Smooth Sign Accuracy (SSA) Framework

• Smooth Sign Accuracy (SSA) is a framework that enhances forecast reliability by balancing target tracking accuracy with controlled sign-change frequency.
• It formalizes an accuracy-smoothness dilemma, optimizing predictive alignment under a holding-time constraint to reduce spurious signal fluctuations.
• Its multivariate extension, M-SSA, leverages cross-sectional information to improve nowcasting and forecasting performance across multiple time series.

Smooth Sign Accuracy (SSA) is a forecasting and signal-extraction framework designed for settings in which conventional mean-squared-error (MSE) optimization is too narrow. Its central premise is that a useful predictor should not only track a target in level, but should also be directionally reliable and should avoid excessive, spurious sign changes. SSA formalizes this requirement as an accuracy-smoothness (AS) dilemma: improving tracking accuracy typically moves a predictor toward the classical MSE solution, which can be noisy, while enforcing smoothness reduces false turning points at some cost in target tracking. The framework therefore maximizes predictive alignment with a target subject to an explicit smoothness constraint, where smoothness is measured by the expected time between zero crossings, or equivalently by the frequency of sign changes (Wildi, 10 Jan 2026). Its multivariate extension, M-SSA, incorporates cross-sectional information across several time series and is used for forecasting, nowcasting, and smoothing on an accuracy-smoothness frontier (Wildi, 14 Feb 2026).

1. Formal criterion and basic setup

SSA is formulated for a linear prediction problem. The target is written as

$z_t := \sum_{k=-\infty}^{\infty} \gamma_k x_{t-k},$

and the causal linear predictor of $z_{t+\delta}$ is

$y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$

where $\delta>0$ denotes forecasting, $\delta=0$ nowcasting, and $\delta<0$ backcasting (Wildi, 10 Jan 2026).

The theory is first developed under $x_t=\epsilon_t$, i.i.d. white noise, with $\epsilon_t$ standardized. In that case,

$$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$

and the classical MSE predictor is

$$y_{t,MSE}=\boldsymbol{\gamma}_\delta' \boldsymbol{\epsilon}_t,$$

with $z_{t+\delta}$0 (Wildi, 10 Jan 2026).

The core SSA criterion fixes a smoothness level and then maximizes predictive alignment with the target. In equivalent form, it is

$z_{t+\delta}$1

where

$z_{t+\delta}$2

is the lag-1 autocorrelation of the predictor, and $z_{t+\delta}$3 is the tridiagonal matrix with $z_{t+\delta}$4 on the first off-diagonals, so that

$z_{t+\delta}$5

The framework denotes this solution by $z_{t+\delta}$6 (Wildi, 10 Jan 2026).

This formulation generalizes MSE-based prediction rather than replacing it. Under white noise, the MSE predictor is recovered by choosing

$z_{t+\delta}$7

The paper also gives an MSE variant of SSA that minimizes squared error subject to the holding-time restriction, making the relationship to classical MSE explicit (Wildi, 10 Jan 2026).

2. Sign accuracy, holding time, and the accuracy-smoothness dilemma

SSA is built on two Gaussian identities that connect correlation, directional performance, and smoothness. For a zero-mean stationary Gaussian predictor $z_{t+\delta}$8, sign accuracy is defined as

$z_{t+\delta}$9

and satisfies

$y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$0

Because $y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$1 is strictly increasing on $y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$2, maximizing target correlation is equivalent to maximizing sign accuracy (Wildi, 10 Jan 2026).

Smoothness is defined through the expected duration between successive zero crossings, the holding time

$y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$3

for stationary Gaussian $y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$4. A higher $y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$5 therefore means a larger holding time, fewer sign changes, and a smoother predictor (Wildi, 10 Jan 2026). This is a distinct notion of smoothness: SSA does not penalize curvature in levels, but regulates the expected zero-crossing rate.

The AS dilemma is the statement that these objectives conflict. On the relevant solution branches, increasing smoothness lowers target correlation and hence lowers sign accuracy; decreasing smoothness does the opposite (Wildi, 10 Jan 2026). A dual interpretation follows: SSA can be read not only as the most accurate predictor at a chosen smoothness level, but also as the smoothest predictor among all linear predictors with a specified target correlation (Wildi, 10 Jan 2026).

This gives $y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$6, or equivalently

$y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$7

a direct behavioral interpretation. Raising $y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$8 suppresses false turning points and enforces a more monotone trajectory; lowering it makes the predictor more reactive but noisier (Wildi, 10 Jan 2026).

3. Parametric solution and filter interpretation

A central analytical result is that the problem has a one-parameter characterization. The matrix $y_t=\sum_{k=0}^{L-1} b_k x_{t-k},$9 has eigenvalues

$\delta>0$0

so the feasible smoothness range is

$\delta>0$1

Thus the admissible first-order autocorrelation is bounded by the extremal eigenvalues of $\delta>0$2 (Wildi, 10 Jan 2026).

If the MSE filter is written spectrally as

$\delta>0$3

then, under the stated regularity conditions, the SSA solution has the form

$\delta>0$4

with

$\delta>0$5

The unknown scalar $\delta>0$6 is chosen so that the induced lag-1 autocorrelation equals the prescribed $\delta>0$7 (Wildi, 10 Jan 2026).

In the time domain, $\delta>0$8 satisfies the reversible second-order difference equation

$\delta>0$9

with boundary conditions

$\delta=0$0

This AR(2)-like representation gives the paper’s main intuition: SSA smooths the target by convolving it with an AR(2)-type filter whose single free parameter $\delta=0$1 controls smoothness (Wildi, 10 Jan 2026).

The frequency-domain transfer function is

$\delta=0$2

Its interpretation is direct. If $\delta=0$3, the induced filter is low-pass and suppresses high-frequency noise; if $\delta=0$4, it is high-pass and increases sign changes; if $\delta=0$5, it is band-pass (Wildi, 10 Jan 2026). In that sense, $\delta=0$6 corresponds to deliberate smoothing relative to MSE, while $\delta=0$7 corresponds to deliberate unsmoothing.

4. Dependent and integrated processes

SSA is extended beyond white noise through the Wold decomposition

$\delta=0$8

with invertible MA representation. The criterion is then solved in the innovation domain and transformed back by deconvolution (Wildi, 10 Jan 2026). A practical consequence emphasized in the paper is that, unlike a fixed benchmark filter whose effective smoothness depends on the data-generating process, SSA maintains the chosen holding time across processes.

The most distinctive extension concerns integrated processes. For $\delta=0$9 that is $\delta<0$0, correlation and zero-crossing rates in levels are not well-defined in the same way, so the framework controls sign changes of the first differences of the predictor instead. The relevant object is the error relative to the level MSE predictor,

$\delta<0$1

together with a cointegration constraint ensuring stationarity of this error (Wildi, 10 Jan 2026).

In the $\delta<0$2 case, the emphasized MSE-based version imposes the holding-time restriction on

$\delta<0$3

so $\delta<0$4 controls sign changes in the predictor’s first differences. The resulting level predictor is described as maximal monotone: among predictors with the same MSE-type tracking performance, it minimizes sign changes of $\delta<0$5 (Wildi, 10 Jan 2026). For $\delta<0$6 processes, the same logic is applied to second differences, yielding predictors with the fewest inflection points / lowest curvature in levels (Wildi, 10 Jan 2026).

Implementation is correspondingly structured. The paper summarizes it as: compute the benchmark MSE predictor, choose the desired smoothness level via $\delta<0$7 or $\delta<0$8, solve for $\delta<0$9 on the monotone branch in the stationary case, transform through MA inversion for dependent processes, and impose cointegration for integrated processes (Wildi, 10 Jan 2026).

5. Multivariate extension: M-SSA

The multivariate extension, M-SSA, lets the predictor for one target series use the full multivariate system. With

$x_t=\epsilon_t$0

a multivariate Wold decomposition

$x_t=\epsilon_t$1

target

$x_t=\epsilon_t$2

and predictor

$x_t=\epsilon_t$3

the white-noise version of the criterion is expressed with

$x_t=\epsilon_t$4

and, for each target component $x_t=\epsilon_t$5,

$x_t=\epsilon_t$6

This is the direct multivariate analog of the univariate SSA problem (Wildi, 14 Feb 2026).

Cross-sectional information enters through the covariance matrix $x_t=\epsilon_t$7, the stacked target coefficients, and the multivariate Wold representation. For target $x_t=\epsilon_t$8, the predictor is built from lagged values of all $x_t=\epsilon_t$9 series, not just its own history (Wildi, 14 Feb 2026). The practical implication is that smoother or leading series can improve the target’s nowcast or forecast while preserving a prescribed holding time.

The M-SSA solution again has a one-parameter form,

$\epsilon_t$0

with $\epsilon_t$1 chosen to satisfy the desired autocorrelation constraint (Wildi, 14 Feb 2026). In time-domain form, the coefficients satisfy a non-stationary and time reversible difference equation with zero boundary conditions.

The framework is used in three application domains. For forecasting, the target is a future value. For nowcasting, $\epsilon_t$2 and the target may be an acausal signal such as a two-sided trend filter. For smoothing, the target is causal, and M-SSA acts as a smoother whose smoothness notion differs from Whittaker-Henderson or Hodrick-Prescott smoothing because it controls zero-crossing frequency rather than curvature (Wildi, 14 Feb 2026).

6. Empirical behavior and comparative results

The empirical case studies are intended to show how the AS frontier behaves in practice. In a quarterly $\epsilon_t$3 customization with $\epsilon_t$4, the MSE nowcast has $\epsilon_t$5. Two alternative filters are reported: $\epsilon_t$6, which smooths relative to MSE, and $\epsilon_t$7, which unsmooths. Their holding times are $\epsilon_t$8 for MSE, $\epsilon_t$9 for $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$0, and $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$1 for $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$2, while the target HP filter has holding time $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$3 (Wildi, 10 Jan 2026).

In the monthly U.S. industrial production example, modeled as approximately ARIMA(1,1,0) after log differencing, the target is a two-sided monthly HP trend with $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$4. The comparison is among the level MSE predictor, the classic one-sided HP concurrent filter, and the proposed $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$5-SSA predictor. Reported in-sample MSEs relative to the two-sided HP target are $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$6 for MSE, $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$7 for $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$8-SSA, and $$y_t=\mathbf b' \boldsymbol{\epsilon}_t,\qquad \boldsymbol{\epsilon}_t=(\epsilon_t,\ldots,\epsilon_{t-(L-1)})',$$9 for HP-C; empirical holding times of differenced predictors are $$y_{t,MSE}=\boldsymbol{\gamma}_\delta' \boldsymbol{\epsilon}_t,$$0, $$y_{t,MSE}=\boldsymbol{\gamma}_\delta' \boldsymbol{\epsilon}_t,$$1, and $$y_{t,MSE}=\boldsymbol{\gamma}_\delta' \boldsymbol{\epsilon}_t,$$2, respectively (Wildi, 10 Jan 2026). The stated interpretation is that $$y_{t,MSE}=\boldsymbol{\gamma}_\delta' \boldsymbol{\epsilon}_t,$$3-SSA materially improves smoothness over MSE and improves accuracy over HP-C.

For multivariate nowcasting, the bivariate INDPRO + CLI example is the clearest illustration of how leading-indicator information changes the trade-off (Wildi, 14 Feb 2026).

Method	Target corr.	HT
HP-C	0.650	11.132
M-SSA	0.736	17.263
MSE	0.744	11.011

These results are paired with the qualitative claim that M-SSA achieves the desired 50% holding-time increase over MSE, loses only a little target correlation relative to MSE, beats HP-C on both target correlation and holding time, and eliminates the lag observed for univariate SSA in this application (Wildi, 14 Feb 2026).

A further contrast with curvature-based smoothers appears in the white-noise smoothing comparison with HP. With $$y_{t,MSE}=\boldsymbol{\gamma}_\delta' \boldsymbol{\epsilon}_t,$$4 and $$y_{t,MSE}=\boldsymbol{\gamma}_\delta' \boldsymbol{\epsilon}_t,$$5, SSA1 matches HP’s holding time and attains higher target correlation, while SSA2 matches HP’s target correlation and attains a larger holding time; HP nevertheless remains superior in RMS second-order differences (Wildi, 14 Feb 2026). This supports the paper’s claim that SSA smoothness is specifically about sign-change frequency rather than curvature.

7. Terminological ambiguity of “SSA”

The abbreviation SSA is overloaded in the literature, and disambiguation is often necessary. In two time-series papers, it refers to Singular Spectrum Analysis, including work on structured Hankel implementations and on forecasting parameter selection [(0911.4498); (Knapik et al., 2024)]. In sign-language processing, SSA denotes sign–subtitle alignment, a temporal localization task over signing video rather than a forecasting criterion (Jang et al., 8 Dec 2025). In optimization, smooth sign transformations have been studied for sign-based optimizers, but that work does not define a concept called Smooth Sign Accuracy (Feoktistov et al., 29 May 2026).

Within the forecasting literature, by contrast, Smooth Sign Accuracy denotes the criterion that maximizes target correlation—and, under Gaussianity, sign accuracy—subject to explicit control of the predictor’s sign-change rate through holding time (Wildi, 10 Jan 2026). Its multivariate extension, M-SSA, preserves that definition while allowing each target to borrow information from multiple series (Wildi, 14 Feb 2026).