On the use of financial analysis tools for the study of Dst time series in the frame of complex systems

Technical analysis is considered the oldest, currently omnipresent, method for financial markets analysis, which uses past prices aiming at the possible short-term forecast of future prices. In the frame of complex systems, methods used to quantitatively analyze specific dynamic phenomena are often used to analyze phenomena from other disciplines on the grounds that are governed by similar dynamics. An interesting task is the forecast of a magnetic storm. The hourly st D is used as a global index for the monitoring of Earth's magnetosphere, which could be either in quiet (normal) or in magnetic storm (pathological) state. This work is the first attempt to apply technical analysis tools on st D time series, aiming at the identification of indications which could be used for the study of the temporal evolution of Earth's magnetosphere state. We focus on the analysis of st D time series around the occurrence of magnetic storms, discussing the possible use of the resulting information in the frame of multidisciplinary efforts towards extreme events forecasting. We employ the following financial analysis tools: simple moving average ( SMA ), Bollinger bands, and relative strength index ( RSI ). Using these tools, we formulate a methodology based on all indications that could be revealed in order to infer the onset, duration and recovery phase of a magnetic storm, focusing on the temporal sequence they occur. The applicability of the proposed methodology is examined on characteristic cases of magnetic storms with encouraging results for space weather forecasting.


Introduction
The relatively new field of complex systems increasingly gains the interest of scientists working on disciplines ranging from physics and engineering to economics, biosciences and social sciences, e.g., [1][2][3][4][5][6][7][8]. The unique characteristic of complex systems is that they may have certain quantitative features that are intriguingly similar, while their dynamics are governed by a set of universal principles [9,10]. Thus, complex systems from different disciplines are often analyzed within similar mathematical frameworks.
There is an apparent paradox in the above mentioned suggestion. How is it possible for a concept as multifaceted as complexity to serve as a unifying direction? When one considers a phenomenon that is "complex"' refers to a system whose phenomenological laws, which describe the global behavior of the system, are not necessarily directly related to the "microscopic" laws that regulate the evolution of its elementary parts [11]. This is a basic reason for our interest in complexity [11][12][13][14][15]. There is a common factor in these seemingly diverse phenomena. The complex systems adopt a pattern of behavior almost completely determined by the collective effects. They exhibit remarkable properties of self-organization and emergence of coherent structure over many scales. The main feature of collective behavior is that an individual unit's action is dominated by the influence of its neighbors; the unit behaves differently from the way it would behave on its own, so that, all units simultaneously alter their behavior to a common pattern. Thus, new features emerge as we move from one scale to another and the science of complexity is about revealing the principles that govern the ways in which new properties appear [11].
Two of the most vivid and richest examples of the dynamics of a complex system at work is that of economic systems (financial markets) [14,[16][17][18][19][20][21][22][23][24] and Earth's magnetoshere [25][26][27][28][29][30][31][32][33][34][35]. Their richness in interactions renders them characteristic examples of complex dynamics. In this paper, we focus on the identification of indications which could be used for the study of the temporal evolution of Earth's magnetosphere state and consequently for the forecast of magnetic storms (MSs). Magnetic storms occur when the accumulated input power from the solar wind to magnetosphere exceeds a certain threshold. Notice, MSs are a main element of space weather: they have severe impacts on both space-borne and ground-based technological systems [36]. Thus, the prediction of a forthcoming intense MS is a major task. Magnetic storm intensity is usually represented by an average of the geomagnetic perturbations measured at four mid-latitude magnetic observatories, known as the st D index [37].
This work aims at enhancing the suggestion that transferring ideas, methods and insights from investigations in hitherto disparate areas, namely, economic and geophysical systems, will cross-fertilize and lead to important new results concerning the dynamics of the corresponding extreme events, i.e., dynamics of economic crises and magnetic storms. A question effortlessly arising is whether the aforementioned notion is groundless or not. A number of analysis methods have been mutually used to p. 3 study the dynamics of financial markets, earthquakes, and magnetosphere, e.g., [17,[38][39][40][41][42][43][44][45][46][47][48][49][50][32][33][34][35]. Several authors have suggested that earthquake dynamics and the dynamics of economic (financial) systems can be analyzed within similar mathematical frameworks, e.g., [10,14,[51][52][53][54], and references therein]. On the other hand, authors have also suggested that earthquake dynamics and magnetic storms dynamics can be analyzed within similar mathematical frameworks, as well e.g., [33][34][35]. Thus, the question whether these two complex systems, namely, financial crises and magnetic storms, can be analyzed within the same mathematical framework seems to be justified. This conclusion is specifically enhanced by the fact that the signature of Discrete Scale Invariance (DSI) [49] characterizes the earthquakes and financial crises [49], and magnetic storm [34], as well. It is important to stress the practical consequence of the presence of the corresponding log-periodic structures. For prediction purposes, it is much more constrained and thus reliable to fit a part of an oscillating data than a simple power law which can be quite general especially in the presence of noise [49].
Technical analysis, primarily employed for the empirical analysis of economical time series, is considered the oldest method for investment analysis with origins dating perhaps before the 1800s [55]. Charles Dow who developed the famous Dow Theory, which was later refined by S. A. Nelson, W. P. Hamilton and R. Rhea in the early 20 th century, is considered the pioneer of modern technical analysis [56]. Nowadays, technical analysis is omnipresent in financial markets analysis. Taylor and Allen [57] as well as Menkhoff [58], nearly two decades later, analyzed data of 692 fund managers in five countries, including the USA, concluding that at least 90% or 87%, respectively, of the involved respondents pay enough attention to technical analysis in order to take investment decisions primarily on short term investments. Note that technical analysis has also been used in other disciplines too such as medicine, e.g., [59,60], data communications, e.g., [61,62], textile engineering, e.g., [63], wireless sensor networks, e.g., [64] and aviation [65]. Technical analysis uses past prices having as target the possible identification of future prices. The efficiency of technical analysis in markets which are characterized by long-term memory, as determined by Hurst exponent, has been recently studied in fifteen of the largest global equity markets [55].
In this paper, we investigate the use of some widely employed tools of technical analysis for the study of the temporal evolution of st D time series. Specifically, we study here whether the simple moving average ( SMA), the Bollinger bands, and the relative strength index ( RSI ) can discriminate in the st D time series the transition from pre-storm (quiet) geomagnetic activity to the generation of a magnetic storm, through the identification of specific indications in particular temporal sequence. Note that, to the best of our knowledge, this is the first time that such a study appears in the literature. Such a study: (i) could be utilized for space weather forecasting purposes, especially if it will be combined with other studies (e.g., of log-periodic corrections) and (ii) may offer ideas in order to provide a physical meaning in the above mentioned empirical tools of technical analysis. p. 4 The remaining of this contribution is organized as follows: The necessary background information on the used technical analysis tools, as well as information about the analyzed time series is provided in Sec. 2. In Sec. 3 we describe the behavior of the considered technical analysis tools when applied on st D time series during quiet magnetospheric conditions as well as during the evolution of a magnetic storm, while a methodology for the use of the specific tools in the study of magnetic storms is introduced as a step-by-step procedure. The analysis of different st D time series including magnetic storms is presented and the obtained results are discussed in Sec. 4. Finally, in Sec. 5, the presented st D analysis method and obtained results are discussed, while the conclusions are summarized.

Data and Analysis Methods
In the following we provide a brief introduction to the main information and equations related to the analysis methods which are used in this paper, as well as a description of the Earth's magnetosphere observables ( st D time series) on which these methods are applied on.

Methods
The main aspects of the simple moving average ( SMA), Bollinger bands, and relative strength index ( RSI ) methods which are widely used in the empirical analysis economic time series known as "technical analysis" are briefly presented in the following. The introduction to these methods is presented in terms of the information which is extracted by each one of them during the analysis of stock market time series, while all necessary mathematical formulas for their application are also given.

Moving Average
The moving average is one of the most popular and widely used tools of technical analysis. It is characterized as an automated trendline and it is considered an objective and reliable tool for stock market analysis [66,67]. The information provided by moving average is threefold [68]: (i) it determines the direction of the trend, (ii) it confirms the change of the trend and (iii) it smoothes the extraneous data which are often misguiding. As it is expected, the moving average responds to price changes with lag, since it is calculated using past prices. The longer the length of the calculation period, the slower it responds to changes [69]. A wide variety of moving averages are used in stock market time series analysis. In this paper we will refer to the simple moving average () SMA which is defined by the following formula [56]: where k SMA is the simple moving average at period k , i C is the closing price for the period i , n is the total number of periods to be included in the moving mean calculation and k is the number of the period being studied within the total number of periods in the database.
The most popular interpretations of the moving average results are the following: (i) upward (or downward) crossing of the moving average curve by the price curve is a sign of upward (or downward) trend [70]; (ii) upward (or downward) slope of moving average curve means upward (or downward) trend of the price; also if the slope is steep, the trend is expected to be strong, while, conversely, if the slope is gentle, the trend is expected to be weak [71]; (iii) the moving average often acts as "support" and "resistance" level of the price trend [72], i.e., if the price values and the moving average are plotted on the same chart, and the chart shows that the price value, within a time period, never seems to be able to rise above a value A or fall below a value B of the moving average, then the resistance level is the value A and the support level is the value B, respectively; (iv) the prices have the tendency to return to the value of moving average [72].

Bollinger Bands
The Bollinger bands were developed by John Bollinger in the 1980s and are a tool of technical analysis that belongs to a wider category of price analysis methods called "trading bands". The trading bands in general consist of curves lying at a distance above and below a measure of central tendency [73,74]. In the specific case of Bollinger bands the measure of central tendency is the simple moving average and the distance of the curves from the moving average depends on the local standard deviation of prices. The standard deviation measures the volatility of prices, since it is a statistical measure that indicates how far prices range from average [75]. Due to the way Bollinger bands are constructed, their distance regulates itself and adapts to the volatility of market prices. When volatility is high, the bands widen, while when it is small the bands narrow approaching the moving average [76]. The Bollinger bands consist of three curves: the Middle Band () MiddleB , the Upper Band () UpperB , and the Lower Band () LowerB , calculated by the following formulas, respectively: . As it is expected, these parameters were selected so that the majority of the prices fall within the bands. It has been argued that, in most markets, about 88% to 89% of the prices are within the bands for 20 n  and 2 K  [73]. This has been further supported by the work of Liu et al. [76] who studied stock market indices (DOW, NASDAQ and S&P500) from 1991 to 2005 concluding that 94% of the daily closing prices were within the Bollinger bands. Moreover, Xu and Yang [77] reached a similar conclusion after studying the indices SPY, QQQ and DIA for the period 2008 to 2011, namely during the economic crises of 2008, and found that more than 95% of the prices lie within Bollinger bands [77].
The most popular interpretations of Bollinger bands results are the following [70]: (i) after the narrowing of the bands, the prices usually have a sharp change; (ii) when prices move outside the bands, this implies that the current trend is going to continue; (iii) when local maximum (or local minimum) which is outside of bands is followed by local maximum (or local minimum) inside the bands, this is an indication of an upcoming reversing of the current trend.; (iv) for a price change which starts at one of the bands it is expected that this will continue changing, maintaining its trend, until it covers all the way to the other band.
Finally, we should mention that originally the Bollinger bands method was exclusively used in finance time series analysis, but over time, this method has been proven to find application in various fields such as textile engineering, data networks, wireless sensor network and aviation, e.g., [62][63][64][65].

Relative Strength Index (RSI)
Relative Strength Index () RSI was developed by W. J. Wilder [78]. It is one of the best known momentum indicators and measures the speed and the magnitude of the direction of price movements [79,77]. The RSI takes values in the interval [0, 100]; generally, values of RSI over 70 suggest probable "overbought" situation, while values below 30 indicate a probable "oversold" situation. Overbought (or oversold) is a term used to describe the situation following a rise (or decline) in share prices, in which some investors believe that prices have risen (or fallen) exceedingly, i.e., the share is overvalued (or undervalued) relative to fundamentals [80]. The specific p. 7 threshold values (70 and 30) are not absolute; other threshold values also appear in the literature [72]. The RSI value is calculated using the following equation [ [56]. Apparently, the smaller the calculation period, n , the more sensitive to price changes is the RSI , while in the opposite case the index is less sensitive [56].
The following interpretations of the RSI analysis results have been proposed [78]: (i) the RSI often reaches a zero slope (local maximum) over 70 or a zero slope (local minimum) below 30, prior to a local maximum or local minimum of the prices, respectively, providing an indication of a possible trend reversal; (ii) the chart of RSI shows often formations, such as "head" and "shoulders" (when three peaks appear successively and the second is higher than the other two, then the first and third are called shoulders and the second is called head) or "triangles" (a triangle occurs as the range between peaks and troughs narrows), that indicate a possible change of trend, formations sometimes not evident in the price chart; (iii) "failure swings" above the level of 70 (or below the level of 30) are very strong evidence of a trend reversal. The term failure swing describes a local maximum (or local minimum) of the RSI curve above 70 (or below 30) that is followed by a second local maximum (or local minimum) below 70 (or above 30); (iv) the RSI shows several times clearer "resistance" and "support" levels of the chart of prices, i.e., if the RSI value is plotted on a chart, and the chart shows that the RSI value, within a time period, never seems to be able to rise above a value A or fall below a value B, then the resistance level is the value A and the support level is the value B [80]; (v) when "divergence" between the prices and the RSI values are observed, namely new high (or low) that is not verified by a new high (or low) of the RSI values, the prices tend to follow the direction of motion of the RSI values.

Data
Magnetic storms are the most prominent global phenomenon of geospace dynamics, interlinking the solar wind, magnetosphere, ionosphere, atmosphere and, occasionally, the Earth's surface [36,81,82]. Magnetic storms produce a number of distinct physical effects in the near-Earth space environment: acceleration of charged particles in space, intensification of electric currents in space and on the ground, impressive aurora displays and global magnetic disturbances on the Earth's surface [83]. The p. 8 latter serve as the basis for storm monitoring via the hourly st D index, which is computed from an average over four mid-latitude magnetic observatories [37];  Table 1, respectively. Informative plots summarizing the analysis results for the full set of analyzed magnetic storms listed in Table 1 are provided as on-line supplementary material.

A Proposed Magnetic Storm Analysis Based on Stock Market Tools
Technical analysis is the process aiming at the possible identification of future prices by analyzing past price data [70], using a set of empirical stock market analysis tools. Crucial factor towards achieving this goal is to identify the direction (upward, downward or "sideways" -horizontal-), duration and strength of the price trend [79]. In order to identify those characteristics of trend, the use and combined interpretation of different tools of technical analysis is required [73,79]. The application of these tools is flexible and can be adapted to any time duration (from a few minutes up to months) [56]; however, the obtained results are considered more reliable in the case of short-term (in the scale of a few hours to a few days) analyses, compared to the ones obtained for long-term (in the scale of a many months to a few years), ones [75]. Based on this fact and given that the duration of a typical magnetic storm ranges in the order of a few hours up to a few days, while the available data are hourly st D values, a short term prediction of the time evolution of the phenomenon seems feasible.
In this paper, we attempt to apply a combination of three technical analysis tools, on hourly st D data variations. Specifically: SMA, which has been calculated for 20 n  hours (cf. Eq. (1) ), is used to identify the trend of st D values and offers, albeit with some lag, quite reliable signs of p. 9 upcoming changes in st D values. Note that, "if the slope is sharp, the trend is strong, and if the slope is shallow, the trend is weak; a flat or choppy SMA indicates a rangebound market" [71].
(ii) The Bollinger bands, calculated for an SMA of 20 n  hours and 2 K  (cf. Eqs. (2) -(4)), are used in order to provide a depiction of the range of variation of the Through the application of these stock market tools on st D time series, our aim is to identify specific indications which could be used for the study of the temporal evolution of the state of Earth's magnetosphere and, accordingly, formulate a methodology that could be employed in order to infer the onset, duration and recovery phase of a magnetic storm. The here proposed methodology was extracted after the analysis of 24 magnetic storms of all classes occurred from 1958 to 2015, shown in Table 1.
In the following Sec. 3.1 and Sec.3.2, we describe the behavior of the considered technical analysis tools during quiet magnetospheric conditions as well as during the evolution of a magnetic storm, respectively, providing thus a method of application of these tools for the study of the temporal evolution of st D time series. Especially in the second case, we provide a detailed description of all the indications that could be employed in order to infer the onset, duration and recovery phase of a magnetic storm, focusing on the temporal sequence in which they occur. The application of the proposed methodology on characteristic cases of magnetic storms is demonstrated in Sec. 4.

Quiet Magnetospheric Conditions
In periods that normal (quiet) magnetospheric conditions prevail, cf. Fig. 1, we observe that all the applied technical analysis methods suggest that no significant downward trend of Dst time series analysis using financial tools p. 10 evolves with nearly horizontal slope. In terms of the Bollinger bands (shown by the two red/dotted curves in Fig. 1.a), the same situation is reflected to that these bands are evolving close to each other, nearly horizontally, indicating low volatility of st D values; in most cases st D values vary within the Bollinger bands, however in some cases short st D curve segments may be found to move slightly outside one of the bands, as shown in Fig. 1.a (close to 25/10/2007 17:00). At the same time, the RSI indicator (shown in Fig. 1.b) mainly varies between the thresholds 30 and 70 (grey horizontal dotted and dashed line, respectively, in Fig. 1.b). However, several times "support" or "resistance" may be found at the threshold levels 30 or 70 respectively, i.e., the RSI value may be shortly found on (or slightly outside) the lower or the higher threshold level, respectively, to return inside immediately after. Note that a short movement of the RSI value in the oversold area (below 30) by itself, i.e., without being part of the sequence of indications described in Sec. 3.2 does not imply an impeding magnetic storm. The described behavior of the considered technical analysis methods can easily be verified in Fig. 1.

Magnetic Storms Evolution in Terms of Financial Analysis Tools
After analyzing the st D time series corresponding to all the magnetic storms included in Table 1, we considered that it is appropriate to classify the indications, which resulted from the application of the employed technical analysis tools and could be used in order to infer the onset, duration and recovery of a magnetic storm, into main and secondary. The ones characterized as main are those which have been found to occur for all the examined cases of magnetic storms, whereas the indications characterized as secondary should be considered as indications supporting the main ones, without however being expected to be found in all cases.    Fig. 3.a and 3.d: The red line shows the failure swing point [84]. Fig. 3.b: The red line which it is called "neckline" connects the lows of the two troughs between the three peaks. The line can slope up or down. The point at which the RSI value crosses the red line is often used as a final indication for the change of trend ("trigger point") [84]. Fig. 3.c.: Horizontal support line is an imaginary line that acts as support for the RSI values. Once RSI values are below this line, it resists any upward movement of the RSI values. Down sloping top trend line is an imaginary line that acts as a layer of resistance for the RSI values [84].
In the following we present a methodology of inferring the evolution, namely the onset, duration and recovery phase, of a magnetic storm by assessing the behavior of the three considered technical analysis tools. We employ all signs, main and secondary, focusing on the temporal sequence in which they occur. The methodology is presented in the form of three consecutive phases, each one consisting of a set of indications organized in steps according to their order of occurrence. An illustrative example of application is given in parallel using a super-storm which occurred on 31/3/2001 with a peak st D value of -387 nT (see Fig. 2.a, as well as line no. 13 of Table 1). p. 14

Phase I: Preparation
(I.a) The first main indication of a possible upcoming magnetic storm is the narrowing of the Bollinger bands (cf. Fig. 2.a). The narrowing of bands indicates very low volatility of the st D values preceding a volatility breakout [73] and a corresponding extension of the bands. Immediately after the narrowing of the Bollinger bands we only know that this will be followed by a greater volatility of  Fig. 2.b.), as well as a "failure swing" is observed (cf. Sec. 2.1.3, see solid cycle area in Fig. 2.b, and enlarged in Fig. 3.a), which are also indications of a continuing downward trend of the st D curve. Finally, during the same period there have been observed, more rarely than the previous indications, formulations in the RSI indicator known as "head and shoulders" or "triangles" (cf. Sec. 2.1.3, see solid cycle area in Fig. 2.b and enlarged in Fig. 3.b and Fig. 3.c, respectively).  Fig. 2.a. This is considered a very strong indication, since it has been clearly identified before each one of the studied magnetic storms. The downward movement of st D usually starts a few hours earlier (cf. Fig. 2.a) because, as we already mentioned, the SMA provides signs of trend change with a lag. Another main indication is the steep downward slope of the RSI indicator ( Fig. 2.b). This is indicating that st D curve is moving with high speed downwards, and consequently the magnetic storm is in full deployment. The next very important indication, of the indications sequence implying the main phase of a magnetic storm, is the downwards crossing of the lower Bollinger band by the st D curve (cf. Fig. 2.a, the moment of crossing is denoted by the vertical dashed line). This usually happens shortly after the downwards crossing of SMA ; however, sometimes these two indications occur simultaneously. The downwards crossing of the lower Bollinger p. 15 band by the st D curve is considered a main indication, since it has been identified in all the analyzed magnetic storm cases.
(II.b) In the majority of them, the st D curve, after crossing the lower Bollinger band, continues to move outside the bands during the entire time period of its downward movement. This behavior, according to technical analysis literature, is a very strong indication for the evolution of the phenomenon, since it implies a continuation of the current (downward in our case) trend [70]. However, since this is not always observed for magnetic storms it has been classified as a secondary indication. Note that in the differentiating cases after the downwards crossing of the lower Bollinger band by the st D curve, the st D curve continues moving for a while outside the bands but soon after it penetrates into the bands; however, it continues its downward movement very close, almost in parallel, to the lower Bollinger band.
(II.c) Another main indication of the deployment of the phenomenon during its main phase is the retreat of the RSI indicator in the oversold area (below 30), see Fig. 2.b. Note that this has several times been observed to occur simultaneously to the downward crossing of the lower Bollinger band by the st D curve mentioned in step (II.a) (observe the vertical dashed line in Fig. 2.a,b). During the same time period we observe the downward slope of the SMA; the steeper the slope the more intensive will be the storm.  Fig. 2.a,b). Specifically, as it is mentioned in technical analysis literature: "when price touches the lower Bollinger band, and RSI is above 30, it is an indication that the trend should continue, if price touches the lower Bollinger band and RSI is below 30 (possibly approaching 20), the trend may reverse itself and move upward" [67]. This means that if we observe a simultaneous crossing (of the lower Bollinger band by the  Fig. 2.b, and enlarged in Fig. 3.d).
(III.d) The final indication informing us for the termination of the phenomenon is the entrance of the RSI indicator to the overbought situation, which implies that the upward trend of the st D curve is strong before it returns to the quiet conditions.

Analysis Results
The applicability of the methodology proposed in Sec. 3 for the analysis of magnetic storms based on stock market (technical analysis) tools is examined in detail on four Potirakis et. al.
Dst time series analysis using financial tools p. 17 characteristic cases of magnetic storms. Moreover, Table 1 shows which of the considered indications (per phase and step) were identified for each one of the 24 analyzed magnetic storms. The corresponding informative plots, which summarize the analyses results for each one of magnetic storms listed in Table 1, are also provided as on-line supplementary material. Table 1. List of the magnetic storms which were analyzed using the methodology proposed in Sec. 3. Apart, from the details of each magnetic storm (date/time of occurrence and peak st D value), the indications (per phase and step) which were identified for each one of them are denoted. The notation used is the defined in Sec. 3 for the phases, steps and the numbering of main and secondary indications. The identified indications are denoted by an "X" mark.
Step I.a Step II.b Step II. Step II.a Step II.c Step III.a p. 19

First Case-Study
The first case we examine is that of the storm occurred on 22/10/1999 (see line no. 10 of Table 1); the analyses results are portrayed in Fig. 4. The evolution of the specific magnetic storm in terms of the phases and corresponding steps described in Sec. 3.2 follows:

Phase I: Preparation
In Fig. 4, we observe a (prolonged) narrowing of Bollinger bands, from 18/10/1999 21:00 until 22/10/1999 01:00. As already mentioned in step (I.a) of Sec. 3.2 this is a first indication for a possible upcoming magnetic storm. Note that the secondary indications described in step (I.b) are not observed in the specific case.

Phase II: Main
Step Step (II.b): As long as the st D curve moves downward outside the Bollinger bands a continuation of the current (downward) trend is implied.
Step (II.c): One hour later, on 22/10/1999 02:00, the RSI indicator retreats below 30 while st D value is -77 nT. During the next five hours, the downward movement of st D continues, while in parallel, the SMA curve retreats with steep slope, concluding the sequence of main indications related to the deployment of the magnetic storm.
Step (II.d): At this point we should note that at the moment when the st D curve downward crossed the lower Bollinger, the RSI indicator had not yet reached its lower threshold value of 30. This, as it has been observed in the majority of the examined magnetic storm cases (cf., Sec. 3.2), means that the downward movement of the st D , and consequently the magnetic storm, is expected to last for long time, provided that the RSI indicator finally retreats below 30, as it indeed happens in our case.

Phase III: Recovery
Step (III.a): On 22/10/1999 07:00, the RSI curve has almost zero slope indicating that the downward trend of st D curve will soon be reversed to upward trend.
Step Step (III.c): After that, we identify two more main indications implying the end of the phenomenon: first the RSI indicator exits the oversold situation on 22/10/1999 14:00, p. 21 while two hours later the upward crossing of the SMA curve by the st D curve happens. Note that the secondary indications of step (III.c) described in Sec 3.2 were not observed in the specific magnetic storm case.
Step (III.d): Finally, on 23/10/1999 02:00 the RSI indicator enters the overbought situation signifying the conclusion of the magnetic storm.

Second Case-Study
Next we examine a storm which occurred on 08/11/2004 (see line no. 18 of Table 1), while the corresponding analyses are shown in Fig. 5.
Step (I.b): At 21:00 of the next day we can see two secondary indications of the preparation phase, namely the head fake and the divergence between the st D curve and the RSI indicator. Moreover, in parallel, the RSI indicator curve forms a failure swing. Although these are secondary indications, they enhance the evidence in favor of an upcoming magnetic storm, since they imply a continuing downward movement of st D .

Phase II: Main
Step Step (II.b): At this point, we observe that the st D curve moves downward outside the Bollinger bands. As long as this is satisfied a continuation of the current (downward) trend is implied. p. 22 Step (II.c): On 08/11/2004 01:00, the RSI indicator curve retreats below 30. During the next six hours the downward movement of st D continues, while in parallel, the SMA curve retreats with steep slope.
Step (II.d): At this point we should take into account that the RSI indicator reached the low threshold value of 30 later than the moment when the  Step (III.c): Then, the RSI indicator gradually moves upwards leading to an indication signifying the end of the magnetic storm: the exit of the RSI indicator from the oversold situation which occurs on 08/11/2004 11:00. Following that, the next main indication is the upward crossing of the SMA curve by the st D curve which takes place on 08/11/2004 13:00. From this point on, although with a certain delay, the SMA curve develops an upwards slope. Five hours later, on 08/11/2004 18:00 a secondary indication, a failure swing, appears in the RSI indicator curve enhancing the evidence in favor of the upward trend of st D curve.
Step (III.d): Finally, the information about the upcoming definite conclusion of the magnetic storm is provided by the entrance of the RSI indicator into the overbought situation on 09/11/2004 01:00. p. 24

Third Case-Study
The third magnetic storm analyzed in terms of technical analysis tools is a storm which occurred on 26/05/1967(see line no. 8 of Table 1), while the corresponding analyses are shown in Fig. 6. For the specific magnetic storm we can identify the following sequence of indications which can be used to infer its evolution by shortterm analysis.
Step (I.b): On 25/5/1967 14:00 we observe that a head fake is formed over the upper Bollinger band, which is a secondary indication warning for an upcoming downward trend of st D curve. Note that this is the only secondary indication that was identified at this step for the specific storm.

Phase II: Main
Step has not yet been observed. As already mentioned in Sec. 3.2, this step corresponds to a secondary indication which is not always observed after a downwards crossing of the lower Bollinger band by the st D curve, so we don't pay any special attention to this fact. After, the second crossing of the lower Bollinger band, we also notice that d the RSI indicator moves with a steep downward slope, which is gradually reducing.
Step Step (II.c): The RSI indicator moves again downwards to the oversold situation to retreat below the 30 level one hour later. At the same time we note that the SMA curve gradually begins to move downwards with steep slope, indicating a strong downwards trend for st D .
Step (II.d): Given that the st D curve downward crossed the lower Bollinger band before the RSI indicator retreats below 30, we expect that the downward movement of st D until it reaches its lowest value will last for long time.

Phase III: Recovery
Step (III.a): On 26/5/1967 02:00 the RSI curve reaches almost zero slope indicating that the downward trend of st D curve is soon expected to be reversed to an upward trend. One hour later, at 03:00, we can see that the st D curve penetrates inside the p. 26 Bollinger bands, which is a secondary indication enhancing the possibility that the st D trend is soon expected to be upward.
Step (III.b): One hour later, on 25/5/1967 04:00 a local maximum of the upper Bollinger band is observed which is a main indication also informing us about the upcoming upwards trend of the st D curve. Next, the st D curve moves towards the SMA curve and away from the lower Bollinger band, while, in parallel, the width of the bands is gradually reducing.
Step (III.c): On 26/5/1967 11:00 two main indications appear simultaneously indicating the end of the phenomenon. Specifically, the st D curve upward crosses the SMA curve, and soon after the SMA develops an upwards slope, while the RSI indicator exits the oversold situation.
Step (III.d): Finally, on 26/5/1967 20:00 we observe the RSI indicator entering the overbought situation which signifies the definite completion of the phenomenon.

Fourth Case-Study
The last magnetic storm case is a relatively recent intense storm which occurred on 25/10/2011 (see line no. 23 of Table 1); the corresponding analyses are shown in Fig.  7. The evolution of the specific magnetic storm in terms of the phases and corresponding steps described in Sec. 3.2 follows:

Phase I: Preparation
Step (I.a): From 20/10/2011 21:00 up to 24/10/2011 19:00 we observe a prolonged narrowing of the Bollinger bands, which, as already mentioned in Sec. 3.2, is a main indication for the preparation of an upcoming magnetic storm. Step

Phase II: Main
Step (II.a): A few hours later, on 24/10/2011 23:00, we can see that two main indications appearing simultaneously, namely the downward crossing of the SMA curve and the lower Bollinger band by the st D . We also observe that the RSI indicator is moving with a steep downward slope towards the oversold (under 30) area, indicating that the st D values move with high speed downwards.
Step Step (II.c): On 24/10/2011 00:00 the RSI indicator enters the oversold situation, while from this point on the SMA curve moves downwards with a steep slope.
Step (II.d): The specific magnetic storm is an example for which step (II.d) does not provide a correct prediction for the duration of the magnetic storm. Although the retreat of the RSI indicator into the oversold situation happened delayed with respect to the moment when the downward crossing of the lower Bollinger band by the st D curve occurred, the duration of the downward movement of the st D curve, and correspondingly the duration of the magnetic storm, was not long. We remind that the specific indication was classified as a secondary one exactly due to the reason that there are magnetic storm cases, even though very few, for which it doesn't lead to a correct inference for their expected duration.

Phase III: Recovery
Step (III.a): On 25/10/2011 02:00, we observe that the RSI indicator almost reaches zero slope, indicating that st D curve will possibly move upwards during the following hours. The st D curve also reaches its local minimum value then, while two hours later the st D curve enters back into the Bollinger bands, one more indication that the phenomenon is leaded to its completion.
Step Step (III.c) On 25/10/2011 10:00 the RSI curve exits the oversold situation, while two hours later the st D curve upward crosses of the SMA curve, which then moves with an upwards slope, enhancing the indications that the phenomenon is coming to its end. Moreover, on 25/10/2011 17:00 a failure swing is observed, which is a secondary indication.
Step (III.d): The final recovery phase indication, signifying the conclusion of the magnetic storm, comes on 25/10/2011 21:00 when the RSI indicator enters the overbought situation.

Discussion -Conclusions
In the frame of complex systems, we studied the time series of st D .in terms of the empirical financial analysis method known as technical analysis, focusing on the temporal evolution of magnetic storms. Specifically, we employed the combination of three very popular tools of technical analysis, the simple moving average ( SMA), the Bollinger bands, and the relative strength index ( RSI ) in order to formulate a methodology of magnetic storm analysis which could be used for space weather forecasting.
This methodology was developed after the analysis of more than 20 cases of magnetic storms, revealing all indications which, in specific temporal sequence of occurrence, provide information about the onset, duration and recovery phase of a magnetic storm. The applicability of the proposed methodology was presented in detail on four characteristic cases of magnetic storms, while the results for the whole set of 24 analyzed magnetic storms verify the repeatability of the proposed indications, rendering the results encouraging for space weather forecasting. However, we focus on the fact that the results of this study enhance the view that quantitative analysis methods can be successfully be transferred between economic and geophysical systems. Our results show that st D time series around the occurrence of magnetic storms can be successfully analyzed by the same empirical tools applied on share price time series for investment analysis.
In general, when a new magnetic storm is about to happen, the sequence of indications described by the proposed methodology are repeated. However, in some special cases such as "double storms" (two magnetic storms occurring close to each other), very intensive, or very long storms, some indications may not be observed, implying the need for further investigation of the proposed use of the considered technical analysis tools. This should mainly focus on the tuning of analysis parameters based on extensive statistics resulting from application of the specific tools for long enough time periods, e.g., for the last two solar cycles. Note that the considered technical analysis methods were applied on lead to an automated magnetic storm forecasting tool with an acceptable success rate (based on extensive statistics), at least for the classes of intense and super storms. Although this is outside the scope of the specific work, it is in our near future plans to elaborate such a study.
The here presented study followed the empirical way of analyzing financial time series for the analysis of st D time series, trying to identify indications, also used in economics, which could be associated with the temporal evolution of a magnetic storm. In a next step, one could focus on the interpretation of the observed behavior of the considered financial tools, in terms of the physics of the magnetosphere. Such a study could, on one hand, increase the effectiveness of the proposed methodology for the study of magnetic storms, and, on the other hand, offer ideas in order to better understand stock market processes.