Checkpoint allocation and release Out-of-order speculative processors need a bookkeeping method to recover from incorrect speculation. In recent years, several microarchitectures that employ checkpoints have been proposed, either extending the reorder buffer or entirely replacing it. This work presents an in-dept-study of checkpointing in checkpoint-based microarchitectures, from the desired content of a checkpoint, via implementation trade-offs, and to checkpoint allocation and release policies. A major contribution of the article is a novel adaptive checkpoint allocation policy that outperforms known policies. The adaptive policy controls checkpoint allocation according to dynamic events, such as second-level cache misses and rollback history. It achieves 6.8% and 2.2% speedup for the integer and floating point benchmarks, respectively, and does not require a branch confidence estimator. The results show that the proposed adaptive policy achieves most of the potential of an oracle policy whose performance improvement is 9.8% and 3.9% for the integer and floating point benchmarks, respectively. We exploit known techniques for saving leakage power by adapting and applying them to checkpoint-based microarchitectures. The proposed applications combine to reduce the leakage power of the register file to about one half of its original value. Modern out-of-order (OOO) processor cores concurrently process many instructions in different stages of their pipelines. Such processors require a mechanism to recover the processor state after the occurrence of an incorrect speculation. There are two well-known approaches for recovering the processor state: the reorder buffer (ROB) [Smith and Pleszkun 1988] and checkpointing [Hwu and Patt 1987]. In recent years, several microarchitectures that employ checkpoints have been proposed, either extending the reorder buffer ("hybrid" microarchitectures that use both recovery approaches, ROB and checkpointing) or entirely replacing the ROB with checkpointing (ROB-free).. Figure 1 presents several key microarchitectures that use checkpoints and maps four sets of goals that checkpoints help accomplish. Reliability hybrids use checkpoints to recover from errors. Resource reclamation hybrids speculatively reuse resources, such as registers and load/store queue entries, and use checkpoints to recover from misspeculations. Memory wall hybrids employ checkpoints to handle long latency events, such as L2 cache misses, by either skipping over the load that caused the miss or predicting the load value. Faster recovery hybrids use checkpoints to recover faster from branch mispredictions without waiting until the mispredicted branch reaches the head of the ROB. These hybrid microarchitectures are further described in the related work section (Section 7). This article focuses on ROB-free microarchitectures in which checkpoints are used extensively for recovering from mispredictions.. A recovery is comprised of two phases: rollback, in which the latest safe check-point preceding the point of misprediction is recovered, and re-execution in which the code segment between the recovered checkpoint and the mispredicting instruction is executed again. Checkpoint-based microarchitectures apply a checkpoint allocation policy-a heuristic algorithm, indicating when the processor state should be saved. Taking a checkpoint consumes two resources, a checkpoint buffer entry and registers that are part of the checkpoint's mapping table. A checkpoint allocation policy should be balanced, taking too few checkpoints will require larger code segments to re-execute on recovery and taking too many checkpoints may consume all available resources bringing the processor to a temporary halt. Easing the pressure on such resources can also be achieved by releasing unneeded checkpoints as soon as they are no longer required (OOO checkpoint release management).. The primary objective of this article is to present a detailed study of checkpoint allocation and release in a checkpoint-based microarchitecture. Specifically, the article makes the following contributions.. (1) It presents a novel adaptive checkpoint allocation policy in which the policy changes according to dynamic events, such as second-level cache misses, rollback history, and the utilization of the register file and the checkpoint buffer. It also compares the performance of the new policy with two previously proposed checkpoint allocation policies. (2) It evaluates the effectiveness of OOO checkpoint release and presents a streamlined scheme that releases OOO only checkpointed registers but does not require the complex OOO checkpoint release logic. (3) It evaluates whether the state of predictors, following wrong-path execution, should be restored, reset, or left as is after rollback. (4) It introduces two new applications of known techniques for saving leakage power. First, it applies a combination of two techniques-early register release and turning free registers off-to the register management mechanism in checkpoint-based microarchitectures. Second, it applies the technique of drowsy caches to checkpointed registers.. The remainder of the article is organized as follows. Section 2 describes the simulation methodology. Section 3 examines the checkpoint buffer organization and studies the benefits of including the state of different predictors in the checkpointed state. Section 4 shows that previously proposed checkpoint allocation policies are suboptimal and introduces an adaptive policy that achieves better results. Section 5 investigates trade-offs involving in-order and OOO checkpoint release. Section 6 presents a technique that reduces the leakage power of checkpointed and released registers. Finally, related work is surveyed in Section 7, and conclusions are drawn in Section 8. Beginning with a highly efficient ROB-free microarchitecture, we have investigated checkpoint buffer organization trade-offs, checkpoint allocation and release policies, and ways to exploit specific features of checkpoint-based microarchitectures to reduce the leakage power of the register file. The motivation for the first part of this research was provided by results showing that an oracle checkpoint allocation policy performs significantly better than the two previously proposed policies, which are based on setting a minimal distance between consecutive checkpoints (MinDist) and allocating them at hard-to-predict branches (LowConfBr). For the integer benchmarks, the oracle policy performs 20.7% and 9.8% better than the MinDist and LowConfBr policies, respectively. A smaller performance gap also exists for the floating-point benchmarks.. We have presented a novel adaptive checkpoint allocation policy, which achieves most of the speed-up potential shown by the oracle policy. The adaptive policy changes according to dynamic events, such as second-level cache misses and rollback history. The adaptive policy is based on the observation that checkpoint allocation policies should only maintain large distances between consecutive checkpoints, either when a very large instruction window is beneficial or when the registers are overutilized and are not reclaimed efficiently due to excessive use of checkpoints. The adaptive checkpoint allocation policy achieves 6.8% and 2.2% speed-up for integer and floating point benchmarks, respectively. The adaptive policy does not require a branch confidence estimator and maintains its benefit for various checkpoint buffer depths and memory-access latencies.. Checkpoints provide an efficient infrastructure for restoring the state of a predictor. Efficiency derives from having a limited set of known points in the program, which the processor can rollback to, regardless of the size of the instruction window. Speculatively updated predictors become contaminated by wrong-path instructions. We have evaluated whether the state of the return address stack (RAS), branch predictor, and confidence estimator should be restored or reset during rollback. Results vary depending on the predictor type. Restoring the index and values of the RAS always boosts performance (4.6% speedup for the integer benchmarks), while restoring the global history of the branch predictor leads to mixed results. As for the confidence estimator, which is used only in the LowConfBr policy, we have found that resetting the global history shift register is better than either restoring it or leaving it as is.. Finally, we have shown that checkpoint-based microarchitectures present opportunities for reducing the leakage power of the register file. A substantial amount of leakage power is saved by releasing registers early and turning them off. Additional power is saved by placing checkpointed registers in drowsy mode. These two techniques add up to save about one half of the leakage power of the register file. Most of this section surveys microarchitectures that use checkpoints, with emphasis on their goal, the way the checkpoints are allocated, used, and released. At the end of this section, we discuss previous work on reducing the leakage power consumption of register files. We begin our survey at the left side of Figure 1 . In IBM's Power6 processor [McGhan 2006 ] certain parts of the pipeline and control are protected using parity and fault screening methods [Racunas et al 2007] . Every cycle, a new checkpoint is created so that the processor state can be restored if a soft or hard error is detected. The checkpoint-based recovery technique is inspired by highly reliable mainframe systems [Meaney et al. 2005] . Cherry [Martinez et al. 2002; ] is a resource-reclamation ROB-checkpoint hybrid that employs a single checkpoint to speculatively reuse resources, such as registers and load/store queue entries, prior to committing the instruction they were allocated to. Cherry is based on a conservative approach to reuse. For example, it does not initiate a reuse beyond unresolved branches. A less conservative recycling approach was studied by Ergin et al. [2006] , who proposed an early register reclamation mechanism implemented using a "checkpointed register file." In a checkpointed register file, every register has a shadow register that can be used to back-up values that may be needed in case of misspeculation. The shadow registers are used in conjunction with the ROB to restore the processor state.. High-frequency processors require a high number of cycles to access the main memory [Wilkes 2001] . Load instructions that miss the cache hierarchy, and their data-dependent instructions clog the ROB, which causes the processor to halt until the miss is resolved. Memory wall hybrids divide into two major categories, depending on the action they take to unclog the ROB. Mutlu et al. [2003] introduced the RunAhead microarchitecture that unclogs the ROB by skipping the long latency load and its data-dependent instructions. RunAhead has two modes: regular and prefetch. It switches to the prefetch mode on a L2 cache miss and returns to the regular mode when that load is resolved. In its basic form, RunAhead does not use checkpoints for speculation, the checkpoint taken on switching from regular to prefetch mode is always restored, flushing all the results calculated during the prefetch mode. The goal of the prefetch mode is to train the caches and branch predictor. Ramirez et al. [2006] compared the runahead and checkpoint-based microarchitectures and found that the latter achieves significantly better performance on a lower power budget.. Memory wall hybrids that belong to the second category unclog the ROB by employing a load value predictor [Lipasti and Shen 1996] . and Ceze et al. [2006] predict the load value upon L2 misses, use 4-and 1-deep checkpoint buffers, respectively, and perform a rollback for load value mispredictions.. Many modern processors have one or two mapping table instances, a speculative copy, used at the rename stage and optionally a nonspeculative copy, updated at the commit stage. These mapping table instances are in sync with the ROB's tail and head pointers, respectively. On recovery, the ROB's tail is adjusted and a new speculative mapping table is required. One way to achieve this is to delay the recovery until the mispredicted instruction reaches the commit stage. Better performance is achieved, however, by initiating the recovery as soon as the misprediction is detected. The new speculative mapping table is created in two phases: (i) one of the two existing table instances is copied and then (ii) it is adjusted by traversing all ROB entries from the reference point to the point of misprediction. Hybrid processors periodically allocate checkpoints, and have more than two mapping table instances that can be used as reference points. This reduces the average number of ROB entries to traverse, which in turn reduces the recovery time.. A faster recovery hybrid approach was implemented in two processors: the MIPS R10000 [Yeager 1996 ] and the Alpha 21264 [Kessler 1999 ]. These processors allocate a checkpoint at every branch. Such aggressive checkpoint allocation policies avoid the need to traverse the ROB but do not scale to large instruction windows. To solve this scalability limitation, Moshovos [2003] describes a more economic checkpoint allocation policy that takes checkpoints only at hardto-predict branches. Following this direction, Akl and Moshovos [2006] presented BranchTap, a microarchitecture that avoids further speculation when the newly fetched instructions are at high risk of being on the wrong path. The adaptive pipeline gating decreases the mean instruction window, so the limited number of checkpoints available is better dispersed and the number of ROBentries that have to be traversed following a rollback is reduced. The authors have also evaluated OOO checkpoint release, reaching similar conclusion to ours.. Checkpoint allocation policies used by ROB-free and faster recovery hybrids share a similar purpose-to reduce the number of instructions that need to be processed following a misprediction. Figure 17 illustrates the recovery process such microarchitectures perform following the detection of the same branch misprediction (BEQ i ). The hybrid performs a rollback to BEQ i and updates the mapping table starting at either CP 2 or CP 3 and adjusting it by traversing Inst g to BEQ i or Inst m to BEQ i . ROB-free processors on the other hand, perform a rollback to Inst g but can immediately reexecute from there. In both cases, the overall recovery penalty depends on the number of instructions that need to be processed (traversed or re-executed). There are also significant differences. Checkpoints in faster recovery hybrids affect neither the instruction window size nor the efficiency of register reclamation. On the other hand, checkpoints in ROB-free processors are an integral part of the bookkeeping method. Allocating them consumes checkpointed registers and can reduce the size of the instruction window. Furthermore, instructions processed through the pipeline carry the tag of the checkpoint they belong to. Releasing checkpoints OOO requires setting new values to these tags.. Reaching ROB-free microarchitectures on the right side of Figure 1 , Hwu and Patt [1987] were the first to propose the use of checkpoints to recover the processor state following a misspeculation. Two main microarchitectures stand at the basis of recent work on coarse-grained bookkeeping: the Kilo-instruction processor [Cristal et al. 2004] and CPR . Both microarchitectures are resource efficient and are designed for an order-of-magnitude larger instruction window. A larger instruction window enables the execution of more instructions, while an L2 miss is processed and better exploits the available ILP. The Kilo-instruction microarchitecture [Cristal et al. 2004 ] has a static checkpoint allocation policy that attempts to balance the depth of the required checkpoint buffer, the re-execution penalty, and the store queue utilization. It allocates a checkpoint at the following points in the program: At the first branch after 64 instructions, at the 64th store instruction and when the number of instructions after the last checkpoint exceeds 512. The MinDist policy, which is used in this article for the purpose of comparison and evaluation of the proposed adaptive policy, is conceptually close to the Kilo-instruction microarchitecture. The CPR microarchitecture ] uses a dedicated branch prediction confidence estimator. CPR allocates a checkpoint at the following points in the program: at a hard-to-predict branch (a branch predicted with a low-confidence estimation level), at the first branch after a rollback, and when the number of instructions after the last checkpoint exceeds 256. The same checkpoint allocation policy approach was used in Srinivasan et al. [2004] and Gandhi et al. [2005] . The baseline microarchitecture used in this article (LowConfBr), also uses a confidence estimator for allocating checkpoints.. Changing the supply voltage of registers in order to reduce the leakage power consumption has been an active research topic in recent years. Ayala et al. [2003] place all registers in a drowsy state by default, awaking registers as by demand prior to the read or write cycle. Their baseline model is an in-order processor using an architectural rather than unified register file. Khasawneh and Ghose [2005] manage the supply voltage, at a coarse-grained level (register zones). One of their proposed schemes also works by waking up drowsy registers on demand. A unified register file is used, thus an extra cycle delay between the scheduler select and register read is required in order to switch to the regular supply voltage. The authors assume that cycle is hidden because they assume the scheduler selects the instruction more than a cycle before it reads its operands.. A different approach is taken by Shieh and Hsu [2006] , who use register access frequency information, obtained by the compiler, to assign infrequently used registers to infrequently used register banks, which are placed in drowsy state. The downside is that high-traffic registers are mapped into the same banks, which may lead to port collisions and performance slowdown. The work closest to the leakage power reduction scheme presented in this article is the work of Goto and Sato [2004] . They switch registers to the drowsy state after they are committed and until they are allocated, provided that they are no longer part of the mapping table. These are the registers that are considered free in a ROB-based microarchitecture with no early register release mechanisms. Our scheme is different in several ways: (i) it has greater potential for reduction (owing to early release of registers); (ii) it has minimal overhead logic for managing the state per register; (iii) it reduces the leakage between the bit lines and the ground line, which in a multiported structure such as the register file often accounts for a large proportion of the leakage power; and (iv) it also reduces the power consumed by checkpointed registers.

B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O 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Decidability of Branching Bisimulation on Normed Commutative Context-Free Processes We investigate normed commutative context-free processes (Basic Parallel Processes). We show that branching bisimilarity admits the bounded response property: in the Bisimulation Game, Duplicator always has a response leading to a process of size linearly bounded with respect to the Spoiler's process. The linear bound is effective, which leads to decidability of branching bisimilarity. For weak bisimilarity, we are able merely to show existence of some linear bound, which is not sufficient for decidability. We conjecture however that the same effective bound holds for weak bisimilarity as well. We suppose that further elaboration of novel techniques developed in this paper may be sufficient to demonstrate decidability.

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Real-time communication over switched ethernet for military applications Full-Duplex Switched Ethernet is a forecasted new technology for advanced military aircraft system interconnection. However, it was not originally developed to meet the requirements of real-time communications. Therefore, in this paper, we analyze traffic shaping and a priority handling approach over switched Ethernet to achieve reliable transmission with bounded delays that conform to the real-time constraints, required by military applications.Categories and Subject DescriptorsComputer communication networks-local and wide area networksGeneral TermsExperimentationFull-Duplex Switched Ethernet, traffic shaping, priority handling, delay boundsThe traditional communication buses for military avionics systems, that have been widely used in various military aircraft for decades, are MIL-STD 1553B buses. These are 1Mbps command/response multiplexer data buses with a centralized system control. Nowadays, the complexity of avionics systems still increases at a high rate and has been pushing the limits of the command/response architecture. In fact, a homogeneous interconnection system is needed to meet new requirements, such as higher transmission speed, decentralization of control, easy maintenance and low cost. So, we analyze the possibility of using general Full-Duplex Switched Ethernet, specially after the successful civil experience of A380's AFDX (Avionics Full-Duplex Switched Ethernet).Copyright is held by the author/owner. This technology is not originally designed to deliver timeconstrained communications, but some of its properties, such as its reduced cost, its flexibility and its expandability for future investment, make this COTS network an attractive candidate to replace the MIL-STD 1553B and the other data buses used in the military aircrafts. Despite its popularity, Full-Duplex Switched Ethernet has a serious drawback when carrying real-time messages. The Full-Duplex Switched Ethernet LAN does not use CSMA/CD arbitration since no collision can happen but, unfortunately, congestion may occur when simultaneous traffic data attempt to share network segments and messages can be lost if buffers overflow. Thus, it is impossible to guarantee predictable delays in delivering packets to the local nodes in the general case and we have to add new mechanisms to avoid information loss and to guarantee hard real-time communication with low latency. To achieve real-time behavior on Ethernet, several approaches and techniques have been experienced by adding a medium access control, for example:• Token-passing methods that consist in circulating a token among the stations to allow them to transmit data; • TDMA where stations transmit messages at predetermined slots in a cyclic fashion with a precise clock synchronization which has been successfully implemented on CAN (TTCAN [4] ); • FTT-Ethernet [3] where a central master periodically distributes a group of tokens that allows slaves to send data for a specific amount of timeIn this paper, we show the possibility of having a reliable transmission with bounded delays over Switched Ethernet, that conform to the real-time constraints of military application, by installing traffic shapers in each local node and by providing a prioritized service to real-time packets. The rest of the paper is organized as follows: Section 2 describes our traffic shaping approach and the priority handling model to provide real-time guarantees over Full-Duplex Switched Ethernet, and evaluates experimental results on a real case traffic. In Section 3, we give an outlook of our future work.Prior to defining our approach, military applications requirements and measures of performance must be identified. A main concern for designing military local-area networks is 266 In this paper, we have shown that traffic shaping and priorities handling can be used to achieve reliable packet transmission with bounded delays that conform to the realtime constraints required by military applications. Future work will target other QoS guarantees, like jitter which is inherently low on 1553B applications. The traditional communication buses for military avionics systems, that have been widely used in various military aircraft for decades, are MIL-STD 1553B buses. These are 1Mbps command/response multiplexer data buses with a centralized system control. Nowadays, the complexity of avionics systems still increases at a high rate and has been pushing the limits of the command/response architecture. In fact, a homogeneous interconnection system is needed to meet new requirements, such as higher transmission speed, decentralization of control, easy maintenance and low cost. So, we analyze the possibility of using general Full-Duplex Switched Ethernet, specially after the successful civil experience of A380's AFDX (Avionics Full-Duplex Switched Ethernet).. This technology is not originally designed to deliver timeconstrained communications, but some of its properties, such as its reduced cost, its flexibility and its expandability for future investment, make this COTS network an attractive candidate to replace the MIL-STD 1553B and the other data buses used in the military aircrafts. Despite its popularity, Full-Duplex Switched Ethernet has a serious drawback when carrying real-time messages. The Full-Duplex Switched Ethernet LAN does not use CSMA/CD arbitration since no collision can happen but, unfortunately, congestion may occur when simultaneous traffic data attempt to share network segments and messages can be lost if buffers overflow. Thus, it is impossible to guarantee predictable delays in delivering packets to the local nodes in the general case and we have to add new mechanisms to avoid information loss and to guarantee hard real-time communication with low latency. To achieve real-time behavior on Ethernet, several approaches and techniques have been experienced by adding a medium access control, for example:. • Token-passing methods that consist in circulating a token among the stations to allow them to transmit data; • TDMA where stations transmit messages at predetermined slots in a cyclic fashion with a precise clock synchronization which has been successfully implemented on CAN (TTCAN [4] ); • FTT-Ethernet [3] where a central master periodically distributes a group of tokens that allows slaves to send data for a specific amount of time. In this paper, we show the possibility of having a reliable transmission with bounded delays over Switched Ethernet, that conform to the real-time constraints of military application, by installing traffic shapers in each local node and by providing a prioritized service to real-time packets. The rest of the paper is organized as follows: Section 2 describes our traffic shaping approach and the priority handling model to provide real-time guarantees over Full-Duplex Switched Ethernet, and evaluates experimental results on a real case traffic. In Section 3, we give an outlook of our future work.

O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O B I I O O O O O O O O O O O O O O O O O O O O


Generalized

structured

additive

regression

based

on

Bayesian

P-splines

Generalized

additive

models

(GAM)

for

modeling

nonlinear

effects

of

continuous

covariates

are

now

well

established

tools

for

the

applied

statistician.

A

Bayesian

version

of

GAM's

and

extensions

to

generalized

structured

additive

regression

(STAR)

are

developed.

One

or

two

dimensional

P-splines

are

used

as

the

main

building

block.

Inference

relies

on

Markov

chain

Monte

Carlo

(MCMC)

simulation

techniques,

and

is

either

based

on

iteratively

weighted

least

squares

(IWLS)

proposals

or

on

latent

utility

representations

of

(multi)categorical

regression

models.

The

approach

covers

the

most

common

univariate

response

distributions,

e.g.

the

binomial,

Poisson

or

gamma

distribution,

as

well

as

multicategorical

responses.

For

the

first

time,

Bayesian

semiparametric

inference

for

the

widely

used

multinomial

logit

model

is

presented.

Two

applications

on

the

forest

health

status

of

trees

and

a

space-time

analysis

of

health

insurance

data

demonstrate

the

potential

of

the

approach

for

realistic

modeling

of

complex

problems.

Software

for

the

methodology

is

provided

within

the

public

domain

package

BayesX.

Generalized

additive

models

(GAM)

provide

a

powerful

class

of

models

for

modeling

nonlinear

effects

of

continuous

covariates

in

regression

models

with

non-Gaussian

responses.

A

considerable

number

of

competing

approaches

is

now

available

for

modeling

and

estimating

nonlinear

functions

of

continuous

covariates.

Prominent

examples

are

smoothing

splines

(e.g.

Hastie

and

Tibshirani,

local

polynomials

(e.g.

Fan

and

Gijbels,

regression

splines

with

adaptive

knot

selection

(e.g.

Friedman

and

Silverman,

Friedman,

Stone,

Hansen,

Kooperberg

and

Truong,

and

P-splines

(Eilers

and

Marx,

Marx

and

Eilers,

Currently,

smoothing

based

on

mixed

model

representations

of

GAM's

and

extensions

is

extremely

popular,

see

e.g.

Lin

and

Zhang

Currie

and

Durban

Wand

and

the

book

by

Ruppert,

Wand

and

Carroll

Indeed,

the

approach

is

very

promising

and

has

several

distinct

advantages,

e.g.

smoothing

parameters

can

be

estimated

simultaneously

with

the

regression

functions..

Bayesian

approaches

are

currently

either

based

on

regression

splines

with

adaptive

knot

selection

(e.g.

Smith

and

Kohn,

Denison,

Mallick

and

Smith,

Biller,

Di

Matteo,

Genovese

and

Kass,

Biller

and

Fahrmeir,

Hansen

and

Kooperberg,

or

on

smoothness

priors

(Hastie

and

Tibshirani

and

own

work).

Inference

is

based

on

Markov

chain

Monte

Carlo

inference

techniques.

A

nice

introduction

into

MCMC

can

be

found

in

Green

This

paper

can

be

seen

as

the

final

in

a

series

of

articles

on

Bayesian

semiparametric

regression

based

on

smoothness

priors

and

MCMC

simulation

techniques,

see

particularly

Fahrmeir

and

Lang

Fahrmeir

and

Lang

and

Lang

and

Brezger

A

very

general

class

of

models

with

structured

additive

predictor

(STAR)

is

proposed

and

MCMC

algorithms

for

posterior

inference

are

developed.

STAR

models

include

a

number

of

model

classes

well

known

from

the

literature

as

special

cases.

Examples

are

generalized

additive

mixed

models

(e.g.

Lin

and

Zhang,

geoadditive

models

(Kammann

and

Wand,

varying

coefficient

models

(Hastie

and

Tibshirani,

or

geographically

weighted

regression

(Fotheringham,

Brunsdon

and

Charlton,

The

approach

covers

the

most

common

univariate

response

distributions

(Gaussian,

binomial,

Poisson,

gamma)

as

well

as

models

for

multicategorical

responses..

In

a

first

paper

Fahrmeir

and

Lang

develop

univariate

Generalized

additive

mixed

models

based

on

random

walk

and

Markov

random

field

priors.

Inference

is

based

on

MCMC

simulation

based

on

conditional

prior

proposals

suggested

by

Knorr-Held

in

the

context

of

dynamic

models.

Fahrmeir

and

Lang

extend

the

methodology

to

models

with

multicategorical

re-sponses.

As

an

alternative

to

conditional

prior

proposals,

more

efficient

MCMC

techniques

that

rely

on

latent

utility

representations

are

proposed,

see

also

Albert

and

Chib

Chen

and

Dey

and

Holmes

and

Held

In

Lang

and

Brezger

Bayesian

versions

of

P-splines

for

modeling

nonlinear

effects

of

continuous

covariates

and

time

scales

are

proposed.

Additionally,

two

dimensional

P-splines

for

modeling

interactions

between

continuous

covariates

are

developed.

P-splines

usually

perform

better

in

terms

of

bias

and

mean

squared

error

than

the

simple

random

walk

priors

used

in

Fahrmeir

and

Lang

and

Fahrmeir

and

Lang

In

fact,

Bayesian

P-splines

contain

random

walk

priors

as

a

special

case.

MCMC

inference

makes

use

of

matrix

operations

for

band

or

more

generally

sparse

matrices

but

is

restricted

to

Gaussian

responses.

The

plan

of

this

paper

is

the

following:.

Models

with

STAR

predictor

are

described

in

a

more

general

form

than

the

previous

papers

and

a

unified

notation

is

used.

We

thereby

start

with

generalized

additive

models

and

gradually

extend

the

approach

to

incorporate

unit-or

cluster

specific

heterogeneity,

spatial

effects

and

interactions

between

covariates

of

different

types..

We

extend

the

inference

techniques

for

Gaussian

responses

in

Lang

and

Brezger

to

situations

with

fundamentally

non-Gaussian

responses..

We

develop

a

number

of

highly

efficient

updating

schemes

with

iteratively

weighted

least

squares

(IWLS)

used

for

fitting

generalized

linear

models

as

the

main

building

block.

The

proposed

updating

schemes

are

much

more

efficient

in

terms

of

mixing

of

the

Markov

chains

and

computing

time

than

conditional

prior

proposals

used

in

Fahrmeir

and

Lang

Related

algorithms

have

been

proposed

by

Gamerman

and

Lenk

and

DeSarbo

for

estimating

Bayesian

generalized

linear

mixed

models.

Compare

also

Rue

and

Knorr-Held

and

Rue

who

develop

efficient

MCMC

updating

schemes

for

spatial

smoothing

of

poisson

responses..

For

the

first

time,

we

present

semiparametric

Bayesian

inference

for

multinomial

logit

models..

For

categorical

responses

we

review

the

state

of

the

art

of

MCMC

inference

techniques

based

on

latent

utility

representations.

A

comparison

with

the

direct

sampling

schemes

(see

the

second

point)

is

made..

Our

Bayesian

approach

for

semiparametric

regression

has

the

following

advantages

compared

to

existing

methodology:.

Extendability

to

more

complex

formulations

A

main

advantage

of

a

Bayesian

approach

based

on

MCMC

techniques

is

its

flexibility

and

extendability

to

more

complex

formulations.

Our

approach

may

also

be

used

as

a

starting

point

for

Bayesian

inference

in

other

model

classes

or

more

specialized

settings.

For

example

Hennerfeind,

Brezger

and

Fahrmeir

build

on

the

inference

techniques

of

this

paper

for

developing

geoadditive

survival

models..

Inference

for

functions

of

the

parameters

Another

important

advantage

of

inference

based

on

MCMC

is

easy

prediction

for

unobserved

covariate

combinations

including

credible

intervals,

and

the

availability

of

inference

for

functions

of

the

parameters

(again

including

credible

intervals).

We

will

give

specific

examples

in

our

second

application..

Estimating

models

with

a

large

number

of

parameters

and

observations.

In

Fahrmeir,

Kneib

and

Lang

we

compare

the

relative

merits

of

the

full

Bayesian

approach

presented

here,

and

empirical

Bayesian

inference

based

on

mixed

model

technology

where

the

smoothing

parameters

are

estimated

via

restricted

maximum

likelihood.

Although

the

standard

methodology

from

the

literature

has

been

improved

the

algorithms

are

still

of

the

order

p

where

p

is

the

total

number

of

parameters.

Similar

problems

arise

if

the

smoothing

parameters

are

estimated

via

GCV,

see

Wood

In

our

Bayesian

approach

based

on

MCMC

techniques

we

can

use

a

divide

and

conquer

strategy

similar

to

backfitting.

The

difference

to

backfitting

is,

however,

that

we

are

able

to

estimate

the

smoothing

parameters

simultaneously

with

the

regression

parameters

with

almost

negligible

additional

effort.

We

are

therefore

able

to

handle

problems

with

more

than

parameters

and

observations.

The

methodology

of

this

paper

is

included

in

the

public

domain

program

BayesX,

a

software

package

for

Bayesian

inference.

It

may

be

downloaded

including

a

detailed

manual

from

As

a

particular

advantage

BayesX

can

estimate

reasonable

complex

models

and

handle

fairly

large

data

sets..

We

will

present

examples

of

STAR

models

in

two

applications.

In

our

first

application

we

analyze

longitudinal

data

on

the

health

status

of

beeches

in

northern

Bavaria.

Important

influential

factors

on

the

health

state

of

trees

are

e.g.

the

age

of

the

trees,

the

canopy

density

at

the

stand,

calendar

time

as

a

surrogate

for

changing

environmental

conditions,

and

the

location

of

the

stand.

The

second

application

is

a

space-time

analysis

of

hospital

treatment

costs

based

on

data

from

a

German

private

health

insurance

company..

The

remainder

of

this

paper

is

organized

as

follows:

The

next

section

describes

Bayesian

GAM's

based

on

one

or

two

dimensional

P-splines

and

discusses

extensions

to

STAR

models.

Section

gives

details

about

MCMC

inference.

In

Section

we

present

two

applications

on

the

health

status

of

trees

and

hospital

treatment

costs.

Section

concludes

and

discusses

directions

for

future

research..

Bayesian

STAR

models

GAM's

based

on

Bayesian

P-splines

Suppose

that

observations

(y

i

x

i

v

i

i

n,

are

given,

where

y

i

is

a

response

variable,

x

i

(x

x

ip

is

a

vector

of

continuous

covariates

and

v

i

(v

v

iq

are

further

(mostly

categorical)

covariates.

Generalized

additive

models

(Hastie

and

Tibshirani,

assume

that,

given

x

i

and

v

i

the

distribution

of

y

i

belongs

to

an

exponential

family,

i.e..

where

b(·),

c(·),

θ

i

and

φ

determine

the

respective

distributions.

A

list

of

the

most

common

distributions

and

their

specific

parameters

can

be

found

e.g.

in

Fahrmeir

and

Tutz

page

The

mean

µ

i

E(y

i

x

i

v

i

is

linked

to

a

semiparametric

additive

predictor

η

i

by.

Here,

h

is

a

known

response

function

and

f

f

p

are

unknown

smooth

functions

of

the

continuous

covariates

and

v

i

γ

represents

the

strictly

linear

part

of

the

predictor.

Note

that

the

mean

levels

of

the

unknown

functions

f

j

are

not

identifiable.

To

ensure

identifiability,

the

functions

f

j

are

constrained

to

have

zero

means.

This

can

be

incorporated

into

estimation

via

MCMC

by

centering

the

functions

f

j

about

their

means

in

every

iteration

of

the

sampler.

To

avoid

that

the

posterior

is

changed

the

subtracted

means

are

added

to

the

intercept

(included

in

v

i

γ)..

For

modeling

the

unknown

functions

f

j

we

follow

Lang

and

Brezger

who

present

a

Bayesian

version

of

P-splines

introduced

in

a

frequentist

setting

by

Eilers

and

Marx

and

Marx

and

Eilers

The

approach

assumes

that

the

unknown

functions

can

be

approximated

by

a

polynomial

spline

of

degree

l

and

with

equally

spaced

knots.

over

the

domain

of

x

j

The

spline

can

be

written

in

terms

of

a

linear

combination

of

M

j

k

j

l

B-spline

basis

functions

(De

Boor,

Denoting

the

m-th

basis

function

by

B

jm

we

obtain.

By

defining

the

n

M

j

design

matrices

X

j

with

the

elements

in

row

i

and

column

m

given

by

X

j

(i,

m)

B

jm

(x

ij

we

can

rewrite

the

predictor

in

matrix

notation

as.

Here,

β

j

(β

β

jM

j

j

p,

correspond

to

the

vectors

of

unknown

regression

coefficients.

The

matrix

V

is

the

usual

design

matrix

for

linear

effects.

To

overcome

the

well

known

difficulties

involved

with

regression

splines,

Eilers

and

Marx

suggest

a

relatively

large

number

of

knots

(usually

between

to

to

ensure

enough

flexibility,

and

to

introduce

a

roughness

penalty

on

adjacent

regression

coefficients

to

regularize

the

problem

and

avoid

overfitting.

In

their

frequentist

approach

they

use

penalties

based

on

squared

r-th

order

differences.

Usually

first

or

second

order

differences

are

enough.

In

our

Bayesian

approach,

we

replace

first

or

second

order

differences

with

their

stochastic

analogues,

i.e.

first

or

second

order

random

walks

defined

by.

with

Gaussian

errors

u

jm

N

τ

j

and

diffuse

priors

β

const,

or

β

and

β

const,

for

initial

values,

respectively.

The

amount

of

smoothness

is

controlled

by

the

variance

parameter

τ

j

which

corresponds

to

the

inverse

smoothing

parameter

in

the

traditional

approach.

By

defining

an

additional

hyperprior

for

the

variance

parameters

the

amount

of

smoothness

can

be

estimated

simultaneously

with

the

regression

coefficients.

We

assign

the

conjugate

prior

for

τ

j

which

is

an

inverse

gamma

prior

with

hyperparameters

a

j

and

b

j

i.e.

τ

j

IG(a

j

b

j

Common

choices

for

a

j

and

b

j

are

a

j

and

b

j

small,

e.g.

b

or

b

j

Alternatively

we

may

set

a

j

b

j

e.g.

a

j

b

j

Based

on

experience

from

extensive

simulation

studies

we

use

a

j

b

j

as

our

standard

choice.

Since

the

results

may

considerably

depend

on

the

choice

of

a

j

and

b

j

some

sort

of

sensitivity

analysis

is

strongly

recommended.

For

instance,

the

models

under

consideration

could

be

re-estimated

with

(a

small)

number

of

different

choices

for

a

j

and

b

j

In

some

situations,

a

global

variance

parameter

τ

j

may

be

not

appropriate,

for

example

if

the

underlying

function

is

highly

oscillating.

In

such

cases

the

assumption

of

a

global

variance

parameter

τ

j

may

be

relaxed

by

replacing

the

errors

u

jm

N

τ

j

in

by

u

jm

N

τ

j

/δ

jm

The

weights

δ

jm

are

additional

hyperparameters

and

assumed

to

follow

independent

gamma

distributions

δ

jm

G(.

This

is

equivalent

to

a

t-distribution

with

ν

degrees

of

freedom

for

β

j

(see

e.g.

Knorr-Held

in

the

context

of

dynamic

models).

As

an

alternative,

locally

adaptive

dependent

variances

as

proposed

in

Lang,

Fronk

and

Fahrmeir

and

Jerak

and

Lang

could

be

used

as

well.

Our

software

is

capable

of

estimating

such

models,

but

we

do

not

investigate

them

in

the

following.

However,

estimation

is

straightforward,

see

Lang

and

Brezger

Lang,

Fronk

and

Fahrmeir

and

Jerak

and

Lang

for

details.

This

paper

proposes

semiparametric

Bayesian

inference

for

regression

models

with

responses

from

an

exponential

family

and

with

structured

additive

predictors.

Our

approach

allows

estimation

of

nonlinear

effects

of

continuous

covariates

and

time

scales

as

well

as

the

appropriate

consideration

of

unobserved

unit-or

cluster

specific

and

spatial

heterogeneity.

Many

well

known

regression

models

from

the

literature

appear

to

be

special

cases

of

our

approach,

e.g.

dynamic

models,

generalized

additive

mixed

models,

varying

coefficient

models,

geoadditive

models

or

the

famous

and

widely

used

BYM-model

for

disease

mapping

(Besag,

York

and

Mollie

The

proposed

sampling

schemes

work

well

and

automatically

for

the

most

common

response

distributions.

Software

is

provided

in

the

public

domain

package

BayesX..

Our

current

research

is

mainly

focused

on

model

choice

and

variable

selection.

Presently,

model

choice

is

based

primarily

on

pointwise

credible

intervals

for

regression

parameters

and

the

DIC.

A

first

step

for

more

sophisticated

variable

selection

is

to

replace

pointwise

credible

intervals

by

simultaneous

probability

statements

as

proposed

by

Besag,

Green,

Higdon

and

Mengersen

and

more

recently

by

Knorr-Held

For

the

future,

we

plan

to

develop

Bayesian

inference

techniques

that

allow

estimation

and

model

choice

(to

some

extent)

simultaneously.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Light stemming approaches for the French, Portuguese, German and Hungarian languages This paper describes and evaluates various general stemming approaches for the French, Portuguese (Brazilian), German and Hungarian languages. Based on the CLEF test-collections, we demonstrate that light stemmers for the French, Portuguese and Hungarian languages perform well, and reasonably well for the German language. Variations in mean average precision among the different stemming approaches are also evaluated and sometimes they are found statistically significant. Stemming is the conflation of word variants into a common stem, and in information retrieval (IR), applying stemmers to index documents (and requests) is assumed to be a good practice. Although N-gram [1] indexing would be an exception, we usually assume that stemming will efficiently conflate several different word variants into a common form. For example, when a query contains the word "horse," it seems reasonable to also retrieve documents containing the related word "horses." Properly designed stemming procedures should also reduce the size of inverted files, and also be helpful in other uses such as text data mining, natural language processing or gathering statistics on a document corpus.. In our approach to stemmer design, we begin by removing only inflectional suffixes. As such, singular and plural word forms (e.g., "dogs" and "dog") or feminine and masculine variants (e.g., "actress" and "actor") will conflate to the same root.. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Stemming schemes that remove only morphological inflections are termed "light" suffix-stripping algorithms. More sophisticated approaches exist for removing derivational suffixes (e.g., '-ment', '-ably', '-ship' in the English language). Stemming procedures [2] , [3] ignore word meanings and tend to make errors, usually due to over-stemming (e.g., "organization" is reduced to "organ") or to under-stemming (e.g., "create" and "creation" do not conflate to the same root). Stemmers are usually designed to work with general text in any given language, yet most studies on their IR performance have involved English language stemmers only. Given the absence of evaluation studies for other European languages, this paper is intended to fill this gap. In this paper, upon analyzing four different languages and various stemming approaches, we demonstrated that the Okapi probabilistic model produces the best retrieval performance. Moreover, the difference between the MAP and other IR models is statistically significant for the French, Portuguese and German corpora.. Empirical evidence clearly shows that a stemming procedure improves retrieval effectiveness when applied to European languages belonging to either the Latin (French, Portuguese), Germanic (German) or Finno-Ugrian (Hungarian) families. From a statistical point of view, the difference in retrieval performance for these four languages is significant for the best nine performing IR models.. From comparing different stemming strategies, it seems a light stemming approach produces better MAP than does a more aggressive stemmer, and for some IR models, the difference between these two stemming schemes could be statistically significant and in favor of a light stemming solution.. For the German and the Hungarian languages, the performance difference between the stemmers is not statistically significant. Most stemming approaches are based on the target language's morphological rules (e.g., [2] , [3] ) and suffix removal is also controlled by quantitative restrictions (e.g., '-ing' is removed when the resulting stem has more than three letters as in "running," but not in "king") or qualitative restrictions (e.g., '-ize' is removed if the resulting stem does not end with "e" as in "seize"). Certain ad hoc spelling correction rules can also be applied to improve conflation accuracy (e.g., "running" gives "run" and not "runn"), particularly when phonetic rules are applied to facilitate easier pronunciation.. Another approach consults an online dictionary to obtain better conflation results [4] , [6] , while Xu & Croft [5] suggest a corpus-based approach that more closely reflects the language at hand rather than its grammar. Few stemming procedures 1 have been suggested for other European languages than English, and those schemes available usually apply to the more popular languages and rely on a dictionary [6] or a deeper morphological analysis [7] .. Following a performance analysis of various English stemmers, Harman [8] showed there were no statistically significant improvements between them. A query-by-query analysis did however reveal that stemming affected performance, even though the number of queries showing improvement was almost equal to the number of queries showing decreased performance. Other studies generally conclude that the use of a stemmer shows a modest improvement, and the difference with an approach ignoring stemming is not always statistically significant.. It was also surprising to note that during recent CLEF evaluation campaigns, 2 only a few stemmers were suggested and compared. For example, Di Nunzio et al. [9] showed that for statistical stemmers the relative retrieval performance may vary for each of the five studied languages. This means that such an approach may work well for one language yet work poorly for another. When compared to statistical stemmers, Porter's stemmers seem to work slightly better. For German, Braschler & Ripplinger [10] showed that for short queries stemming may enhance mean average precision by 23%, compared to 11% for longer queries. Finally, Tomlinson [11] evaluated the differences between Porter's stemmer [3] and the lexical stemmer (based on a dictionary of the corresponding language). For Finnish and German, he found that the lexical stemmer tended to produce statistically better results, while for seven other languages performance differences were insignificant.. Based on these facts, the rest of this paper will address the following questions: 1) Does stemming affect IR performance for European languages other than English? 2) For these languages, are light stemming approaches less effective than more complex suffix-stripping algorithms?

O O O O O O B B O B O O O O O O O O O O O O O B O B O B O O O O O O O O O O O O O O B O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O B O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Short-term site-wide maintenance scheduling Preventive maintenance is essential for every chemical production site to prevent failure and accidents, however it upsets material and utility flows inside the site and also causes production loss. In order to minimize the loss, maintenance of each plant unit has to be carefully scheduled together with considerations on site-wide material and utility balances.This will involve both production and utility systems, and indeed is a very complicated problem. A scheduling strategy is then employed to handle the problem efficiently. It divides the scheduling into two steps, long-term and short-term. Long-term maintenance scheduling determines the combination of plant shutdown in a period of two to five years.Based upon the long-term schedule, a short-term maintenance scheduling optimizes the exact timing of plant shutdown, overhaul, inspection and startup within a maintenance period of four to ten weeks. Short-term maintenance scheduling involves pre-set utility and material demand profiles during a plant shutdown, overhaul and startup making it a very challenging task. In this paper, a multi-period mixed integer linear programming (MILP) model, a Site-Model, is proposed as an aid to optimize short-term site-wide maintenance schedule. A special formulation is also developed to deal with the pre-set utility and material demand profiles in the Site-Model. A chemical production site contains variety of production and utility plants. All of them require regular maintenance to enhance their reliability and to avoid emergency shutdowns.. Nevertheless, plant maintenance is unfavorable on production suspension, huge capital and human resources involvement. It also affects the material and utility balances between the production and the utility units. An effective maintenance schedule should hence be able to balance all above factors and provide a feasible and economical result.. In this paper, a multi-period mixed integer linear programming (MILP) model, called. Site-Model, is proposed for solving this type of maintenance scheduling problems. A site-model includes all the major material and utility balances of the production and utility plants as well as their interconnections and constraints.. To optimize a maintenance schedule, it requires tackling both long-term scheduling and short-term scheduling (Cheung and Hui, 2003) . In a chemical production site, not every plant has to be maintained every year. There are always some plants being operated while others are down for maintenance. Due to the limitations on the availability of materials, utilities and skilled workers, the production site usually defines two to three maintenance periods in a year. Each period lasts for four to ten weeks. Production plants and utility facilities are then divided into groups and maintained separately in different maintenance periods. A long-term maintenance schedule provides grouping for maintenance plants and assigns a maintenance period to each group (rough maintenance timing) over a two to five year time span. It takes into account the basic material and utility balances during the whole scheduling time horizon, and to determine long-term production and inventory strategies as well as materials and utilities import/export policies.. Superimposed on the long-term maintenance schedule, short-term scheduling extracts the grouping results of maintenance plants from every maintenance period. It then further optimizes them by deciding the detail timing of the maintenance main operations (i.e.. shutdown, overhaul, inspection and startup) of all the maintenance plants during a maintenance period (Figure 1 ). The main operations are normally completed by a relatively short period (four to ten weeks), however their utility and labor demands could be significantly fluctuated. The main objective of short-term scheduling is thus to guarantee a feasible operation, smooth utility and labor demands, and minimize production losses during the maintenance period.. Only a small number of literature addresses maintenance scheduling about chemical productions. Some of them focused on the short-term scheduling of multipurpose batch plants with integration of maintenance and production (Dedopoulos and Shah, 1995 and Pistikopoulos et al., 2001) . Kim and Han (2001) have discussed maintenance scheduling of utility systems by assuming a fixed configuration of utility and production systems and fixed utility demands. Hui and Natori (1996) included maintenance feasibility as a consideration in a case study of utility system expansion. Tan and Kramer (1997) scheduled maintenance for chemical processes. So far, maintenance schedules for a chemical production complex including the interaction between production and utility systems have not been investigated in any of earlier works.. The idea of applying a site-model for long-term site-wide maintenance scheduling has previously been discussed by Cheung and Hui (2003) . This paper will then focus on the short-term maintenance scheduling of a chemical production complex, which contains both production and utility plants.

O O B B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O B B O O O O O B O O O O O O O O O O O O O O B B O O O O O O B O O O O O O B O O O O O O O B B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B B O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O B O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O B O O O B O O O O O O O O O O O O O B O O O B O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O B B O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B B O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O


Optimal policies in continuous time inventory control models with limited supply Abstract In this paper, we deal with the problem of a fixed number of units of a certain perishable commodity over a continuous time horizon. Airline seats, hotel rooms, advertising space in newspapers, and some seasonal products that must be sold before the end of the season are examples of such commodities that cannot be carried over and are not storable for consumers. This paper considers such a problem of continuous time perishable inventory control by applying semi-Markov decision processes over a finite time horizon. It is shown that there is an optimal policy that is simple and stationary. Furthermore, some analytical properties of this optimal policy and its value function are explored under certain specific assumptions. Consider a certain perishable commodity whose supply is fixed. A company wants to sell a fixed number of units of this commodity. On the other hand, consumers have different (nonhomcgeneous) preferences for the commodity. In such a circumstance, the company can set different prices on the identical commodity; that is, the law of one price does not hold in this market. Such commodities are called price differentiation products. Examples of price differentiation products might include airline seats, hotel rooms, concert tickets, cabins on cruise ships, and seasonal goods such as winter coats that have to be sold before the end of the season. The date and place for selling such commodities is fixed, it is not possible to use these commodities either before or after the specified date, and they are perishable goods that cannot be carried over. It is possible, however, to sell these commodities beforehand.. In this paper, we develop a framework of inventory control for perishable commodities with limited supply by explicitly incorporating the existence of multiple prices for that commodity. It is shown that such an inventory control model can be formulated as a semi-Markov decision process over a finite continuous time horizon. Our main result is to show that, under some specific assumptions, there is an optimal policy and an optimal value function corresponding to it. In addition, some analytical properties are explored. The problem of selling a fixed number of units of a certain perishable commodity in order to maximize the expected revenue is known as the yield management problem in the airline industry (see [i-3] ), h w ere most studies on yield management have focused on either pricing policy or static control problems with two fares (see [1, . Feng and Gallego [S] consider the problem of deciding the optimal timing of a single-price change from a given initial price to either a lower or higher price. Gallego and Ryzin [5] investigate the problem of dynamic pricing policies for selling a given stock of items by a deadline. Lee and Hersh [9] develop a discrete-time dynamic programming model for finding an optimal booking policy for airline seat inventory control. Brumelle and Walczak [lo] model an airline seat allocation for a single-leg flight with multiple fares that is similar to ours but has different assumptions. The present study differs from these earlier studies in two regards. First, the planning horizon is continuous times; second, ours is a dynamic model that deals with a multiple-price selling policy for a perishable product that is not necessarily restricted to airline seats.. In Section 2, we develop the problem of selling a fixed number of units of a perishable commodity over a continuous time planning horizon. We apply a semi-Markov decision process approach to the problem so as to maximize the expected total revenue. Section 3 discusses what analytical properties an optimal policy and its value function possess. In particular, we are interested in the development of value function concavity and optimal policy monotonicity with respect to their arguments; these tell us that an optimal selling curve decreases as the time draws nearer to the end of the planning horizon. Finally, the last section is a conclusion with additional comments.

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A hybrid dynamic forecast model for analyzing celebrity endorsement effects on consumer attitudes This study investigates the time-varying effects of celebrity endorsements on consumer purchase attitudes toward promoted products using a novel dynamic hierarchical multi-attribute attitude forecast model. The induced direct and indirect effects via constructs of product attributes and net product value are then incorporated into the proposed conceptual model, which is formulated with a discrete-time nonlinear stochastic system. An empirical study of product categories of sport shoes and notebook computers demonstrates the feasibility of the proposed methodology. The analytical results demonstrate the capability of the proposed model to forecast consumer attitudes toward a promoted product and reveal the potential heterogeneity in patterns of attitude changes characterized by product attributes, price, and endorser performance as perceived by consumers. Furthermore, we infer that celebrity endorsement can significantly influence consumer purchase attitudes via both direct and indirect effects through productattribute construct. Despite the capability of the proposed model to forecast celebrity endorsement effects on consumer attitudes toward a promoted product and attitude changes, several interesting findings may provide significant implications for further discussion.. First, the celebrity endorsement effect on consumer-expected product value remains unclear. Therein, the socioeconomic characteristics of target customers and product type may be causes. For instance, in the sport shoes case, target consumers may be students positioned within a consumer group that has low income and high consumption intentions. Thus, most of students are affordable at the retail price, and willing to pay for sport shoes when its fits with their demand and preferences. Therefore, ''price'' may not be their major concern; other the factors related to product attributes and endorser performance may also be important. Additionally, the proposed conceptual framework can be applied to other product categories (e.g., durable goods), as it involves product-attribute construction, which facilitates the incorporation of different product attributes into the framework for characterizing promoted products. Therefore, we infer that different generalizations may be based on study case, which is aimed at different consumer groups (e.g., female consumers) associated with specific product categories (e.g., luxurious dresses and cosmetics products) endorsed by celebrities. Nevertheless, the proposed dynamic forecast model is applicable to diverse cases for determining the time-varying celebrity endorsement effects on consumer attitudes.. Second, the phenomenon of negative celebrity endorsement effects may be noteworthy. The performance of celebrity endorsement may also be influenced by the endorser's performance, which can be positive or negative, particularly when endorsers are superstars in their professionals [9] . Thus, updated performance can be reassessed using the two criteria and five sub-criteria. The resulting direct endorsement effect can then be further applied to estimate consumer attitudes toward product-attribute and net product value constructs, followed by determination of a consumer's overall attitude toward a promoted product. Briefly, the proposed model assesses perceived performance of an endorser, and traces consumer attitude changes to determine the induced impact on demand markets during a promotional period.. Third, celebrity endorsement can influence consumer evaluations of product attributes and the associated belief strengths. Such a phenomenon may be of particular interest under the condition in that certain product attributes are perceived as highly congruent with celebrity endorser characteristics. For example, among the many product attributes, ''function'' and ''style'' may be relatively sensitive to the perceived celebrity performance, as consumer evaluations and belief strengths were significantly changed by the comparison of the forecast results with and without celebrity endorsement in the sport shoes case. Thus, we infer that most consumers may agree with the congruence between endorser characteristics (e.g., expertise and likeability of Wang) and the attributes of the sport shoes, thus contributing to a high degree of attitude change in the product-attribute construct, compared to the forecast results obtained under no endorsement.. Fourth, despite the fact that the endorsement by Wang may have a greater promotional effect than that of Michael Jordan in the study cases, some other factors are worthy of consideration. From a strategic perspective, product positioning and segmentation should also be considered when planning a celebrity endorsement strategy. For instance, if the promoted sports shoes are aimed at global markets, an endorsement by Jordan will likely be more successful than an endorsement by Wang. However, Wang may have the advantage when product promoted is aimed mainly at the local market. Furthermore, endorsement cost relative to induced benefit is also noteworthy when assessing alternatives to celebrity endorsements.. An interesting direction for future research is to incorporate diverse promotional strategies into the proposed framework to obtain insights into hybrid promotional effects on consumer behavior and demand markets. Additionally, the proposed model permits the continuous updating of estimates of consumer attitudes as endorser performance is updated. Thus, collaborative advertising strategies can be utilized in response to attitude changes of target consumers during a promotional period. Furthermore, the aggregated output in terms of E k ij (n) can help marketers assess celebrity endorsement projects, and the proposed model can be furthered by incorporating other novel endorsement assessment approaches into its framework. Extensions of study scope, including the model to domains of purchase intention and behavior, also warrant further research to either measure the overall effect of advertising expenditures on sales or assess the effectiveness of a celebrity endorsement on a firm's profitability; however, these are challenging issues [11] . Study cases for different product categories endorsed by different celebrities may also interest marketers for strategic marketing.. In summary, this study will benefit by adding additional insights into celebrity endorsement effects on consumer purchase attitudes, and by additional studies for promotional strategic planning and management.. weights which are the same as those obtained from Saaty's λ − max technique, termed as the largest eigenvalue technique.. In addition, the geometric mean technique may also satisfy the condition of the absence of rank reversal [32] . Accordingly, given a t × t pair-wise comparison matrix associated with a given layer l (D .3) , the relative weights associated with the criteria of layer 2 and sub-criteria of layer 3 can then be readily approximated, and used in Eq. (3) to estimate E k ij (n). Based on this conceptual framework, we argue that a consumer's (denoted by i) instantaneous purchase attitude (Θ k ij (n)) toward a promoted product (denoted by j) advertised by celebrity k on a given purchase occasion n is composed of deterministic (θ k ij (n)) and stochastic (ε(n)) terms; thus,. where ε(n) is independent of θ k ij (n), and possibly caused by unknown extrinsic variables. Now, this work rationalizes the structure of θ k ij (n) using a multi-attribute attitude form composed of three main constructs, i.e., celebrity endorsement (E k ij (n)), product attributes (A k ij (n)), and price considerations (P k ij (n)), based on the proposed conceptual framework. Thus, θ. where A k ij (n), P k ij (n), and E k ij (n) are the instantaneous evaluations of consumer i with respect to the collection of attributes of product j, net product value, and celebrity endorser k on a given purchase occasion n, respectively; and (n) , and E k ij (n), respectively. As consumer purchase attitudes may change over time [12] , these consumer evaluations and belief strengths are defined as dynamic forms to rationalize the time-varying features of these variables and parameters.. According to Lavidge and Steiner [26] , A k ij (n) and P k ij (n) are likely affected by attributed performance of celebrity endorsement in any one of four phases (awareness, knowledge, liking, and preference) utilized in the consumer purchase decision process. For instance, a consumer may prefer product A over B due to admiration for the celebrity endorsing product A, who thereby increases consumer preference for product A. If the celebrity is well known and has considerable experience and expertise associated with the promoted product, consumer-expected product price or quality may increase, leading to a cognitive response that increases consumer belief in a celebrity endorsed product [8] . Accordingly, we propose that A k ij (n) and P k ij (n) can be functions of E k ij (n), which are derived as follows:

O O B I I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Square integrability of Gaussian bells on time scales We will consider a generalization E(x) of the continuum Gaussian bell e −x 2 /2 on a time scale T. A crucial question is whether this Gaussian bell is square integrable on the corresponding time scale. However, this is not the case in general. We will establish sufficient and necessary conditions for the time scale T to satisfy E ∈ L 2 (T).

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