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Convolutions - Gaussian and Kinetic Decay

In addition to the EMG model, many kinetic decay models have closed form solutions for integer ratios of powers that differ by 1. The following models consist of Gaussian Ä Kinetic Decay for order 0, 1/2, 2/3, 2/4, 4/5, 5/6, 6/7, 7/8, and 9/10. These are significant since catalyzed kinetics have such orders.

As closed form solutions, even the rather onerous orders nearer 1 are computationally fast relative to
models which require internally iterative special functions, but there can be precision issues, especially
in the orders from 5/6 upward. This is evident in the order 9/10 plots, but can occur with any of the
models as the a_{3} t or time constant
approaches 0.

Also note the similarity illustrated at the end of this topic where kinetic and a half-Gaussian probabilisitic decays are compared.

Gaussian Ä Kinetic Decay Order 0

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 0

Built in model: KMG[0]

User-defined peaks and view functions: KMG[0](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 1/2

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 1/2

Built in model: KMG[1/2]

User-defined peaks and view functions: KMG[1/2](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 2/3

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 2/3

Built in model: KMG[2/3]

User-defined peaks and view functions: KMG[2/3](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 3/4

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 3/4

Built in model: KMG[3/4]

User-defined peaks and view functions: KMG[3/4](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 4/5

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 4/5

Built in model: KMG[4/5]

User-defined peaks and view functions: KMG[4/5](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 5/6

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 5/6

Built in model: KMG[5/6]

User-defined peaks and view functions: KMG[5/6](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 6/7

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 6/7

Built in model: KMG[6/7]

User-defined peaks and view functions: KMG[6/7](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 7/8

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 7/8

Built in model: KMG[7/8]

User-defined peaks and view functions: KMG[7/8](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 9/10

a_{0} = Area

a_{1} = Center (mean)

a_{2} = Width

a_{3}
= Kinetic Time Constant t,
order 9/10

Built in model: KMG[9/10]

User-defined peaks and view functions: KMG[9/10](x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Kinetic Decay Order 1 (EMG)

a_{0} = Area

a_{1} = Center (as mean of Gaussian peak that is convolved by the a_{3} exponential)

a_{2} = Width (SD of the Gaussian peak that is convolved by the a_{3} exponential)

a_{3} = The first order kinetic or exponential decay width convolving the Gaussian (can be negative
to model fronted peaks)

Built in model: EMG

User-defined peaks and view functions: EMG(x,a_{0},a_{1},a_{2},a_{3})

Gaussian Ä Half-Gaussian Decay (GMG)

a_{0} = Area

a_{1} = Center (as mean of Gaussian peak that is convolved by the a_{3} half-Gaussian)

a_{2} = Width (SD of the Gaussian peak that is convolved by the a_{3} half-Gaussian)

a_{3} = The first order probabilisitic half-Gaussian width convolving the Gaussian (can be negative
to model fronted peaks)

Built in model: GMG

User-defined peaks and view functions: GMG(x,a_{0},a_{1},a_{2},a_{3})

Similarity to a Half-Gaussian Probabilisitic Decay

A half-Gaussian probabilistic decay tends to be quite similar to a kinetic decay with an order in the vicinity of approx 0.55.

The following graph consists of four plots, a fronted GMG and 1/2 order KMG, and a tailed GMG and 1/2 order KMG, the parameters adjusted to produce as close a match as possible. These are very different models which can produce similar shapes.