fuller_Test_func(series , column_name):
print (f'Dickey-Fuller test results for columns: {column_name}')
dftest = adfuller(series, autolag='AIC')
dfoutput = pd.Series(dftest[0:4], index=['Test Statistic','p-value','No Lags Used','Number of observations used'])
for key,value in dftest[4].items():
dfoutput['Critical Value (%s)'%key] = value
print (dfoutput)
if dftest[1] <= 0.05:
print("Conclusion:====>")
print("Reject the null hypothesis")
print("The data is stationary")
else:
print("Conclusion:====>")
print("The null hypothesis cannot be rejected")
print("The data is not stationary")
```
```python theme={null}
Augmented_Dickey_Fuller_Test_func(df["y"],'Ads')
```
```text theme={null}
Dickey-Fuller test results for columns: Ads
Test Statistic -7.089634e+00
p-value 4.444804e-10
No Lags Used 9.000000e+00
...
Critical Value (1%) -3.462499e+00
Critical Value (5%) -2.875675e+00
Critical Value (10%) -2.574304e+00
Length: 7, dtype: float64
Conclusion:====>
Reject the null hypothesis
The data is stationary
```
### Autocorrelation plots
The important characteristics of Autocorrelation (ACF) and Partial
Autocorrelation (PACF) are as follows:
Autocorrelation (ACF): 1. Identify patterns of temporal dependence: The
ACF shows the correlation between an observation and its lagged values
at different time intervals. Helps identify patterns of temporal
dependency in a time series, such as the presence of trends or
seasonality.
1. Indicates the “memory” of the series: The ACF allows us to determine
how much past observations influence future ones. If the ACF shows
significant autocorrelations in several lags, it indicates that the
series has a long-term memory and that past observations are
relevant to predict future ones.
2. Helps identify MA (moving average) models: The shape of the ACF can
reveal the presence of moving average components in the time series.
Lags where the ACF shows a significant correlation may indicate the
order of an MA model.
Partial Autocorrelation (PACF): 1. Identify direct dependence: Unlike
the ACF, the PACF eliminates the indirect effects of intermediate lags
and measures the direct correlation between an observation and its
lagged values. It helps to identify the direct dependence between an
observation and its lag values, without the influence of intermediate
lags.
1. Helps to identify AR (autoregressive) models: The shape of the PACF
can reveal the presence of autoregressive components in the time
series. Lags in which the PACF shows a significant correlation may
indicate the order of an AR model.
2. Used in conjunction with the ACF: The PACF is used in conjunction
with the ACF to determine the order of an AR or MA model. By
analyzing both the ACF and the PACF, significant lags can be
identified and a model suitable for time series analysis and
forecasting can be built.
In summary, the ACF and the PACF are complementary tools in time series
analysis that provide information on time dependence and help identify
the appropriate components to build forecast models.
```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");
# Grafico
plot_pacf(df["y"], lags=30, ax=axs[1],color="lime")
axs[1].set_title('Partial Autocorrelation')
#plt.savefig("Gráfico de Densidad y qq")
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
```python theme={null}
from statsmodels.tsa.seasonal import seasonal_decompose
a = seasonal_decompose(df["y"], model = "additive", period=12)
a.plot();
```
### Multiplicative
```python theme={null}
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 `Seasonal Exponential Smoothing Optimized Model`.
2. Data to test our model
For the test data we will use the last 30 hours to test and evaluate the
performance of our model.
```python theme={null}
train = df[df.ds<='2017-09-20 17:00:00']
test = df[df.ds>'2017-09-20 17:00:00']
```
```python theme={null}
train.shape, test.shape
```
```text theme={null}
((186, 3), (30, 3))
```
## Implementation of SeasonalExponentialSmoothingOptimized with StatsForecast
### Load libraries
```python theme={null}
from statsforecast import StatsForecast
from statsforecast.models import SeasonalExponentialSmoothingOptimized
```
### Building 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 = 24 # Hourly data
horizon = len(test) # number of predictions
models = [SeasonalExponentialSmoothingOptimized(season_length=season_length)]
```
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](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='h')
```
### Fit the Model
```python theme={null}
sf.fit(df=train)
```
```text theme={null}
StatsForecast(models=[SeasESOpt])
```
Let’s see the results of our
`Seasonal Exponential Smoothing Optimized Model`. We can observe it with
the following instruction:
```python theme={null}
result=sf.fitted_[0,0].model_
result
```
```text theme={null}
{'mean': array([161532.05 , 161051.69 , 135531.64 , 105600.39 , 96717.39 ,
82608.34 , 80224.33 , 78075.98 , 85233.23 , 100179.336,
122245.62 , 118087.57 , 109614.81 , 104729.91 , 104895.02 ,
115862.96 , 130370.98 , 144231.89 , 149036.73 , 149072.73 ,
148110.77 , 148760.73 , 149767.53 , 150561.8 ], dtype=float32),
'fitted': array([ nan, nan, nan, nan, nan,
nan, nan, nan, nan, nan,
nan, nan, nan, nan, nan,
nan, nan, nan, nan, nan,
nan, nan, nan, nan, nan,
nan, nan, nan, nan, nan,
nan, nan, nan, nan, nan,
nan, nan, nan, nan, nan,
nan, nan, 163840. , 166235. , 139520. ,
105895. , 96780. , 82520. , 80125. , 75335. ,
85105. , 102080. , 125135. , 118030. , 109225. ,
102475. , 102240. , 115840. , 130540. , 144325. ,
148970. , 149150. , 148040. , 148810. , 149830. ,
150570. , 162030.27 , 163222.1 , 137347.33 , 103835.8 ,
96733.95 , 82522.45 , 80086.9 , 75132.05 , 85074.36 ,
100452.66 , 121044.03 , 118001.6 , 109242.15 , 102349.03 ,
102321.49 , 115768.25 , 130501. , 144286.1 , 149005. ,
149121.25 , 148039.8 , 148799.25 , 149789.2 , 150557.16 ,
161740.55 , 162812.36 , 136965.22 , 112853.91 , 96768.61 ,
82573.375, 80164.38 , 88707.87 , 85164.8 , 100944.15 ,
117929.875, 118111.086, 109563.58 , 103815.7 , 104036.375,
115942.47 , 130508.39 , 144268.03 , 149088.7 , 149155.03 ,
148096.75 , 148823.2 , 149797.77 , 150525.92 , 160582.38 ,
159756.83 , 134514.39 , 117874.29 , 96767.92 , 82683.74 ,
80253.336, 89338.625, 85232.055, 100619.03 , 114659.62 ,
118224.67 , 109881.99 , 105514.21 , 106070.33 , 116194.74 ,
130678.805, 144436.45 , 149261.16 , 149331.28 , 148247.19 ,
148908.03 , 149890.33 , 150620.88 , 161557.95 , 161701.48 ,
136228.19 , 113004.195, 96773.695, 82673.66 , 80245.91 ,
78459.88 , 85267.016, 100517.48 , 120224.3 , 118155.68 ,
109777.57 , 105240.35 , 105717.734, 116058.445, 130571.32 ,
144349.64 , 149169.75 , 149255.28 , 148186.9 , 148872.55 ,
149844.78 , 150618.77 , 161553.17 , 160112.8 , 135912.81 ,
107124.39 , 96756.41 , 82642.07 , 80226.8 , 74707.9 ,
85226.28 , 100202.98 , 119687.805, 118105.27 , 109668.59 ,
104848.68 , 105212.516, 115939.71 , 130463.1 , 144266.05 ,
149142.61 , 149154.38 , 148177.3 , 148833.17 , 149860.03 ,
150673.38 ], dtype=float32)}
```
Let us now visualize the fitted values 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}
fitted=pd.DataFrame(result.get("fitted"), columns=["fitted"])
fitted["ds"]=df["ds"]
fitted
```
| | fitted | ds |
| --- | ------------- | ------------------- |
| 0 | NaN | 2017-09-13 00:00:00 |
| 1 | NaN | 2017-09-13 01:00:00 |
| 2 | NaN | 2017-09-13 02:00:00 |
| ... | ... | ... |
| 183 | 148833.171875 | 2017-09-20 15:00:00 |
| 184 | 149860.031250 | 2017-09-20 16:00:00 |
| 185 | 150673.375000 | 2017-09-20 17:00:00 |
```python theme={null}
sns.lineplot(df, x="ds", y="y", label="Actual", linewidth=2)
sns.lineplot(fitted,x="ds", y="fitted", label="Fitted", linestyle="--" )
plt.title("Ads watched (hourly data)");
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.
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}
# Prediction
Y_hat = sf.forecast(df=train, h=horizon, fitted=True)
Y_hat
```
| | unique\_id | ds | SeasESOpt |
| --- | ---------- | ------------------- | ------------- |
| 0 | 1 | 2017-09-20 18:00:00 | 161532.046875 |
| 1 | 1 | 2017-09-20 19:00:00 | 161051.687500 |
| 2 | 1 | 2017-09-20 20:00:00 | 135531.640625 |
| ... | ... | ... | ... |
| 27 | 1 | 2017-09-21 21:00:00 | 105600.390625 |
| 28 | 1 | 2017-09-21 22:00:00 | 96717.390625 |
| 29 | 1 | 2017-09-21 23:00:00 | 82608.343750 |
```python theme={null}
values=sf.forecast_fitted_values()
values.head()
```
| | unique\_id | ds | y | SeasESOpt |
| - | ---------- | ------------------- | -------- | --------- |
| 0 | 1 | 2017-09-13 00:00:00 | 80115.0 | NaN |
| 1 | 1 | 2017-09-13 01:00:00 | 79885.0 | NaN |
| 2 | 1 | 2017-09-13 02:00:00 | 89325.0 | NaN |
| 3 | 1 | 2017-09-13 03:00:00 | 101930.0 | NaN |
| 4 | 1 | 2017-09-13 04:00:00 | 121630.0 | NaN |
```python theme={null}
sf.plot(train, Y_hat)
```
### 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, 30 hours ahead.
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}
forecast_df = sf.predict(h=horizon)
forecast_df
```
| | unique\_id | ds | SeasESOpt |
| --- | ---------- | ------------------- | ------------- |
| 0 | 1 | 2017-09-20 18:00:00 | 161532.046875 |
| 1 | 1 | 2017-09-20 19:00:00 | 161051.687500 |
| 2 | 1 | 2017-09-20 20:00:00 | 135531.640625 |
| ... | ... | ... | ... |
| 27 | 1 | 2017-09-21 21:00:00 | 105600.390625 |
| 28 | 1 | 2017-09-21 22:00:00 | 96717.390625 |
| 29 | 1 | 2017-09-21 23:00:00 | 82608.343750 |
## 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=)`, 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=df,
h=horizon,
step_size=30,
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.
```python theme={null}
crossvalidation_df
```
| | unique\_id | ds | cutoff | y | SeasESOpt |
| --- | ---------- | ------------------- | ------------------- | -------- | ------------- |
| 0 | 1 | 2017-09-18 06:00:00 | 2017-09-18 05:00:00 | 99440.0 | 141401.750000 |
| 1 | 1 | 2017-09-18 07:00:00 | 2017-09-18 05:00:00 | 97655.0 | 152474.250000 |
| 2 | 1 | 2017-09-18 08:00:00 | 2017-09-18 05:00:00 | 97655.0 | 152482.796875 |
| ... | ... | ... | ... | ... | ... |
| 87 | 1 | 2017-09-21 21:00:00 | 2017-09-20 17:00:00 | 103080.0 | 105600.390625 |
| 88 | 1 | 2017-09-21 22:00:00 | 2017-09-20 17:00:00 | 95155.0 | 96717.390625 |
| 89 | 1 | 2017-09-21 23:00:00 | 2017-09-20 17:00:00 | 80285.0 | 82608.343750 |
## 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.
```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 | SeasESOpt |
| - | ---------- | ------ | ----------- |
| 0 | 1 | mae | 6694.042188 |
| 1 | 1 | mape | 0.060392 |
| 2 | 1 | mase | 0.827062 |
| 3 | 1 | rmse | 8118.297509 |
| 4 | 1 | smape | 0.028961 |
## References
1. [Changquan Huang • Alla Petukhina. Springer series (2022). Applied
Time Series Analysis and Forecasting with
Python.](https://link.springer.com/book/10.1007/978-3-031-13584-2)
2. Ivan Svetunkov. [Forecasting and Analytics with the Augmented
Dynamic Adaptive Model (ADAM)](https://openforecast.org/adam/)
3. [James D. Hamilton. Time Series Analysis Princeton University Press,
Princeton, New Jersey, 1st Edition,
1994.](https://press.princeton.edu/books/hardcover/9780691042893/time-series-analysis)
4. [Nixtla SeasonalExponentialSmoothingOptimized
API](../../src/core/models.html#seasonalexponentialsmoothingoptimized)
5. [Pandas available
frequencies](https://pandas.pydata.org/pandas-docs/stable/user_guide/timeseries.html#offset-aliases).
6. [Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting
Principles and Practice (3rd
ed)”](https://otexts.com/fpp3/tscv.html).
7. [Seasonal periods- Rob J
Hyndman](https://robjhyndman.com/hyndsight/seasonal-periods/).