Leave-One-Out Cross-Validation (LOOCV) Algorithm

A Leave-One-Out Cross-Validation (LOOCV) Algorithm is a Leave-K-Out Cross-Validation Algorithm where each cross-validation fold includes only one test item.

• Context:
  • It can (typically) be a Time-Consuming Algorithm.
  • It can (typically) achieve a High Variance.
  • …
• Example(s):
  • …
• Counter-Example(s):
  • Ten-Fold Cross-Validation Algorithm.
• See: Cross-Validation Algorithm, Jackknife Resampling.

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References

2021

• (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Cross-validation_(statistics)#Leave-one-out_cross-validation Retrieved:2021-4-15.
  • Leave-one-out cross-validation (LOOCV) is a particular case of leave-p-out cross-validation with p = 1.The process looks similar to jackknife; however, with cross-validation one computes a statistic on the left-out sample(s), while with jackknifing one computes a statistic from the kept samples only.

    LOO cross-validation requires less computation time than LpO cross-validation because there are only [math]\displaystyle{ C^n_1=n }[/math] passes rather than [math]\displaystyle{ C^n_p }[/math] . However, [math]\displaystyle{ n }[/math] passes may still require quite a large computation time, in which case other approaches such as k-fold cross validation may be more appropriate.

    Pseudo-Code-Algorithm:

    Input:

    x, {vector of length N with x-values of incoming points}

    y, {vector of length N with y-values of the expected result}

    interpolate( x_in, y_in, x_out ), { returns the estimation for point x_out after the model is trained with x_in-y_in pairs}

    Output:

    err, estimate for the prediction error}

    Steps:

 err ← 0
 for i ← 1, ..., N do
   // define the cross-validation subsets
   x_in ← (x[1], ..., x[i − 1], x[i + 1], ..., x[N])
   y_in ← (y[1], ..., y[i − 1], y[i + 1], ..., y[N])
   x_out ← x[i]
   y_out ← interpolate(x_in, y_in, x_out)
   err ← err + (y[i] − y_out)^2
 end for
 err ← err/N

2020

• https://towardsdatascience.com/understanding-8-types-of-cross-validation-80c935a4976d
  • QUOTE: ... 2. Leave-one-out cross-validation:

     Leave-one-out cross-validation (LOOCV) is an exhaustive cross-validation technique. It is a category of LpOCV with the case of p=1.
    
    (Source), LOOCV operations

  • For a dataset having n rows, 1st row is selected for validation, and the rest (n-1) rows are used to train the model. For the next iteration, the 2nd row is selected for validation and rest to train the model. Similarly, the process is repeated until n steps or the desired number of operations.

    Both the above two cross-validation techniques are the types of exhaustive cross-validation. Exhaustive cross-validation methods are cross-validation methods that learn and test in all possible ways. They have the same pros and cons discussed below:

  • Pros:
    • Simple, easy to understand, and implement.
  • Cons:
    • The model may lead to a low bias.
    • The computation time required is high.

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