Generic Models

PhyTorch provides a comprehensive library of generic curve-fitting models that can be applied across various scientific domains. These models use the same unified API as all PhyTorch models.

Linear Model

Simple linear regression:

$$y = a \cdot x + b$$

Usage

from phytorch import fit
from phytorch.models.generic import Linear
import numpy as np

data = {
    'x': np.array([1, 2, 3, 4, 5]),
    'y': np.array([2.1, 3.9, 6.2, 7.8, 10.1])
}

result = fit(Linear(), data)
print(f"Slope: {result.parameters['a']:.2f}")
print(f"Intercept: {result.parameters['b']:.2f}")

result.plot()

Parameters

Parameter	Description	Units
a	Slope	-
b	Intercept	-

Sigmoidal Model

Rational sigmoid curve for S-shaped responses:

$$y = \frac{y_{max}}{1 + \left|\frac{x}{x_{50}}\right|^s}$$

Usage

from phytorch import fit
from phytorch.models.generic import Sigmoidal

data = {
    'x': np.array([-3, -2, -1, 0, 1, 2, 3]),
    'y': np.array([0.5, 1.2, 2.5, 5.0, 7.5, 9.0, 9.8])
}

result = fit(Sigmoidal(), data)
result.plot()

Parameters

Parameter	Description	Units
ymax	Maximum response	-
x50	Half-saturation point	-
s	Steepness parameter	-

Rectangular Hyperbola

Saturating hyperbola commonly found in resource-limited and biological processes:

$$y = \frac{y_{max} \cdot x}{x_{50} + x}$$

This model describes processes such as enzyme kinetics (Michaelis-Menten), light response curves, and nutrient uptake.

Usage

from phytorch import fit
from phytorch.models.generic import RectangularHyperbola

# Saturating response data
data = {
    'x': np.array([0.5, 1, 2, 5, 10, 20, 50]),
    'y': np.array([2.5, 4.5, 7.5, 12, 15, 17, 19])
}

result = fit(RectangularHyperbola(), data)
print(f"ymax: {result.parameters['ymax']:.2f}")
print(f"x50: {result.parameters['x50']:.2f}")

Parameters

Parameter	Description	Units
ymax	Maximum asymptotic value	-
x50	Half-saturation constant	-

Non-rectangular Hyperbola

More flexible saturation curve:

$$y = \frac{\alpha x + y_{max} - \sqrt{(\alpha x + y_{max})^2 - 4\theta\alpha x y_{max}}}{2\theta}$$

Usage

from phytorch import fit
from phytorch.models.generic import NonrectangularHyperbola

result = fit(NonrectangularHyperbola(), data)

Parameters

Parameter	Description	Units
ymax	Maximum response	-
alpha	Initial slope	-
theta	Curvature (0-1)	-

Arrhenius Model

Temperature response following Arrhenius kinetics:

$$y = y_{ref} \cdot \exp\left[\frac{H_a}{R} \left(\frac{1}{T_{ref}} - \frac{1}{x}\right)\right]$$

where $R = 0.008314$ kJ/(mol·K) and $T_{ref} = 298.15$ K.

Usage

from phytorch import fit
from phytorch.models.generic import Arrhenius

# Temperature in Kelvin
data = {
    'x': np.array([283, 288, 293, 298, 303, 308]),
    'y': np.array([50, 70, 95, 125, 160, 200])
}

result = fit(Arrhenius(), data)
print(f"Activation energy: {result.parameters['Ha']:.1f} kJ/mol")

Parameters

Parameter	Description	Typical Range	Units
yref	Value at reference T	-	-
Ha	Activation energy	30-100	kJ/mol

Peaked Arrhenius Model

Temperature response with high-temperature deactivation:

$$y = y_{max} \cdot f_{arr}(x) \cdot f_{peak}(x)$$

where:

$$f_{arr}(x) = \exp\left[\frac{H_a}{R}\left(\frac{1}{T_{ref}} - \frac{1}{x}\right)\right]$$ $$f_{peak}(x) = \frac{1 + \exp\left[\frac{H_d}{R}\left(\frac{1}{T_{opt}} - \frac{1}{T_{ref}}\right) - \ln\left(\frac{H_d}{H_a} - 1\right)\right]}{1 + \exp\left[\frac{H_d}{R}\left(\frac{1}{T_{opt}} - \frac{1}{x}\right) - \ln\left(\frac{H_d}{H_a} - 1\right)\right]}$$

with $R = 0.008314$ kJ/(mol·K) and $T_{ref} = 298.15$ K.

Usage

from phytorch import fit
from phytorch.models.generic import PeakedArrhenius

# Temperature response with optimum
data = {
    'x': np.linspace(280, 320, 20),
    'y': measured_response
}

result = fit(PeakedArrhenius(), data)
print(f"Optimal temperature: {result.parameters['Topt']:.1f} K")

Parameters

Parameter	Description	Units
ymax	Maximum at Topt	-
Ha	Activation energy	kJ/mol
Hd	Deactivation energy	kJ/mol
Topt	Optimal temperature	K

Gaussian Model

Bell-shaped curve:

$$y = a \cdot \exp\left[-\frac{(x - \mu)^2}{2\sigma^2}\right]$$

Usage

from phytorch import fit
from phytorch.models.generic import Gaussian

result = fit(Gaussian(), data)
print(f"Peak location: {result.parameters['mu']:.2f}")
print(f"Width: {result.parameters['sigma']:.2f}")

Parameters

Parameter	Description	Units
a	Peak height	-
mu	Peak location	-
sigma	Width	-

Weibull Distribution

Weibull probability density function:

$$y = \frac{k}{\lambda}\left(\frac{x - x_0}{\lambda}\right)^{k-1}\exp\left[-\left(\frac{x - x_0}{\lambda}\right)^k\right]$$

Usage

from phytorch import fit
from phytorch.models.generic import Weibull

result = fit(Weibull(), data)

Parameters

Parameter	Description	Units
x0	Location parameter	-
lambda	Scale parameter	-
k	Shape parameter	-

Beta Distribution

Flexible distribution on bounded interval:

$$y = a \cdot \frac{(x - x_{min})^{\alpha-1}(x_{max} - x)^{\beta-1}}{B(\alpha, \beta)(x_{max} - x_{min})^{\alpha+\beta-1}}$$

Usage

from phytorch import fit
from phytorch.models.generic import Beta

result = fit(Beta(), data)

Parameters

Parameter	Description	Units
a	Amplitude	-
alpha	Shape parameter 1	-
beta	Shape parameter 2	-
xmin	Lower bound	-
xmax	Upper bound	-

Custom Parameter Bounds

All models support custom parameter constraints:

from phytorch import fit, FitOptions

options = FitOptions(
    bounds={
        'ymax': (0, 100),
        'x50': (-10, 10)
    }
)

result = fit(Sigmoidal(), data, options)

Model Selection

Use information criteria for model comparison:

# Fit multiple models
models = [Linear(), Sigmoidal(), RectangularHyperbola()]
results = [fit(model, data) for model in models]

# Compare R²
for i, result in enumerate(results):
    print(f"Model {i+1}: R² = {result.r_squared:.4f}")

References

• Michaelis, L., & Menten, M. L. (1913). The kinetics of invertase action. Biochem. z, 49, 333-369.
• Arrhenius, S. (1889). On the reaction velocity of the inversion of cane sugar by acids. Z. Phys. Chem, 4, 226-248.
• Thornley, J. H. M., & Johnson, I. R. (1990). Plant and Crop Modelling: A Mathematical Approach to Plant and Crop Physiology. Oxford University Press.