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Table of Contents

• Introduction
• ETS Models
• ETS Estimation
• Model Selection
• Loading libraries and data
• Explore data with the plot method
• Split the data into training and testing
• Implementation of AutoETS with StatsForecast
• Cross-validation
• Model evaluation
• References

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Introduction

Automatic forecasts of large numbers of univariate time series are often needed in business. It is common to have over one thousand product lines that need forecasting at least monthly. Even when a smaller number of forecasts are required, there may be nobody suitably trained in the use of time series models to produce them. In these circumstances, an automatic forecasting algorithm is an essential tool. Automatic forecasting algorithms must determine an appropriate time series model, estimate the parameters and compute the forecasts. They must be robust to unusual time series patterns, and applicable to large numbers of series without user intervention. The most popular automatic forecasting algorithms are based on either exponential smoothing or ARIMA models.

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Exponential smoothing

Although exponential smoothing methods have been around since the 1950s, a modelling framework incorporating procedures for model selection was not developed until relatively recently. Ord, Koehler, and Snyder (1997), Hyndman, Koehler, Snyder, and Grose (2002) and Hyndman, Koehler, Ord, and Snyder (2005b) have shown that all exponential smoothing methods (including non-linear methods) are optimal forecasts from innovations state space models.

Exponential smoothing methods were originally classified by Pegels’ (1969) taxonomy. This was later extended by Gardner (1985), modified by Hyndman et al. (2002), and extended again by Taylor (2003), giving a total of fifteen methods seen in the following table.

Some of these methods are better known under other names. For example, cell (N,N) describes the simple exponential smoothing (or SES) method, cell (A,N) describes Holt’s linear method, and cell (Ad,N) describes the damped trend method. The additive Holt-Winters’ method is given by cell (A,A) and the multiplicative Holt-Winters’ method is given by cell (A,M). The other cells correspond to less commonly used but analogous methods.

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Point forecasts for all methods

We denote the observed time series by $y_1,y_2,...,y_n$. A forecast of $y_{t+h}$ based on all of the data up to time $t$ is denoted by $\hat y_{t+h|t}$. To illustrate the method, we give the point forecasts and updating equation for method (A,A), the he Holt-Winters’ additive method:

where $m$ is the length of seasonality (e.g., the number of months or quarters in a year), $\ell_{t}$ represents the level of the series, $b_t$ denotes the growth, $s_t$ is the seasonal component, $\hat y_{t+h|t}$ is the forecast for $h$ periods ahead, and $h_{m}^{+} = [(h − 1) mod \ m] + 1$. To use method (1), we need values for the initial states $\ell_{0}$, $b_0$ and $s_{1−m}, . . . , s_0$, and for the smoothing parameters $\alpha, \beta^{*}$ and $\gamma$. All of these will be estimated from the observed data.

Equation (1c) is slightly different from the usual Holt-Winters equations such as those in Makridakis et al. (1998) or Bowerman, O’Connell, and Koehler (2005). These authors replace (1c) with $s_{t} = \gamma^* (y_{t}-\ell_{t})+ (1-\gamma^*)s_{t-m}.$

If $\ell_{t}$ is substituted using (1a), we obtain

$s_{t} = \gamma^*(1-\alpha) (y_{t}-\ell_{t-1}-b_{t-1})+ [1-\gamma^*(1-\alpha)]s_{t-m},$

Thus, we obtain identical forecasts using this approach by replacing $\gamma$ in (1c) with $\gamma^{*} (1-\alpha)$. The modification given in (1c) was proposed by Ord et al. (1997) to make the state space formulation simpler. It is equivalent to Archibald’s (1990) variation of the Holt-Winters’ method.

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Innovations state space models

For each exponential smoothing method in Other ETS models , Hyndman et al. (2008b) describe two possible innovations state space models, one corresponding to a model with additive errors and the other to a model with multiplicative errors. If the same parameter values are used, these two models give equivalent point forecasts, although different prediction intervals. Thus there are 30 potential models described in this classification.

Historically, the nature of the error component has often been ignored, because the distinction between additive and multiplicative errors makes no difference to point forecasts.

We are careful to distinguish exponential smoothing methods from the underlying state space models. An exponential smoothing method is an algorithm for producing point forecasts only. The underlying stochastic state space model gives the same point forecasts, but also provides a framework for computing prediction intervals and other properties.

To distinguish the models with additive and multiplicative errors, we add an extra letter to the front of the method notation. The triplet (E,T,S) refers to the three components: error, trend and seasonality. So the model ETS(A,A,N) has additive errors, additive trend and no seasonality—in other words, this is Holt’s linear method with additive errors. Similarly, ETS(M,Md,M) refers to a model with multiplicative errors, a damped multiplicative trend and multiplicative seasonality. The notation ETS(·,·,·) helps in remembering the order in which the components are specified.

Once a model is specified, we can study the probability distribution of future values of the series and find, for example, the conditional mean of a future observation given knowledge of the past. We denote this as $\mu_{t+h|t} = E(y_{t+h | xt})$, where xt contains the unobserved components such as $\ell_t$, $b_t$ and $s_t$. For $h = 1$ we use $\mu_t ≡ \mu_{t+1|t}$ as a shorthand notation. For many models, these conditional means will be identical to the point forecasts given in Table Other ETS models, so that $\mu_{t+h|t} = \hat y_{t+h|t}$. However, for other models (those with multiplicative trend or multiplicative seasonality), the conditional mean and the point forecast will differ slightly for $h ≥ 2$.

We illustrate these ideas using the damped trend method of Gardner and McKenzie (1985).

Each model consists of a measurement equation that describes the observed data, and some state equations that describe how the unobserved components or states (level, trend, seasonal) change over time. Hence, these are referred to as state space models.

For each method there exist two models: one with additive errors and one with multiplicative errors. The point forecasts produced by the models are identical if they use the same smoothing parameter values. They will, however, generate different prediction intervals.

To distinguish between a model with additive errors and one with multiplicative errors (and also to distinguish the models from the methods), we add a third letter to the classification of in the above Table. We label each state space model as ETS(⋅,.,.) for (Error, Trend, Seasonal). This label can also be thought of as ExponenTial Smoothing. Using the same notation as in the above Table, the possibilities for each component (or state) are: Error ={ A,M }, Trend ={N,A,Ad} and Seasonal ={ N,A,M }.

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ETS(A,N,N): simple exponential smoothing with additive errors

Recall the component form of simple exponential smoothing:

If we re-arrange the smoothing equation for the level, we get the “error correction” form,

where $e_{t}=y_{t}-\ell_{t-1}=y_{t}-\hat{y}_{t|t-1}$ s the residual at time $t$.

The training data errors lead to the adjustment of the estimated level throughout the smoothing process for $t=1,\dots,T$. For example, if the error at time $t$ is negative, then $y_t < \hat{y}_{t|t-1}$ and so the level at time $t-1$ has been over-estimated. The new level $\ell_{t}$ is then the previous level $\ell_{t-1}$ adjusted downwards. The closer $\alpha$ is to one, the “rougher” the estimate of the level (large adjustments take place). The smaller the $\alpha$, the “smoother” the level (small adjustments take place).

We can also write $y_t = \ell_{t-1} + e_t$, so that each observation can be represented by the previous level plus an error. To make this into an innovations state space model, all we need to do is specify the probability distribution for $e_t$. For a model with additive errors, we assume that residuals (the one-step training errors) $e_t$ are normally distributed white noise with mean 0 and variance $\sigma^2$. A short-hand notation for this is $e_t = \varepsilon_t\sim\text{NID}(0,\sigma^2)$; NID stands for “normally and independently distributed”.

Then the equations of the model can be written as

We refer to (2) as the measurement (or observation) equation and (3) as the state (or transition) equation. These two equations, together with the statistical distribution of the errors, form a fully specified statistical model. Specifically, these constitute an innovations state space model underlying simple exponential smoothing.

The term “innovations” comes from the fact that all equations use the same random error process, $\varepsilon_t$. For the same reason, this formulation is also referred to as a “single source of error” model. There are alternative multiple source of error formulations which we do not present here.

The measurement equation shows the relationship between the observations and the unobserved states. In this case, observation $y_t$ is a linear function of the level $\ell_{t-1}$, the predictable part of $y_t$, and the error $\varepsilon_t$, the unpredictable part of $y_t$. For other innovations state space models, this relationship may be nonlinear.

The state equation shows the evolution of the state through time. The influence of the smoothing parameter $\alpha$ is the same as for the methods discussed earlier. For example, $\alpha$ governs the amount of change in successive levels: high values of $\alpha$ allow rapid changes in the level; low values of $\alpha$ lead to smooth changes. If $\alpha=0$, the level of the series does not change over time; if $\alpha=1$, the model reduces to a random walk model, $y_t=y_{t-1}+\varepsilon_t$.

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ETS(M,N,N): simple exponential smoothing with multiplicative errors

In a similar fashion, we can specify models with multiplicative errors by writing the one-step-ahead training errors as relative errors

$\varepsilon_t = \frac{y_t-\hat{y}_{t|t-1}}{\hat{y}_{t|t-1}}$

where $\varepsilon_t \sim \text{NID}(0,\sigma^2)$. Substituting $\hat{y}_{t|t-1}=\ell_{t-1}$ gives $y_t = \ell_{t-1}+\ell_{t-1}\varepsilon_t$ and $e_t = y_t - \hat{y}_{t|t-1} = \ell_{t-1}\varepsilon_t$.

Then we can write the multiplicative form of the state space model as

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ETS(A,A,N): Holt’s linear method with additive errors

For this model, we assume that the one-step-ahead training errors are given by

$\varepsilon_t=y_t-\ell_{t-1}-b_{t-1} \sim \text{NID}(0,\sigma^2)$

Substituting this into the error correction equations for Holt’s linear method we obtain

where for simplicity we have set $\beta=\alpha \beta^*$.

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ETS(M,A,N): Holt’s linear method with multiplicative errors

Specifying one-step-ahead training errors as relative errors such that

$\varepsilon_t=\frac{y_t-(\ell_{t-1}+b_{t-1})}{(\ell_{t-1}+b_{t-1})}$

and following an approach similar to that used above, the innovations state space model underlying Holt’s linear method with multiplicative errors is specified as

where again $\beta=\alpha \beta^*$ and $\varepsilon_t \sim \text{NID}(0,\sigma^2)$.

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Estimating ETS models

An alternative to estimating the parameters by minimising the sum of squared errors is to maximise the “likelihood”. The likelihood is the probability of the data arising from the specified model. Thus, a large likelihood is associated with a good model. For an additive error model, maximising the likelihood (assuming normally distributed errors) gives the same results as minimising the sum of squared errors. However, different results will be obtained for multiplicative error models. In this section, we will estimate the smoothing parameters $\alpha, \beta, \gamma$ and $\phi$ and the initial states $\ell_0, b_0, s_0,s_{-1},\dots,s_{-m+1}$, by maximising the likelihood.

The possible values that the smoothing parameters can take are restricted. Traditionally, the parameters have been constrained to lie between 0 and 1 so that the equations can be interpreted as weighted averages. That is, $0< \alpha,\beta^*,\gamma^*,\phi<1$. For the state space models, we have set $\beta=\alpha\beta^*$ and $\gamma=(1-\alpha)\gamma^*$. Therefore, the traditional restrictions translate to $0< \alpha <1, 0 < \beta < \alpha$ and $0< \gamma < 1-\alpha$. In practice, the damping parameter $\phi$ is usually constrained further to prevent numerical difficulties in estimating the model.

Another way to view the parameters is through a consideration of the mathematical properties of the state space models. The parameters are constrained in order to prevent observations in the distant past having a continuing effect on current forecasts. This leads to some admissibility constraints on the parameters, which are usually (but not always) less restrictive than the traditional constraints region (Hyndman et al., 2008, pp. 149-161). For example, for the ETS(A,N,N) model, the traditional parameter region is $0< \alpha <1$ but the admissible region is $0< \alpha <2$. For the ETS(A,A,N) model, the traditional parameter region is $0<\alpha<1$ and $0<\beta<\alpha$ but the admissible region is $0<\alpha<2$ and $0<\beta<4-2\alpha$.

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Model selection

A great advantage of the ETS statistical framework is that information criteria can be used for model selection. The AIC, AIC_c and BIC, 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

$AIC_c = AIC + \frac{2k(k+1)}{T-k-1}$

and the Bayesian Information Criterion (BIC) is

$\text{BIC} = \text{AIC} + k[\log(T)-2]$

Three of the combinations of (Error, Trend, Seasonal) can lead to numerical difficulties. Specifically, the models that can cause such instabilities are ETS(A,N,M), ETS(A,A,M), and ETS(A,Ad,M), due to division by values potentially close to zero in the state equations. We normally do not consider these particular combinations when selecting a model.

Models with multiplicative errors are useful when the data are strictly positive, but are not numerically stable when the data contain zeros or negative values. Therefore, multiplicative error models will not be considered if the time series is not strictly positive. In that case, only the six fully additive models will be applied.

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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 numpy as np
import pandas as pd

import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.graphics.tsaplots import plot_acf
from statsmodels.graphics.tsaplots import plot_pacf

plt.style.use('fivethirtyeight')
plt.rcParams['lines.linewidth'] = 1.5
dark_style = {
    'figure.facecolor': '#212946',
    'axes.facecolor': '#212946',
    'savefig.facecolor':'#212946',
    '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': '#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)

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Read Data

df = pd.read_csv("https://raw.githubusercontent.com/Naren8520/Serie-de-tiempo-con-Machine-Learning/main/Data/Esperanza_vida.csv", usecols=[1,2])
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            float64
unique_id     object
dtype: object

We need to convert the ds from object type to datetime.

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

fig, axs = plt.subplots(nrows=1, ncols=2)

plot_acf(df["y"],  lags=20, ax=axs[0],color="fuchsia")
axs[0].set_title("Autocorrelation");

# Plot
plot_pacf(df["y"],  lags=20, 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 statsmodels.tsa.seasonal import seasonal_decompose 
a = seasonal_decompose(df["y"], model = "add", period=1)
a.plot();

Breaking down a time series into its components helps us to identify the behavior of the time series we are analyzing. In addition, it helps us to know what type of models we can apply, for our example of the Life expectancy data set, we can observe that our time series shows an increasing trend throughout the year, on the other hand, it can be observed also that the time series has no seasonality.

By looking at the previous graph and knowing each of the components, we can get an idea of which model we can apply: * We have trend * There is no seasonality

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Split the data into training and testing

Let’s divide our data into sets

1. Data to train our model.
2. Data to test our model.

For the test data we will use the last 6 years to test and evaluate the performance of our model.

train = df[df.ds<='2013-01-01'] 
test = df[df.ds>'2013-01-01']

train.shape, test.shape

((54, 3), (6, 3))

sns.lineplot(train,x="ds", y="y", label="Train")
sns.lineplot(test, x="ds", y="y", label="Test")
plt.show()

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Implementation of AutoETS with StatsForecast

from statsforecast import StatsForecast
from statsforecast.models import AutoETS

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Instantiate Model

sf = StatsForecast(models=[AutoETS(model="AZN")], freq='YS')

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Fit the Model

sf.fit(df=train)

StatsForecast(models=[AutoETS])

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Model Prediction

y_hat = sf.predict(h=6)
y_hat

sf.plot(train, y_hat)

Let’s add a confidence interval to our forecast.

y_hat = sf.predict(h=6, level=[80,90,95])
y_hat

sf.plot(train, y_hat, level=[95])

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Forecast method

Memory Efficient Exponential Smoothing predictions.

This method avoids memory burden due from object storage. It is analogous to fit_predict without storing information. It assumes you know the forecast horizon in advance.

y_hat = sf.forecast(df=train, h=6, fitted=True)
y_hat

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In sample predictions

Access fitted Exponential Smoothing insample predictions.

sf.forecast_fitted_values()

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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.

from functools import partial

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

evaluate(
    y_hat.merge(test),
    metrics=[ufl.mae, ufl.mape, partial(ufl.mase, seasonality=1), ufl.rmse, ufl.smape],
    train_df=train,
)

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References

1. Nixtla Automatic Forecasting
2. Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting Principles and Practice (3rd ed)”.