# Using Hazard and Surrogate Functions for Understanding Memory and Forgetting

## Abstract

**:**

## 1. Introduction

## 2. Fundamentals for the Hazard Analysis of Memory and Forgetting

#### 2.1. Linking Forgetting to the Product Failure Context

#### 2.2. Initial Conditions and Rationality Constraints for Memory Reliability Modeling

#### 2.3. Probability and Subprobability Hazard Models

#### 2.4. Weibull Distribution and Hazard

#### 2.5. A Surrogate Function and Proofs of Monotonicity

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

#### 2.6. Using Hazard to Create Novel Probability Models

#### 2.7. Hazard for Mixture Models

**Theorem**

**3.**

**Proof.**

#### 2.8. Disjunctive Hazard Systems

**Theorem**

**4.**

**Proof.**

#### 2.9. Conjunctive Hazard Systems

**Theorem**

**5.**

**Proof.**

## 3. Hazard Functions for the Candidate Models for Memory Retention

#### 3.1. Modified-Hyperbolic Model

#### 3.2. Modified Exponential Model and the Multiple-Store Model

#### 3.3. Modified Logarithm Model

#### 3.4. Modified Power Model

#### 3.5. Modified Single-Trace Fragility Theory

#### 3.6. Modified Anderson-Schooler Model

#### 3.7. Ebbinghaus Model

#### 3.8. Trace Susceptibility Theory

#### 3.9. Two-Trace Hazard Model

#### 3.10. Scale Invariance and the Number of Fitting Parameters

## 4. Some Empirical Evidence about Memory Hazard

#### 4.1. Using a Surrogate Function for Assessing the Hazard Function

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

#### 4.2. Evidence for the Two-Trace Hazard Model

^{−2}for the previously reported 30 participants in the short-term time domain; consequently by 5 min the value for ${\theta}_{{S}_{1}}$ would be less than ${10}^{-140}$. Thus, Sloboda only fitted ${\theta}_{{S}_{2}}=1-b(1-{e}^{-a{t}^{c}})$ to the long-term storage probability values. The mean and standard deviation for the fits of the model for the a, b, and c parameters, respectively, are: 1.9(1.1) min

^{−2}, $0.991\left(0.009\right)$, and $0.238\left(0.043\right)$. The resulting fit is excellent for each of the nine participants; the mean correlation between the model predicted values and storage probabilities is $0.947$.

## 5. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MCMC | Markov chain Monte Carlo |

MPT | multinomial processing tree |

PPM | population parameter mapping |

## Appendix A. PPM Details for Model 7B and the Software for θ S Estimation

**rbeta( )**.

**storage_7B( )**was created for this task. Some comments are included in the source code shown below.

- storage_7B<-function(prior=rep(1,8),freq,samples=50000){
- #Function is to find thetaS for model 7B given recognition
- #memory data. The~data is stored in a 8 cell
- #vector for frequencies for old and new recognition.
- #The first four cells correspond to the respective
- # frequencies for no_sure, no_unsure, yes_unsure, yes_sure,
- #for the old recognition tests, and~next four cells are
- #corresponding frequencies for the new recognition tests.
- #The only coherence condition is that thetaS is nonnegative.
- #The prior vector is the default of c(1,1,1,1,1,1,1,1) that is
- #joint flat Dirichlet prior. If~the user has an informative
- #prior, then the prior vector needs to be inputted.
- #The default value for samples is 50000, but~the user
- #can enter their preferred value for the number of
- #Monte Carlo samples.
- if (length(freq)!=8)
- {stop("data vector must have length 8 for the data frequencies.")}
- thetaS=rep(0,samples)
- postphin=prior+freq
- alpha4=postphin[4]
- alpha3=postphin[3]
- alphaTo=postphin[1]+postphin[2]+postphin[3]+postphin[4]
- alphaTn=postphin[5]+postphin[6]+postphin[7]+postphin[8]
- alpha8=postphin[8]
- alpha7=postphin[7]
- b4=rbeta(samples,alpha4,alphaTo-alpha4)
- b3=rbeta(samples,alpha3,alphaTo-alpha3-alpha4)
- b8=rbeta(samples,alpha8,alphaTn-alpha8)
- b7=rbeta(samples,alpha7,alphaTn-alpha7-alpha8)
- phi4=b4
- phi3=(1-b4)*b3
- phi8=b8
- phi7=(1-b8)*b7
- c=0
- for (i in 1:samples){
- tstry=phi4[i]-((phi3[i]*phi8[i])/phi7[i])
- if (tstry>=0){
- c=c+1
- thetaS[c]=tstry} else {c=c}
- }
- thetaS=thetaS[1:c]
- probcoh=c/samples
- outlist<-list(c=c,probcoh=probcoh,thetaS=thetaS)
- #outlist is the list of output values and vectors
- }

**A<- storage_7B(freq=c(36,294,265,101,123,369,188,16))**

**A**, which resides in the R workspace. The user can see the number of coherent mapping from the command

**A$c**. The proportion of coherent samples is obtained via the command

**A$probcoh**. The vector of ${\theta}_{S}$ values can be further examined by the standard base-R commands. For example, the posterior median of the distribution can be found from the command

**median(A$thetaS)**. Additionally, the command

**quantile(A$thetaS,prob=0.975)**results in the value for the $97.5$-percentile. If the user wants to have the results from two different conditions resident in the R workspace at the same time then a different name should be employed when calling the function again with different data (e.g,

**A1<- storage_7B( )**).

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**Table 1.**Nine candidate models for the memory survivor function $S\left(t\right)$ are shown along with the corresponding hazard function. The parameters a, b, c and d are positive. The cumulative functions for the two respective traces of the two-trace hazard model are ${F}_{1}\left(t\right)=1-exp(-d\phantom{\rule{0.166667em}{0ex}}{t}^{2})$ and ${F}_{2}\left(t\right)=b(1-exp(-a{t}^{c})$.

Model Name | Survivor Function | Hazard Function |
---|---|---|

modified hyperbolic | $1-b+\frac{b}{at+1}$ | $\frac{ab}{{(at+1)}^{2}[1-b+\frac{b}{at+1}]}$ |

modified exponential | $1-b+b{e}^{-at}$ | $\frac{a}{\frac{1-b}{b}{e}^{at}+1}$ |

modified logarithm | $1-b\frac{log(t+1)}{log(U+1)}$ | $\frac{b}{(t+1)log\frac{(U+1)}{{(t+1)}^{b}}}$ |

modified power | $1-b+\frac{b}{{(t+1)}^{c}}$ | $\frac{cb}{(1-b){(t+1)}^{c+1}+b(t+1)}$ |

modified single-trace fragility | $1-d+\frac{d{e}^{-at}}{{(1+bt)}^{c}}$ | $\frac{da+\frac{cbd}{1+bt}}{d+{e}^{at}(1-d){(1+bt)}^{c}}$ |

modified Anderson-Schooler | $1-d+\frac{db}{b+{t}^{c}}$ | $\frac{dbc}{{t}^{1-c}[(1-d){(b+{t}^{c})}^{2}+db(b+{t}^{c})]}$ |

Ebbinghaus | $\frac{b}{b+{[log(t+1)]}^{c}}$ | $\frac{c{[log(t+1)]}^{c-1}}{(t+1)(b+{[log(t+1)]}^{c})}$ |

trace susceptibility theory | $1-b+b{e}^{-a{t}^{2}}$ | $\frac{2abt{e}^{-a{t}^{2}}}{1-b+b{e}^{-a{t}^{2}}}$ |

two-trace hazard theory | $1-{F}_{1}\left(t\right){F}_{2}\left(t\right)$ | $\frac{2dt{e}^{-d{t}^{2}}{F}_{2}\left(t\right)+bac{t}^{c}{e}^{-a{t}^{c}}{F}_{1}\left(t\right)}{1-{F}_{1}\left(t\right){F}_{2}\left(t\right)}$ |

**Table 2.**The nine candidate models for the memory survivor function $S\left(t\right)$ that are time-scale invariant. The parameters a, b, c and d are positive. The cumulative functions for the two respective traces of the two-trace hazard model are ${F}_{1}\left(t\right)=1-exp(-d\phantom{\rule{0.166667em}{0ex}}{t}^{2})$ and ${F}_{2}\left(t\right)=b(1-exp(-a{t}^{c})$.

Model Name | Survivor Function | # Parameters |
---|---|---|

modified hyperbolic | $1-b+\frac{b}{at+1}$ | 2 |

modified exponential | $1-b+b{e}^{-at}$ | 2 |

modified logarithm | $1-b\frac{log(at+1)}{log(aU+1)}$ | 3 |

modified power | $1-b+\frac{b}{{(at+1)}^{c}}$ | 3 |

modified single-trace fragility | $1-d+\frac{d{e}^{-at}}{{(1+bt)}^{c}}$ | 4 |

modified Anderson-Schooler | $1-d+\frac{db}{b+{\left(at\right)}^{c}}$ | 4 |

Ebbinghaus | $\frac{b}{b+{[log(at+1)]}^{c}}$ | 3 |

trace susceptibility theory | $1-b+b{e}^{-a{t}^{2}}$ | 2 |

two-trace hazard theory | $1-{F}_{1}\left(t\right){F}_{2}\left(t\right)$ | 4 |

**Table 3.**The frequencies labels are shown for the various response categories for the experimental task that is associated with model 7B from [23]. Participants responded either yes or no to the test probe and rated their decision either as sure or unsure. The test probes were either old or new corresponding to the probe, respectively, being a memory target or a novel item. For each outcome in the Table there is also a population proportion ${\varphi}_{i}$, $i=1,\cdots ,8$.

Test Type | No Sure | No Unsure | Yes Unsure | Yes Sure |
---|---|---|---|---|

old | ${n}_{1}$ | ${n}_{2}$ | ${n}_{3}$ | ${n}_{4}$ |

new | ${n}_{5}$ | ${n}_{6}$ | ${n}_{7}$ | ${n}_{8}$ |

**Table 4.**The table provides the response frequencies from experiment 3 in [25] for each temporal delay.

Time | $({\mathbf{n}}_{1},\cdots ,{\mathbf{n}}_{4})$ | $({\mathbf{n}}_{5},\cdots ,{\mathbf{n}}_{8})$ |
---|---|---|

1.33 | (13, 4, 9, 670) | (658, 28, 4, 6) |

2.33 | (33, 21, 39, 603) | (610, 69, 6, 11) |

5 | (81, 79, 165, 371) | (378, 223, 76, 19) |

13 | (42, 194, 275, 185) | (163, 341, 177, 15) |

31 | (36, 294, 265, 101) | (123, 369, 188, 16) |

Time in s. | Storage Estimate | u(t) | 95 % u(t) Interval |
---|---|---|---|

1.33 | 0.938 | 0.041 | [0.002, 0.088] |

2.33 | 0.764 | 0.098 | [0.061, 0.178] |

5 | 0.470 | 0.104 | [0.091, 0.116] |

13 | 0.230 | 0.059 | [0.054, 0.062] |

31 | 0.111 | 0.028 | [0.027, 0.030] |

**Table 6.**The table provides the response frequencies from experiment 2 in [17] for each temporal delay.

Time | $({\mathbf{n}}_{1},\cdots ,{\mathbf{n}}_{4})$ | $({\mathbf{n}}_{5},\cdots ,{\mathbf{n}}_{8})$ |
---|---|---|

1.33 | (1, 0, 1, 898) | (880, 6, 9, 5) |

2.33 | (8, 3, 13, 876) | (858, 26, 15, 1) |

5 | (49, 80, 142, 629) | (646, 179, 53, 22) |

13 | (74, 226, 283, 317) | (434, 329, 99, 38) |

32.67 | (63, 281, 314, 242) | (296, 412, 143, 49) |

76 | (61, 322, 309, 208) | (244, 462, 166, 28) |

Time in s. | Storage Estimate | u(t) | 95 % u(t) Interval |
---|---|---|---|

1.33 | 0.994 | 0.001 | [0.000, 0.007] |

2.33 | 0.969 | 0.011 | [0.000, 0.017] |

5 | 0.630 | 0.073 | [0.062, 0.084] |

13 | 0.230 | 0.059 | [0.054, 0.064] |

32.67 | 0.148 | 0.026 | [0.024, 0.028] |

76 | 0.172 | 0.011 | [0.010, 0.011] |

**Table 8.**The table provides the response frequencies from experiment 1 in [17] for each temporal delay.

Time | $({\mathbf{n}}_{1},\cdots ,{\mathbf{n}}_{4})$ | $({\mathbf{n}}_{5},\cdots ,{\mathbf{n}}_{8})$ |
---|---|---|

2.33 | (5, 5, 6, 224) | (224, 13, 1, 0) |

3.67 | (10, 29, 48, 153) | (187, 52, 1, 0) |

6.33 | (26, 55, 95, 64) | (114, 103, 20, 3) |

11.67 | (9, 83, 106, 42) | (45, 133, 59, 3) |

Time in s. | Storage Estimate | u(t) | 95 % u(t) Interval |
---|---|---|---|

2.33 | 0.908 | 0.031 | [0.000, 0.071] |

3.67 | 0.551 | 0.116 | [0.075, 0.233] |

6.33 | 0.194 | 0.124 | [0.102, 0.144] |

11.67 | 0.147 | 0.071 | [0.060, 0.077] |

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Chechile, R.A.
Using Hazard and Surrogate Functions for Understanding Memory and Forgetting. *AppliedMath* **2022**, *2*, 518-546.
https://doi.org/10.3390/appliedmath2040031

**AMA Style**

Chechile RA.
Using Hazard and Surrogate Functions for Understanding Memory and Forgetting. *AppliedMath*. 2022; 2(4):518-546.
https://doi.org/10.3390/appliedmath2040031

**Chicago/Turabian Style**

Chechile, Richard A.
2022. "Using Hazard and Surrogate Functions for Understanding Memory and Forgetting" *AppliedMath* 2, no. 4: 518-546.
https://doi.org/10.3390/appliedmath2040031