Robbins-Monro 適測方法的延展及其於心理物理實驗閾 值估計的應用

The generalized Robbins-Monro process and its application to psychophysical experiments for threshold estimation

楊昊紘(Hau-Hung Yang)

指導教授 ： 徐永豐

國立臺灣大學/理學院/心理學研究所/碩士(2022年)

https://doi.org/10.6342/NTU202201404

摘要

在傳統心理物理學研究中，閾值及其背後表徵一直以來都深受研究者重視，因此如何測量特定閾值機率所對應的刺激強度是一個重要的議題。過去文獻中研究者發展了多個適測方法來測量閾值，其可分為變動步距的適測方法與固定步距的適測方法。而適測方法的特色在於該類方法會以受試者在先前的回答(通常為是/否的判斷)決定下一回合的刺激強度，其中廣為人知的變動步距適測方法是透過應用Robbins-Monro 過程發展而來。過去的文獻中的Robbins-Monro 適測方法僅以是/否的判斷作為受試者的反應，未嘗試使用其他變數拓展適測方法。本研究透過加入其他反應變數如反應時間與信心程度來拓展Robbins-Monro 適測方法，並證明拓展後的方法仍具有一致性以及在何種條件下新的方法應更為有效。此外本研究也以蒙地卡羅模擬實驗來驗證拓展的Robbins-Monro 適測方法在模擬的情境下有更快的收斂速度，並在之後討論模擬研究中使用的適測方法與先前定理中提到的條件的關聯。

關鍵字

適測方法 ； 隨機逼近法 ； 加速隨機逼近法 ； 閾值測量 ； 信心程度 ； 反應時間 ； 心理計量函數

並列摘要

In classical psychophysics, the study of threshold and underlying representations is of theoretical interest, and the relevant issue of finding the stimulus (intensity) corresponding to a certain threshold level is an important topic. In the literature, researchers have developed various adaptive (also known as｀up-down＇) methods, including the fixed step-size and variable step-size methods, for the estimation of threshold. A common feature of this family of methods is that the stimulus to be assigned to the current trial depends upon the participant＇s response in the previous trial(s), and very often the Yes-No (binary) response format is adopted.A well-known earlier work of the variable step-size adaptive methods is the application of the Robbins-Monro process. However, previous studies have paid little attention to other facets of response variables (in addition to the Yes-No response variable) that could be used to facilitate the performance of the Robbins-Monro process. This thesis concerns the generalization of the Robbins-Monro process by incorporating other response variables, such as response time and response confidence, into the process. I first proved the consistency of the generalized method and explored possible requirements, under which the proposed method achieves (at least) the same efficiency as the original method does. I then conducted a Monte Carlo simulation study to explore some properties of the sampling distribution of the estimator from the generalized method and compared its performance with the original method. The results showed that the generalized Robbins-Monro process can improve the speed of convergence. I also discussed the issue of relative efficiency (in terms of MSE), focusing on the relationship between the implemented generalized algorithms in the simulation and the conditions specified in the theorems .

並列關鍵字

adaptive method ； stochastic approximation ； accelerated stochastic approximation ； threshold ； response confidence ； response time ； psychometric function

參考文獻

Abdellaoui, M., Bleichrodt, H., L’Haridon, O., van Dolder, D. (2016). Measuring loss aversion under ambiguity: A method to make prospect theory completely observable. Journal of Risk and Uncertainty, 52(1), 1–20. https : / / doi . org / 10 . 1007 /s11166-016-9234-y

Bedggood, P., Ahmad, A., Chen, A., Lim, R., Maqsudi, S., Metha, A. (2020). Are you sure? The relationship between response certainty and performance in visual detection using a perimetry-style task. Journal of Vision, 20(8), 27–27. https://doi.org/10.1167/jov.20.8.27

Chung, K. L. (1954). On a stochastic approximation method. Annals of Mathematical Statistics, 25(3), 463–483. https://doi.org/10.1214/aoms/1177728716

Derman, C. (1957). Non-parametric up-and-down experimentation. Annals of Mathematical Statistics, 28(3), 795–798. https://doi.org/10.1214/aoms/1177706895

Dixon, W. J., Mood, A. M. (1948). A method for obtaining and analyzing sensitivity data. Journal of the American Statistical Association, 43(241), 109–126. https : //doi.org/10.1080/01621459.1948.10483254