Error Definition

Biological   organisms   cannot   repeat   a   movement  exactly  the  same  way  across  practice  trials or  attempts.  In  other  words,  there  is  an  inherent amount  of  imprecision  in  the  motor  systems  of animals, including humans. This variability can be considered a source of error. Second, in any attempt to  solve  a  new  motor  problem,  whether  a  young athlete pole vaulting or a dog attempting to catch a tennis ball for the first time, there is inaccuracy— in other words, error—as the organism figures out how to solve these motor problems.

Definition and Measurement

Error  is  defined  as  the  difference  between  an observable   behavior   and   a   desired   behavior. More formally, error is the difference between an observed score and a desired score, called a target:

ei  = Xi – T

where e is the error, X is the score on trial i, and T is the target score. Traditionally error has been classified as bias, precision, and accuracy.

Bias means that a set of scores tends to produce either  a  negative  or  a  positive  error.  Bias  is  also called algebraic error, as it has a magnitude and a direction (positive or negative) on each and every trial.  The  average  algebraic  error,  or  bias,  over  a series of trials has been called constant error.

Precision is a measure of response consistency. In other words, does the individual perform identically  every  time?  Variability  is  the  usual  measure  of  imprecision.  Variability  is  considered  the average distance a set of scores is from the mean score.  Typically,  the  measure  of  precision  is  just the standard deviation of the set of scores. Thus, a smaller standard deviation means an increase in consistency (precision) compared to a larger standard  deviation.  Historically,  variable  error  is  the name of this error score.

Accuracy,   the   most   commonly   used   and important  measure  of  error,  is  defined  as  the average  distance  a  set  of  scores  is  from  a  target value. Traditionally, absolute error, defined as the average  absolute  difference  (computed  on  a  trial by trial basis, and then averaged) between a score and a target, was the accuracy measure of choice. Absolute  error  has  several  statistical  issues  that have resulted in researchers abandoning this measure.  A  much  better  measure  of  accuracy  is  root mean  squared  error  (RMSE).  This  measure  uses the  one-dimensional  distance  formula  and  then takes the average of the set of distances. RMSE is a combination of bias and variability.

So far in this discussion of error, we have focused on  measuring  performance  as  the  outcome  of  a single trial and then describing the average of a set of  trials.  In  other  words,  these  measures  are outcome scores. Error also can be used to capture the process of movement. The movement of the effector over time (a trajectory) provides a rich source of  information  concerning  the  underlying  movement control processes. For example, a movement trajectory can be compared to an ideal movement trajectory and deviations from the ideal would be considered an error. The standard measure of error for a continuous trajectory is RMSE. The error is summed up across samples across a single trajectory.  The  larger  the  RMSE,  the  less  accurate  the trajectory is to the template or standard.

The  trajectory  RMSE  measure  can  be  slightly altered  to  ask  an  important  theoretical  question in motor control. Are trajectories scalable in space and time? In other words, does the production of a  skilled  action  show  evidence  for  an  ideal  template  that  might  exhibit  spatial  variability  and temporal variability, while the movement appears to  be  derived  from  a  common  invariant  trajectory?  In  speech  motor  control,  Anne  Smith,  Lisa Goffman,  Howard  Zelaznik,  Goangshiuan  Ying, and Clare McGillem developed a measure known as  the  spatial–temporal  index  (STI).  First,  the trajectory  time  base  is  changed  to  a  percentage. Therefore,  all  movements  have  a  temporal  scale from  0  to  100.  Second,  the  displacement  values are  placed  on  a  normal  scale  (they  usually  range from –1.5 to +1.5). At each 2-percentage interval, the standard deviation of the normalized displacements is computed. The 50 standard deviations are either summed or averaged. The STI captures how closely  a  set  of  trajectories  follows  the  average spatial–temporal  trajectory.  The  average  normalized trajectory purportedly represents the invariant template. The STI measure does not make any theoretical statements about the cause of any spatial– temporal invariance. The measure has been shown to capture development of speech motor patterns and  the  effects  of  practice  on  speech  learning. Although the assumptions of linear scaling might not  be  fully  justified,  the  measure  appears  to  be fairly robust in spite of these issues.

The Value of Errors

The   previous   section   presented   the   descriptive  aspects  of  error.  Now,  attention  is  focused on  whether  errors  are  good,  bad,  or  indifferent. Obviously,  at  high  levels  of  performance  errors generally are considered bad. Missing the putt to lose the championship is not a good thing. In competition, errors might seem to always be bad. But, that clearly does not have to be the case. A good baseball  pitcher  deliberately  may  throw  a  bad pitch so that the batter might swing and miss, or the batter might be expecting a pitch in the future series of pitches. Is home plate the target on every pitch? No! Thus, errors in pitching are difficult to measure. However, errors in pitching can be measured  in  practice.  The  target  changes,  but  during the game, error in pitching can only be measured if we know the goal of the pitcher.

What  about  errors  during  practice?  For  high-level performers errors might be necessary for an expert to figure out the nuances of a difficult task. For example, when concert pianists need to learn a new fingering sequence, they experiment with the best  method  to  get  it  done.  In  other  words,  they are  making  deliberate  errors  to  help  them  search for the solution to this new problem. New learners need and use errors to guide the learning process. James A. Adams, one of the first, if not the first, to propose a major role of errors in learning, believed that  errors  provided  motivation  and  information that  learners  use  to  change  their  attempt  on  the next  trial.  Ann  Gentile,  although  not  explicit  on this matter, believes that learning is a search process. The learner is attempting to find a reasonable solution to the motor learning problem. She calls this  getting  the  idea  of  the  movement.  After  this approximate solution, the learning then works at refining  the  solution.  One  can  define  getting  the idea  when  the  learner  is  consistently  producing an  acceptable  level  of  error  and  then  moves  to  a deliberate  attempt  to  reduce  error.  What  would be  the  consequences  of  a  performer  not  being able to control and produce error? One would be an  inflexible  robot.  Errors  provide  the  flexibility for  individuals  to  respond  to  their  environment.

Human  beings  would  not  be  functional  if  they only  possessed  fixed  action  patterns,  such  as  stereotypical reflexes.

Errors  also  are  organized.  Robert  Sessions Woodworth  in  1899,  Paul  M.  Fitts  in  1954, and  more  recently  Richard  A.  Schmidt,  Howard Zelaznik, Brian Hawkins, James S. Frank, and John T. Quinn in 1979 described this organization. The organization is called the speed-accuracy trade-off. With some exceptions the cost of increasing movement  speed  (decreasing  duration,  increasing  distance, or the combination of the two) is a decrease in  movement  accuracy  in  the  spatial  dimension. Home  run  hitters  tend  to  strike  out  more  often than  singles  hitters.  Why?  A  home  run  in  baseball requires increased bat speed. The cost of that increase  is  paid  in  the  currency  of  an  increase  in spatial variability. There is a clear exception to this speed–accuracy trade-off in the temporal domain. Temporal variability decreases as the timed interval decreases  (in  other  words,  doing  the  task  faster). Therefore, in tasks that require both spatial as well as temporal accuracy performers need to learn the optimal movement speed to minimize spatial error but maximize temporal accuracy.

What is the source of error and variability? One way to view errors and variability in performance is via the classic view of test theory. A performance score  is  composed  of  two  components,  the  true score and a source of random variations called an error.  The  true  score  represents  the  mean  of  an infinite number of trials. In other words, the true score is a construct, and thus it represents the individual’s level of performance.

Recently,  scholars  in  motor  behavior  have begun  to  discover  that  error  over  a  series  of  trials  does  have  a  random  component.  However, error  also  has  a  predictive  component.  In  other words,  a  performance  score  in  series  predicts  a performance score in later in the time series. These relations are called long-term correlations, and are thought  to  be  the  results  of  the  underlying  nonlinear  dynamics  of  human  performance.  These long-term  correlations  are  shown  in  fluctuations of  step–cycle  duration,  reaction  time,  and  cycle duration in repetitive timing tasks. There are some very  exciting  hints  that  these  long-term  fluctuations are crucial to healthy behavior, and the lack of these correlations might be the signature of diseased states. Fluctuations, which are not random, but structured, are a reliable signature of normal biological function.

The observation that errors have a nonrandom component  might  be  the  reason  for  the  success of  sport  psychologists  working  with  high-level performers.  Obviously  if  errors  in  performance were  solely  random,  the  only  way  to  improve performance  would  be  to  dramatically  increase the amount of deliberate practice. However, most high-level performers already practice many hours per  day.  If  performers  acquire  techniques  to  control  the  nonrandom  component  of  performance fluctuations, performance will improve.

References:

  1. Adams, J. A. (1971). A closed loop theory of motor learning. Journal of Motor Behavior, 3, 111–150.
  2. Fitts, P. M. (1954). The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, 47, 381–391.
  3. Gentile, A. (1972). A working model of skill acquisition with application to teaching. Quest, 17, 3–23.
  4. Schmidt, R. A., Zelaznik, H. N., Hawkins, B., Frank, J. S., & Quinn, J. T. J. (1979). Motor-output variability: A theory for the accuracy of rapid motor acts. Psychological Review, 86, 415–451.
  5. Schutz, R. W., & Roy, E. A. (1973). Absolute error—devil in disguise. Journal of Motor Behavior, 5, 141–153.
  6. Smith, A., Goffman, L., Zelaznik, H. N., Ying, G. S., & McGillem, C. (1995). Spatiotemporal stability and patterning of speech movement sequences. Experimental Brain Research, 104, 493–501.
  7. Woodworth, R. S. (1899). The accuracy of voluntary movement. Psychological Review, 3, 1–114.

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