From Com­plic­ated to Simple, From Brute­force to Gen­eral

A New Dir­ec­tion to Ap­proach MF CT

Guide writ­ten by Hack­zzzzzz. Con­tri­bu­tions from the Amaz­ing Com­munity and those in the ac­know­ledge­ments at bot­tom..

Mod­i­fied From [LONG POST 1/​2] From Com­plic­ated to Simple, From Brute­force to Gen­eral - A New Dir­ec­tion to Ap­proach MF CT and Post 2/​2

Re­vised by LE★Baldy

TLDR:

• There are mainly 3 re­set pat­terns used in MF CT de­pend­ing on the $\rho$ of pub­lic­a­tion, namely $e4.5/1v_2$ re­set, $e9/2v_2$ re­set, and $e6/1v_4$ re­set.
• A def­in­ite cri­terion of $c_2$ and $a_2$ by com­par­ing their cost and the cost for buy­ing all $v$ vari­ables be­fore a re­set is found, a sim­ilar cri­terion with ex­cel­lent con­fid­ence has been found for $\delta$, there is a mod­er­ate con­fid­ence of cri­terion when com­par­ing $c_1$ cost and the cost for buy­ing all $v$ vari­ables be­fore a re­set, and $a_1$ has no re­la­tion­ship with re­set as it does not im­pact $\rho$ growth after a suf­fi­ciently long time when $I$ reached cap, which was de­term­ined by the level $a_2$.
• A pub­lic­a­tion table, like Basal Prob­lem (BaP CT), is not likely but still pos­sible in MF CT, where the the­ory pub­lished spe­cific po­s­i­tion may shorten total com­ple­tion time.

Ab­stract #

Mag­netic Field (MF CT) is a cus­tom the­ory which is in­spired by elec­tro­mag­net­ism in phys­ics. Des­pite the simple phys­ics concept im­ple­men­ted in MF CT, the the­ory has ad­di­tional re­set fea­tures which com­plic­ated the the­ory in terms of the re­set pat­tern, its in­flu­ence on other vari­ables (namely $c_1$, $c_2$, $a_1$, $a_2$, and $\delta$) and the length of a pub­lic­a­tion. Hence, a gen­er­al­ized in­vest­ig­a­tion throughout the span of 9 months was con­duc­ted to ex­plore the un­dis­covered cri­teria con­cern­ing the above-men­tioned as­pects of MF CT. Do note that the con­ver­sion of $\rho$ and $\tau$ in MF CT is in 1 to 1 ra­tio, hence $\rho$ (pre­ferred) and $\tau$ de­scribed in MF CT may be used in­ter­change­ably in sub­sequent sec­tions.

The fol­low­ing ab­bre­vi­ations will be used throughout the sub­sequent sec­tion:

• Re­set pat­tern – A rule in which a particle re­set oc­curs when the rule is sat­is­fied
• $e4.5/1v_2$ re­set – A re­set pat­tern of which a particle re­sets when an ad­di­tional level of $v_2$, and all $v_1$, $v_3$, $v_4$ level with cost lower than that of ad­di­tional level of $v_2$ are all pur­chased, $e4.5$ is in­dic­ated for the gain of rho between re­set as the cost of every $v_2$ re­set dif­fer by a mag­nitude of $e4.5$
• $e9/2v_2$ re­set – A re­set pat­tern of which a particle re­sets when 2 ad­di­tional levels of $v_2$, and all $v_1$, $v_3$, $v_4$ level with cost lower than that of 2nd ad­di­tional level of $v_2$ are all pur­chased, $e9$ is in­dic­ated for the gain of rho between re­set as the cost of every 2 $v_2$ re­set dif­fer by a mag­nitude of $e9$
• $e6/1v_4$ re­set – A re­set pat­tern of which a particle re­sets when an ad­di­tional level of $v_4$, and all $v_1$, $v_2$, $v_3$ level with cost lower than that of ad­di­tional level of $v_4$ are all pur­chased, $e6$ is in­dic­ated for the gain of rho between re­set as the cost of every $v_4$ re­set dif­fer by a mag­nitude of $e6$
• Re­set cost – The total cost of pur­chas­ing $v_1$, $v_2$, $v_3$, and $v_4$ for a re­set
• $\rho$ of pub­lic­a­tion – The value of $\rho$ of last pub­lic­a­tion, i.e., the “In­put” in sim 3.0
• $\rho$ gain – The value of the gain of $\rho$ in the cur­rent pub­lic­a­tion
• “Be­fore” – The last level of des­ig­nated vari­able pur­chased be­fore the re­set
• “After” – The first level of des­ig­nated vari­able pur­chased after the re­set

Meth­od­o­logy #

Data col­lec­tion #

A series of MF CT pub­lic­a­tion was sim­u­lated in sim 3.0 from $0\rho$ to $e600\rho$ with step in­ter­val $1$ us­ing best over­all strategy (MFdCoast Depth: 0 $c_1$: xxx, MFd2­Coast Depth: 0 $c_1$: xxx and MFd3­Coast Depth: 0 $c_1$:xxx) in depth 0 (No brute­force). Due to the pur­pose of this in­vest­ig­a­tion and my tech­nical dif­fi­culties, in­vest­ig­a­tion in­volving a higher depth was not used. Then, data set entries con­sist of the de­tail of re­set po­s­i­tions, vari­able pur­chase po­s­i­tions, gain of $\rho$ of each of the 601 pub­lic­a­tions are ob­tained.

Data ana­lysis and se­lec­tion cri­teria #

Data sets re­lated to re­sets are isol­ated from the crude list to in­vest­ig­ate the pat­tern of re­set in each of the pub­lic­a­tion. The ${\log }_{10}\rho$ gain between re­set is cal­cu­lated and the re­set data sets are ex­cluded if the ${\log }_{10}\rho$ gain yields a res­ult of $15$ or above, in which most of the data set are caused by rapid changes of $\rho$ and the re­set pat­tern can­not be fol­lowed strictly by the sim­u­lator due to its ma­nip­u­la­tion lim­it­a­tion at the very start of a pub­lic­a­tion.

Sub­sequently, data sets re­lated to re­sets, as well as the data entry which con­sist of a $c_1$ “Be­fore”, a re­set, and a $c_1$ “After” is isol­ated for de­term­in­a­tion of a cri­terion to com­pare the cost of $c_1$ levels and the cost of re­set us­ing ra­tio (i.e., “re­l­at­ive cost ra­tio” between $c_1$, and re­set cost be­fore a re­set.). The set of entry is in­cluded only if all three above-men­tioned ele­ments is in­cluded (i.e. a $c_1$ “Be­fore”, a re­set, and a $c_1$ “After”). For ex­ample, with the ref­er­ence of Table 1 where the $c_1$ “Be­fore”, a re­set, and the $c_1$ “After” had been isol­ated. The re­l­at­ive cost ra­tio of $c_1$ “Be­fore” com­pared to re­set cost can be cal­cu­lated by $\frac{2.52e264}{1.45e265}=0.17379$, while the re­l­at­ive cost ra­tio of $c_1$ “After” com­pared to re­set cost can be cal­cu­lated by $\frac{5.04e264}{1.45e265}=0.34759$.

Class: new­words;
ID: table-1;

To de­term­ine the strength of a cost cri­terion as a cutoff, a Re­ceiver-op­er­at­ing Char­ac­ter­istic Curve (ROC) is plot­ted us­ing a series of cutoffs and the Area Un­der Curve (AUC) of the ROC is cal­cu­lated. The ex­act cutoff po­s­i­tion will be then de­term­ined by ex­plor­ing the list of “Be­fore” and “After” data with the aid of a box and whisker dia­gram. The ap­proach is re­peated for $c_2$, $a_1$, $a_2$, and $\delta$. More de­tailed ex­plan­a­tions on the nature and in­ter­pret­a­tion of graphs will be ex­plained sub­sequently.

Lastly, the re­set pat­tern and the ${\log }_{10}\rho$ gain of each of the 601 pub­lic­a­tions will be re­trieved for ana­lysis of pub­lic­a­tion be­ha­vior and ex­plore the re­la­tion­ship between $\rho$ of pub­lic­a­tion and the re­set pat­tern used, which will be use­ful for pre­dict­ing the pat­tern used in a pub­lic­a­tion $\rho$ without re­fer­ring to the sim­u­lator.

Re­ceiver-op­er­at­ing Char­ac­ter­istic (ROC) Curve #

A Re­ceiver-op­er­at­ing Char­ac­ter­istic (ROC) curve is a com­mon graph­ical plot visu­al­iz­ing the per­form­ance of a bin­ary clas­si­fier by plot­ting True Pos­it­ive Rate (Sens­it­iv­ity; $\frac{\text{no. of true pos­it­ive}}{\text{Total no. of pos­it­ive}}$) against False Pos­it­ive Rate (1 – Spe­cificity; $\frac{\text{no. of false pos­it­ive}}{\text{Total no. of neg­at­ive}}$) across all clas­si­fic­a­tion thresholds, mainly in ma­chine learn­ing and the med­ical field. It demon­strates the trade-off between sens­it­iv­ity and spe­cificity, with higher curves in­dic­at­ing bet­ter clas­si­fic­a­tion ac­cur­acy.

With the aid of the above concept, we can design a threshold which is to ef­fect­ively and cor­rectly sep­ar­ate the list of vari­ables pur­chased be­fore and after a re­set by their re­l­at­ive cost ra­tio (e.g., $c_1$ cost di­vided by re­set cost). Take an ex­ample of $c_1$ as the vari­able and $0.1$ as the re­l­at­ive cost ra­tio threshold, the in­ter­pret­a­tion of res­ult can be ref­er­enced by Table 2 and a set of data re­gard­ing “True Pos­it­ive Rate” $(y-axis)$ and “False Pos­it­ive Rate” $(x-axis)$, or a point of ROC Curve, can be ob­tained.

Class: strat_sep­ar­ated;
ID: table-2;

Next, re­peat the cal­cu­la­tion with a vari­ety of re­l­at­ive cost ra­tio threshold and ob­tain a set of data points re­gard­ing “True Pos­it­ive Rate” and “False Pos­it­ive Rate”. Plot­ting the dots in a graph will yield a ROC Curve. The Area Un­der the Curve (AUC) of a ROC curve is cal­cu­lated to meas­ure over­all per­form­ance. An AUC of a ROC of $1$ in­dic­ates a per­fect iden­ti­fic­a­tion of “Pos­it­ive” and “Neg­at­ive” ex­ists, while the AUC of a ROC of $0.5$ is no bet­ter than ran­dom guess­ing. The lar­ger the AUC of a ROC, the bet­ter the per­form­ance of a clas­si­fic­a­tion threshold, if ap­pro­pri­ately set. In sum­mary, one can in­ter­pret a ROC Curve as in­dic­ated in Graph 1.

There are three ap­proaches to de­term­ine the pre­ferred threshold when no def­in­ite threshold can be es­tab­lished (i.e. AUC of ROC Curve less than $1$), namely Youden’s In­dex ap­proach, min­imal dis­tance ap­proach (MDA), and weighted ap­proach. The Youden’s In­dex is cal­cu­lated by Sens­it­iv­ity $+$ Spe­cificity $–1$ (i.e., Sens­it­iv­ity $–$ “False Pos­it­ive rate”). Then the threshold will be de­term­ined when Youden’s In­dex reached max­imum.

The min­imal dis­tance ap­proach cal­cu­lates the dis­tance between dot plots of ROC Curve and an ima­gin­ary ideal situ­ation (i.e., $(0,1)$ on a ROC plot) via. the for­mula $\sqrt{(1-\text{Sens­it­iv­ity}{)}^2+(1-\text{Spe­cificity}{)}^2}$, The threshold will be de­term­ined when the dis­tance between the two men­tioned point is at min­imum. The fi­nal ap­proach takes ac­count of weighted factors in each situ­ation, such as the time loss due to “in­cor­rect” vari­able pur­chases. Since the factors and the weight­ing of each factor is sub­ject­ive and can dif­fer from user to user, this method will be omit­ted in sub­sequent con­sid­er­a­tions of re­l­at­ive cost ra­tio threshold.

Box and whisker dia­gram #

A Box and whisker dia­gram is an­other com­mon graphic plot visu­al­iz­ing the dis­per­sion of a sets of dis­crete data, as well as the range, Lower Quart­ile (Q1), Me­dian (Q2), Up­per Quart­ile (Q3) and out­liers, if any, of a data set in stat­ist­ical ana­lysis. In this in­vest­ig­a­tion, a box and whisker dia­gram will il­lus­trate the spread of data set, as well as giv­ing an ap­prox­im­ate pic­ture of how a re­l­at­ive cost ra­tio threshold per­form when com­pared to sim 3.0. With a box and whisker dia­gram, one can in­ter­pret as be­low in Graph 2.

Res­ults #

A cu­mu­lat­ive num­ber of $83,736$ data set entries were ob­tained from sim v3.0. The data was fur­ther re­fined based on the aim of dif­fer­ent in­vest­ig­a­tions. Fur­ther de­tails are presen­ted in cor­res­pond­ing sec­tions.

Re­set pat­tern vs. ${\log }_{10}\rho$ of pub­lic­a­tion #

A total of $20,032$ data sets re­lated to re­sets are isol­ated from the crude list cor­res­pond­ing to the $\rho$ of pub­lic­a­tion to in­vest­ig­ate the pat­tern of re­set in each of the pub­lic­a­tion. There are mainly three types of re­set pat­terns used in dif­fer­ent MF CT pub­lic­a­tion $(n=601)$, they are $e4.5/1v_2$ re­set $(233/601,38.77\mathrm{\%})$, $e9/2v_2$ re­set $(274/601,45.59\mathrm{\%})$, and $e6$/$1v_4$ re­set $(94/601,15.64\mathrm{\%})$. The use of three types of re­set pat­terns in their re­spect­ive $\rho$ of pub­lic­a­tion is presen­ted be­low (Graph 3), ex­clud­ing out­liers, one can sum­mar­ize that $e4.5/1v_2$ re­set are the main pat­tern used be­fore $e220\rho$, then the pat­tern gradu­ally shifts to $e9/2v_2$ re­set and be­come the main­stream pat­tern after $e265\rho$. The pat­tern con­tin­ues un­til $e480$, when $e6/1v_4$ re­set star­ted to be in­creas­ingly used un­til $e600\rho$. The trend of us­age of re­set pat­tern is il­lus­trated in Graph 4.

$c_1$ cost vs. re­set cost #

A total of $1,121$ sets of data entries con­sist­ing of a $c_1$ “Be­fore”, a re­set, and a $c_1$ “After” were iden­ti­fied from the crude data set and the re­l­at­ive cost ra­tio of $c_1$ “Be­fore” and $c_1$ “After” were cal­cu­lated and com­pared. A ROC Curve were plot­ted (Graph 5) and AUC were cal­cu­lated to be $0.731$ in­dic­at­ing a mod­er­ate-strength threshold, if ap­pro­pri­ately set, ex­ists for $c_1$ cost vs. re­set cost with a con­sid­er­able num­ber of in­con­sist­en­cies. The ac­cur­acy of identi­fy­ing $c_1$ “Be­fore” and $c_1$ “After” us­ing a series of re­l­at­ive cost ra­tio thresholds ran­ging from $0.01$ to $2.00$ with in­ter­val of $0.01$ $(x-axis)$ and the cor­res­pond­ing Youden’s In­dex and MDA $(y-axis)$ are dis­played in Graph 6 and Graph 7 re­spect­ively.

While Youden’s In­dex achieves max­imum when $c_1$ re­l­at­ive cost ra­tio threshold is set at $0.19$ (Sim­ilar value in­ter­val $0.17–0.23$), dis­tance from ideal via. MDA reaches min­imum at $c_1$ re­l­at­ive cost ra­tio threshold of $0.29$ (Sim­ilar value in­ter­val $0.19–0.67$). Since Youden’s In­dex meas­ures the greatest ver­tical dis­tance between ROC curve and ran­dom guess­ing line (i.e., straight line con­nect­ing $(0,0)$ and $(1,1)$ of a ROC curve), us­ing the lower $c_1$ re­l­at­ive cost ra­tio threshold cal­cu­lated with Youden’s In­dex Ap­proach en­sures more $c_1$ “After” to be iden­ti­fied $(1,024/1,121,91.35\mathrm{\%})$ while the ac­cur­acy of identi­fy­ing $c_1$ “Be­fore” is sig­ni­fic­antly lower $(475/1,121,42.37\mathrm{\%})$. On the other hand, us­ing threshold from MDA usu­ally en­sures a bal­anced res­ult, es­pe­cially for skewed data. In this case, both identi­fy­ing $c_1$ “Be­fore” $(590/1,121,52.63\mathrm{\%})$ and $c_1$ “After” $(830/1,121,74.04\mathrm{\%})$ are con­sid­er­ably ap­pro­pri­ate with the threshold of $0.29$ cal­cu­lated above.

A not­able men­tion is the min­imum dis­tance cal­cu­lated in MDA re­mains at a low plat­eau from $c_1$ re­l­at­ive cost ra­tio $0.19$ to $0.67$, in re­sponse to this, two ad­di­tional $c_1$ re­l­at­ive cost ra­tios of $0.40$ and $0.50$ are eval­u­ated. For $c_1$ re­l­at­ive cost ra­tio of $0.40$, ac­curacies of identi­fy­ing $c_1$ “Be­fore” and $c_1$ “After” be­ing $66.37\mathrm{\%}(744/1,121)$ and $56.74\mathrm{\%}(636/1,121)$, while those for $c_1$ re­l­at­ive cost ra­tio of $0.50$ are $75.02\mathrm{\%}(841/1,121)$ and $51.20\mathrm{\%}(574/1,121)$ re­spect­ively. A box and whisker dia­gram com­par­ing $c_1$ “Be­fore” and $c_1$ “After” is il­lus­trated be­low as sup­ple­ment­ary in­form­a­tion (Graph 8).

$c_2$ cost vs. re­set cost #

A total of $1,056$ sets of data entries con­sist­ing of a $c_2$ “Be­fore”, a re­set, and a $c_2$ “After” were iden­ti­fied from the crude data set and the re­l­at­ive cost ra­tio of $c_2$ “Be­fore” and $c_2$ “After” were cal­cu­lated and com­pared. A ROC Curve was plot­ted (Graph 9) and AUC were cal­cu­lated to be $1.000$ in­dic­at­ing a per­fect threshold ex­ists for $c_2$ cost vs. re­set cost. With ref­er­ence to a box and whisker dia­gram com­par­ing $c_2$ “Be­fore” and $c_2$ “After” (Graph 10), a sub­sequent threshold of re­l­at­ive cost ra­tio was set to be $1.0$ and the ac­cur­acy of identi­fy­ing $c_2$ “Be­fore” and $c_2$ “After” were $100\mathrm{\%}$ and $100\mathrm{\%}$ re­spect­ively. This res­ult can be in­ter­preted as a threshold of re­l­at­ive cost ra­tio $1.0$ is def­in­ite for $c_2$ cost vs re­set cost.

$a_1$ cost vs. re­set cost #

A total of $1,095$ sets of data entries con­sist­ing of a $a_1$ “Be­fore”, a re­set, and a $a_1$ “After” were iden­ti­fied from the crude data set and the re­l­at­ive cost ra­tio of $a_1$ “Be­fore” and $a_1$ “After” were cal­cu­lated and com­pared. A ROC Curve were plot­ted (Graph 11) and AUC were cal­cu­lated to be $0.865$ in­dic­at­ing a mod­er­ate-to-good-strength threshold, if ap­pro­pri­ately set, ex­ists for $a_1$ cost vs. re­set cost with a con­sid­er­able num­ber of in­con­sist­en­cies. Since the re­l­at­ive cost ra­tio of $a_1$ spans across a con­sid­er­able range of mag­nitudes, the re­l­at­ive cost ra­tio has been con­ver­ted to ${\log }_{10}$ to fa­cil­it­ate easier com­par­is­ons. The ac­cur­acy of identi­fy­ing $a_1$ “Be­fore” and $a_1$ “After” us­ing a series of re­l­at­ive cost ra­tio thresholds ran­ging from $-5.00$ to $2.00$ with in­ter­val of $0.01$ $(x-axis)$ and the cor­res­pond­ing Youden’s In­dex and MDA $(y-axis)$ are dis­played in Graph 12 and Graph 13 re­spect­ively. A sim­ilar meth­od­o­logy de­scribed in “$c_1$ cost vs re­set cost” sec­tion was ad­op­ted. Both Youden’s In­dex Ap­proach and MDA gives a con­sensus of an $a_1$ re­l­at­ive cost ra­tio threshold at $-1.05{\log }_{10}$, which in­dic­ates ${10}^{-1.05}=0.112$ for the ac­tual $a_1$ re­l­at­ive cost ra­tio threshold. In this case, both ac­curacies in identi­fy­ing $a_1$ “Be­fore” $(917/1,095,83.74\mathrm{\%})$ and $a_1$ “After” $(798/1,095,72.88\mathrm{\%})$ are re­l­at­ively good with the threshold of $0.112$ cal­cu­lated above. With a box and whisker dia­gram com­par­ing $a_1$ “Be­fore” and $a_1$ “After” il­lus­trated be­low (Graph 14), it may be more sens­ible to set a threshold of ${10}^{-1.30}=0.05$, in which achieve an ac­cur­acy of $77.17\mathrm{\%}(845/1,095)$ for identi­fy­ing $a_1$ “Be­fore” and $77.63\mathrm{\%}(850/1,095)$ for identi­fy­ing $a_1$ “After”.

$a_2$ cost vs. re­set cost #

A total of $1,107$ sets of data entries con­sist­ing of a $a_2$ “Be­fore”, a re­set, and a $a_2$ “After” were iden­ti­fied from the crude data set and the re­l­at­ive cost ra­tio of $a_2$ “Be­fore” and $a_2$ “After” were cal­cu­lated and com­pared. A ROC Curve was plot­ted (Graph 15) and AUC were cal­cu­lated to be $1.000$ in­dic­at­ing a per­fect threshold ex­ists for $a_2$ cost vs. re­set cost. With ref­er­ence to a box and whisker dia­gram com­par­ing $a_2$ “Be­fore” and $a_2$ “After” (Graph 16), a sub­sequent threshold of re­l­at­ive cost ra­tio was set to be $1.0$ and the ac­cur­acy of identi­fy­ing $a_2$ “Be­fore” and $a_2$ “After” were $100\mathrm{\%}$ and $100\mathrm{\%}$ re­spect­ively. This res­ult can be in­ter­preted as a threshold of re­l­at­ive cost ra­tio $1.0$ is def­in­ite $a_2$ cost vs. re­set cost.

$\delta$ cost vs. re­set cost #

A total of $941$ sets of data entries con­sist­ing of a $\delta$ “Be­fore”, a re­set, and a $\delta$ “After” were iden­ti­fied from the crude data set and the re­l­at­ive cost ra­tio of $\delta$ “Be­fore” and $\delta$ “After” were cal­cu­lated and com­pared. A ROC Curve was plot­ted (Graph 17) and AUC were cal­cu­lated to be $0.999$ in­dic­at­ing an ex­cel­lent threshold, if ap­pro­pri­ately set, ex­ists for $\delta$ cost vs. re­set cost with oc­ca­sional in­con­sist­en­cies. With ref­er­ence to a box and whisker dia­gram com­par­ing $\delta$ “Be­fore” and $\delta$ “After” (Graph 18), a sub­sequent threshold of re­l­at­ive cost ra­tio was set to be $1.0$ and the ac­cur­acy of identi­fy­ing $\delta$ “Be­fore” and $\delta$ “After” were $100\mathrm{\%}$ and $98.62\mathrm{\%}$ re­spect­ively. This res­ult can be in­ter­preted as a threshold of re­l­at­ive cost ra­tio $1.0$ is ex­cel­lent for $\delta$ cost vs. re­set cost. The ac­cur­acy of identi­fy­ing $\delta$ “Be­fore” and $\delta$ “After” us­ing other re­l­at­ive cost ra­tio thresholds and the cor­res­pond­ing Youden’s In­dex and MDA are dis­played in Table 3. Both Youden’s In­dex and MDA are co­her­ent in set­ting the re­l­at­ive cost ra­tio threshold as $1.0$ is more ideal than any other thresholds. The in­con­sist­ency of fail­ure to identi­fy­ing $\delta$ “After” was ret­ro­spect­ively found to be in some re­sets between $e86$ to $e95$ and $e152$ pub­lic­a­tions. Us­ing the re­l­at­ive cost ra­tio threshold as $1.0$ for $\delta$ may res­ult in time dif­fer­ence com­pared to the sim­u­la­tion res­ult.

Class: new­words;
ID: table-3;

$\rho$ gain vs $\rho$ of pub­lic­a­tion #

Throughout the 9 months of in­vest­ig­a­tions, sev­eral hy­po­theses con­cern­ing the re­la­tion­ship between $\tau$ gain, $\tau$ of pub­lic­a­tions, and re­set pat­terns have been pro­posed. First part of this in­vest­ig­a­tion was to eval­u­ate the ef­fect of re­set pat­tern on the ${\log }_{10}\rho$ gain of the pub­lic­a­tion, it was hy­po­thes­ized that the rho dif­fer­ence between re­sets has some in­flu­ence on the ${\log }_{10}\rho$ gain of the pub­lic­a­tion, which was dis­proved by a graph plot­ting $\frac{{\log }_{10}(\rho \text{ gain})}{{\log }_{10}(\text{re­set pat­tern }\rho )}$ $(y-axis)$ against ${\log }_{10}(\rho \text{ of pub­lic­a­tion})$ $(x-axis)$ (Graph 19). The graph re­veals no vis­ible re­la­tion­ship between re­set pat­tern and ${\log }_{10}(\rho \text{ gain})$ of the pub­lic­a­tion, des­pite sev­eral at­tempts us­ing arith­metic mean, geo­met­ric mean, and root-mean-square (RMS) of data sets.

The second part of this sec­tion in­vest­ig­ates the ef­fect of ${\log }_{10}\tau$ of pub­lic­a­tion on ${\log }_{10}\tau$ gain in a pub­lic­a­tion and, hence, in­vest­ig­ates the pos­sib­il­ity of a pub­lic­a­tion table where one can com­plete MF CT at a shorter time with a series of set pub­lic­a­tion rhos. To verify the above in­vest­ig­a­tion, the ${\log }_{10}\tau$ gain was ob­tained and plot­ted against ${\log }_{10}\rho$ of pub­lic­a­tion for MF CT and a sub­sequent mov­ing-av­er­age of ${\log }_{10}\tau$ gain of the pre­vi­ous 20 $\tau$ was also plot­ted (Graph 20). Sim­ilar graphs were also plot­ted for Basal Prob­lem (BaP CT; Graph 21) where a pub­lic­a­tion table is veri­fied to be ef­fect­ive in short­en­ing the com­ple­tion time of BaP CT, and for Wei­er­strass Sine Product (WSP CT; Graph 22) where there is no pub­lic­a­tion table for com­par­ison.

One can ap­pre­ci­ate the mov­ing-av­er­age plot for BaP CT is re­l­at­ively spiky, while that for WSP CT is re­l­at­ively a straight line with slight up-and-down des­pite highly fluc­tu­at­ing ${\log }_{10}\tau$ gain val­ues. It is also no­tice­able that ${\log }_{10}\tau$ gain fluc­tu­ates more for pub­lic­a­tion that goes across a mile­stone when com­pared to other $\tau$ of pub­lic­a­tion when the pub­lic­a­tion does not go across a mile­stone.

With the graph plot­ted for MF CT com­par­ing to the two con­trolled graphs, one can ap­pre­ci­ates the mov­ing-av­er­age plot for MF CT re­mains re­l­at­ively straight but in­cludes some spikes that has lar­ger amp­litudes than those in WSP CT but smal­ler amp­litudes than those in BaP CT, with large fluc­tu­ation for pub­lic­a­tion that goes across a mile­stone. With the ob­ser­va­tion, one can con­clude a pub­lic­a­tion table, like BaP CT, is less likely but still pos­sible for MF CT. Find­ing the des­ig­nated pub­lic­a­tion po­s­i­tion was out of the scope of this in­vest­ig­a­tion, sub­sequent in­vest­ig­a­tion would be needed if any other evid­ence sug­gests a pub­lic­a­tion table.

Con­clu­sion and Dis­cus­sion #

Con­clu­sion #

The above in­vest­ig­a­tions il­lus­trate the fact that a gen­er­al­ized re­set pat­tern across dif­fer­ent ${\log }_{10}\rho$ of pub­lic­a­tion can be found in MF CT (with a con­sid­er­able por­tion of un­ex­plained ex­cep­tional cases). By in­vest­ig­at­ing the re­la­tion­ship of re­l­at­ive cost ra­tio of $c_1$, $c_2$, $a_1$, $a_2$, and $\delta$ “Be­fore” and “After” to re­set cost, us­ing a re­l­at­ive cost ra­tio threshold of $1.0$ on $c_2$, $a_2$, and $\delta$ is ex­cel­lent in dif­fer­en­ti­at­ing “Be­fore” and “After” for a re­set. Mean­while, the strength of a re­l­at­ive cost ra­tio threshold of $c_1$ and $a_1$ is lim­ited, and a range of the re­l­at­ive cost ra­tio of $c_1$ and $a_1$ was sug­ges­ted with vari­able ac­curacies which need fur­ther evid­ence-based in­vest­ig­a­tions to dis­cover un­der­ly­ing cri­teria. The ${\log }_{10}\rho$ gain of the pub­lic­a­tion has no re­la­tion­ship with re­set pat­terns, and a pub­lic­a­tion table is less likely but still pos­sible for MF CT.

Eval­u­at­ing the ef­fect of rho of pub­lic­a­tion on re­set pat­terns #

By re­call­ing the for­mula of MF CT, one can sim­plify the for­mula for rho dot as fol­low,

Where $\alpha$, $\beta$, and $\gamma$ are ex­po­nents that only changes when cor­res­pond­ing mile­stones are used.

The un­der­ly­ing reas­ons for the trans­ition of $e4.5/1v_2$ re­set to $e9/2v_2$ re­set from $e220\rho$ to $e260\rho$ are prob­ably due to the in­crease dom­in­ance of $t$ (via. $x$) in growth of $\dot{\rho }$ in re­spond to lengthened time between re­sets and mile­stone 5 and 6 (at $e275\rho$ and $e325\rho$ re­spect­ively) which in­creases the ex­po­nent of $x$, hence length­en­ing the re­set time to be­ne­fit ad­di­tional $\dot{\rho }$ due to $t$ by “com­bin­ing” two $e4.5/1v_2$ re­sets into one single $e9/2v_2$ re­set. As MF CT pro­gresses, value of $v_4$ gradu­ally in­creases and sig­ni­fic­antly ac­count for the growth of $v$ with the ef­fect of mile­stone 7 and 8 (at $e425$ and $e475\rho$ re­spect­ively), such that the ef­fect $v_4$ starts to over­come that of $v_2$ (See Graph 23), the re­set pat­tern gradu­ally trans­its again from $e9/2v_2$ re­set to $e6/1v_4$ re­set. However, the above hy­po­thesis has not been veri­fied and re­quire fur­ther veri­fic­a­tion with ac­count for the ef­fect of $t$ on $\dot{\rho }$. The ex­ist­ence of out­lier has also not been ex­plained in this in­vest­ig­a­tion; fur­ther eval­u­ation is needed and un­der­ly­ing cri­teria may be yet to be dis­covered.

Eval­u­at­ing the Ef­fect of time of re­set #

The above sim­u­la­tions were sim­u­lated in close to real-life situ­ations when a game with proper $c_1$, $c_2$, $a_1$, $a_2$, and $\delta$ levels be­ing pur­chased based on the cri­teria provided by MFd, MFd2, and MFd3. However, the above in­vest­ig­a­tions were con­duc­ted in a static man­ner, while the real game in MF CT is a dy­namic game pro­gres­sion with $t_s$ in­flu­en­cing $x$ (to­gether with $v_2$) and, hence, $\dot{\rho }$. With the de­riv­a­tion of for­mula in pre­vi­ous sec­tion, one can con­clude $\dot{\rho }$ is pro­por­tional to $t_s^{\beta }$, where $\beta$ de­pends on the num­ber of mile­stone (i.e., $\beta =3.2$ be­fore Mile­stone 5, $\beta =3.3$ between Mile­stone 5 and 6 and $\beta =3.4$ af­ter­ward). This may be the part of the ex­plan­a­tion why the trans­ition point from $e9/2v_2$ re­set to $e6/1v_4$ re­set at a later-than-ex­pec­ted rho. Fur­ther evid­ence will be needed in or­der to verify this hy­po­thesis.

Eval­u­at­ing the Ef­fect of strategy used in pub­lic­a­tion #

The MF CT pub­lic­a­tions were sim­u­lated by sim 3.0 us­ing the best rate out of the three strategies de­veloped by play­ers – MFdCoast Depth: 0 $c_1$: xxx $(50/601,8.32\mathrm{\%})$, MFd2­Coast Depth: 0 $c_1$: xxx $(236/601,39.27\mathrm{\%})$, and MFd3­Coast Depth: 0 $c_1$:xxx $(315/601,52.41\mathrm{\%})$. The use of three types of strategy in their re­spect­ive $\rho$ of pub­lic­a­tion is presen­ted be­low (Graph 24). To­gether with the trend of us­age of re­set pat­tern is il­lus­trated in Graph 25, one can sum­mar­ize that MFd2 and MFd3 were al­tern­at­ively used throughout MF CT, with MFd3 fi­nally dom­in­ate over MFd2 after $e470\rho$. The use of MFd were low un­til a gradual rise after $e560\rho$, re­pla­cing MFd2.

In this in­vest­ig­a­tion, it is un­clear that whether the strategy ad­op­ted in a pub­lic­a­tion altered the res­ults of above in­vest­ig­a­tions, es­pe­cially on $c_1$ and $a_1$ re­l­at­ive cost ra­tio due to the three dis­crete buy­ing cri­teria used in the three strategies. With re­spond to this, the $c_1$ and $a_1$ data sets were fur­ther di­vided into three groups based on the strategy used in the pub­lic­a­tion, their ROC Curves were plot­ted, their AUC of ROC Curve, Youden’s In­dex ap­proach, and MDA were eval­u­ated and com­pared, and the sig­ni­fic­ance of the ef­fect of buy­ing strategies for $c_1$ and $a_1$ were com­pared in Table 4 and Table 5 re­spect­ively. Since the scale of $c_1$ cost for each $c_1$ level is $2$ and the scale of $a_1$ cost for each $a_1$ level is $25$ throughout MF CT, A box and whisker dia­gram com­par­ing only $c_1$ “Be­fore” and $a_1$ “Be­fore” for three strategies have been plot­ted (Graph 26 and Graph 27).

Class: strat_sep­ar­ated;
ID: table-4;

Class: strat_sep­ar­ated;
ID: table-5;

By com­par­ing $c_1$ and $a_1$ re­l­at­ive cost ra­tio of pub­lic­a­tion us­ing MFd2 with those us­ing MFd, one may ob­serve the $c_1$ re­l­at­ive cost ra­tio threshold used in pub­lic­a­tion of MFd2 is higher than that of MFd while that of $a_1$ is lower, and is stat­ist­ic­ally sig­ni­fic­ant, re­flect­ing the dif­fer­ence in $c_1$ and $a_1$ buy­ing cri­teria al­ters the $c_1$ and $a_1$ re­l­at­ive cost ra­tio threshold. The com­par­ison between MFd and MFd3 is in­sig­ni­fic­ant as they share the same $c_1$ buy­ing cri­teria in their re­spect­ive pub­lic­a­tions. The reason for lower $c_1$ re­l­at­ive cost ra­tio threshold and higher $a_1$ re­l­at­ive cost ra­tio threshold in pub­lic­a­tion us­ing MFd3 is yet to be known. To sum­mar­ize, the ad­jus­ted $c_1$ and $a_1$ re­l­at­ive cost ra­tio should be ad­jus­ted ac­cord­ingly de­pend on the strategy used in the pub­lic­a­tion, as de­scribed in Table 6. However, the un­der­ly­ing mech­an­ism(s) for de­term­in­a­tion of strategy used in a pub­lic­a­tion has not been found in these in­vest­ig­a­tions, which is es­sen­tial for de­term­in­ing the $c_1$ and $a_1$ re­l­at­ive cost ra­tio threshold.

Class: strat_sep­ar­ated;
ID: table-6;

Ac­know­ledge­ment #

Lastly, I would like to give a huge thanks to the fol­low­ing people/​group of people for as­sist­ing the veri­fic­a­tion of hy­po­thesis and fur­ther find­ings on MF CT:

• Mathis S.

  • For design­ing the MF CT, sug­gest­ing some hy­po­thesis to be veri­fied and design­ing sim­u­la­tion for MF CT via. sim 3.0 for data re­triev­als.

• Hotab, ba­sic­ally, i am little cat

  • For design­ing sim­u­la­tions for MF CT for data re­triev­als and re­fine the hy­po­thesis upon test­ing.

• Maimai, Black Seal

  • For sug­gest­ing the strategies used in MF CT.

• All other people

  • For provid­ing ex­per­i­mental data and provid­ing sup­port whenever I need them.