### 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(); ```

### 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. $y(t) = Level * Trend * seasonality * Noise$ ```python theme={null} from statsmodels.tsa.seasonal import seasonal_decompose from plotly.subplots import make_subplots import plotly.graph_objects as go def plotSeasonalDecompose( x, model='additive', filt=None, period=None, two_sided=True, extrapolate_trend=0, title="Seasonal Decomposition"): result = seasonal_decompose( x, model=model, filt=filt, period=period, two_sided=two_sided, extrapolate_trend=extrapolate_trend) fig = make_subplots( rows=4, cols=1, subplot_titles=["Observed", "Trend", "Seasonal", "Residuals"]) for idx, col in enumerate(['observed', 'trend', 'seasonal', 'resid']): fig.add_trace( go.Scatter(x=result.observed.index, y=getattr(result, col), mode='lines'), row=idx+1, col=1, ) return fig ``` ```python theme={null} plotSeasonalDecompose( df["y"], model="additive", period=24, title="Seasonal Decomposition") ```

## Split the data into training and testing Let’s divide our data into sets 1. Data to train our `Croston Classic Model`. 2. Data to test our model For the test data we will use the last 500 hours to test and evaluate the performance of our model. ```python theme={null} train = df[df.ds<='2023-01-31 19:00:00'] test = df[df.ds>'2023-01-31 19:00:00'] ``` ```python theme={null} train.shape, test.shape ``` ```text theme={null} ((9500, 3), (500, 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="--",linewidth=2) sns.lineplot(test, x="ds", y="y", label="Test", linewidth=2, color="yellow") plt.title("Store visit"); plt.show() ```

## Implementation of CrostonClassic with StatsForecast To also know more about the parameters of the functions of the `CrostonClassic Model`, they are listed below. For more information, visit the [documentation](../../src/core/models.html#crostonclassic) ```text theme={null} alias : str Custom name of the model. ``` ### Load libraries ```python theme={null} from statsforecast import StatsForecast from statsforecast.models import CrostonClassic ``` ### 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 = 24 # Hourly data horizon = len(test) # number of predictions models = [CrostonClassic()] ``` 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='h') ``` ### Fit the Model ```python theme={null} sf.fit(df=train) ``` ```text theme={null} StatsForecast(models=[CrostonClassic]) ``` Let’s see the results of our `Croston Classic Model`. We can observe it with the following instruction: ```python theme={null} result=sf.fitted_[0,0].model_ result ``` ```text theme={null} {'mean': array([27.41841685]), 'fitted': array([ nan, 0. , 5. , ..., 30.61961, 30.61961, 30.61961], dtype=float32), 'sigma': np.float32(49.5709)} ``` ### 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, 25 week 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} Y_hat = sf.forecast(df=train, h=horizon) Y_hat ``` | | unique\_id | ds | CrostonClassic | | --- | ---------- | ------------------- | -------------- | | 0 | 1 | 2023-01-31 20:00:00 | 27.418417 | | 1 | 1 | 2023-01-31 21:00:00 | 27.418417 | | 2 | 1 | 2023-01-31 22:00:00 | 27.418417 | | ... | ... | ... | ... | | 497 | 1 | 2023-02-21 13:00:00 | 27.418417 | | 498 | 1 | 2023-02-21 14:00:00 | 27.418417 | | 499 | 1 | 2023-02-21 15:00:00 | 27.418417 | ```python theme={null} sf.plot(train, Y_hat, max_insample_length=500) ```

### 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, 500 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 | CrostonClassic | | --- | ---------- | ------------------- | -------------- | | 0 | 1 | 2023-01-31 20:00:00 | 27.418417 | | 1 | 1 | 2023-01-31 21:00:00 | 27.418417 | | 2 | 1 | 2023-01-31 22:00:00 | 27.418417 | | ... | ... | ... | ... | | 497 | 1 | 2023-02-21 13:00:00 | 27.418417 | | 498 | 1 | 2023-02-21 14:00:00 | 27.418417 | | 499 | 1 | 2023-02-21 15:00:00 | 27.418417 | ## 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: ![](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=)`, forecasting every second hour `(step_size=50)`. 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, 500 hours 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=50, n_windows=5) ``` 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 | CrostonClassic | | ---- | ---------- | ------------------- | ------------------- | ---- | -------------- | | 0 | 1 | 2023-01-23 12:00:00 | 2023-01-23 11:00:00 | 0.0 | 23.655830 | | 1 | 1 | 2023-01-23 13:00:00 | 2023-01-23 11:00:00 | 0.0 | 23.655830 | | 2 | 1 | 2023-01-23 14:00:00 | 2023-01-23 11:00:00 | 0.0 | 23.655830 | | ... | ... | ... | ... | ... | ... | | 2497 | 1 | 2023-02-21 13:00:00 | 2023-01-31 19:00:00 | 60.0 | 27.418417 | | 2498 | 1 | 2023-02-21 14:00:00 | 2023-01-31 19:00:00 | 20.0 | 27.418417 | | 2499 | 1 | 2023-02-21 15:00:00 | 2023-01-31 19:00:00 | 20.0 | 27.418417 | ## 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 | CrostonClassic | | - | ---------- | ------ | -------------- | | 0 | 1 | mae | 33.704756 | | 1 | 1 | mape | 0.632593 | | 2 | 1 | mase | 0.804074 | | 3 | 1 | rmse | 45.262709 | | 4 | 1 | smape | 0.767960 | # 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 CrostonClassic API](../../src/core/models.html#crostonclassic) 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/).