Nonlinear regression models

Polynomial Multiple Regression Model

Fig. 3.1. Two-dimensional designation black box models on diagrams

If a black box has, for example, two inputs, and the dependence of the output on the inputs resembles a quadratic one, then it is advisable to choose the following hypothesis:

Y = A ₀ + A ₁ · X ₁ + A ₂ · X ₂ + A ₃ · X ₁ · X ₂ + A ₄ · X ₁ · X ₁ + A ₅ · X ₂ · X ₂ .

Denote: Z ₁ = X ₁ · X ₂ ; Z ₂ = X ₁ · X ₁ ; Z ₃ = X ₂ · X ₂ and substitute these expressions in the previous formula:

Y = A ₀ + A ₁ · X ₁ + A ₂ · X ₂ + A ₃ · Z ₁ + A ₄ · Z ₂ + A ₅ · Z ₃ .

Thus, this task is reduced to a linear multiple model. And the black box model now looks like the one shown in fig. 3.2.

Fig. 3.2. Converted black box model

Multiplicative regression model

Fig. 3.3. Designation of a multi-dimensional model black box diagrams

Y = A ₀ · X ₁ ^{A ₁} · X ₂ ^{A ₂} ·… · X _m ^{A _m} .

Let us count the left and right sides of this equation:

ln ( Y ) = ln ( A ₀ ) + A ₁ · ln ( X ₁ ) + A ₂ · ln ( X ₂ ) +… + A _m · ln ( X _m ).

Denote:

W = ln ( Y ), B ₀ = ln ( A ₀ ), Z ₁ = ln ( X ₁ ), Z ₂ = ln ( X ₂ ), ..., Z _m = ln ( X _m ).

We get:

W = B ₀ + A ₁ · Z ₁ + A ₂ · Z ₂ +… + A _m · Z _m .

That is, the transition to a linear multiple model is again made.

Reverse regression model

Fig. 3.4. Designation of a multi-dimensional model black box diagrams

Y = k / ( A ₀ + A ₁ X ₁ + ... + A _m X _m ).

Replace: W = 1 / Y , a _i = A _i / k . And we turn to the linear multiple model:

W = a ₀ + a ₁ · X ₁ +… + a _m · X _m .

Exponential model

Fig. 3.5. Designation of a multi-dimensional model black box diagrams

Y = e ^{B ₀ + B ₁ X ₁ + B ₂ X ₂ + ... + B _m X _m} .

Let us count the left and right sides of the equation:

ln ( Y ) = B ₀ + B ₁ · X ₁ + B ₂ · X ₂ +… + B _m · X _m .

Perform the replacement W = ln ( Y ) and get:

W = B ₀ + B ₁ · X ₁ + B ₂ · X ₂ +… + B _m · X _m .

Next, we use the expression for the linear multiple model.