目的関数 - Ionworks Studio

Objective functions define how the discrepancy between model predictions and experimental data is quantified during parameter fitting.

Cost Function Formulations

Residual Form (Array)

The residual vector represents the difference between model predictions and data:

r(x) = y_{\text{model}}(x) - y_{\text{data}}

where

x

represents the parameter vector.

Canonical Form (Scalar)

The canonical cost function aggregates the residuals into a single scalar value:

\Phi(x) \propto r(x)^\top r(x) = \sum_i^N r_i(x)^2

This sum-of-squares formulation is fundamental to least-squares optimization.

Common Cost Functions

Sum Squared Error
Mean Squared Error
Root Mean Squared Error
Wasserstein Distance

The basic sum of squared residuals:

\Phi_{\text{SSE}}(x) = \sum_i^N r_i^2 = r^\top r

This can be represented in both array and scalar forms, making it compatible with all optimization algorithms.

The Wasserstein distance (also known as earth mover’s distance) measures the distance between two probability distributions. Intuitively, it is the minimum amount of “work” required to transform one distribution into the other.For one-dimensional samples, it is computed by sorting both arrays and comparing them point-wise:

\Phi_{\text{W}}(x) = \frac{1}{N} \sum_i^N \left| \tilde{y}_{\text{model},i}(x) - \tilde{y}_{\text{data},i} \right|

where

\tilde{y}

denotes the sorted samples.Use Wasserstein when you care about matching the distribution of values (for example, a histogram of cell capacities or impedance magnitudes) rather than point-wise agreement in time. Because the inputs are sorted before comparison, the result is invariant to the ordering of the model and data arrays.Weighted point-cloud mode. When one variable supplies positions and another supplies weights — for example matching a dQ/dV curve where peaks should align in voltage — pass both position_variable and weight_variable to iws.costs.Wasserstein. The cost then computes a single weighted Wasserstein-1 distance per objective:

\Phi_{\text{W,weighted}}(x) = W_1\!\left(p_{\text{model}}, p_{\text{data}}, w_{\text{model}}, w_{\text{data}}\right)

where positions

p

come from position_variable and the (sign-stripped, renormalised) weights

w

come from weight_variable. Both must be set together. See Pipelines → Data Fitting → Objective Functions for a configuration example.

Maximum Likelihood Estimation (MLE)

Under the assumption of independent, identically distributed Gaussian errors, the MLE cost function is:

r_{\text{MLE}}(x) = y_{\text{model}} - y_{\text{data}}

\Phi_{\text{MLE}}(x) = \sum_i^N r_i^2 = r^\top r

This is equivalent to the sum squared error formulation, providing a probabilistic interpretation of least-squares fitting.

Usage in ionworkspipeline

import ionworkspipeline as iwp

# Default: Sum Squared Error
objective = iwp.objectives.FittingObjective(
    options={"model": model, "data": data},
    cost=iwp.costs.SumSquaredError()
)

# Mean Squared Error
objective = iwp.objectives.FittingObjective(
    options={"model": model, "data": data},
    cost=iwp.costs.MeanSquaredError()
)

# Root Mean Squared Error (scalar-only)
objective = iwp.objectives.FittingObjective(
    options={"model": model, "data": data},
    cost=iwp.costs.RMSE()
)

# Wasserstein distance (distributional fit)
objective = iwp.objectives.FittingObjective(
    options={"model": model, "data": data},
    cost=iwp.costs.Wasserstein()
)

For most optimization algorithms, the default sum-squared-error formulation provides the best compatibility and performance. Use MSE or RMSE when you need interpretable, scale-independent metrics.

The cost functions and usage examples above are not exhaustive. See the API reference for full details on costs and objectives.

​Cost Function Formulations

​Residual Form (Array)

​Canonical Form (Scalar)

​Common Cost Functions

​Maximum Likelihood Estimation (MLE)

​Usage in ionworkspipeline

Cost Function Formulations

Residual Form (Array)

Canonical Form (Scalar)

Common Cost Functions

Maximum Likelihood Estimation (MLE)

Usage in ionworkspipeline