Mean signed deviation

<h2 id="definition">Definition</h2>
<p>The mean signed difference is derived from a set of <i>n</i> pairs,  
  
    
      
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    {\displaystyle ({\hat {\theta }}_{i},\theta _{i})}
  
, where 
  
    
      
        
          
            
              
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    {\displaystyle {\hat {\theta }}_{i}}
  
 is an estimate of the parameter 
  
    
      
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    {\displaystyle \theta }
  
 in a case where it is known that 
  
    
      
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    {\displaystyle \theta =\theta _{i}}
  
.  In many applications, all the quantities 
  
    
      
        
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    {\displaystyle \theta _{i}}
  
 will share a common value. When applied to <a href="/facts/Forecasting/pwFtGyzM">forecasting</a> in a <a href="/facts/Time_series_analysis/fSXPR817">time series analysis</a> context, a forecasting procedure might be evaluated using the mean signed difference, with  
  
    
      
        
          
            
              
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    {\displaystyle {\hat {\theta }}_{i}}
  
 being the predicted value of a series at a given <a href="/facts/Lead_time/GDl9TkTW">lead time</a> and 
  
    
      
        
          θ
          
            i
          
        
      
    
    {\displaystyle \theta _{i}}
  
 being the value of the series eventually observed for that time-point. The mean signed difference is defined to be  
</p>

MSD
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    {\displaystyle \operatorname {MSD} ({\hat {\theta }})={\frac {1}{n}}\sum _{i=1}^{n}{\hat {\theta _{i}}}-\theta _{i}.}

<h2 id="use-cases">Use Cases</h2>
<p>The mean signed difference is often useful when the estimations 
  
    
      
        
          
            
              
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    {\displaystyle {\hat {\theta _{i}}}}
  
 are biased from the true values 
  
    
      
        
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    {\displaystyle \theta _{i}}
  
 in a certain direction. If the estimator that produces the 
  
    
      
        
          
            
              
                θ
                
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    {\displaystyle {\hat {\theta _{i}}}}
  
 values is unbiased, then 
  
    
      
        MSD
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    {\displaystyle \operatorname {MSD} ({\hat {\theta _{i}}})=0}
  
. However, if the estimations 
  
    
      
        
          
            
              
                θ
                
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    {\displaystyle {\hat {\theta _{i}}}}
  
 are produced by a <a href="/facts/Bias_of_an_estimator/oxIvEgmd">biased estimator</a>, then the mean signed difference is a useful tool to understand the direction of the estimator's bias.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bias_of_an_estimator/oxIvEgmd">Bias of an estimator</a></li>
<li><a href="/facts/Deviation_(statistics)/F7cqnhtF">Deviation (statistics)</a></li>
<li><a href="/facts/Mean_absolute_difference/G7HtnNdD">Mean absolute difference</a></li>
<li><a href="/facts/Mean_absolute_error/kzu4tbtv">Mean absolute error</a></li></ul>

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

<ol>
<li id="fn:1"><p>Harris, D. J.; Crouse, J. D. (1993). "A Study of Criteria Used in Equating". Applied Measurement in Education. 6 (3): 203. doi:10.1207/s15324818ame0603_3. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Willmott, C. J. (1982). "Some Comments on the Evaluation of Model Performance". Bulletin of the American Meteorological Society. 63 (11): 1310. Bibcode:1982BAMS...63.1309W. doi:10.1175/1520-0477(1982)063<1309:SCOTEO>2.0.CO;2. <a href="https://doi.org/10.1175%2F1520-0477%281982%29063%3C1309%3ASCOTEO%3E2.0.CO%3B2" target="_blank">https://doi.org/10.1175%2F1520-0477%281982%29063%3C1309%3ASCOTEO%3E2.0.CO%3B2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Mean signed deviation open-in-new

Mean signed deviation