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In statistics, Linear Template Fit (LTF) is a method for simulation-based parameter estimation applicable to both univariate and high-dimensional multivariate analyses, in cases where the underlying model function cannot be evaluated continuously but is only available at a discrete set of reference values of the parameters of interest^[1]. The LTF combines a linear regression with a (generalized) least squares approach to produce a fully closed-form analytic expression for the best estimator, enabling direct and computationally efficient parameter determination.

The method addresses situations where the residuals cannot be expressed analytically or are too computationally expensive to evaluate repeatedly, as is often the case in iterative minimization algorithms. In the Linear Template Fit, the residuals are estimated from the random variables and from a linear approximation of the underlying true model, while the true model needs to be provided for at least n+1 distinct reference values (where n is the number of estimators). The true distribution is then approximated by a linear regression using pre-computed templates, yielding estimators that are determined directly from the data. This closed-form structure enables fully analytic error propagation, allowing uncertainty quantification that is valuable in complex physics analyses.

The Linear Template Fit is used in particle physics, where it has been employed to determine the W-boson's mass,^[2] the top quark's mass and width,^[3] and the strong coupling constant in quantum chromodynamics.^[1]

The Linear Template Fit considers a least squares problem with m observations (data points) d_i, i = 1, ..., m, which are assumed to follow a normal (Gaussian) probability distribution. The objective function is written in terms of a χ² function:

$${\displaystyle \chi ^{2}({\boldsymbol {\alpha }})=(\mathbf {d} -{\boldsymbol {\lambda }}({\boldsymbol {\alpha }}))^{\mathsf {T}}V^{-1}(\mathbf {d} -{\boldsymbol {\lambda }}({\boldsymbol {\alpha }})),}$$

where:

• ${\displaystyle \mathbf {d} }$ is the m-vector of observed data values
• ${\displaystyle {\boldsymbol {\lambda }}({\boldsymbol {\alpha }})}$ is the m-vector of the theoretical model predictions, dependent on the n-dimensional parameter vector ${\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})}$
• ${\displaystyle V}$ is the m×m covariance matrix incorporating all uncertainty sources

The best estimators ${\displaystyle {\hat {\boldsymbol {\alpha }}}}$ are found by minimizing ${\displaystyle \chi ^{2}}$.

In many practical applications, particularly in high energy physics, the model ${\displaystyle {\boldsymbol {\lambda }}({\boldsymbol {\alpha }})}$ is computationally intensive to evaluate, or may not even be available for arbitrary values of ${\displaystyle {\boldsymbol {\alpha }}}$. Instead, model predictions are only provided at a finite set of j reference values ${\displaystyle {\boldsymbol {\dot {\alpha }}}_{(j)}}$. These predictions are called templates:

$${\displaystyle \mathbf {y} _{(j)}={\boldsymbol {\lambda }}({\boldsymbol {\dot {\alpha }}}_{(j)}),\quad j=1,\ldots ,j_{\text{max}}.}$$

The Linear Template Fit exploits only these pre-computed template predictions.

The first step of the Linear Template Fit is to construct a continuous approximation of the model from the discrete templates. In every bin i, the model ${\displaystyle \lambda _{i}(\alpha )}$ is approximated by a linear function:

$${\displaystyle \lambda _{i}(\alpha )\approx y_{i}(\alpha ;{\hat {\theta }}_{0}^{(i)},{\hat {\theta }}_{1}^{(i)})={\hat {\theta }}_{0}^{(i)}+{\hat {\theta }}_{1}^{(i)}\alpha ,}$$

where ${\displaystyle {\hat {\theta }}_{0}^{(i)}}$ and ${\displaystyle {\hat {\theta }}_{1}^{(i)}}$ are determined by linear regression of the template values ${\displaystyle y_{(j),i}}$ at the reference values ${\displaystyle {\dot {\alpha }}_{j}}$.

The regression matrix M is constructed from the reference values and a column of ones:

$${\displaystyle M={\begin{pmatrix}1&{\dot {\alpha }}_{1}\\1&{\dot {\alpha }}_{2}\\\vdots &\vdots \\1&{\dot {\alpha }}_{j}\end{pmatrix}}.}$$

The best estimators for the regression parameters for the i-th bin are given by the least squares solution:

$${\displaystyle {\hat {\boldsymbol {\theta }}}_{(i)}=M^{+}\mathbf {y} _{(i)},}$$

where ${\displaystyle M^{+}}$ is the Moore–Penrose pseudoinverse of M. A key simplification in the Linear Template Fit is that the bin-wise regression can be treated as unweighted. This is because all templates are typically generated using the same methodology (e.g., the same Monte Carlo statistics), leading to approximately equal uncertainties across templates that cancel in the regression. Thus:

$${\displaystyle M^{+}=(M^{\mathsf {T}}M)^{-1}M^{\mathsf {T}}.}$$

The matrix ${\displaystyle M^{+}}$ is a 2×j matrix that is universal — it depends only on the reference values ${\displaystyle {\dot {\alpha }}_{j}}$, not on the template values themselves, and is therefore the same in every bin i.

The pseudoinverse ${\displaystyle M^{+}}$ is decomposed into two j-dimensional column vectors:

$${\displaystyle M^{+}={\begin{pmatrix}{\boldsymbol {\bar {m}}}^{\mathrm {T} }\\{\boldsymbol {\tilde {m}}}^{\mathrm {T} }\end{pmatrix}}}$$

where ${\displaystyle {\boldsymbol {\bar {m}}}}$ and ${\displaystyle {\boldsymbol {\tilde {m}}}}$ are j-vectors derived from the columns of ${\displaystyle (M^{+})^{\mathsf {T}}}$.

The template matrix Y is an m×j matrix constructed from the column vectors of all template distributions:

$${\displaystyle Y={\begin{pmatrix}\mathbf {y} _{(1)}&\mathbf {y} _{(2)}&\cdots &\mathbf {y} _{(j)}\end{pmatrix}}.}$$

Substituting the regression coefficients into the linear approximation (eq. 4) and using the vectors ${\displaystyle {\boldsymbol {\bar {m}}}}$, ${\displaystyle {\boldsymbol {\tilde {m}}}}$ and matrix Y, the model is expressed as:

$${\displaystyle {\boldsymbol {\lambda }}(\alpha )\approx \mathbf {y} (\alpha )=Y{\boldsymbol {\bar {m}}}+Y{\boldsymbol {\tilde {m}}}\,\alpha .}$$

Substituting this into the χ² objective function (eq. 1):

$${\displaystyle \chi ^{2}(\alpha )=(\mathbf {d} -Y{\boldsymbol {\bar {m}}}-Y{\boldsymbol {\tilde {m}}}\,\alpha )^{\mathsf {T}}W(\mathbf {d} -Y{\boldsymbol {\bar {m}}}-Y{\boldsymbol {\tilde {m}}}\,\alpha ),}$$

where ${\displaystyle W=V^{-1}}$ is the inverse covariance matrix.

Since χ² is quadratic in ${\displaystyle \alpha }$, the best estimator is found at the stationary point ${\displaystyle \partial \chi ^{2}/\partial \alpha =0}$, yielding the master formula of the Linear Template Fit:

$${\displaystyle {\hat {\alpha }}=\left((Y{\boldsymbol {\tilde {m}}})^{\mathsf {T}}WY{\boldsymbol {\tilde {m}}}\right)^{-1}(Y{\boldsymbol {\tilde {m}}})^{\mathsf {T}}W(\mathbf {d} -Y{\boldsymbol {\bar {m}}}).}$$

Introducing the generalized inverse matrix ${\displaystyle F=\left((Y{\boldsymbol {\tilde {m}}})^{\mathsf {T}}WY{\boldsymbol {\tilde {m}}}\right)^{-1}(Y{\boldsymbol {\tilde {m}}})^{\mathsf {T}}W}$, the estimator simplifies to ${\displaystyle {\hat {\alpha }}=F(\mathbf {d} -Y{\boldsymbol {\bar {m}}})}$, and this matrix enables fully analytic error propagation and uncertainty analysis. The variance of the best estimator is:

$${\displaystyle \sigma _{\hat {\alpha }}^{2}=FVF^{\mathsf {T}}=(F^{\mathsf {T}}WF)^{-1}.}$$ |14}}

Given that the data follow a normal distribution and the linear approximation holds, ${\displaystyle {\hat {\alpha }}}$ is a best linear unbiased estimator (BLUE) according to the Gauss–Markov theorem.

For models depending on k parameters ${\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{k})}$, the Linear Template Fit generalizes naturally. The regression matrix M becomes a j × (n+k) matrix,

$${\displaystyle M={\begin{pmatrix}1&{\dot {\alpha }}_{(1),1}&\cdots &{\dot {\alpha }}_{(1),k}\\1&{\dot {\alpha }}_{(2),1}&\cdots &{\dot {\alpha }}_{(2),k}\\\vdots &\vdots &\ddots &\vdots \\1&{\dot {\alpha }}_{(j),1}&\cdots &{\dot {\alpha }}_{(j),k}\end{pmatrix}},}$$

and the pseudoinverse decomposition becomes

$${\displaystyle M^{+}={\begin{pmatrix}{\boldsymbol {\bar {m}}}^{\mathrm {T} }\\{\tilde {M}}^{\mathrm {T} }\end{pmatrix}},}$$

where ${\displaystyle {\tilde {M}}}$ is now a j × k matrix. The linearized model as a function of all parameters ${\displaystyle {\boldsymbol {\alpha }}}$ is:

$${\displaystyle {\boldsymbol {\lambda }}({\boldsymbol {\alpha }})\approx Y{\boldsymbol {\bar {m}}}+Y{\tilde {M}}\,{\boldsymbol {\alpha }},}$$

and after analytic minimization, the closed-form expression for the best estimator $\hat{\boldsymbol{\alpha}}$ of the parameters of interest becomes:

$${\displaystyle {\hat {\boldsymbol {\alpha }}}=\left((Y{\tilde {M}})^{\mathsf {T}}WY{\tilde {M}}\right)^{-1}(Y{\tilde {M}})^{\mathsf {T}}W(\mathbf {d} -Y{\boldsymbol {\bar {m}}}).}$$

Introducing the generalized inverse matrix ${\displaystyle F=\left((Y{\tilde {M}})^{\mathsf {T}}WY{\tilde {M}}\right)^{-1}(Y{\tilde {M}})^{\mathsf {T}}W}$, the estimator simplifies to ${\displaystyle {\hat {\boldsymbol {\alpha }}}=F(\mathbf {d} -Y{\boldsymbol {\bar {m}}})}$, and this matrix enables fully analytic error propagation and uncertainty analysis.

The Linear Template Fit can be formulated in further variants:

• Systematic uncertainties with full bin-to-bin correlations can be included by treating them as nuisance parameters ${\displaystyle {\boldsymbol {\epsilon }}}$ that are also determined analytically in closed form. This allows for detailed insights into possible biases from each uncertainty component.
• The log-normal variant reformulates the equations for log-normal-distributed quantities instead of normally distributed ones. Log-normal uncertainties represent normally distributed relative uncertainties, which is often a reasonable assumption for systematic effects such as luminosity uncertainties. This variant applies the logarithm to both data and template values before performing the fit, and considers relative uncertainties in the covariance matrices.
• When detector effects such as resolution or acceptance need to be accounted for, the template matrix can be modified by incorporating a detector response matrix A, i.e., ${\displaystyle Y\rightarrow AY}$, where A represents the detector migration and Y contains the templates at particle level. This "forward folding" approach avoids the need for iterative or regularized unfolding.

A key advantage of the Linear Template Fit is its closed-form analytic expression, which enables comprehensive and straightforward error propagation of all uncertainty components. Each uncertainty source can be propagated separately to the fit results using standard linear error propagation ${\displaystyle {\hat {\mathbb {V} }}_{\hat {\mathbf {a} }}={\mathcal {F}}\,\mathbb {V} \,{\mathcal {F}}^{\mathsf {T}}}$. Fully bin-to-bin correlated systematic uncertainties are propagated as ${\displaystyle {\boldsymbol {\sigma }}_{\hat {\mathbf {a} }}^{(\mathbf {s} _{(\ell )})}={\mathcal {F}}\,\mathbf {s} _{(\ell )}}$.

Uncertainties in the templates can likewise be propagated to the best estimator. In addition, uncertainties that are not included in the fit can be propagated, for example, theoretical uncertainties that cannot be constrained by data in the fit.

When the model exhibits significant non-linearity in the parameters, the linear approximation may be insufficient. The quadratic template fit extends the method using second-degree polynomials for the parameter dependence of the model.^[1]

In each bin i, the model is approximated as:

$${\displaystyle y_{i}(\alpha ;{\boldsymbol {\hat {\theta }}}_{(i)})={\hat {\theta }}_{0}^{(i)}+{\hat {\theta }}_{1}^{(i)}\alpha +{\hat {\theta }}_{2}^{(i)}\alpha ^{2},}$$ |26}}

which requires at least three reference values for a univariate problem (or at least n²+2n for n parameters in the multivariate case).

Since the χ² function becomes of order ${\displaystyle {\mathcal {O}}(4)}$ in ${\displaystyle \alpha }$, no closed-form solution exists. The quadratic template fit employs an iterative algorithm:

1. The Linear Template Fit is performed to obtain an initial estimator ${\displaystyle {\hat {\boldsymbol {\alpha }}}_{(0)}}$.
2. The Newton algorithm is employed with a few m iterations to obtain improved estimators ${\displaystyle {\hat {\boldsymbol {\alpha }}}_{(m)}={\hat {\boldsymbol {\alpha }}}_{(m-1)}+\Delta {\boldsymbol {\alpha }}_{(m)}}$, where the Hesse matrix is analytically calculable.
3. The best estimator and error calculation are obtained using linearized approximations in the equations of the Linear Template Fit.

The first step provides a starting point close to the minimum. Since the starting point is already in the vicinity of the minimum, the Hesse matrix is commonly positive definite, and the Newton method has excellent convergence for nearly quadratic functions. The algorithm typically converges in just a few iterations.

Beyond correcting for non-linearity, the quadratic template fit also provides an important consistency check: agreement between the linear and quadratic estimators validates that the underlying model is sufficiently linear within the region of interest.

The quality of a Linear Template Fit result can be ensured by appropriate selection of the template reference values: the best estimator should lie within the interval spanned by the templates, and the spacing between reference points should be comparable to or smaller than the expected uncertainty of the estimator. Several cross-checks are available to validate the result. An alternative estimator can be obtained by fitting a parabola to the χ² values computed at each template reference point; its consistency with the LTF result serves as a diagnostic for the linearity of the problem. Additionally, the expected distance to the minimum (EDM) from a Newton-step evaluation quantifies non-linear effects, and an independent fit using the quadratic template fit provides a further check — agreement between all three estimators indicates a well-constrained, sufficiently linear fit.

The Linear Template Fit has found various applications in high-energy particle physics and other fields. Some examples are:

• W-boson mass measurement: The CMS collaboration used LTF to extract the W-boson mass from hadronic jet mass distributions in boosted W boson decays at ${\displaystyle {\sqrt {s}}=13}$ TeV, exploiting the closed-form analytic structure for detailed uncertainty quantification^[2].
• Top quark mass measurement: CMS determined the top quark mass from the differential tt¯ production cross section as a function of the jet mass in hadronic decays of boosted top quarks at ${\displaystyle {\sqrt {s}}=13}$ TeV, where LTF provided a computationally efficient alternative to iterative numerical minimization^[3].
• Top quark mass and width determination: Amoroso et al. applied LTF to simultaneously determine the top-quark mass and width including NLO parton-shower effects, demonstrating the method's capability for multi-parameter fits with correlated systematic uncertainties^[4].
• Strong coupling constant determination: The original LTF paper demonstrated the method by extracting ${\displaystyle \alpha _{s}(m_{Z})=0.1159\pm 0.0014_{\text{(exp)}}\pm 0.0011_{\text{(pdf)}}\pm 0.0001_{\text{(NP)}}}$ from inclusive jet cross sections at ${\displaystyle {\sqrt {s}}=7}$ TeV, where templates were available only at discrete values of ${\displaystyle \alpha _{s}}$ due to the computational cost of NLO QCD calculations^[1].

• Least squares
• Linear regression
• Ordinary least squares
• Weighted least squares
• Generalized least squares
• Nonlinear least squares
• Maximum likelihood estimation

Category:Least squares Category:Regression analysis Category:Statistical estimation Category:Statistical dispersion