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to h. SinceGsatisfies ZUTA, there is an observed node v∈ch(h)such that v∈ch(h)andv/\e}atio\slash∈/uniontext ℓ≻hch(ℓ). Hence, pa (v)\S={h}. By Corollary 4.4, there further exist two disjoint sets /tildewiderW,/tildewideU⊆ V\{v}with|/tildewiderW|=|/tildewideU|=|H|such that there is an intersection-free matching between /t... | https://arxiv.org/abs/2502.02986v1 |
Conditions (i) and (ii) are easily checked, and an intersec tion-free matching of UandWthat avoids Sis given by v2i←hi←wi. Moreover, since pa ({v}∪U)\S= pa({v2i−1,v2i})\S={hi}, there can not exist an intersection-free matching of {v}∪U)and {v}∪W). Thus by Theorem 4.7we can determine generic sign-identifiability of all l... | https://arxiv.org/abs/2502.02986v1 |
ch(ℓ)is also a/tildewideB-first ordering on ch (ℓ)that satisfies condition (ii). Now, consider a latent node ℓ∈jpa(B)\(S∪h)withℓ≺ZUTAh. Note that≺ℓis a/tildewideB-first- ordering on ch (ℓ)\{uh}. Now, we extend this ordering to an ordering /tildewide≺ℓon the whole set of children ch(ℓ). We define it as the block-ordering /t... | https://arxiv.org/abs/2502.02986v1 |
φG:RD−→R(|V| 2) Λ/ma√sto−→od(ΛΛ⊤) has fibers of the form φ−1 G(φG(Λ)) ={/tildewideΛ∈RD:/tildewideΛ = ΛΨ forΨ∈{±1}|H|×|H|diagonal} for almost all Λ∈RD. Proof. LetG= (V∪H,D)be a factor analysis graph. Take a generic tuple (Ω,Λ)∈ΘG, and assume that the fiber φ−1 G(φG(Λ))is of the form as in the statement. Now, consider anot... | https://arxiv.org/abs/2502.02986v1 |
one can use any computer algebra system such as SINGULAR ( Decker et al. ,2024), Macaulay2 ( Grayson and Stillman ,2024) or SageMath ( Stein,2024). A factor analysis graph is generically sign-identifiable if its degree of sign-identi fiability is 1. If the degree of sign-identifiability is infinite, then the real fiber Fφ(Λ... | https://arxiv.org/abs/2502.02986v1 |
F. (2024). Weak identification in low-dimensional fa ctor models with one or two factors. The Review of Economics and Statistics , pages 1–45. 38 de Oliveira Santos, R., Gorgulho, B. M., de Castro, M. A., Fisb erg, R. M., Marchioni, D. M., and Baltar, V. T. (2019). Principal component analysis and facto r analysis: diffe... | https://arxiv.org/abs/2502.02986v1 |
A. S., Waters, A. E., Studer, C., and Baraniuk, R. G. (2014 ). Sparse factor analysis for learning and content analytics. Journal of Machine Learning Research , 15:1959–2008. Lauritzen, S. L. (1996). Graphical models , volume 17 of Oxford Statistical Science Series . Clarendon Press. Lee, T.-H. and Seregina, E. (2023).... | https://arxiv.org/abs/2502.02986v1 |
K. (2017). Sp arse exploratory factor analysis. Psychometrika , 82(3):778–794. Whitney, H. (1957). Elementary structure of real algebraic varieties. Annals of Mathematics. Second Series , 66:545–556. Williams, B. (2020). Identification of the linear factor mode l.Econometric Reviews , 39(1):92–109. Wilson, E. B. and Wor... | https://arxiv.org/abs/2502.02986v1 |
Parametric Scaling Law of Tuning Bias in Conformal Prediction Hao Zeng* 1Kangdao Liu* 1 2Bingyi Jing1Hongxin Wei1 Abstract Conformal prediction is a popular framework of uncertainty quantification that constructs predic- tion sets with coverage guarantees. To uphold the exchangeability assumption, many conformal predic... | https://arxiv.org/abs/2502.03023v1 |
us to investi- gate the influence of parameter tuning on the coverage gap in the absence of a hold-out set. In this work, we reveal a previously unrecognized phe- nomenon: the tuning bias - the coverage gap introduced by using the same dataset for tuning and calibration, is neg- ligible for simple parameter tuning in c... | https://arxiv.org/abs/2502.03023v1 |
used proce- dure, split conformal prediction (Papadopoulos, 2008) initi- ates with a calibration step. For each sample (xi, yi)from the calibration set Dcal:={(xi, yi)}n i=1, we compute the non-conformity score si:=S(xi, yi)for a score function S:X ×Y → R. The non-conformity score function Smea- sures the strangeness o... | https://arxiv.org/abs/2502.03023v1 |
used for conformal prediction calibration. Given the transformation parameter λ, we can calculate the threshold ˆtin Eq. (2)with parameter λ. If we use the same set for both parameter tuning and conformal prediction calibration, it will destroy the data exchange- ability assumption and probably introduce an additional ... | https://arxiv.org/abs/2502.03023v1 |
the large number of parameters requiring tuning. This tuning could potentially result in overfitting, where the transformed score distribution is overly adapted to the hold-out set, significantly reducing its performance on the test sample. As the size of the calibration set increases, the tuning bias converges to zero... | https://arxiv.org/abs/2502.03023v1 |
an increment of 1000. The parameter tuning method used here is vector scaling with full parameter space. For the size of the calibration set, the tuning bias decreases with the size of the calibration set increasing, as shown in Figure 2 (b). The tuning bias decreases about 50% from the size of 6000 to 10000. As the si... | https://arxiv.org/abs/2502.03023v1 |
defined as: ˆtλ D=Q((1−α)(1+1 /n))(Sλ D), (4) where Qp(S)denotes the p-th empirical quantile of non- empty set S:Qp(S) = inf {q:|{s∈ S:s≤q}| ≥p|S|}. We formulate the parameter tuning using the same set as the calibration set for conformal prediction as follows: the tuning parameter ˆλis selected by minimizing a pre-spe... | https://arxiv.org/abs/2502.03023v1 |
finite, we bound the tun- ing bias using classical concentration inequality, Dvoret- zky–Kiefer–Wolfowitz inequality (Dvoretzky et al., 1956), on the empirical process with the finite union of probability: 5 Parametric Scaling Law of Tuning Bias in Conformal Prediction Proposition 4.2. For a finite parameter space Λ, w... | https://arxiv.org/abs/2502.03023v1 |
Corollary 4.4. For a classification problem, and the hold- out set size n, the tuning bias of selection of score func- tions (Yang & Kuchibhotla, 2024) with the number of candi- dates Mis bounded by: TuningBias (bC)≤r log(2M) 2n+1√ 2np log(2M). The proof of Corollary 4.4 is provided in Appendix F. For the score aggrega... | https://arxiv.org/abs/2502.03023v1 |
a more complex parameter space compared to temperature scaling, the tuning bias of vector scaling is larger than that of temperature scaling: Corollary 4.7. For a classification problem with Kclasses, the calibration set size nand the same pre-trained model, the tuning bias of temperature scaling is smaller than that o... | https://arxiv.org/abs/2502.03023v1 |
For a binary classification problem, we haveTuningBias (bCTS) = 0≤TuningBias (bCVS), where bCTSandbCVSare the prediction sets of models tuned by tem- perature scaling and vector scaling, respectively. We provide the proof of the above proposition in Appendix I. From the example, we demonstrate that designing tuning alg... | https://arxiv.org/abs/2502.03023v1 |
tuning and calibration. Learnability As classical learnability theory, a learnable model could be regarded as a risk minimization model with a specific hypothesis class (van der Vaart & Wellner, 1996; Vapnik, 1991; 1999; Vapnik & Chervonenkis, 1971; Ver- shynin, 2018). And further, the constrained risk minimiza-tion mo... | https://arxiv.org/abs/2502.03023v1 |
on Learning Representations , 2022. Barber, R. F., Cand `es, E. J., Ramdas, A., and Tibshirani, R. J. Conformal prediction beyond exchangeability. The Annals of Statistics , 51:816–845, 2023. Dabah, L. and Tirer, T. On temperature scaling and confor- mal prediction of deep classifiers. arXiv preprint arXiv: 2402.05806 ... | https://arxiv.org/abs/2502.03023v1 |
pp. 20810–20851. PMLR, 2023. Liu, J., Bai, M., Jiang, N., and Yu, D. Structural risk mini- mization of rough set-based classifier. Soft Computing , 24:2049–2066, 2020. Liu, K., Zeng, H., Huang, J., Zhuang, H., V ong, C. M., and Wei, H. C-adapter: Adapting deep classifiers for efficient conformal prediction sets. In The... | https://arxiv.org/abs/2502.03023v1 |
University Press, Cambridge, 1th edition, 2018. ISBN 978-1-108-41519-4. doi: 10.1017/9781108231596. V ovk, V . Conditional validity of inductive conformal predic- tors. In Proceedings of the Asian Conference on Machine Learning , pp. 475–490. PMLR, 2012. V ovk, V ., Gammerman, A., and Shafer, G. Algorithmic Learning in... | https://arxiv.org/abs/2502.03023v1 |
(NLL) to obtain the optimal parameter ℓNLL,p(λ) =−1 nX (x,y)∈D callog(pλ(y|x)), where pλ(y|x)is the transformed output probability from temperature scaling or vector scaling. The transformed score function Sλ(X, y)is a function of p(y|X)based on some base score function, such as APS (Romano et al., 2020) defined as: SA... | https://arxiv.org/abs/2502.03023v1 |
Ba, 2017) with a batch size of 256 and a learning rate of 0.1. The model is tuned for 10 epochs, and the only parameter, T, is set to 1×10−4by default. In our empirical study, we explore the application of C-Adapter using hold-out data. Conformal Training Conformal Training (ConfTr) (Stutz et al., 2022) is a training f... | https://arxiv.org/abs/2502.03023v1 |
are shown in Figures 5a and 5b, using APS and THR as the score functions, respectively. C. Some Useful Lemmas and Corollaries In this section, we provide some useful lemmas and corollaries that are used in the main text. Lemma C.1. (Dvoretzky et al., 1956; Massart, 1990, Dvoretzky–Kiefer–Wolfowitz Inequality) Let x1,x2... | https://arxiv.org/abs/2502.03023v1 |
500 1000 1500 2000 2500 Size of Calibration Set0.00.51.01.52.0CovGap (%) CIFAR-100 same hold-out 1000 2000 3000 4000 5000 Size of Calibration Set0.00.51.0CovGap (%) Image/glyph1197et same hold-out (a) RAPS 500 1000 1500 2000 2500 Size of Calibration Set0.51.01.52.0CovGap (%) CIFAR-10 same hold-out 500 1000 1500 2000 25... | https://arxiv.org/abs/2502.03023v1 |
third line is because when t≥a >0,1 t≤1 a. Lemma C.3. For a score function Sλwith one-dimensional parameter space Λ =R, consider the class: H={1{Sλ(x, y)≤t} |λ∈R, t∈R}. IfSλ(x, y)is continuous, bounded over λfor any fixed (x, y), then VC(H)≤2. Specifically, if Sλ(x, y)is distinct over X × Y for any fixed λ, then VC (H)... | https://arxiv.org/abs/2502.03023v1 |
cannot be achieved at a single λvalue. This phenomenon is because the continuity of Sλ(x, y)overRdensures that the ordering relationships between d+ 2points cannot be arbitrary with only ddegrees of freedom. Therefore, d+ 2points cannot be shattered, and VC(H)≤d+ 1. Lemma C.6 (Empirical Process Bound via VC Dimension) ... | https://arxiv.org/abs/2502.03023v1 |
u)du+Z∞ q log(2|Λ|) 2nP(RΛ,Dcal> u)du ≤r log(2|Λ|) 2n+Z∞ q log(2|Λ|) 2n2|Λ|exp(−2nu2)du =r log(2|Λ|) 2n+2|Λ|√ 2nZ∞ √ log(2|Λ|)exp(−t2)dt ≤r log(2|Λ|) 2n+2|Λ|√ 2n·1 2|Λ|p log(2|Λ|) =r log(2|Λ|) 2n+1√ 2np log(2|Λ|) where the second inequality is due to Lemma C.2. F. Proof of Corollary 4.3 and 4.4 Proof. It is a direct ap... | https://arxiv.org/abs/2502.03023v1 |
Ordinal Patterns Based Change Point Detection Annika Betken1,Giorgio Micali1, and Johannes Schmidt-Hieber1 a.betken@utwente.nl, g.micali@utwente.nl, a.j.schmidt-hieber@utwente.nl Abstract The ordinal patterns of a fixed number of consecutive values in a time series is the spatial ordering of these values. Counting how ... | https://arxiv.org/abs/2502.03099v1 |
For a comprehensive review of automated sleep stage classification methods see Zhang et al. (2024). However, these methods typically lack interpretability and theoretical justification, may inherit biases present in training data, and are often sensitive to small perturbations of the input; see Lipton (2018). To addres... | https://arxiv.org/abs/2502.03099v1 |
coefficients ajand i.i.d. random variables ( Zj)j∈Z. We also assume that E[Zj] = 0 and Var( Zj) =σ2 Zfor all j,implying thatXtis centered. By Kolmogorov’s three-series theorem, Xtexists almost surely ifP∞ j=0a2 j< 2 ∞; see Wu (2002). Since the innovations Zjare not assumed to be Gaussian, the process Xt can be non-Gaus... | https://arxiv.org/abs/2502.03099v1 |
, n +r−1. A common approach to estimate the probability p(u0, . . . , u r−1) := P(Xt≤u0, Xt+1≤u1, . . . , X t+r−1≤ur−1) is by using the relative frequency of this event in the sample, expressed as: bpn(u0, . . . , u r−1) :=1 nnX t=11 Xt≤u0, . . . , X t+r−1≤ur−1 . (3) By taking the expectation, we conclude that this i... | https://arxiv.org/abs/2502.03099v1 |
with variance σ2:= Var ( 1(X1≤u)) + 2P∞ j=1Cov (1(X1≤u),1(X1+j≤u)). In particular, for τ= 1, we obtain √n bpn(u)−p(u)D− → N (0, σ2). (8) The proof of Theorem 1 is provided in Appendix A. The key ingredient is to verify the conditions of a modified version of Theorem 10 in Furma´ nczyk (2007). We now discuss the case ... | https://arxiv.org/abs/2502.03099v1 |
the column vector ( ξt−ξt, ξt+1−ξt, . . . , ξ t+r−ξt)⊤.Multiplying this vector from the left with the ( r+ 1)×(r+ 1) permutation matrix that occurs in the middle of the matrix product that defines Vπ,reorders the entries and gives ( ξt+π0−ξt, ξt+π1−ξt, . . . , ξ t+πr−ξt)⊤.Thus, VπXt+1= (ξπ1−ξπ0, . . . , ξ πr−ξπr−1)⊤.He... | https://arxiv.org/abs/2502.03099v1 |
for ordinal pattern probabilities for a class of processes whose increments exhibit long-range dependence. Theorem 4 (Long-Range Dependence) .Let(ξt)t≥0be a time series whose increments Xt= ξt−ξt−1form a linear process Xt=P∞ j=0bjZt−jwith bjj→∞∼jd−1ford∈(0,1/2). IfZ1 admits a density fsuch that f∈L∞(R)∩C1(R)andf′∈L∞(R)... | https://arxiv.org/abs/2502.03099v1 |
mild conditions on the subordinated function gand the finite second moment of the innovations Z1, Furma´ nczyk concluded that1√nP⌊nτ⌋ j=1g(Xj) =⇒(B(τ))τ∈[0,1]. A similar setting was discussed in Wu (2002), who expanded on the results of Ho and Hsing (1997). In their Theorem 4.1, Ho and Hsing presented a central limit t... | https://arxiv.org/abs/2502.03099v1 |
pointwise multivariate reduction principle tailored to the ordinal patterns map. Importantly, our result does not hold uniformly but for each fixed point. This choice is motivated by the fact that, as discussed, the estimator for the ordinal pattern probabilities (13) is the empirical sum process evaluated at 0. This s... | https://arxiv.org/abs/2502.03099v1 |
+p(1,2,0) +p(1,0,2). Likewise, the probability to observe the all-raising pattern (0 ,1,2) or the all-falling pattern (2,1,0) is 1 −q. It follows that the permutation entropy PeEn = qlog4 q+ (1−q) log2 1−q only depends on q.The corresponding estimator of qfor an epoch ( ξt, . . . , ξ t+m+1) ofm+ 2 observations is ˆqm:=... | https://arxiv.org/abs/2502.03099v1 |
the mean directly corresponds to a change in the parameter ρ(1), signaling a transition to a different sleep stage. 3.1 Change-point detection via turning rate analysis In this section, we address the problem of testing whether a given time series ξ0, . . . , ξ n+1 exhibits stationarity in its increments X1, . . . , X ... | https://arxiv.org/abs/2502.03099v1 |
. . , X n+1forms a Gaussian time series, test (26) becomes equivalent to testing for a change in the autocorrelation parameter ρ(1), as shown in (22). In this context, a significant shift in the mean of the turning rate series directly corresponds to a change in ρ(1). For EEG time series, such a shift is indicative of ... | https://arxiv.org/abs/2502.03099v1 |
.4 to 0.7 after 2500 observations. The value SCnbis computed for both series. Figure 3 shows the distribution of the test statistics based on 1000 repetitions. Figure 3: Histogram of SCnbforn= 5000 and 1000 simulations of MA(1) without (left) and with (right) change of ρ(1). In the right plot, the autoregressive parame... | https://arxiv.org/abs/2502.03099v1 |
. 5 Discussion and outlook The main theoretical contribution of this work corresponds to the establishment of central limit theorems for relative frequencies in linear processes; see Section 2.1. An intriguing question is whether one can extend these limit theorems to estimators of the form1 nPn t=11(Xt∈A) for more gen... | https://arxiv.org/abs/2502.03099v1 |
Measures. John Wiley & Sons. Brockwell, P. and Davis, R. (2006). Introduction toTime Series andForecasting. Springer Texts in Statistics. Springer New York. Chambon, S., Lajnef, T., Thorey, Y., Koechlin, T., Bertrand, C., and Maquet, P. (2018). Deep learning methods for automatic classification of sleep stages in polys... | https://arxiv.org/abs/2502.03099v1 |
ordinal pattern based entropies. Communications inNonlinear Science andNumerical Simulation, 84:105–156. Parlitz, U., Berg, S., Luther, S., Schirdewan, A., Kurths, J., and Wessel, N. (2012). Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics. Computers in Biology andMedicine, 42(3):31... | https://arxiv.org/abs/2502.03099v1 |
the truncated linear process Xt,jasXt,j:=Pj i=0AiZt−i. Specifically, Xt,0= A0Zt, and Xt=Xt,∞. Further, let Rt,j=Xt−Xt,j=P∞ i=j+1AiZt−i. For fixed u= (u0, . . . , u r−1)⊤, pj(u) =P(Xt,j≤u), p(u) =P(X1≤u). (33) pjonly depends on jas (Xt,j)t≥1forms a stationary process. Moreover, we define a function g:Rr→Ras Lipschitz if... | https://arxiv.org/abs/2502.03099v1 |
Lemma 4. LetXt=P∞ j=0AjZt−jbe a multivariate linear process satisfyingP∞ j=0∥Aj∥<∞ and Assumption 1. Then, for Ut,jdefined in (34) (i)E[U2 t,0]≤P(X1≤u), (ii)E[U2 t,j]≤ C∥Aj+1∥2. Proof. (i) follows from expending the square, E (1(Xt≤u)−E[1(Xt≤u)|Ft−1])2 =P(Xt≤u)−E E[1(Xt≤u)|Ft−1]2 ≤P(X1≤u). To prove (ii), notice tha... | https://arxiv.org/abs/2502.03099v1 |
+i, i), E[U1+j,jU1+i,i] =E[U1,jU1+i−j,i], and σ2 u=u−1X i,j=0E[U1+j,jU1+i,i] =u−1X i,j=0E[U1,jU1+i−j,i]u→∞− − − →∞X i,j=0E[U1,jU1+i−j,i] =:σ2. Moreover, by decomposition (34), ( 1(Xt≤u)−p(u)) (1(Xt′≤u)−p(u)) =P∞ i,j=0Ut,jUt′,i, ∞X i,j=0E[U1,jU1+i−j,i] =X s∈Z∞X i,j=0E[U1,jU1+s,i] =X s∈ZCov (1(Xs≤u),1(X1+s≤u)). For condi... | https://arxiv.org/abs/2502.03099v1 |
function f. For r≥1, we define Xt= (Xt, . . . , X t+r−1)⊤. Iffadmits a bounded derivative, then (49) is satisfied for Xtwiths0:=r. Proof of Proposition 2. It is enough to show that for any x= (x0, . . . , x r−1)⊤, there exists a constant C>0 independent of xsuch that, for all j≥r |pj(x0, . . . , x r−1)| ≤ C ,|∇pj(x0, .... | https://arxiv.org/abs/2502.03099v1 |
. . , r −1. This follows since K(ℓ) ℓs a submatrix of B−1and since Bis a lower triagular Toeplitz matrix. It can be shown that B−1 is of the form B−1= a−1 0 ∗a−1 0......... ∗ ··· ∗ a−1 0 . Every submatrix obtained by removing rows and columns corresponding to the same indexes, meaning K−(i1,...,ih) −(i1,...,i... | https://arxiv.org/abs/2502.03099v1 |
(ii) Var T(3) n(x) =O(n) (iii) Var T(2) n(x) =( O(n) for d∈(0,1/4) O(n4d+ζ) for d∈[1/4,1/2). Proof for (i): Defining Kt,j(x) :=pj−1(x−Rt,j−1)−pj(x−Rt,j) +1{j≥r}∇pj−1(x−Rt,j)AjZt−j, we have T(1) N(x) =Pn t=1P∞ j=1Kt,j(x).From Lemma 8, E Kt,j(x)Kt′,j′(x) = 0 whenever j′̸=t′−t+j. For the product E Kt,j(x)Kt′,t′−t+j... | https://arxiv.org/abs/2502.03099v1 |
∥Aj′∥(jj′)d−1/2+ζ/2. 30 Taking into account that j′= (t′−t) +j, and that Aj∼jd−1A∞, we find Var(T(2) n)≤CnX t=1nX t′=t∞X j=1∥Aj∥ · ∥Aj′∥(jj′)d−1/2+ζ/2 ≤CnX t=1nX t′=t∞X j=1(j((t′−t) +j))d−1+ζ/2(j((t′−t) +j))d−1/2+ζ/2 ≤CnnX t=1∞X j=1(j((n−t) +j))d−1+ζ/2(j((n−t) +j))d−1/2+ζ/2 =Cnn−1X t=0∞X j=1(j(t+j))d−1+ζ/2(j(t+j))d−1/2... | https://arxiv.org/abs/2502.03099v1 |
F t−j⊂ ··· ⊂ F t−1⊂ F t. Consider Xt=P∞ j=0AjZt−j, where Aj∈Rr×r. Then, for all x∈Rrthe following hold (i)1({Xt≤x}) =P(Xt≤x|Ft), 32 (ii)P(Xt≤x)a.s= lim j→∞P(Xt≤x|Ft−j), (iii)P(Xt≤x|Ft−j) =pj(x−Rt,j). Proof. LetK−j=E[1({Xt≤x})|Ft−j] =P(Xt≤x|Ft−j) for all j∈N. (i)XtisFtmeasurable, and the previous theorem yields P(Xt≤x|F... | https://arxiv.org/abs/2502.03099v1 |
∇pj−1 x−Rt,j−1+λAj(Zt−j−t) Aj(Zt−j−t)dG(t), g′′(λ) =−1 2ZD ∇2pj−1 x−Rt,j−1+λAj(Zt−j−t) Aj(Zt−j−t), Aj(Zt−j−t)E dG(t). Moreover, since g∈C2([0,1]), the mean value theorem yields, g(1) = g(0) + g′(0) +1 2g′′(λ∗) for some value λ∗∈(0,1) yields pj−1(x−Rt,j−1)−pj(x−Rt,j) =0−Z ∇pj−1 x−Rt,j−1 Aj(Zt−j−t)dG(t) −1 2ZD ∇2pj... | https://arxiv.org/abs/2502.03099v1 |
increments of Theorem 4, then m n1/2+dnbX j=1(ˆqj,m−q)D−→ N (0, σ2)asnb→ ∞ (orn→ ∞ ). (59) where σ2=P γ1,γ2∈T(∇pγ2(0))⊤Vγ1V V⊤ γ2∇pγ1(0),with V=Γ(d)2 Γ(2d+2) cos( πd)EandVγthe matrices defined in (24). Proof of Theorem 9. We set h(Xt) :=P γ∈T1(VγXt≤0)−q . (i) Using (25), m(ˆqj,m−q) =m−1X i=0X γ∈T1 Π(ξ(j−1)(m+2)+ i, ξ(... | https://arxiv.org/abs/2502.03099v1 |
the first term n−(1 2+d)Pn j=1h(Xj) as a consequence of Theorem 8. For γ∈ T, letpγ(x) =P(VγX1≤x). Then, n−(1 2+d) nX j=1h(Xj)−q+X γ∈T((∇pγ(0)Vγ)n¯Xn =n−(1 2+d) nX j=1X γ∈T(1(VγXj≤0)−pγ(0)) +X γ∈T(∇pγ(0)Vγ)nX j=1Xj ≤X γ∈Tn−(1 2+d) nX j=11(VγXj≤0)−pγ(0) +∇pγ(0)VγnX j=1Xj P− →0. 39 The last relation is due to the reductio... | https://arxiv.org/abs/2502.03099v1 |
arXiv:2502.03174v1 [math.ST] 5 Feb 2025Robust Label Shift Quantification Alexandre Lecestre1 1MISTEA, INRAE, Montpellier, France February 6, 2025 Abstract In this paper, we investigate the label shift quantification problem. We propose robust estimators of the label distribution whic h turn out to coincide with the Maxim... | https://arxiv.org/abs/2502.03174v1 |
label distribution β∗∈ Wkare given by α∗:= (Ds(X×{i}))i∈[k]andβ∗:= (Dt(X×{i}))i∈[k], where Wk={x∈[0,1]k;x1+···+xk= 1} is the simplex. The simplex Wkis identified as the class of all probability distri- butionsover[ k]hereandintherestofthepaper. Theliteratureaddressesseve ral challenges within the context of label shift.... | https://arxiv.org/abs/2502.03174v1 |
Distri- bution Feature Matching (DFM) which is a general framework includin g KMM and BBSE. Dussap et al. (2023) also extend the classical framewor k to consider thecontaminated label shift setting where the covariate distribution of the test sample is of the form β∗ 0Q0+β∗ 1Q∗ 1+···+β∗ kQ∗ k, (1) withβ∗∈ Wk+1, modelli... | https://arxiv.org/abs/2502.03174v1 |
narei.i.d.with common distribu- tionP∗of the form P∗:=β∗ 1Q∗ 1+···+β∗ kQ∗ k, whereandβ∗∈ Wk. Moreover, Q∗ 1,...,Q∗ karelinearlyindependent in the space of signed measures on ( X,X). This assumption means that the observations are i.i.d. with a common distribution which can be written as a finite mixture. The linear inde... | https://arxiv.org/abs/2502.03174v1 |
of PX, andQbe the associated set of densities with respect to a σ-finite product measure on ( X,X), such that Q={q·dµ:q∈ Q}. We defineρ-estimators on Qas follows. We denote by ψthe function defined by ψ:[0,+∞]→[−1,1] x /ma√sto→x−1 x+1. (2) 5 Forx= (x1,...,xn)∈Xnandq,q′∈ Q, we define T(x,q,q′) :=n/summationdisplay i=1ψ/pare... | https://arxiv.org/abs/2502.03174v1 |
respect to Mmix(q1,...,q k). This resultisprovenin SectionA.1. Itimplies thatallthe resultswewillg ive for our estimator are also valid for the MLE ˆβMLE. One difference is that the MLE might not exist while ρ-estimators are always well-defined. For instance, we do not need to assume that the considered densities are bou... | https://arxiv.org/abs/2502.03174v1 |
k, such that for all β∈ Wkand allξ>0, C(Q)||β−ˆβ||2 1≤n−1n/summationdisplay j=1h2/parenleftBigg Pj,k/summationdisplay i=1βiQi/parenrightBigg +klogn+ξ n,(8) with probability at least 1−e−ξ. •LetQ1,...,Qkbe linearly independent distributions in PX. There is a positive constant C(Q)depending only on Q1,...,Qksuch that for... | https://arxiv.org/abs/2502.03174v1 |
we have C(Q)||β∗−ˆβ||2 1≤|I| n+klogn+ξ n, with probability at least 1 −e−ξ, for allξ >0. As long as the proportion of outliers|I|/nis small compared to n−1klogn, the performance of the estimator is still of the same order as in the ideal case without contamination. T hose last two cases are a good illustration of adver... | https://arxiv.org/abs/2502.03174v1 |
ributions is necessary in order to correctly define our estimator. Therefor e, we define the class of measures P∗(f,α) :=/braceleftbigg ν∈P(f,α);distributions f1·ν,...,f k·ν are linearly independent/bracerightbigg , and we make the following assumption. Assumption 4.1. The class of distributions P∗(f,α) is not empty. Not... | https://arxiv.org/abs/2502.03174v1 |
confusion matrixM(f)is invertible. If f=fα, this claim is equivalent to the linear independence of Q∗ 1,...,Q∗ k. The second claim is always satisfied for f=fα since α=M(fα)α. This result is proven in Section B.3. We have the following deviation in- equality. Corollary 4.7. Under Assumptions 2.1 and 4.5, for all ξ>0we h... | https://arxiv.org/abs/2502.03174v1 |
the set of all subsets of [ k]. Definition 4.8. Calibration Let (X,Y) be a couple of random variables with distribution Π ∈PX×Y. We say that a predictor fiscanonically calibrated, with respect to Π, if fi(X) =E[ /BDY=i|f(X)], for alli∈[k]. We say that a predictor fismarginally calibrated, with respect to Π, if fi(X) =E[... | https://arxiv.org/abs/2502.03174v1 |
terobustnessto outliers and contamination for the proposed method, which includes Maximum Likelihood Label Shift. Our findings support and extend the numeric al study of Saerens et al. (Section 4), confirming the robustness properties of MLLS and further strengthening the theoretical foundation for their use. This work c... | https://arxiv.org/abs/2502.03174v1 |
is proven in Section C.1. A.1 Proof of Proposition 3.1 It is a direct consequence of Corollary 1 of Baraud and Birg´ e (2018 ) since Mmix(q1,...,q k) :=/braceleftBiggk/summationdisplay i=1βiqi;β∈ Wk/bracerightBigg is a convex set of densities. A.2 Proof of Lemma 3.3 LetF1,...,F kbe distributions in PX. Letλbe aσ-finite ... | https://arxiv.org/abs/2502.03174v1 |
Σ= 0, where w/\e}atio\slash= 0 is given by wi=vi/αifor alli. •IfQ∗ 1,...,Q∗ kare linearly independent. Let v∈Rkbe such that 0 = v1fα 1·Q∗ Σ+···+vkfα k·Q∗ Σor equivalently 0 =k/summationdisplay i=1vifα i(x), forQ∗ Σ-almost all x. SincePα≪Q∗ Σwe have 0 =k/summationtext i=1vifα i(x) forPα- almost allxor equivalently 0 =k/... | https://arxiv.org/abs/2502.03174v1 |
we have DF/parenleftbig P,P/parenrightbig ≤91√ 2V/bracketleftBig 9.11+log+/parenleftBign V/parenrightBig/bracketrightBig . wherelog+(x) = max(0,logx)for allx>0andDFis theρ-dimension function introduced in Section C.2. ThisresultisproveninSectionC.2.1. FromLemma2.6.15invan der Vaart A.W. & Wellner J.A. (1996), the class... | https://arxiv.org/abs/2502.03174v1 |
rSRD: An Rpackage for the Sum of Ranking Differences statistical procedure Balázs R. Sziklai∗1,2, Attila Gere3, Károly Héberger4, and Jochen Staudacher5 1HUN-REN Centre for Economic and Regional Studies, Budapest, Hungary 2Department of Operations Research and Actuarial Sciences, Corvinus University of Budapest, Budape... | https://arxiv.org/abs/2502.03208v1 |
that is, tied values are replaced by the arithmetic mean of their corresponding ranks. After rank transformation, the distances between 2 the solutions and the reference are computed. SRD uses the L1-norm. We discuss the advantage of this distance metric as well as possible alternatives in Section 2.3. SRD values are n... | https://arxiv.org/abs/2502.03208v1 |
homepage offers a MATLAB code2. Attila Gere implemented a platform independent shiny application3. Finally, Klára Kollárné Hunek and Károly Héberger de- veloped an MS Excel macro4. While these existing packages are undoubt- edly valuable, they often lack thorough documentation, and none of them provide the comprehensiv... | https://arxiv.org/abs/2502.03208v1 |
1927) and Placket-Luce (Luce, 1959; Plackett, 1975). The Thurstone model was developed in experimental psychology to han- dle cases when subjects have to arrange a series of stimuli in absolute rank order according to the sensation it prompted. The distribution of sensations 5 from a particular stimulus is assumed to b... | https://arxiv.org/abs/2502.03208v1 |
suggests that L1-norm is a sensible choice, comparable to other commonly used distance metrics. Héberger and Škrbić found that SRD (which uses the L1-norm) is slightly stricter than Spearman rho and Kendall tau, that is, it rejects more models Héberger and Škrbić (2012). Sipos et al. on the other hand compared SRD, Ken... | https://arxiv.org/abs/2502.03208v1 |
purposes. The rSRD package implements two additional statistical tests: the Dietterich t-test (Dietterich, 1998) and Alpaydin’s F-test (Alpaydin, 1999), both popular cross-validation tools in machine learning. Sziklai et al. (2024) compared the performance of all three tests on synthetic and real data under various par... | https://arxiv.org/abs/2502.03208v1 |
their corresponding ranks. 9 For instance, if mij=mkjare tied for the 6th and 7th places, they are each replaced by a rank of 6.5. Let us denote the rank-transformed matrix by R= [rij]. The SRD score of the jth solution is the distance between the jth column and the reference column in L1-norm SRD j=n/summationdisplay ... | https://arxiv.org/abs/2502.03208v1 |
a large absolute difference in item scores, they might still rank items in the exact same way. Discordant tastes will result in different rankings. If the tastes have no relation to each other, we expect the corresponding ranking to show no pattern. Therefore, the distance between the solution and the reference ranking... | https://arxiv.org/abs/2502.03208v1 |
very close SRD scores are not necessarily similar to each other. Rankings can differ from the reference in different sections and still be of the same distance. Cross-Validation combined with Statistical Testing (CVST) is designed to determine whether two solutions are inher- ently the same or not. Consequently, the nu... | https://arxiv.org/abs/2502.03208v1 |
belonging to the same ideological family. One way to identify potential allies is to analyze the language they are using. In the following ex- ample, the texts of the amendments of the Committee on Industry, Research and Energy (ITRE) are analyzed in the 2014-2019 legislative term to profile MEPs. Expressions are class... | https://arxiv.org/abs/2502.03208v1 |
generating the SRD distribution, Buzek’s profile would have fallen to the right of the threshold6. Second, MEP Kaljurand’s profile is so different from Rego’s, that it cannot even be considered random, but rather Rego’s exact opposite. Reverse orderings are always revelatory, in our case it probably indicates that Kalj... | https://arxiv.org/abs/2502.03208v1 |
pg Fouls pg pts Bayern19.8 38 64.8 86 14.5 2.2 9 77Muenchen Bayer13.5 66 53.7 81.8 11.8 1.9 10.7 64Leverkusen Borussia13.3 62 59.4 84 10.4 2.1 10.6 69Dortmund RB Leipzig 12.9 49 56.5 83.1 10.2 1.9 10.6 58 SC Freiburg 13.6 34 48.6 76.2 6.6 1.7 11.5 55 Borussia14.8 67 54.1 82 10.9 2.2 10.6 45M.Gladbach 1. FC Koeln 13.8 6... | https://arxiv.org/abs/2502.03208v1 |
games. Even if the objects are on the same scale, it might be beneficial to standard- ize the values. For instance, in the MEP profiles, not every representative was equally productive. MEPs who submitted more amendments inevitably generated more phrases in each category, thus the absolute numbers are a bit misleading.... | https://arxiv.org/abs/2502.03208v1 |
not change the input, only returns with a new matrix. rSRD provides a function for detailed SRD calculation, displaying the rankingtransformation, thedistancecalculation, andtheraw(unnormalized) SRD scores. R> SRD_input = utilsCreateReference(SRD_input, method = "mixed", ref) R> utilsDetailedSRD(SRD_input) A A_Rank A_D... | https://arxiv.org/abs/2502.03208v1 |
vector is fixed. ’r’Therearenoties. Boththecolumnvectorandthereferencearegenerated randomly. ’t’Ties occur with a fixed probability specified by the user for both the solution vectors and the reference vector. 23 ’p’Ties occur with a fixed probability specified by the user for the solution vectors, the reference vector... | https://arxiv.org/abs/2502.03208v1 |
resent the underlying solution space. Suppose there are two solutions with tie frequencies xandy, wherex << y. Then we may consider generating a solution using the fixed tie probabilityx+y 2(options ’t’ or ’p’). However, if the distance between xandyis too large, there is a chance that none of the generated vectors dis... | https://arxiv.org/abs/2502.03208v1 |
setting the output_to_file parameter to false. Some information is only saved to the CSV file, the function does not return the number and indices of the removed rows. Sec- ondly, thefunctionorderssolutionbasedonthemedianofthecomputedSRD scores. Thus, RY cards, which was the second column of our dataframe, was relayed ... | https://arxiv.org/abs/2502.03208v1 |
plotHeatmapSRD(profiles_df, output_to_file = TRUE, color = utilsColorPalette) The color palette can be easily customized. The size of the palette in- dicates how many categories the [0,1]interval is divided. For instance, the following code changes the color range from orange to green. R> myPalette <- c("#eb9c34", "#eb... | https://arxiv.org/abs/2502.03208v1 |
a seed parameter into the cross-validation function would make the package more convenient. Acknowledgement This work was supported by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the K type funding scheme, K 134260 (Héberger), K 146320... | https://arxiv.org/abs/2502.03208v1 |
Learning: Data Mining, Inference, and Prediction (2nd ed.). Springer Series in Statistics. Springer New York. Héberger, K. (2010, January). Sum of ranking differences compares methods or models fairly. TrAC Trends in Analytical Chemistry 29 (1), 101–109. Héberger, K. and B. Škrbić (2012). Ranking and similarity for qua... | https://arxiv.org/abs/2502.03208v1 |
two things. The American Journal of Psychology 15 (1), 72–101. Sziklai, B. R., M. Baranyi, and K. Héberger (2024, Aug). Does cross- validation work in telling rankings apart? Central European Journal of Operations Research . Sziklai, B. R., P. Biró, and L. Csató (2022). The efficacy of tournament designs. Computers & O... | https://arxiv.org/abs/2502.03208v1 |
CARROT: A Cost Aware Rate Optimal Router Seamus Somerstep†□Felipe Maia Polo†□Allysson Flavio Melo de Oliveira‡◦ Prattyush Mangal‡Mírian Silva‡◦△Onkar Bhardwaj‡◦ Mikhail Yurochkin *‡◦Subha Maity *♠ □Department of Statistics, University of Michigan ‡IBM Research◦MIT-IBM Watson AI Lab △Federal University of Minas Gerais ♠... | https://arxiv.org/abs/2502.03261v2 |
model cost by creating binary routers that select between a large, costly model and a cheap, small model. However, they do not predict the cost of individual queries and, as we shall see, the reduced flexibility of binary routing leads to performance degradation in practice. A recent work, Hu et al. [2024], introduces ... | https://arxiv.org/abs/2502.03261v2 |
we use zero- shot prompting and corresponding chat templates to represent practical use cases and collect input and output token counts to allow flexibility when studying cost-performance trade-offs. The importance of a carefully curated dataset when studying routing cannot be overstated. In fact, on prior datasets suc... | https://arxiv.org/abs/2502.03261v2 |
works implicitly assume that the models in the ensemble have similar expertise , and thus it is beneficial to aggregate their predictions, whereas in our case, models may have complementary expertise , and averaging their outputs might be detrimental because most of them may not be suitable for an input. Therefore, we ... | https://arxiv.org/abs/2502.03261v2 |
to RP(g, µ)to estimate the oracle router at a particular µ, this approach is not scalable, any small change in µwould require refitting a new router. Given this, we develop a plug-in approach which lets us estimate the oracle routers at every value of µ. The key intuition lies within an explicit form of the g⋆ µthat we... | https://arxiv.org/abs/2502.03261v2 |
is a standard assumption in minimax investigations for non-parametric classification problems [Audibert and Tsybakov, 2007, Cai and Wei, 2019, Kpotufe and Martinet, 2018, Maity et al., 2022]. Next, we place Hölder smoothness conditions on the functions Φ⋆ m. This controls the difficulty of their estimation. For a tuple... | https://arxiv.org/abs/2502.03261v2 |
the actual risk RP(µ, g), we establish a lower bound on the excess risk: EP(µ, g) =RP(µ, g)− R P(µ, g⋆ µ), (3.6) that compares the risk of a proposed router to the oracle one. We denote Γ ={g:X → [M]}as the class of all routers. For an n∈Nwe refer to the map An:Zn→Γ, which takes the dataset Dn as an input and produces ... | https://arxiv.org/abs/2502.03261v2 |
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