StaFIR: Convex Learning of Stationarity-Aware Causal Filters

Lorena Egger; Mathis Linger

cs.LGOct 5, 2026

StaFIR: Convex Learning of Stationarity-Aware Causal Filters

Authors: Lorena Egger, Mathis Linger

Organizations: Drakai Capital Paris

Abstract

Reducing nonstationarity in a persistent time series entails deciding how much of its temporal dependence to remove. In finance, fractional differencing is often tuned using the Augmented Dickey--Fuller (ADF) test, limiting the search to a one-parameter family of lag profiles and addressing input preservation only indirectly. We propose StaFIR, a causal finite-impulse-response filter with a learned nonnegative mixture of exponential lag profiles. Its convex learning objective balances empirical stationarity with similarity to the input. We evaluate StaFIR on ARFIMA--GARCH controlled settings and rolling financial series, including a realized-volatility forecasting task. The experiments show that StaFIR adjusts its filtering strength to persistence while limiting unnecessary transformation in stationary regimes. In downstream forecasting, there is no clear accuracy difference from fixed half-order differencing, while StaFIR achieves higher measured similarity to the raw signal. A complementary direct forecasting experiment finds that greater input similarity is associated with smaller forecasting penalties, although the raw representation remains stronger.

Figures & tables

Figure 1 : Held-out stationarity gain

\Delta S_{\mathrm{stat}}

and similarity cost

C_{\mathrm{sim}}

on the synthetic designs. Dashed lines indicate fixed methods and solid lines adaptive methods. The two panels left of the separator vary the fractional-integration order

d_{\mathrm{true}}

on both sides of the stationarity boundary

d_{\mathrm{true}}=0.5

, marked by the fine vertical line. The two panels to the right vary GARCH persistence over

p_{\mathrm{garch}}\in[0,0.99]

. Lines average over 30 seeds.

Figure 2 : Mean held-out stationarity gain

\Delta S_{\mathrm{stat}}

and similarity cost

C_{\mathrm{sim}}

across rolling origins for the four log-price series, on shared axes. The dashed curve is the path traced by fixed-order fractional differencing as

d

varies. Stationarity gain is the selection criterion of the contextual methods; Table 1 gives additional held-out diagnostics, and Appendix E.3 gives paired bootstrap intervals.

Method	ADF rejection	PP rejection	KPSS non-rejection	LW $\widehat{d}$
FD(0.5)	100	100	31	0.46 [0.44, 0.48]
FD-adf	38	43	11	0.82 [0.80, 0.85]
FD-context	71	72	26	0.61 [0.56, 0.65]
StaFIR-context	77	80	10	0.53 [0.49, 0.58]

Table 1 : Conventional held-out stationarity diagnostics on the rolling log-price series. Entries are pooled rates (%) of rejection of the ADF and PP unit-root nulls and non-rejection of the KPSS stationarity null. LW

\widehat{d}

is the local Whittle estimate of the residual memory of the held-out output, with a 95% stationary-bootstrap interval. The raw series pool at

\widehat{d}=0.96

and first differences at

-0.04

. Estimation details are in Appendix E.3 .

Method	Similarity	$h=1$	$h=5$	$h=20$	$h=40$	$h=80$
raw $=$ FD-adf	100.0	$0.00$	$0.00$	$0.00$	$0.00$	$0.00$
1-diff	53.9	$6.13\pm 0.89$	$27.12\pm 4.92$	$42.84\pm 10.75$	$49.19\pm 10.34$	$87.01\pm 18.43$
EWMA	91.8	$1.32\pm 0.60$	$8.54\pm 2.78$	$13.53\pm 4.86$	$21.71\pm 7.01$	$31.82\pm 7.44$
FD(0.5)	79.4	$\mathbf{-0.86\pm 0.55}$	$0.77\pm 1.93$	$-1.83\pm 2.90$	$0.06\pm 3.67$	$0.94\pm 2.85$
FD-context	93.9	$-0.34\pm 0.34$	$0.21\pm 1.10$	$-0.80\pm 1.45$	$0.34\pm 1.79$	$0.97\pm 1.25$
StaFIR-context	93.5	$-0.83\pm 0.30$	$-2.25\pm 0.81$	$-3.28\pm 1.09$	$-3.07\pm 1.38$	$\mathbf{-1.84\pm 1.33}$

Table 2 : Downstream QLIKE relative to raw (%, equal-weight mean over four series

\pm

one paired stationary-bootstrap standard error, with bootstrap details in Appendix D.6 ). Lower is better. Bold marks the lowest point estimate at each horizon. FD-adf equals raw because it selects

d=0

throughout. Similarity is measured on the realized-variance inputs. All preprocessing rows use the common forecasting head. log-HAR is an external task-specific reference with a different direct forecasting architecture.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Adaptive	Parameter	Selection rule
raw , 1-diff	no	–	fixed references
EWMA , rolling-z	no	window $K+1=101$	fixed
FD(0.5)	no	order $d=0.5$	fixed a priori
FD-adf	yes	order $d$	smallest order with ADF rejection
FD-context	yes	order $d$	contextual rule of Appendix C.2
StaFIR-context	yes	$(\lambda_{s},\lambda)$	contextual rule of Appendix C.2

Appendix

Table 3 : Compared preprocessing methods.

Panel of Figure 3 — Figure 3 : Rolling evaluation design for the market series. Each 2,000-observation window consists of training (1,280), validation (320), and evaluation (400) segments, with a step of 63 observations between recalibrations. Every later window re-selects the penalty pair and the adaptive fractional order and re-fits the weight vector $w$ on its own calibration segment. The fixed FD(0.5) baseline is unchanged.

Component	Package	Routine	Specification
ADF test	statsmodels v 0.14.6	statsmodels.tsa.stattools.adfuller	Unit-root null. Constant only, regression = "c" , with one fixed lag, maxlag = 1 and autolag = None . Parameters from baseline configuration in Lopez de Prado [2018] . We report the statistic, $p$ -value, and $5\%$ rejection indicator.
PP test	arch v 8.0.0	arch.unitroot.PhillipsPerron	Unit-root null. Constant only, trend = "c" . Bandwidth / lag length is chosen automatically with lags = None . We report the statistic, $p$ -value, selected bandwidth / lag, and $5\%$ rejection indicator.
KPSS test	statsmodels v 0.14.6	statsmodels.tsa.stattools.kpss	Level-stationarity null. Constant only, regression = "c" . Lag length is selected using nlags = "auto" . We report the statistic, interpolated $p$ -value, selected lag length, and $5\%$ rejection indicator.
SOCP solver	cvxpy v 1.9.2 with clarabel v 0.11.1	CLARABEL through the cvxpy modeling layer	Solves the filter SOCP in Section 3 . We record solver status, objective value, iterations, feasibility, and runtime. Settings are max_iter = 50 and tol_gap_abs = tol_gap_rel = tol_feas = $10^{-6}$ .

Appendix

Table 4: Implementation details for tests and solver.

Series	Underlying	Obs.	Sample span	Origins	Origin span
sp500	S&P 500 index ( SPX Index )	6,539	2000-01-03 – 2025-12-31	73	2006-05-15 – 2024-05-22
eurusd	EUR/USD rate ( EURUSD Curncy )	6,783	2000-01-03 – 2025-12-31	76	2006-02-17 – 2024-03-29
cds_idx_sp	On-the-run Crossover five-year index spread	5,307	2005-08-30 – 2025-12-31	53	2011-10-17 – 2024-05-07
xbx_index	CoinDesk Bitcoin Price Index ( XBX )	4,077	2014-11-03 – 2025-12-31	33	2019-03-21 – 2024-09-26

Appendix

Table 5 : Daily log-price series after cleaning.

Method	Stationarity	Similarity	ADF rejection	PP rejection	KPSS non-rejection	Fail
			Test outcomes (%)
raw	65.3 $\pm$ 0.6	–	61	63	19	0/3000
1-diff	80.4 $\pm$ 0.6	50.3 $\pm$ 0.3	100	100	99	0/3000
EWMA	70.1 $\pm$ 0.5	82.5 $\pm$ 0.8	85	88	62	0/3000
rolling-z	70.0 $\pm$ 0.5	83.1 $\pm$ 0.8	83	86	54	0/3000
FD(0.5)	78.2 $\pm$ 0.5	74.4 $\pm$ 0.4	100	100	71	0/3000

Appendix

Table 6 : Synthetic-data evaluation. The score columns are mean

\pm

sample SD across independent root seeds after averaging the scenario grid within each seed, and are conditional on successful fits. The three test columns report decision rates over the scenarios at which the test returned a decision, pooled over the whole grid: rejection of the ADF and PP unit-root nulls and non-rejection of the KPSS stationarity null. Fail reports failed fits over attempted fits, and a failed fit is never counted as a failed test. Figure 1 summarizes the same comparison on the score columns.

$d_{\mathrm{true}}$	Mean $S_{\mathrm{stat}}$	KPSS non-rejection (%)
0.00	80.4	97.3
0.20	79.1	71.7
0.40	75.3	47.0
0.49	73.1	37.7

Appendix

Table 7 : Diagnostics on raw inputs in the stationary companion grid. Each row averages 30 seeds and 10 equally spaced GARCH persistence settings in

[0.5,0.99]

, for 300 evaluations on untransformed held-out segments of 400 observations. KPSS uses a constant, automatic lag selection, and a 5% significance level. No evaluation in these rows failed or lacked a KPSS decision. All four fractional-integration orders lie below the theoretical covariance-stationarity boundary.

Memory increases ( d_up )
	Stationarity gain			Similarity cost	Effect turnover $T_{y}$
Method	Pre	Trans.	Post	Trans.	Pre	Trans.	Post
FD-adf	0.00 (0.00)	2.91 (0.26)	3.27 (0.35)	2.03 (0.18)	0.000 (0.000)	0.171 (0.015)	0.206 (0.017)
FD-context	4.73 (0.60)	11.21 (0.43)	14.62 (0.92)	19.62 (0.74)	0.093 (0.023)	1.159 (0.261)	1.669 (0.406)
StaFIR-context	4.60 (0.50)	9.47 (0.42)	13.46 (0.97)	16.08 (0.69)	0.174 (0.021)	1.189 (0.192)	1.325 (0.234)
Memory decreases ( d_down )

Appendix

Table 8 : Response to a within-series regime shift. Origins are grouped by whether the rolling calibration window is before, across, or after the scheduled break. Entries are means over independent seeds with standard errors in parentheses. Similarity cost is shown for the transition phase, on which the comparison of Appendix E.2 rests. Fixed transformations are omitted because they do not recalibrate.

Method	ADF rejection	PP rejection	KPSS non-rejection	$M$	$V$	$G$	$\rho_{\mathrm{P}}$	$\rho_{\mathrm{P},\mathcal{T}}$	dCor	$S_{\mathrm{stat}}$	$S_{\mathrm{sim}}$	$\overline{T}_{y}$
	Test outcomes (%)			Stationarity components			Similarity components			Aggregate score
raw	7	6	5	63.5	61.3	65.7	1.00	1.00	1.00	57.6	100.0	–
1-diff	100	100	97	94.3	74.9	72.3	0.07	0.19	0.16	71.9	43.0	–
EWMA	42	44	34	70.6	64.0	65.6	0.48	0.78	0.52	61.3	71.7	–
rolling-z	30	31	34	70.1	65.7	66.6	0.48	0.80	0.53	62.2	72.4	–
FD(0.5)	100	100	31	77.1	72.2	70.3	0.56	0.80	0.54	67.7	73.9	–

Appendix

Table 9 : Rolling log-price diagnostics pooled over 235 origins. Test columns report ADF and PP rejection rates and KPSS non-rejection rates (%);

M,V,G

are the stationarity components and

\rho_{\mathrm{P}},\rho_{\mathrm{P},\mathcal{T}},\mathrm{dCor}

the similarity components. Aggregate scores are on a 0–100 scale.

\overline{T}_{y}

is mean effect turnover over 231 predecessor pairs; “–” denotes fixed transforms. Pooling is descriptive because origins overlap.

Baseline	$\Delta S_{\mathrm{stat}}$	$\Delta C_{\mathrm{sim}}$	$\Delta\|\widehat{d}\,\|$	Paired origins
XO 5Y
FD-context	0.9 [-0.9, 2.5]	-2.2 [-7.0, 2.4]	-0.08 [-0.15, -0.02]	53
FD(0.5)	-2.2 [-4.4, -0.3]	-9.3 [-13.3, -5.1]	0.03 [-0.04, 0.11]	53
1-diff	-6.3 [-8.4, -4.3]	-40.5 [-44.2, -36.9]	0.38 [0.32, 0.46]	53
FD-adf	8.5 [6.3, 10.6]	15.9 [12.1, 19.7]	-0.41 [-0.51, -0.32]	53
raw	9.8 [7.9, 11.6]	16.8 [13.1, 20.4]	-0.48 [-0.55, -0.40]	53

Appendix

Table 10 : Paired rolling log-price comparisons. Entries are mean within-origin differences with percentile 95% stationary-bootstrap intervals (10,000 replications; expected block length 7 origins). Positive stationarity differences favor StaFIR-context ; positive similarity-cost differences indicate additional preservation cost. The final panel gives equal weight to the four series.

\Delta|\widehat{d}\,|

is the paired difference in absolute local Whittle residual memory (Table 1 ); negative values favor StaFIR-context .

Figure 4 : Frequency and magnitude of material transformation changes for the three adaptive methods on the log-price series. The horizontal axis,

P(T_{y}>0.01)

, is the share of consecutive origin pairs whose effect turnover

T_{y}

exceeds 0.01. On its log scale, the vertical axis reports the median

T_{y}

among only those origin pairs that exceed the threshold. Whiskers extend to the conditional 90th percentile.

Figure 5 : Per-series forecast performance on the log five-day realized-variance series. At each forecast horizon

h

, points report QLIKE relative to raw , in percent. Whiskers are one paired stationary-bootstrap standard error. FD-context , StaFIR-context , and FD(0.5) share the common forecasting model. log-HAR is an external task-specific reference with a different, direct forecasting model. FD-adf coincides with raw and is omitted.

Horizon	vs. FD-context	vs. FD(0.5)	vs. raw AR(110)
$h=1$	$-0.49\;[-0.98,\phantom{-}0.02]$	$\phantom{-}0.06\;[-0.91,\phantom{-}1.06]$	$\mathbf{-3.92\;[-5.35,-2.47]}$
$h=5$	$\mathbf{-2.45\;[-4.54,-0.30]}$	$-2.64\;[-6.15,\phantom{-}1.20]$	$\mathbf{-8.24\;[-12.03,-4.05]}$
$h=20$	$-2.54\;[-5.38,\phantom{-}0.31]$	$-0.36\;[-5.53,\phantom{-}5.47]$	$-4.46\;[-11.01,\phantom{-}1.98]$
$h=40$	$-3.40\;[-6.32,\phantom{-}0.05]$	$-2.41\;[-7.68,\phantom{-}5.20]$	$-3.07\;[-9.15,\phantom{-}3.53]$
$h=80$	$-2.72\;[-5.38,\phantom{-}0.46]$	$-2.35\;[-6.76,\phantom{-}3.61]$	$-2.43\;[-7.40,\phantom{-}2.40]$

Appendix

Table 11 : Paired downstream QLIKE comparisons with recursive reintegration for StaFIR-context (%, equal-weight mean over four series with paired stationary-bootstrap marginal 95% intervals). Each percentage uses the named comparator’s QLIKE as its denominator. Negative values favor StaFIR-context . Bold intervals exclude zero. The raw AR(110) control matches the

K+10=110

observations of raw history used by the

K=100

FIR and its ten-lag head.

Horizon	vs. raw	vs. FD(0.5)	vs. FD-context
$h=1$	$\boldsymbol{+25.51\;[+17.98,+35.07]}$	$\boldsymbol{-43.84\;[-49.11,-37.47]}$	$\boldsymbol{-7.18\;[-12.24,-1.54]}$
$h=5$	$\boldsymbol{+8.35\;[+5.03,+12.19]}$	$\boldsymbol{-15.91\;[-20.99,-10.17]}$	$+0.62\;[-3.01,+4.50]$
$h=20$	$\boldsymbol{+5.93\;[+3.35,+9.31]}$	$\boldsymbol{-9.42\;[-13.52,-5.45]}$	$-0.23\;[-2.08,+1.83]$
$h=40$	$\boldsymbol{+4.39\;[+1.87,+7.35]}$	$-3.83\;[-8.30,+0.21]$	$+1.68\;[-0.14,+3.32]$
$h=80$	$+0.57\;[-0.96,+1.86]$	$-1.31\;[-3.76,+0.92]$	$\boldsymbol{+0.85\;[+0.003,+1.87]}$

Appendix

Table 12 : Direct-head sensitivity: QLIKE differences for StaFIR-context versus comparators (%, marginal 95% stationary-bootstrap intervals). Four series-level percentages, normalized by each comparator’s mean QLIKE, are averaged equally. Negative values favor StaFIR-context ; bold intervals exclude zero. Forecasts use ten transformed lags to predict raw log realized variance.

Method	Stationarity	Similarity	ADF rejection	PP rejection	KPSS non-rejection	Fail
			Test outcomes (%)
StaFIR-context	73.2 $\pm$ 0.9	88.2 $\pm$ 1.1	95	97	31	0/3000
StaFIR-context-7-scale	73.3 $\pm$ 0.9	88.1 $\pm$ 1.1	96	97	32	0/3000
StaFIR-context-free-lags	73.1 $\pm$ 0.9	88.1 $\pm$ 1.1	95	97	34	0/3000
StaFIR-context-no-2nd-order	70.7 $\pm$ 0.8	91.6 $\pm$ 1.1	87	90	38	0/3000

Appendix

Table 13 : Synthetic constraint-ablation evaluation. Scores are mean

\pm

sample SD across seeds after averaging the scenario grid; test columns report pooled ADF and PP rejection rates and KPSS non-rejection rates (%), conditional on successful fits. Fail counts unsuccessful fits over attempts.

Method	Stationarity	Similarity	Effect turnover $T_{y}$	ADF rejection	KPSS non-rejection	Fail
				Test outcomes (%)
StaFIR-context	66.3 $\pm$ 3.5	82.7 $\pm$ 5.5	0.725 $\pm$ 0.158	78	10	0/235
StaFIR-context-no-2nd-order	64.4 $\pm$ 3.3	85.4 $\pm$ 5.2	0.984 $\pm$ 0.245	62	14	0/235
StaFIR-context-7-scale	66.4 $\pm$ 3.5	82.3 $\pm$ 5.5	0.715 $\pm$ 0.171	80	17	0/235
StaFIR-context-free-lags	66.1 $\pm$ 3.6	82.7 $\pm$ 5.4	0.734 $\pm$ 0.145	78	13	0/235

Appendix

Table 14 : Rolling log-price constraint ablations. Score and turnover entries are the equal-weight mean

\pm

sample SD of the four per-series rolling means. Test entries are equal-weight means of the four per-series ADF unit-root rejection rates and KPSS stationarity non-rejection rates. Fail counts failed fits over all 235 attempted origins.

	$\zeta$	$\gamma_{\mathrm{sel}}$	$S_{\mathrm{stat}}$	$S_{\mathrm{sim}}$	Joint test outcome	$T_{y}$	Reference reached	Score shortfall
S&P 500
	0.00	0.20	58.6	100.0	0.00	0.019	0.93	0.00
	0.00	0.40	58.6	100.0	0.00	0.019	0.93	0.00
	0.00	0.80	58.6	100.0	0.00	0.019	0.93	0.00
	0.90	0.20	61.3	91.3	0.04	0.742	0.16	3.79
$\ast$	0.90	0.40	61.7	90.1	0.04	0.900	0.18	2.40

Appendix

Table 15 : Sensitivity of the contextual selection rule to the reference fraction

\zeta

and the shortfall curvature

\gamma_{\mathrm{sel}}

. Each row averages the successful rolling origins of the indicated series. The asterisk marks the configuration used in the main experiments,

(\zeta,\gamma_{\mathrm{sel}})=(0.9,0.4)

. Stationarity and similarity are 0–100 scores. Joint test outcome is the share of origins with rejection of both ADF and PP unit-root nulls and non-rejection of the KPSS null.

T_{y}

is mean effect turnover. Reference reached is the share of origins at which the selected candidate’s validation score is at least

S_{\min}

. Score shortfall is the mean of

[S_{\min}-S_{\mathrm{stat}}(y^{\mathrm{val}})]_{+}

	Wall adaptation		Host inf.	State	Adaptation CPU (s)				Peak RSS abs./+ $\Delta$ (MiB)		Selection failures
Method	select (s)	carried (ms)	p50/p95 ( $\mu$ s)	(KiB)	select	carried	cycle	/origin	select	carried	Synthetic	Real
StaFIR-context	1.56	203	4.0/4.1	1.59	1.562	0.219	1.562	1.562	200.3/+15.96	196.6/+12.38	0/3000	0/469
FD-adf	0.061	$<$ 1	4.0/4.1	1.59	0.062	0.000	0.062	0.062	185.9/+1.00	184.9/+0.05	0/3000	0/469
FD-context	0.12	$<$ 1	4.0/4.1	1.59	0.125	0.000	0.125	0.125	188.2/+3.70	184.7/+0.04	0/3000	0/469

Appendix

Table 16 : Deployment cost of the adaptive representations. Select and carried are median wall times for hyperparameter selection and a fixed-setting re-fit; inference is single-update p50/p95 latency. FIR path and numeric state are shared. CPU times are in seconds; peak RSS is absolute memory/increase above process start. Failures count unsuccessful selections.

				Share of origins (%)
Selection period	Per-origin refit (s)	Similarity	Effect turnover $T_{y}$	Joint test outcome	Selection change
FD-adf
$M{=}1$	0.06	96.8 $\pm$ 3.2	0.099 $\pm$ 0.123	10	72
$M{=}2$	0.03	96.7 $\pm$ 3.2	0.064 $\pm$ 0.129	10	37
$M{=}4$	0.02	96.7 $\pm$ 3.2	0.048 $\pm$ 0.128	9	22
$M{=}8$	0.01	96.8 $\pm$ 3.1	0.034 $\pm$ 0.116	9	12

Appendix

Table 17 : Adaptation frequency across selection periods. Hyperparameters are selected every

M

origins (

M{=}\infty

: first origin only); standardization and filters are re-fitted at every origin. Per-origin refit is amortized time; similarity and

T_{y}

are mean

\pm

sample SD over successful origins. Joint outcome requires ADF/PP rejection and KPSS non-rejection; selection change counts hyperparameter changes between consecutive origins (both percentages).

Comparator schedule	$h{=}1$	$h{=}5$	$h{=}20$	$h{=}40$	$h{=}80$
$M{=}2$	$-0.20[-0.47,0.08]$	$\mathbf{-1.18[-1.98,-0.37]}$	$\mathbf{-2.35[-4.29,-0.73]}$	$-1.32[-3.48,0.55]$	$-0.35[-2.27,2.04]$
$M{=}4$	$\mathbf{-0.48[-0.83,-0.11]}$	$\mathbf{-2.17[-3.18,-1.20]}$	$\mathbf{-4.35[-6.90,-2.35]}$	$\mathbf{-3.25[-6.34,-0.86]}$	$-1.95[-4.53,0.75]$
$M{=}8$	$-0.38[-0.85,0.07]$	$-0.82[-2.27,0.73]$	$\mathbf{-3.26[-5.64,-0.05]}$	$\mathbf{-5.51[-8.44,-0.89]}$	$-3.38[-5.88,0.19]$
$M{=}\infty$	$\mathbf{-0.93[-1.34,-0.51]}$	$-1.95[-3.78,0.01]$	$\mathbf{-5.42[-8.18,-2.78]}$	$\mathbf{-4.70[-7.60,-1.98]}$	$-2.14[-4.56,0.68]$

Appendix

Table 18 : Paired stationary-bootstrap QLIKE contrasts for StaFIR-context at

M=1

versus less frequent selection schedules. Entries show the percentage change in mean QLIKE at

M=1

relative to each comparator schedule, averaged equally across series at each forecast horizon

h

. Brackets give 95% intervals; negative values favor

M=1

. Bold intervals exclude zero.

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Open paper AskPDF arXiv

Time series forecasting relies on historical patterns, but real-world series often exhibit non-stationarity and regime shifts that challenge fully parametric forecasters. Inspired by Retrieval-Augmented Generation (RAG), recent work augments forecasters by retrieving relevant historical segments and using them as external evidence at inference time. However, due to the intrinsic non-stationarity of real-world time series, a highly similar past segment does not necessarily imply a similar future, rendering similarity-only retrieval brittle and prone to redundancy. We propose Stationarity-Aware Retrieval-Augmented Time Series Forecasting (SARAF), a framework that adaptively balances relevance and diversity in retrieval. SARAF first forms a candidate pool via temporal similarity with time-aligned enhancement, then applies a diversity-aware selection strategy to cover heterogeneous historical regimes, with the diversification strength automatically modulated by dataset-level stationarity. Moreover, SARAF uses stationarity-aware aggregation to fuse the retrieved futures. Extensive experiments on eight real-world datasets show that SARAF achieves competitive forecasting performance and improves average accuracy and robustness over strong baselines, with particularly clear benefits under challenging non-stationary settings. Code: https://github.com/ShiqiaoZhou/SARAF.

Abstract

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