# Survey on Log-Normally Distributed Market-Technical Trend Data

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Definition 1**

## 2. Detection of Dow-Trends

**Definition 2**

**Definition 3**

_{abs}is given by:

## 3. Retracements

#### 3.1. Distribution of the Retracement

^{μ}, and the mean is given by:

_{i}denoting the n measured retracements. Furthermore, the p-value calculated with the Anderson–Darling test (recommended test by Stephens in [15], Chapter “Test based on EDF statistics”) being applied to the logarithmic transformed data is checked. The so-obtained values are summarized in Table 1.

**Observation 4**

#### 3.2. Delay after a Retracement

_{abs}after a retracement is recognized (as defined in (4)) with the retracement X itself, both must have the same unit. Therefore, the delay will also be considered in units of the last movement. It will be denoted as random variable D

_{X}:

_{X}may somehow depend on the preceding retracement X. The notation with subscript X is only used to denote the delay after a retracement and to distinguish it from other delays to come.

**Observation 5**

#### 3.3. Fibonacci Retracements

#### 3.4. Duration of the Retracement

## 4. Movement and Correction

#### 4.1. Distribution of Relative Movements and Corrections

**Observation 6**

#### 4.2. Delay after Relative Movements and Corrections

**Observation 7**

#### 4.3. Period of Movements and Corrections

## 5. Mathematical Model of Trading Systems

**Lemma 8**

## 6. Conclusions and Outlook

## Author Contributions

## Conflicts of Interest

## References

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^{1.}Published under R. Brenner’s birth name Kempen.^{2.}Published under R. Brenner’s birth name Kempen.^{3.}Published under R. Brenner’s birth name Kempen.

**Figure 1.**(

**a**) Example of a down-trend in a historical setup as Dow used it (freely adapted from Russel [5]); (

**b**) sketch of an up-trend in the sense of Dow.

**Figure 3.**Daily chart of the Adidas stock between September 2012 and August 2013 with the MinMax-process by Maier-Paape (

**a**) based on the integral moving average convergence divergence (MACD) stop and reverse (SAR)-process (

**b**) with two different scalings: one and three. The lines in the charts indicate the last detected extremum.

**Figure 4.**Measured and log-normal fit density of the retracement X in an up-trend and down-trend with scaling one for S&P100 stocks. Each dataset is visualized with a histogram from zero to five with a bin size of 0.11.

**Figure 5.**Measured and log-normal fit density of the retracement X in an up-trend with scaling 1.5 and three for S&P100 stocks. Each dataset is visualized with a histogram from zero to five with a bin size of 0.11.

**Figure 6.**Measured and log-normal fit density of the delay in an up-trend with scaling one for S&P100 stocks. The data are visualized with a histogram from zero to five with a bin size of 0.11.

**Figure 7.**More detailed (finer) histogram of Figure 4 with scaling one from zero to two with a bin size of 0.01.

**Figure 8.**Measured and log-normal fit density of the retracement duration in up- and down-trends (

**a**and

**b**respectively) with scaling one for S&P100 stocks. The data are visualized with a histogram with a bin size of one.

**Figure 9.**Contour plot of the joint density of the retracement value and its duration in up- and down-trends (left and right resp.) with scaling one for S&P100 stocks.

**Figure 10.**Measured and log-normal fit density of the relative movement

**(a)**and relative correction

**(b)**in an up-trend with scaling one for S&P100 stocks. The data are visualized via histogram from zero to one with a bin size of 0.01.

**Figure 11.**(

**a**) Measured and log-normal fit density of the relative delay after a relative movement M (see 2) and (

**b**) after a relative correction C (see 3) in an up-trend with scaling one for S&P100 stocks. The data are visualized with a histogram from zero to one with a bin size of 0.01.

**Figure 12.**Evolution of the period T for scalings between 0.5 and five with a step size of 0.1 for S&P100 stocks. (

**a**) period T for up-trends. (

**b**) Fit parameters of T.

**Figure 13.**Setup of a basic anti- and pro-cyclic trading system for up-trends (

**a**and

**b**, respectively).

**Table 1.**Parameters of the log-normal fit for the retracement X in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 2.**Parameters of the log-normal fit for the delay D

_{X}in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 3.**Correlation between the logarithms of the retracement X (retr.) and delay D

_{X}(after retr.) in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 4.**Parameters of the log-normal fit for the retracement duration in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 5.**Correlation between the logarithms of the retracement X (retr.) and its duration Y in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 6.**Parameters of the log-normal fit for the relative movement in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 7.**Parameters of the log-normal fit for the relative correction in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 8.**Parameters of the log-normal fit for the relative delay D

_{M}after a movement in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 9.**Parameters of the log-normal fit for the relative delay D

_{C}after a correction in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 10.**Correlation between the logarithms of the relative (rel.) movement M and relative delay D

_{M}in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

**Table 11.**Correlation between the logarithms of the relative (rel.) correction C and relative delay D

_{C}in up- and down-trends. (

**a**) S&P100 data. (

**b**) Eurostoxx50 data.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Brenner, R.; Maier-Paape, S.
Survey on Log-Normally Distributed Market-Technical Trend Data. *Risks* **2016**, *4*, 20.
https://doi.org/10.3390/risks4030020

**AMA Style**

Brenner R, Maier-Paape S.
Survey on Log-Normally Distributed Market-Technical Trend Data. *Risks*. 2016; 4(3):20.
https://doi.org/10.3390/risks4030020

**Chicago/Turabian Style**

Brenner, René, and Stanislaus Maier-Paape.
2016. "Survey on Log-Normally Distributed Market-Technical Trend Data" *Risks* 4, no. 3: 20.
https://doi.org/10.3390/risks4030020