Machine learning and deep learning method can achieve good performance in a challenging time series forecasting problems. However, in many prediction problems, classical methods, such as SARIMA and exponential smoothing (Exponential Smoothing) , easily than a more complex method. Therefore, before exploring more advanced methods, it is necessary to understand how classical time-series forecasting methods, but also to evaluate it. This article describes the original time series forecasting and classical methods.

1. Simple prediction method

At any time series forecasting problems, establish baseline (baseline) are essential. Performance benchmarks can learn about the practical implementation of all other models on the same issue.

1.1 predict performance benchmarks (Forecast Performance Baseline)

Predict the performance benchmark to provide a point of comparison. This is a reference point for all the same problems other modeling techniques. If a model is at or below benchmark performance, you should adjust or abandon the technology. Technology for generating a prediction to calculate baseline performance must be easy to implement, and specific details of the problem to be very simple. The goal is to obtain time series forecasting problems as soon as possible benchmark performance in order to better understand the data collection and the development of more advanced models. A good way to make a simple prediction has three characteristics:

Simple: an exercise or intelligent method does not require.
Quick: a small-implemented method of predictive computation speed.
May be repeated: a method for determining, meaning that it produce the desired output given the same input.

1.2 prediction strategy (Forecasting Strategy)

Simple prediction strategy is to predict the nature of those problems hardly make assumptions or make assumptions completely, and can quickly implement strategies and calculations. If a model can be better than a simple prediction strategy performance, you can say it very tactfully. Simple prediction strategy has two major themes:

Simple, direct use or observations.
The average, or use statistical calculations previous observations.

1.3 simple prediction strategy (Naive Forecasting Strategy)

Simple prediction involves direct observations before use as a predictor, without making any changes. It is commonly known as persistent predict because previous observations is persistent. For seasonal data, this simple method can be slightly adjusted. In this case, the observations can be simultaneously performed in one cycle retained. This can be further extended to every possible offset test history data, historical data can be used to save the predicted values. For example, a given sequence:

[1, 2, 3, 4, 5, 6, 7, 8, 9]

We can observe the last value (relative index -1) kept 9, the second or the last previous observation (relative index -2) remains 8, and so on.

1.4 average prediction strategy (Average Forecast Strategy)

Step above simple prediction is a priori value averaging strategy. All previous observations are collected and averaged, the average or the median, no other data processing. In some cases, we may want to shorten the average value of the previous time step used in the calculation of the last few observations. We can extend this to the n possible circumstances of each observation a priori set of tests included in the calculation of the average. For example, given the series:

[1, 2, 3, 4, 5, 6, 7, 8, 9]

We can average the last observed value (9), the last two observations (8,9), and so on. For seasonal data, we may want the same time period and the predicted times of the last n previous observations averaged. For example, having assumed that the period of the series of three steps:

[1, 2, 3, 1, 2, 3, 1, 2, 3]

We can use the window size of 3, take the last observed value (-3 or 1), the last two observations - average (1 and 3, or (3 × 2), or 1), and so on.

2. Since the regression method (Autoregressive Methods)

Auto-regressive integrated moving average (ARIMA) is one of the most widely used univariate time series forecasting methods. Although this method may have a tendency of data processing, but it is time series seasonal component is not supported . Called for an extension of ARIMA seasonal auto-regressive integrated moving average (SARIMA) , which supports seasonal ingredients sequence of direct modeling .

2.1 ARIMA

** auto-regressive integrated moving average model (Autoregressive Integrated Moving Average Model) ** is a kind of statistical analysis and forecasting models for time series data. It is explicitly meets a set of criteria in the time-series data structure, thus provides a simple yet powerful way to perform time series forecasting skilled. ARIMA autoregressive integrated moving average of abbreviations. It is a simple autoregressive moving average (ARMA) promotion, added the concept of difference. The acronym meaning:

AR: autoregressive. A model that uses the dependencies between some hysteresis observed values and observed values.
I: Differential. With the differential value of the original observation (e.g. by subtracting a value from the observation value observed in a time step) that the stationary time series.
MA: Moving Average. A model which is applied with hysteresis by using the observation value of the correlation between the residuals of the moving average of the observed values.

Each of these components are explicitly specified as a parameter in the model. ARIMA (p, d, q) using standard notation, wherein the parameter is replaced with an integer value, in particular a quick indication of the ARIMA model is used. One problem is that it does not support ARIMA seasonal data. This is a time series having a repetition period. ARIMA seasonal data is not desired, i.e. seasonal components have been removed, may be seasonally adjusted by the method of seasonal differences.

The establishment of a specific item number and item number of types of linear regression models included, and data prepared by a certain degree of difference, so smooth, that is, eliminate the negative impact on the regression model structure and seasonal trends. A value of 0 may be used for the parameters, the parameter indicates that the element does not use the model. In this way, ARIMA model can be configured to perform the functions of the ARMA model can even be configured as a simple AR, I or MA model. The basic process employed ARIMA time series model is assumed to generate the observed ARIMA process. This seems obvious, but needs help to stimulate confirm the model assumptions in the original observations and model predictions of residuals.

2.2 What is seasonal ARIMA

Seasonal auto-regressive integrated moving average (SARIMA) is an extension of ARIMA, it explicitly supports univariate time series data seasonal ingredients . It adds three new super autoregressive parameters to specify a seasonal component (AR), a difference (I) and moving average (MA), the seasonal period and the additional parameters.

2.3 How to configure SARIMA

Configuring SARIMA trends and seasonal factors need to choose a sequence of ultra-parameter.

There are three elements of the trend needs to be configured. They are the same as the ARIMA model; specifically:

p: autoregressive number of items;
d: making a difference frequency and sequence made stationary (order);
q: a number of moving average;

There are four elements of a portion of seasonal seasonal elements, but not necessarily, arranged on ARIMA; P, D, Q follows the same definition, but the seasonal component applicable to time series.

m (some data are expressed in s): Time period of the sequence (4 quarter, year 12, etc.)

Mathematical expression SARIMA model are:
$SARIMA(p,d,q)(P,D,Q)m$
wherein the parameter specifies a particular hypertext model. Important that, m the influence parameters P, D and Q parameters. For example, m is 12 months of data showing the annual seasonal cycle. $A P=1$ the first offset seasonal observations using the model, e.g. $t-(m×1)$ or $t-12$ 。 $A P=2$ , uses the last two observations offset seasonal $t-(m×1)，t-(m×2)$ . Similarly, D 1 is the calculated first order seasonal differences, $Q=1$ uses a model order error (e.g., moving average).

Trend elements can be selected through careful analysis ACF (autocorrelation function (determined value q)) and FIG PACF (partial autocorrelation function (determined value p)), to view the last time step (e.g., 2, 3) correlation. Similarly, FIG PACF and ACF analysis, the correlation lag time step to the specified value by viewing the model season season.

3. Exponential Smoothing (Exponential Smoothing Methods)

Exponential smoothing is a univariate data time series forecasting method , which can be extended to support data systems with or seasonal trend component. It can be used as an alternative to the popular Box-Jenkins-ARIMA method family.

3.1 What is the exponential smoothing?

Exponential smoothing is a time series data univariate prediction method. Time series methods, such as Box-Jenkins-ARIMA methods family, developed a model, which is a weighted linear prediction lag and recent observation or. The method of exponential smoothing forecasting prediction is similar to a weighted sum, but explicitly model the right past observations using exponentially decreasing observation weights past values. Specifically, the ratio is based on past observations weighted exponentially decreases.

Generated using exponential smoothing forecasting method is a weighted average of past observations, along with observations of growth, weight decays exponentially. In other words, the closer observation, related to the higher weight.

The method of exponential smoothing method may be considered at the same level, it may be replaced popular prediction method Box-Jenkins-ARIMA time series. In general, these methods are sometimes called ETS model , refer to the explicit modeling errors, and seasonal trend. The method of exponential smoothing forecasting time series are mainly three. An assumption there is no simple way to structure the system, an extension to explicitly deal with trends, as well as the most advanced way to add seasonal support.

3.2 Single exponential smoothing

Single exponential smoothing, referred to the SES , also known as the simple exponential smoothing, is a non-trend or seasonal univariate time series forecasting method. It requires a called $\alpha$ parameter, also referred to as the smoothing factor or smoothing coefficient. This parameter controls the rate previously observed values influence index decay time step. $\alpha$ is usually set to a value between 0 and 1. Large value means that the model focuses on the recent past observation, and give more consideration to the history of small value means making a prediction.

Value close to 1 indicate rapid learning (ie, only the most recent value of impact prediction), and close to zero values indicate slow learning (past observations have a significant impact on the forecast).

Super parameters:

$\alpha$ : the level of smoothing factors.

3.3 pairs of exponential smoothing

Double exponential smoothing is stretched exponential smoothing, it adds support for univariate time series trend . In addition to controlling the level of smoothing factors $\alpha$ external parameters, but also adds an additional smoothing factor to control the impact attenuation trends, called $\beta$ . This approach allows changes in trends in different ways: addition and multiplication, respectively, depending on the trend is linear or exponential. Additive trend having a double exponential smoothing Holt commonly referred to as a linear trend model (Holt's linear trend model), to a method named inventor Charles Holt.

Add trends : a linear trend double exponential smoothing.
By trends : a double exponential smoothing exponential trend.

For longer range (multi-step) predicted that trend is likely to continue to be impractical. Thus, inhibition of the trend over time may be useful. Damping refers to the size tends to be smaller in the future time, and then gradually decreased to a straight line (no trend).

As with the trend model itself, the same principles can be used to suppress the trend, especially linear or exponential inhibitory effect of adding or multiplying. A damping coefficient φ (p or [Phi]) is used to control the rate of inhibition.

Adder inhibition: inhibited linear trend.
Multiplication inhibition: inhibition of this trend index.

Super parameters:

$\alpha$ : the level of smoothing factor;
$\beta$ : smoothing factor trend;
Trend Type: addition or multiplication;
Type of inhibition: addition or multiplication;
$\phi$ : damping coefficient;

3.4 triple exponential smoothing

Triple exponential smoothing is extended exponential smoothing, which explicitly adds support for seasonal single variable time series . This method is sometimes referred to as Holt-Winters exponential smoothing, two contributors to the method of Charles Holt and Peter Winters named. apart from $\alpha$ and $\beta$ smoothing factor, add a new parameter $\gamma$ to control seasonal components. As with the trend, seasonal addition or multiplication process can be modeled as a linear or exponential variation of seasonal.

Seasonal addition: triplet seasonal exponential smoothing linear.
Seasonal multiplication: Index Seasonal triple exponential smoothing.

Triple exponential smoothing exponential smoothing is most advanced variant, configure, can also create single and double exponential smoothing model exponential smoothing model.

As a method for adaptive, Holt-Winter exponential smoothing allowable levels, trends and seasonality (level, trend, seasonality) patterns change over time.

In addition, in order to ensure proper modeling of seasonal, you must specify the time step during seasonal (period) long. For example, if the series is monthly data, and the seasonal cycle is repeated every year, then the cycle = 12.

Super parameters:

Alpha( $\alpha$ ): the level of smoothing factors.
Beta( $\beta$ ): Trends smoothing factor.
Trend Type: addition or multiplication.
Dampen Type: addition or multiplication.
Phi( $\phi$ ): damping coefficient.
Gamma( $\gamma$ ): Seasonal smoothing factor.
Seasonality Type: addition or multiplication.
Period: Step seasonal time period.

3.5 How to configure exponential smoothing?

You can explicitly specify all super model parameters. This is a challenge for experts and beginners. Instead, a numerical optimization is generally used to search for and provide support for the smooth model coefficients (alpha, beta, gamma and Phi), thereby obtaining a minimum error.

For the unknown parameters included in the exponential smoothing, estimated from observations in a more robust, more objective method for determining the value. [... any unknown parameters and initial values of exponential smoothing method may be estimated by minimizing the SSE (square error).

Specify the type of trend and seasonal variations of parameters, such as they are added or multiplied, and whether they should be suppressed, must be clearly specified.

Reference:
https://blog.csdn.net/qifeidemumu/article/details/88782550
https://baike.baidu.com/item/ARIMA model / 10,611,682 = fr aladdin?
Https://machinelearningmastery.com/findings-comparing -classical-and-machine-learning- methods-for-time-series-forecasting /