MATLAB CURVE FITTING TOOLBOX - RELEASE NOTES Guide de l'utilisateur télécharger pdf (Page 111)

Parametric Fitting

3-35

To understand the quantities associated with each type of prediction interval,

recall that the data, fit, and residuals (random errors) are related through the

formula

data = fit + residuals

Suppose you plan to take a new observation at the predictor value x

n+1

. Call

the new observation y

n+1

) and the associated error e

n+1

. Then y

n+1

)

satisfies the equation

where f

n+1

) is the true but unknown function you want to estimate at x

n+1

The likely values for the new observation or for the estimated function are

provided by the nonsimultaneous prediction bounds.

If instead you want the likely value of the new observation to be associated

with any predictor value, the previous equation becomes

The likely values for this new observation or for the estimated function are

provided by the simultaneous prediction bounds.

The types of prediction bounds are summarized below.

The nonsimultaneous and simultaneous prediction bounds for a new

observation and the fitted function are shown below. Each graph contains three

curves: the fit, the lower confidence bounds, and the upper confidence bounds.

The fit is a single-term exponential to generated data and the bounds reflect a

95% confidence level. Note that the intervals associated with a new observation

Table 3-3: Types of Prediction Bounds

Type of Bound Associated Equation

Observation Nonsimultaneous y

n+1

)

Simultaneous y

n+1

(x), globally for any x

Function Nonsimultaneous f

n+1

)

Simultaneous f(x), simultaneously for all x

n 1+

()fx

n 1+

()e

n 1+

x() fx() e+=

1 2 ... 106 107 108 109 110 111 112 113 114 115 116 ... 215 216

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