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Table of Contents

• Introduction
• Croston Optimized Model
• Loading libraries and data
• Explore data with the plot method
• Split the data into training and testing
• Implementation of CrostonOptimized with StatsForecast
• Cross-validation
• Model evaluation
• References

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Introduction

The Croston Optimized model is a forecasting method designed for intermittent demand time series data. It is an extension of the Croston’s method, which was originally developed for forecasting sporadic demand patterns.

Intermittent demand time series are characterized by irregular and sporadic occurrences of non-zero demand values, often with long periods of zero demand. Traditional forecasting methods may struggle to handle such patterns effectively.

The Croston Optimized model addresses this challenge by incorporating two key components: exponential smoothing and intermittent demand estimation.

1. Exponential Smoothing: The Croston Optimized model uses exponential smoothing to capture the trend and seasonality in the intermittent demand data. This helps in identifying the underlying patterns and making more accurate forecasts.

2. Intermittent Demand Estimation: Since intermittent demand data often consists of long periods of zero demand, the Croston Optimized model employs a separate estimation process for the occurrence and size of non-zero demand values. It estimates the probability of occurrence and the average size of non-zero demand intervals, enabling better forecasting of intermittent demand.

The Croston Optimized model aims to strike a balance between over-forecasting and under-forecasting intermittent demand, which are common challenges in traditional forecasting methods. By explicitly modeling intermittent demand patterns, it can provide more accurate forecasts for intermittent demand time series data.

It is worth noting that there are variations and adaptations of the Croston Optimized model, with different modifications and enhancements made to suit specific forecasting scenarios. These variations may incorporate additional features or algorithms to further improve the accuracy of the forecasts.

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Croston Optimized method

The Croston Optimized model can be mathematically defined as follows:

1. Initialization:
   • Let $(y_t)$ represent the intermittent demand time series data at time $t$.
   • Initialize two sets of variables: $(p_t)$ for the probability of occurrence and $(q_t)$ for the average size of non-zero demand intervals.
   • Initialize the forecast $(F_t)$ and forecast error $(E_t)$ variables as zero.
2. Calculation of $(p_t)$ and $(q_t)$:
   • Calculate the intermittent demand occurrence probability $(p_t)$ using exponential smoothing: $[p_t = \alpha + (1 - \alpha)(p_{t-1}),]$ where $(\alpha)$ is the smoothing parameter (typically set between 0.1 and 0.3).
   • Calculate the average size of non-zero demand intervals $(q_t)$ using exponential smoothing: $[q_t = \beta \cdot y_t + (1 - \beta)(q_{t-1}),]$ where $(\beta)$ is the smoothing parameter (typically set between 0.1 and 0.3).
3. Forecasting:
   • If $(y_t > 0)$ (non-zero demand occurrence):
     • Calculate the forecast $(F_t)$ as the previous forecast $(F_{t-1})$ divided by the average size of non-zero demand intervals $(q_{t-1})$: $[F_t = \frac{{F_{t-1}}}{{q_{t-1}}}]$
     • Calculate the forecast error $(E_t)$ as the difference between the actual demand $(y_t)$ and the forecast $(F_t)$: $[E_t = y_t - F_t]$
   • If $(y_t = 0)$ (zero demand occurrence):
     • Set the forecast $(F_t)$ and forecast error $(E_t)$ as zero.
4. Updating the model:
   • Update the intermittent demand occurrence probability $(p_t)$ and the average size of non-zero demand intervals $(q_t)$ using exponential smoothing as described in step 2.
5. Repeat steps 3 and 4 for each time point in the time series.

The Croston Optimized model leverages exponential smoothing to capture the trend and seasonality in the intermittent demand data, and it estimates the occurrence probability and average size of non-zero demand intervals separately to handle intermittent demand patterns effectively. By updating the model parameters based on the observed data, it provides improved forecasts for intermittent demand time series.

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Some properties of the Optimized Croston Model

The optimized Croston model is a modification of the classic Croston model used to forecast intermittent demand. The classic Croston model forecasts demand using a weighted average of historical orders and the average interval between orders. The optimized Croston model uses a probability function to forecast the mean interval between orders.

The optimized Croston model has been shown to be more accurate than the classical Croston model for time series with irregular demand. The optimized Croston model is also more adaptable to different types of intermittent time series.

The optimized Croston model has the following properties:

• It is accurate, even for time series with irregular demand.
• It is adaptable to different types of intermittent time series.
• It is easy to implement and understand.
• It is robust to outliers.

The optimized Croston model has been used successfully to forecast a wide range of intermittent time series, including product demand, service demand, and resource demand.

Here are some of the properties of the optimized Croston model:

• Precision: The optimized Croston model has been shown to be more accurate than the classic Croston model for time series with irregular demand. This is because the optimized Croston model uses a probability function to forecast the average interval between orders, which is more accurate than the weighted average of historical orders.
• Adaptability: The optimized Croston model is also more adaptable to different types of intermittent time series. This is because the optimized Croston model uses a probability function to forecast the mean interval between orders, allowing it to accommodate different demand patterns.
• Ease of Implementation and Understanding: The optimized Croston model is easy to implement and understand. This is because the optimized Croston model is a modification of the classical Croston model, which is a well-known and well-understood model.
• Robustness: The optimized Croston model is also robust to outliers. This is because the optimized Croston model uses a probability function to forecast the mean interval between orders, which allows it to ignore outliers.

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Loading libraries and data

Tip

Statsforecast will be needed. To install, see instructions.

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

import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
import plotly.graph_objects as go
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)

import pandas as pd
df=pd.read_csv("https://raw.githubusercontent.com/Naren8520/Serie-de-tiempo-con-Machine-Learning/main/Data/intermittend_demand2")

df.head()

The input to StatsForecast is always a data frame in long format with three columns: unique_id, ds and y:

• The unique_id (string, int or category) represents an identifier for the series.

• The ds (datestamp) column should be of a format expected by Pandas, ideally YYYY-MM-DD for a date or YYYY-MM-DD HH:MM:SS for a timestamp.

• The y (numeric) represents the measurement we wish to forecast.

df["unique_id"]="1"
df.columns=["ds", "y", "unique_id"]
df.head()

print(df.dtypes)

ds           object
y             int64
unique_id    object
dtype: object

We can see that our time variable (ds) is in an object format, we need to convert to a date format

df["ds"] = pd.to_datetime(df["ds"])

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Explore Data with the plot method

Plot some series using the plot method from the StatsForecast class. This method prints a random series from the dataset and is useful for basic EDA.

from statsforecast import StatsForecast

StatsForecast.plot(df)

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Autocorrelation plots

Autocorrelation (ACF) and partial autocorrelation (PACF) plots are statistical tools used to analyze time series. ACF charts show the correlation between the values of a time series and their lagged values, while PACF charts show the correlation between the values of a time series and their lagged values, after the effect of previous lagged values has been removed.

ACF and PACF charts can be used to identify the structure of a time series, which can be helpful in choosing a suitable model for the time series. For example, if the ACF chart shows a repeating peak and valley pattern, this indicates that the time series is stationary, meaning that it has the same statistical properties over time. If the PACF chart shows a pattern of rapidly decreasing spikes, this indicates that the time series is invertible, meaning it can be reversed to get a stationary time series.

The importance of the ACF and PACF charts is that they can help analysts better understand the structure of a time series. This understanding can be helpful in choosing a suitable model for the time series, which can improve the ability to predict future values of the time series.

To analyze ACF and PACF charts:

• Look for patterns in charts. Common patterns include repeating peaks and valleys, sawtooth patterns, and plateau patterns.
• Compare ACF and PACF charts. The PACF chart generally has fewer spikes than the ACF chart.
• Consider the length of the time series. ACF and PACF charts for longer time series will have more spikes.
• Use a confidence interval. The ACF and PACF plots also show confidence intervals for the autocorrelation values. If an autocorrelation value is outside the confidence interval, it is likely to be significant.

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();

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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

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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: $y(t) = level + Trend + seasonality + noise$

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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.

$y(t) = Level * Trend * seasonality * Noise$

from plotly.subplots import make_subplots

from statsmodels.tsa.seasonal import seasonal_decompose

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

plotSeasonalDecompose(
    df["y"],
    model="additive",
    period=24,
    title="Seasonal Decomposition")

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Split the data into training and testing

Let’s divide our data into sets

Let’s divide our data into sets 1. Data to train our Croston Optimized 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.

train = df[df.ds<='2023-01-31 19:00:00'] 
test = df[df.ds>'2023-01-31 19:00:00']

train.shape, test.shape

((9500, 3), (500, 3))

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

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.xlabel("Hours")
plt.show()

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Implementation of CrostonOptimized with StatsForecast

To also know more about the parameters of the functions of the CrostonOptimized Model, they are listed below. For more information, visit the documentation.

alias : str
    Custom name of the model.

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Load libraries

from statsforecast import StatsForecast
from statsforecast.models import CrostonOptimized

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Instantiating Model

Import and instantiate the models. Setting the argument is sometimes tricky. This article on Seasonal periods by the master, Rob Hyndmann, can be useful for season_length.

season_length = 24 # Hourly data 
horizon = len(test) # number of predictions

# We call the model that we are going to use
models = [CrostonOptimized()]

We fit the models by instantiating a new StatsForecast object with the following parameters:

models: a list of models. Select the models you want from models and import them.

• freq: a string indicating the frequency of the data. (See pandas’ available frequencies.)

• n_jobs: n_jobs: int, number of jobs used in the parallel processing, use -1 for all cores.

• fallback_model: a model to be used if a model fails.

Any settings are passed into the constructor. Then you call its fit method and pass in the historical data frame.

sf = StatsForecast(df=df,
                   models=models,
                   freq='H', 
                   n_jobs=-1)

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Fit the Model

# fit the models
sf.fit()

StatsForecast(models=[CrostonOptimized])

Let’s see the results of our Croston optimized Model. We can observe it with the following instruction:

result=sf.fitted_[0,0].model_
result

{'mean': array([23.606695], dtype=float32)}

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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, 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. Depending on your computer, this step should take around 1min.

Y_hat = sf.forecast(horizon)
Y_hat

Y_hat=Y_hat.reset_index()
Y_hat

Y_hat1 = pd.concat([df,Y_hat])
Y_hat1

fig, ax = plt.subplots(1, 1)
plot_df = pd.concat([df, Y_hat1]).set_index('ds')
plot_df['y'].plot(ax=ax, linewidth=2)
plot_df[ "CrostonOptimized"].plot(ax=ax, linewidth=2, color="yellow")
ax.set_title(' Forecast', fontsize=22)
ax.set_ylabel("Store visit (Hourly data)", fontsize=20)
ax.set_xlabel('Hours', fontsize=20)
ax.legend(prop={'size': 15})
ax.grid(True)

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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.

forecast_df = sf.predict(h=horizon) 
forecast_df

We can join the forecast result with the historical data using the pandas function pd.concat(), and then be able to use this result for graphing.

pd.concat([df, forecast_df]).set_index('ds')

df_plot= pd.concat([df, forecast_df]).set_index('ds').tail(5000)
df_plot

Now let’s visualize the result of our forecast and the historical data of our time series.

plt.plot(df_plot['y'],label="Actual", linewidth=2.5)
plt.plot(df_plot['CrostonOptimized'], label="Croston Optimized", color="yellow") # '-', '--', '-.', ':',

plt.title("Store visit (Hourly data)");
plt.xlabel("Hourly")
plt.ylabel("Store visit")
plt.legend()
plt.show();

Let’s plot the same graph using the plot function that comes in Statsforecast, as shown below.

sf.plot(df, forecast_df)

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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:

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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=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.

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: 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.

crossvalidation_df

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Model Evaluation

We can now compute the accuracy of the forecast using an appropiate accuracy metric. Here we’ll use the Root Mean Squared Error (RMSE). To do this, we first need to install datasetsforecast, a Python library developed by Nixtla that includes a function to compute the RMSE.

!pip install datasetsforecast

from datasetsforecast.losses import rmse

The function to compute the RMSE takes two arguments:

1. The actual values.
2. The forecasts, in this case, Croston Optimized Model.

rmse = rmse(crossvalidation_df['y'], crossvalidation_df["CrostonOptimized"])
print("RMSE using cross-validation: ", rmse)

RMSE using cross-validation:  48.08823

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Acknowledgements

We would like to thank Naren Castellon for writing this tutorial.

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References

1. Changquan Huang • Alla Petukhina. Springer series (2022). Applied Time Series Analysis and Forecasting with Python.
2. Ivan Svetunkov. Forecasting and Analytics with the Augmented Dynamic Adaptive Model (ADAM)
3. James D. Hamilton. Time Series Analysis Princeton University Press, Princeton, New Jersey, 1st Edition, 1994.
4. Nixtla Parameters.
5. Pandas available frequencies.
6. Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting principles and practice, Time series cross-validation”..
7. Seasonal periods- Rob J Hyndman.