Logarithmic Correlation Coefficient:
Measuring Quality of Trend Lines

By Mark P Martin,
Emporium Software

Various traders and market analysts state that the slope or steepness of a trend line is an indication of the quality of the trend. John Murphy writes that "...most important up trend lines tend to approximate an average slope of 45 degrees."  Gann studies try to scale the chart in such a way that an "ideal" trend line would have a 1 x 1, or a 45 degree slope.  Steven B. Achelis writes of Gann studies in Technical Analysis from A to Z,

"W. D. Gann (1878-1955) designed several unique techniques for studying price charts. Central to Gann's techniques was geometric angles in conjunction with time and price. Gann believed that specific geometric patterns and angles had unique characteristics that could be used to predict price action.

“All of Gann's techniques require that equal time and price intervals be used on the charts, so that a rise/run of 1 x 1 will always equal a 45 degree angle.

“Gann believed that the ideal balance between time and price exists when prices rise or fall at a 45 degree angle relative to the time axis. This is also called a 1 x 1 angle (i.e., prices rise one price unit for each time unit).”

First, with regard to the 45-degree angle discussion, consider a hypothetical example with some extremes to illustrate a point.  Suppose some stock goes from 101 to 102 over a five-year period, and the stock has very smooth, consistent movements (albeit very small) with very little variation away from the trend line.  One could construct a chart with 101 at the bottom of the Y-axis and 102 at the top, with 10 increments of 0.10 point each, and five years divided up into 10 increments of 6-months each on the X-axis.  If the chart were perfectly square, then the chart would rise steadily from the lower left-hand corner to the upper right-hand corner, and if the daily (or weekly, or whatever) price changes were consistent, then the chart would form a nice straight 45-degree angle.  According to Gann theory, this would be a good high-quality trend line.  Gann theory goes on to describe how once the chart falls off of a trend line, it finds support/resistance levels at another trend line that is defined by a different ratio, say 1 x 2 or 1 x 3, or 2 x 1 or some integral ratio.  If you don't use a square chart (even if you use square increments), or if you don't fully scale the axes to the minimum and maximum values, or if you leave an empty buffer zone at the right or at the bottom, then the idea of an ideal 45-degree angle breaks down.

I propose that the scaling if the chart is irrelevant, that the "45-degree rule" is constructed for visual inspection on this ideal square chart.  On the other hand, mathematicians and statisticians have already devised a means of measuring the quality of the relationship of one variable (price, in this case) to another (time).  That measure is the correlation coefficient.  Correlation coefficient has been in use since before W.D. Gann was charting stocks.  Karl Pearson published a paper on statistics describing correlation in 1895. The formula for correlation coefficient is available in virtually all spreadsheets and most charting packages, and it's in any statistics text.

The formula for the Pearson product-moment correlation coefficient (r) of X to Y over N (X, Y) value-pairs is:
   

The value of r ranges from -1.0 to +1.0.  A value close to +1.0 indicates a good positive correlation with positive slope; a value close to -1.0 indicates a good negative correlation with negative slope; a value close to 0 indicates a poor correlation with no definable slope.

To use this to measure the quality of trend of a stock, use the closing price (C) for Y, and the date for X.  Most software will translate the date in calculation to the "Julian date", which is the number of days that have passed since the calendar began.  Since Julian date values can be in the millions, these values will break the formula for any 32-bit software doing integer math.  Nearly all home computers are 32-bit.  If using the (Julian) date for the X values doesn't work, then find the number of days for each price point since the beginning of the chart by subtracting the first date from each of the remaining dates.  Again, most software internally converts dates to Julian date values before doing the calculation.  A slightly less accurate way to get the X values is to create an incremental cumulative sum of 1 (or some other fixed increment) for each price point.  This is less accurate because it doesn't account for weekends and holidays, or overnight hours in the case of an intra-day chart. But since the use of correlation coefficient is usually relative, if you use the same method consistently, the inaccuracies will not be noticeable.

The previous discussion assumes that price movement is modeled on a linear curve.  In financial markets, this is not the case.  Financial markets are best modeled with an exponential formula.  In the case of a fixed-rate money-market account that is compounded daily at rate R, the formula for any future value at time t is:
  
If R is an annual interest rate, then you have to further annualize the rate:

where t is in days.
Anyway, Principle P is an exponential function of time, t.  The same model applies to stock prices.  If you look at a 100-year chart of the DJIA, you will see that it changes much more in recent years than in the earlier years.   To properly view such a chart, the Y-axis should be on a logarithmic scale, where you will see increments from 10 to 100, to 1000, etc., and various logarithmic increments in-between.  When plotted on this scale, the 100-year DJIA will appear more like a straight line than the "hockey-stick" curve on a linear chart.   

Examine a chart of the NASDAQ from Jan. 1991 through Mar. 2000.  On a linear chart, it's a hockey-stick curve, but on a logarithmic scale, it's more of a straight-line.  A similar effect can be obtained by plotting the logarithm of the price on a linear chart.  The logarithm of NASDAQ from 1/1991 to 3/2000 on a linear chart is a close to a linear curve.

To apply the exponential nature of the price chart to the correlation calculation, you need only to take the logarithm of the prices (the Y value) before performing the correlation calculation.  It doesn't matter whether you use a natural or a base-10 logarithm, because the correlation formula looks at the differences in values relative to the corresponding X values. Either log will produce the exact same correlation value.  I've computed a 9-year (2250 day) correlation coefficient of the NASDAQ at the top of the bull market, on 3/27/2000.
Using linear prices, r = 0.857
Using logarithmic prices, r = 0.968
The correlation for the exponential model (logarithmic prices) is greater than for the linear model, which shows that the exponential model is a better fit for the data given.  Although the difference in correlation values will be less on a smaller time-scale, this demonstrates that the exponential model is more appropriate for correlation analysis as well as for linear regressions, volatility, and most other statistical calculations.

In practice, this particular use of correlation is similar to many other quality-of-trend indicators.  Use it as a filter for entry (and optionally for exit) rules.  Only enter a trade when correlation is at some relatively high value.  The value of correlation that would indicate a significant trend would be subject to some experimentation.  In pure statistical analysis, the significance of correlation is determined by another complicated formula, which is usually listed in tables of values in the appendix of statistics texts.  For example, with 100 data points, there is a 99 percent confidence level in a correlation of 0.2540.  This means that with 100 data points, if the correlation were 0.2540, there would be a 1% chance that the relationship is due to randomness instead of an actual relationship.  In practice, any correlation that is greater than 0.95 could be thought of as a reliable relationship or trend.  I also like to apply correlation analysis to the equity curve of trading system backtests. Again, I use an exponential model by taking the logarithm of the daily equity values before doing a correlation.  It's been my experience that even with short time periods, if the growth that the system produces is extreme, it's always exponential in nature, and a logarithmic correlation and a logarithmic linear regression is more accurate.  When optimizing a system, I often use a logarithmic linear-regression slope, although lately I've been considering the possibilities of maximizing logarithmic correlation coefficient.  One of the users of Pikker has written a system that in backtesting over the past five years has an equity correlation of 0.98. That system also has exponential growth, and a very high-quality equity curve on a logarithmic scale.  

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Copyright © 2003, Mark P Martin, mark@emporium-sw.com
Emporium Software, http://www.emporium-sw.com
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