Add factor of 2.0 to covariance calculation to compensate for 1/2 in objective function

diluculo · diluculo · commit ea749799254d · 2025-03-21T18:04:42.000+09:00
diff --git a/src/Numerics/Optimization/NonlinearMinimizationResult.cs b/src/Numerics/Optimization/NonlinearMinimizationResult.cs
@@ -57,7 +57,10 @@ void EvaluateCovariance(IObjectiveModel objective)
                 return;
             }
 
-            Covariance = Hessian.PseudoInverse() * objective.Value / objective.DegreeOfFreedom;
+            // The factor of 2.0 compensates for the 1/2 factor in the objective function definition
+            // F(p) = 1/2 * ∑{ Wi * (yi - f(xi; p))^2 }
+            // Without this compensation, the covariance and standard errors would be underestimated by a factor of 2
+            Covariance = 2.0 * Hessian.PseudoInverse() * objective.Value / objective.DegreeOfFreedom;
 
             if (Covariance != null)
             {

Original file line number	Diff line number	Diff line change
`@@ -57,7 +57,10 @@ void EvaluateCovariance(IObjectiveModel objective)`
`57`	`57`	`return;`
`58`	`58`	`}`
`59`	`59`
`60`		`- Covariance = Hessian.PseudoInverse() * objective.Value / objective.DegreeOfFreedom;`
	`60`	`+ // The factor of 2.0 compensates for the 1/2 factor in the objective function definition`
	`61`	`+ // F(p) = 1/2 * ∑{ Wi * (yi - f(xi; p))^2 }`
	`62`	`+ // Without this compensation, the covariance and standard errors would be underestimated by a factor of 2`
	`63`	`+ Covariance = 2.0 * Hessian.PseudoInverse() * objective.Value / objective.DegreeOfFreedom;`
`61`	`64`
`62`	`65`	`if (Covariance != null)`
`63`	`66`	`{`