Zero lag exponential moving average

<h2 id="history">History</h2>
<p>The indicator was created by John Ehlers and Ric Way around 2010.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="formula">Formula</h2>
<p>The formula for a given N-Day <i>period</i> and for a given <i>data</i> series is:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>

Lag
                  
                
              
              
                
                =
                
                  
                    
                      P
                      e
                      r
                      i
                      o
                      d
                      −
                      1
                    
                    2
                  
                
              
            
            
              
                
                  
                    EmaData
                  
                
              
              
                
                =
                
                  
                    Data
                  
                
                +
                (
                
                  
                    Data
                  
                
                −
                
                  
                    Data
                  
                
                (
                
                  Lag days ago
                
                )
                )
              
            
            
              
                
                  
                    ZLEMA
                  
                
              
              
                
                =
                
                  
                    EMA
                  
                
                (
                
                  
                    EmaData
                  
                
                ,
                
                  
                    Period
                  
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\textit {Lag}}&={\frac {Period-1}{2}}\\{\textit {EmaData}}&={\textit {Data}}+({\textit {Data}}-{\textit {Data}}({\text{Lag days ago}}))\\{\textit {ZLEMA}}&={\textit {EMA}}({\textit {EmaData}},{\textit {Period}})\end{aligned}}}

<p>The idea is do a regular <a href="/facts/Exponential_moving_average/tWY74500">exponential moving average</a> (EMA) calculation but on a de-lagged data instead of doing it on the regular data. Data is de-lagged by removing the data from "lag" days ago thus removing (or attempting to) the cumulative effect of the moving average.
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Paper by Ehlers and Way <a href="http://www.mesasoftware.com/papers/ZeroLag.pdf" target="_blank">http://www.mesasoftware.com/papers/ZeroLag.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Formula from Open Source package "Chart" <a href="http://user42.tuxfamily.org/chart/manual/Zero_002dLag-Exponential-Moving-Average.html" target="_blank">http://user42.tuxfamily.org/chart/manual/Zero_002dLag-Exponential-Moving-Average.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Formula description for R <a href="https://search.r-project.org/CRAN/refmans/TTR/html/MovingAverages.html" target="_blank">https://search.r-project.org/CRAN/refmans/TTR/html/MovingAverages.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Zero lag exponential moving average open-in-new

Zero lag exponential moving average