Trend or time series tendency is a somewhat conditional concept. A trend refers to a regular, non-random component of a time series (usually monotonic), which can be calculated by a specific, unambiguous rule. Trend analysis and determining the duration of a trend are one of the main aspects in a trader’s work. Therefore, mathematical modeling of the probability of the existence and monotonicity of a trend, as well as estimating its probable duration, is an important task [1,3].
Figure 1. Trend of the weekly dollar/ruble chart.
After identifying a linear trend, it is necessary to determine how significant it is. This is done using correlation coefficient analysis. The fact is that a correlation coefficient different from zero and thus the presence of a real trend (positive or negative) may turn out to be random, related to the specifics of the considered time series segment. In other words, when analyzing another set of experimental data (for the same time series), it may turn out that the estimate obtained at that time is much closer to zero than the original one (and may even have a different sign), and it becomes difficult to speak about a real trend.
Special techniques have been developed in mathematical statistics for testing the significance of a trend. One of them is based on the use of the Student’s t-distribution.
Figure 2. Trend of the weekly euro/dollar chart.
In addition to the linear trend, trends of more complex structures must also be considered. It is usually not possible to specify a function that can describe this trend in advance. Therefore, in practice, several simple functional dependencies (with parameters) are often tried, and for each of them, it is evaluated how successfully a function of a certain type can describe the tendency of the time series being studied. With the help of a computer, these calculations do not take much time, and sometimes they can even be performed automatically, selecting the optimal trend among several predefined types. However, it is not always the case that among the functions considered there is one that effectively describes the tendency of the given time series. In such cases, other ways must be taken. In particular, various transformations of the elements of the time series are carried out (logarithmation, differentiation – forming differences between adjacent elements of the series, integration – summing consecutive elements of the series, etc.) in order to try to obtain a time series with a clearly expressed linear trend. If this is achieved, the methods described above are applied to the resulting series, and then the inverse transformation is used to return to the original series. It should be emphasized once again that the form of the trend is not uniquely defined by the series itself and is a conditional object, used to better understand the characteristics of the process under consideration.
It is difficult to model and automate the detection of a trend in a time series. However, if the trend is monotonic (steadily increasing or decreasing), it is usually possible to analyze such a series. If the time series contains significant error, the first step in extracting the trend is smoothing. Smoothing always includes some method of local averaging of data, where non-systematic components mutually cancel each other. The most general smoothing method is the moving average, in which each element of the series is replaced by a simple or weighted average of neighboring elements, where L is the sample width. Instead of the average, the median of values falling within the sample can be used. The main advantage of median smoothing compared to moving average smoothing is that the results become more resistant to outliers (those present within the sample). Thus, if there are outliers (for example, due to measurement errors), median smoothing usually leads to smoother or, at least, more “reliable” curves, compared to the moving average with the same sample. The main disadvantage of median smoothing is that in the absence of clear outliers, it results in more “toothed” curves (than moving average smoothing) and does not allow the use of weighting coefficients.
Relatively rarely, when the measurement error is very large, least squares smoothing weighted by distance, or exponentially weighted moving average is used. All these methods filter noise and transform the data into a relatively smooth curve. Time series with a relatively small number of observations and systematically arranged points can be smoothed using splines. However, many monotonic time series can be well approximated by a linear function. If there is an obvious monotonic nonlinear component, the data should be transformed first to eliminate the nonlinearity. Usually, logarithmic, exponential, or (less commonly) polynomial data transformation is used for this purpose.
General superficial knowledge of the subject of economic theory gives rise to contempt for specialized knowledge about it – a statement by Nobel Laureate M. Friedman [2]. From a practical point of view, every time the future course of events is predicted based on the expectation of continuing something previous (a chain of misfortunes or successes, a tendency toward deterioration or improvement, etc.), it is essentially, in one form or another, and to some extent, a bet on the law of inertia. It is not surprising that it has long been discovered in the movement of stock prices. Any development of events can be represented as an arbitrary combination of two states – the inertia of rest or inertia of motion, which once arose under the influence of a certain impulse of any nature: macroeconomics, psychology, the will of chance, etc., and now, having left the period of rest, continues. In the 1960s, a whole series of scientific works appeared, in which mathematical justification for the existence of a trend was provided, with the wide usage of the term “economic inertia,” understood as the impossibility for the entire economic mechanism to sharply change course. By analyzing the market participant behavior model, it is possible to conclude that the market cannot immediately switch its sentiments from bullish to bearish and vice versa, hence the concept of “market inertia” arises. The huge mass of the market does not allow it to maneuver quickly, and any maneuvers must be started in advance, otherwise they simply will not succeed on the FOREX market. Constantly performing “turns”, the market can move anywhere, but this movement can be predicted thanks to “market inertia.” Where exactly the market is moving at the moment, it can be quite difficult to predict. The task becomes simpler if we assume that there is a trend on the market. Then, with a certain degree of accuracy, it is possible to understand where the currency rate is likely to be at the next moment. A sharp maneuver is possible in this case, but for its implementation, a huge force is needed. This can also explain the inertia inherent in the FOREX market.
FOREX Forecasting Method SSA
One of the powerful methods for analyzing time series is the method called SSA – Singular Spectrum Analysis. This method is used for analyzing and forecasting time series as a filter. We briefly describe the principle of operation of this method.</
FAQ
What is a trend in time series analysis?
A trend is a regular, non-random component of a time series, often described by a simple function like linear or quadratic, representing the overall behavior of the data.
How is the significance of a trend tested?
The significance of a trend is tested using statistical methods such as correlation coefficient analysis and the Student’s t-distribution to determine if the trend is likely real or due to random variation.
What is Singular Spectrum Analysis (SSA)?
SSA is a method used for analyzing and forecasting time series by decomposing the data into components, including trends, and can help identify patterns for prediction purposes.
