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a/docs/index.rst b/docs/index.rst index 02275d3..cc111b3 100644 --- a/docs/index.rst +++ b/docs/index.rst @@ -2,6 +2,7 @@ .. include:: examples.rst +.. include:: motivation.rst Full API documentation ====================== diff --git a/docs/introduction.rst b/docs/introduction.rst index 86c420f..96a36ff 100644 --- a/docs/introduction.rst +++ b/docs/introduction.rst @@ -14,7 +14,7 @@ The easiest way right now is to install it straight from Github using Pip: pip install -e git://github.com/better/convoys#egg=convoys -Motivation +Background ---------- Predicting conversions is a really important problem for ecommerce, online advertising, and many other applications. @@ -30,7 +30,7 @@ First of all, you can not learn from users that are younger than X. You also can not learn from users that convert *after* X. For an excellent introduction to this problem (and distributions like the `Weibull distribution `_), here's a blog post about `implementing a recurrent neural network to predict churn `_. -Survival analysis to the rescue +Survival analysis saves the day ------------------------------- Luckily, there is a somewhat similar field called `survival analysis `_. diff --git a/docs/motivation.rst b/docs/motivation.rst new file mode 100644 index 0000000..2ebac5b --- /dev/null +++ b/docs/motivation.rst @@ -0,0 +1,10 @@ +Motivation +========== + +A perfectly valid question is: Convoys really only implements one model: a generalized Gamma distribution multiplied by basically logistic regression. That seems like a very specialized distribution. In order to justify this choice, let's first look at a handful of conversion charts from real data at `Better `_: + +.. image:: images/conversion.gif + +The legend, labels of the axes, and title are all removed so that no business secrets are revealed. The solid lines with shaded area are all the generalized Gamma fits, whereas the dashed lines are the Kaplan-Meier fits. Note that the fit is very good! In fact, we have observed that almost any conversion metric can be modeled reasonably well with the generalized Gamma model (multiplied by logistic regression). + +Empirically, this model seems to hold up pretty well. 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so that no business secrets are revealed. The solid lines with shaded area are all the generalized Gamma fits, whereas the dashed lines are the Kaplan-Meier fits. Note that the fit is very good! In fact, we have observed that almost any conversion metric can be modeled reasonably well with the generalized Gamma model (multiplied by logistic regression). Empirically, this model seems to hold up pretty well. + +Some more mathematical justification +------------------------------------ + +A simple toy problem also demonstrates why we would expect to get a time-dependent distribution (like the Exponential distribution) multiplied by a logistic function. Consider the `continuous-time Markov chain `_ with three states: undecided, converted, or died. + +.. image:: images/convoys-markov-chain.png + :height: 300px + :width: 300px + +Everyone starts out "undecided" but either converts or dies. However, we *only observe the conversions,* not the deaths. + +We can solve for the distribution by thinking of this as a partial differential equation. The distribution over the three states will be given as :math:`P(t) = e^{tA}` where :math:`A` is the `generator matrix `_ + +.. math:: + A = \left( { + \begin{array}{ccc} + -(\lambda_1 + \lambda_2) & \lambda_1 & \lambda_2 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 + \end{array} + } \right) From 613940fbec12aa36af1549422e393b5538bf823d Mon Sep 17 00:00:00 2001 From: Erik Bernhardsson Date: Mon, 3 Dec 2018 21:16:09 -0500 Subject: [PATCH 3/6] simplify and expand --- docs/motivation.rst | 24 ++++++++++++++++-------- 1 file changed, 16 insertions(+), 8 deletions(-) diff --git a/docs/motivation.rst b/docs/motivation.rst index 57c797d..1dc9b65 100644 --- a/docs/motivation.rst +++ b/docs/motivation.rst @@ -20,13 +20,21 @@ A simple toy problem also demonstrates why we would expect to get a time-depende Everyone starts out "undecided" but either converts or dies. However, we *only observe the conversions,* not the deaths. -We can solve for the distribution by thinking of this as a partial differential equation. The distribution over the three states will be given as :math:`P(t) = e^{tA}` where :math:`A` is the `generator matrix `_ +We can solve for the distribution by thinking of this as a partial differential equation. The solution turns out to be quite simple: .. math:: - A = \left( { - \begin{array}{ccc} - -(\lambda_1 + \lambda_2) & \lambda_1 & \lambda_2 \\ - 0 & 0 & 0 \\ - 0 & 0 & 0 - \end{array} - } \right) + P_{\mbox{converted}}(t) = \frac{\lambda_1}{\lambda_1 + \lambda_2}(1 - \exp(-(\lambda_1 + \lambda_2)t)) + +As you can see, the solution is an exponential distribution (the :math:`1 - \exp(-(\lambda_1 + \lambda_2)t)` part) multiplied by a constant factor (the :math:`\lambda_1/(\lambda_1 + \lambda_2)` part). + +Turning it into a regression problem +------------------------------------ + +Note that :math:`\lambda_1` and :math:`\lambda_2` are positive numbers. For each observation :math:`z`, let's write it as a linear combination :math:`\lambda_1 = \exp(a^Tz)` and :math:`\lambda_2 = \exp(b^Tz)` where :math:`a, b` are two unknown vectors. + +With this transformation, the probability of conversion becomes + +.. math:: + P_{\mbox{converted}}(t) = \frac{1}{1 + \exp(-(a-b)^Tz)} + +This is the `sigmoid function `_ which means that we are basically doing logistic regression in the limit where :math:`t \rightarrow \infty`. From 082ad2fe3c4a29cd78c6139a3c61ebfc1d7d0404 Mon Sep 17 00:00:00 2001 From: Erik Bernhardsson Date: Mon, 3 Dec 2018 21:19:16 -0500 Subject: [PATCH 4/6] minor --- docs/motivation.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/motivation.rst b/docs/motivation.rst index 1dc9b65..d97cc66 100644 --- a/docs/motivation.rst +++ b/docs/motivation.rst @@ -35,6 +35,6 @@ Note that :math:`\lambda_1` and :math:`\lambda_2` are positive numbers. For each With this transformation, the probability of conversion becomes .. math:: - P_{\mbox{converted}}(t) = \frac{1}{1 + \exp(-(a-b)^Tz)} + P_{\mbox{converted \rightarrow \infty}}(t) = \frac{1}{1 + \exp(-(a-b)^Tz)} -This is the `sigmoid function `_ which means that we are basically doing logistic regression in the limit where :math:`t \rightarrow \infty`. +This is the `sigmoid function `_ which means that we are doing logistic regression in the limit where :math:`t \rightarrow \infty`. From 586eb4cbca3c7aa0f5cc7c2fa79e3976cfc521c7 Mon Sep 17 00:00:00 2001 From: Erik Bernhardsson Date: Tue, 4 Dec 2018 12:42:24 -0500 Subject: [PATCH 5/6] more work on distributions --- ...chain.png => exponential-markov-chain.png} | Bin docs/images/gamma-markov-chain.png | Bin 0 -> 110366 bytes docs/images/probability-distributions.png | Bin 0 -> 60548 bytes docs/motivation.rst | 40 ++++++++++++++---- 4 files changed, 31 insertions(+), 9 deletions(-) rename docs/images/{convoys-markov-chain.png => exponential-markov-chain.png} (100%) create mode 100644 docs/images/gamma-markov-chain.png create mode 100644 docs/images/probability-distributions.png diff --git a/docs/images/convoys-markov-chain.png b/docs/images/exponential-markov-chain.png similarity index 100% rename from docs/images/convoys-markov-chain.png rename to docs/images/exponential-markov-chain.png diff --git a/docs/images/gamma-markov-chain.png b/docs/images/gamma-markov-chain.png new file mode 100644 index 0000000000000000000000000000000000000000..8823d20381554998a3963cf7574a85b9c7829be2 GIT binary patch literal 110366 zcmeFZWmHw&_Xes*P(Y;xBqRl-yW=QGmvn<79S1mc9}5rxK}uRB1SF&zR1`%*r5i!I zK|=Dbea^x6S9jbm_v8N^!|@{Sy<)Bz&zkd@J6uac@ge~=!KqWHE-EX@Yo9uWiv#|Y z;GG4(IsZ{@7yR!uQd{xHsiIEWdGHr}S0y9lsZ(64(4W)U(g}H|PQlLG>l%6(s;fy@ zA)I+`-$mT9=J9cM1y7$kCFLUless3>xXs|>?Bs%!@R4RhKOq5rhJMY<#DIRp!%>>a zP+f~b4&iRiAj~7g!^b2;z`(#D<$l*jLR($|`#SiWG?Sf&hpPlHueY~1kGCKX!rhja zUtC+401A#yfxCw-QLy19^t|OjeGkJ!qY>Vi3ytM z?|+zedf40iGm{GvyDhLmUg%$V`FZ$w|Bek_m4bdNp>2&sIC($%u_$OuTGAO6o@ z|1%rrEjfg^ zIlVNBpG0f@Tx?C{22Ygh1BOuIsOkrf)2ukRjaZ)I#$R%k?#G+lV|cT;Y+;1xyT3KI zb&DJq_v4o3v=#|5BOHGEbj?c~B{&H_<48^S@oHMyWq;Q0Ug2l+^9w8EJ5m|lX5$8W zt2+h7s(b2s1_{@2@Q4}UA*W8m{_ltX6T<&#!v9&p|Cbe%=QqFBzi2x0kE9+kl-x45 zb;y^@+PkM)HCOa(zac7U%Au!R%*5!GOkx7B@>ckY@!=k?{h(YPkBac#iPJ&l;3?sW zJRH@)Kk-4->?NK<+gCnYkbhaiCQgO_{+xD0U7bww5eIf^)FXWG z`jfs#|Gdt{$L}26q-&Atpb2yB`Ylnk=QvGjKsgrgpm}dC>i1rsZen)+$z**d@EfiT z_|AX5pXCD_^SEunSNTWIehu&Nmg@%VjM8AcX8EFez5B^@aff6W11Y@5z&McHs@`+5 z5Dnr1;fwD2-qI|G++UX)t=~ouMofH^D8#--LJp>*7=;eS^tM?zEIFyUS6@2y|<{_pJRM0^dke; z2O*~qhK$%Te-l*SC`o*O{=}W5NJzuZ1*bk;SkL>kdWs{GGHjf_p1qeomzs#{7G``D zA;nYTrHT89xs&aXkPq-LJ8I+?YYJZ>{1GSM_j}H1*tx31Yv6WhQSEesiOn-M&Cbe! zI}t@gn||Huq8*>2=`$i}L|xLTQN^Y;?rF7mK0aQ`%S~@f_n8cL`VxE9EZE;%_Gou% zP>x8tM~~k3SB%s0P<2SL{WOt%*SPA|a2}7E9(EHFYVaE+9|)^|l4AXlf|lin^5SaJ zwJcpCO;7owVPty}{~%A@QtWcgZ2I68|D_o96cO$e2~P(D*xEzhfZHNoO9SL8n{D}e z1*Nh_>(ORk-rbC#us%=0^?9Xne`TqtZl!Id>G;U0BUzaFlIci#pnCpn4z&THOh*0< zj6QgTG>0818l4lPW;zLu%@(+m;mt8P2f4bUe-lHKEvzF^Sgo9de7^aeGjv;Pr-}^N8%3cH>D>#IsL_=>admaZVm2EA%UN= zyA$^tzP#fss%G-vo6ikJwsLlVn)I6N%Jmy|bs4DqZ>4YJb%hhG$Lda@$UkUzKMUk! zOY_j4F8#_=z-7@WHOg4l=^h=;LQ~IPz4+ffqg!8sjYJh0oIZ%X`4CVYWwB!#CjR1ojPbm=T`2vlj@jzn zAJuMC9BiqsH~M=W3k-@om96X@9>9}Y>qwVuvRxbP7d5IUxM; z_?d3Mq+5BH%)V~p!8EVImEePQg&H7y+GgfUxOei+ddh&Rts@FWXP`$|mTt-WtH$-C=$k|clV|5!!r z`6rs|4WhklI&|)L?CO>)3j#KtWq1frDB1jeWN+ctYe`!t_GnS?T^ws8bz}>PqHa;c_K?j?@lqWM=H@g-Roq28pKit9|38+_ivL7;KhQhBD@P*h zZr2;5GLC*TZ?iKxw?xcXQDS{=?V%DDwX??UHA(*ez3OH>0@|nBo+kNnDl6^j(scre zv3h~&{1$8{CXvZKR~<$2&D?2Me)wN8H1jlUKgW6O*+M3nm`<)nVZct^c;Oqfy2|!H*WLB1sncL7YgY17 zX*|xIs20<3alrxKK50>_s6FR*5B}Q*`V=i^%Qr6O5ppOCI`q75yOx`(?Ys20)%Y6Q z)jm?ZmelYXGNfI}W|4oF_&BD~+DQ+kzWg(N4Pw|TH-D%*zIhCoEsg_xZk(@IFmb2D7rcvUB|NY_x zhPBdiC;R+|WV|$pdbeqs@m+5M12SupGLfdj*Xkk3tSS+t}*riU- zrDi%IHaFry|1}stWv*t1xzCYT*H+Rd|IVfVbS-d2{G~*NeWm<|&#ob7+ViB24>r?c z9R)fmNW#P~BawW_8vm76TljzN*5g65n|)kYwlRCSK+wT%TYuwr1%)yZewhyLov2Ty z+!DogvS-$fwEj5~?6RG?*E)r2h4Uy>Qq|HVD%>|aM7#vX~bscs5ho#L~1X`IV%eb@WVtVMvjC_2LDKe}k1SZGwnTDMZW z`1q5Mpyvo{Z$>ut7y18QN-2yc>vNZ8#kWg3;Is-;y~ljhf?}SpSB1mdr<1yG&Ia$; zUX>BVE=LP)sf&`S{d;Ur-ZWR2*RAI6wK3I=25feeMex+JOCGF!i16qSg{$J3B0Gc! z!3Hz9&1%c->W@<-8d-beZyJ1yM!N`){E!EhxmSFrSTXRejf0OLmm)m!RRm7;W~ZdB z?BTp-*?e~{<$MtlDuFZ!APO&FuQo(dYw+*Y!7&mFOo2wGe*A^$o z5Q~!4tk4ozo48NZ=~6q-`lZ>OR`Qk*oPl(dztyUUl52f1yh|$}0qbQmAA#juq>4Df z)687NRrsy(s9=M;O|7&@HLtB+16!ROwQE1j=waNP#7La7*$vv8^}p@u zQT$e)?n2?X((e1qr6|c2c7c*n?c6k-2C-hBJ8w@g-FYRvWP+20%@70#FCYD4VodcK zvL76@%}9&9vdkLIWWHCP0dGH?ijw3psrPa70%}cpd15?A!z9y4oFnaoWvO@pN+MR>91upF4ZqSi|uHp zi70P9W=h+_SYsNH*#N%cAQeXRC;@_cLENpkOsJazYuy<~9x+9G#B)?A=4dH*KIM1M z@n`0B5erldXE4GM2Nw%*vE&o`yhtj>p~XAc?P(OzDr?Q6AUi&>G74s zvNF5-o>h18(Ka>51rS`P#F&CA2KEYBST>R0I7lC?_IuHCMt)NC%qXj=riwq?d0J#U z|KR(F=Od)-SV-53m9{AJj%uEDJV`^Va?hcGy*yU3r(qvhO9F>}6zh!Uj8SMxt+y=g zj~@q7XzGLL@R`!E%tn1Xxxb!u|9f;$-1z#jG4o{~t$ZTN`8P!9HQ6bGHF-(pf0D2dS)gexfvL#Iu+S0%H7lCG8h?aNubhnP0e0z31n#)|`{IR`n-*B|?hmXk5?J%NFc?<`{77Pkc5ZmN2h zbwcCw%ngcPd7;JfEgu|Lq2I3-b*x>eTaJ@|!_AvH+UZRdN0th9NtHRhjR=J+;w^#8 zEo7`^k}F|#d&sXZ^0spGj>M}H6)+>cgayf!+$2gZ!8#mmd^NT2BgqRczPFYuhh5B? 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by basically logistic regression. That seems like a very specialized distribution. In order to justify this choice, let's first look at a handful of conversion charts from real data at `Better `_: .. image:: images/conversion.gif + :align: center The legend, labels of the axes, and title are all removed so that no business secrets are revealed. The solid lines with shaded area are all the generalized Gamma fits, whereas the dashed lines are the Kaplan-Meier fits. Note that the fit is very good! In fact, we have observed that almost any conversion metric can be modeled reasonably well with the generalized Gamma model (multiplied by logistic regression). @@ -14,27 +15,48 @@ Some more mathematical justification A simple toy problem also demonstrates why we would expect to get a time-dependent distribution (like the Exponential distribution) multiplied by a logistic function. Consider the `continuous-time Markov chain `_ with three states: undecided, converted, or died. -.. image:: images/convoys-markov-chain.png - :height: 300px - :width: 300px +.. image:: images/exponential-markov-chain.png + :align: center + :height: 200px + +Everyone starts out "undecided" but either converts or dies with rates :math:`\lambda_1` and :math:`\lambda_2. However, we *only observe the conversions,* not the deaths. We can solve for the distribution by thinking of this as a partial differential equation: -Everyone starts out "undecided" but either converts or dies. However, we *only observe the conversions,* not the deaths. +.. math:: + \frac{\partial P_{\text{converted}}(t)}{\partial t} = \lambda_1 P_{\text{undecided}}(t) \\ + \frac{\partial P_{\text{dead}}(t)}{\partial t} = \lambda_2 P_{\text{undecided}}(t) -We can solve for the distribution by thinking of this as a partial differential equation. The solution turns out to be quite simple: +The solution turns out to be quite simple: .. math:: - P_{\mbox{converted}}(t) = \frac{\lambda_1}{\lambda_1 + \lambda_2}(1 - \exp(-(\lambda_1 + \lambda_2)t)) + P_{\text{converted}}(t) = \frac{\lambda_1}{\lambda_1 + \lambda_2}\left(1 - \exp(-(\lambda_1 + \lambda_2)t)\right) As you can see, the solution is an exponential distribution (the :math:`1 - \exp(-(\lambda_1 + \lambda_2)t)` part) multiplied by a constant factor (the :math:`\lambda_1/(\lambda_1 + \lambda_2)` part). Turning it into a regression problem ------------------------------------ -Note that :math:`\lambda_1` and :math:`\lambda_2` are positive numbers. For each observation :math:`z`, let's write it as a linear combination :math:`\lambda_1 = \exp(a^Tz)` and :math:`\lambda_2 = \exp(b^Tz)` where :math:`a, b` are two unknown vectors. +Note that :math:`\lambda_1` and :math:`\lambda_2` are positive numbers. For each observation :math:`z`, let's set :math:`\lambda_1 = \exp(a^Tz)` and :math:`\lambda_2 = \exp(b^Tz)` where :math:`a, b` are two unknown vectors. With this transformation, the probability of conversion becomes .. math:: - P_{\mbox{converted \rightarrow \infty}}(t) = \frac{1}{1 + \exp(-(a-b)^Tz)} + P_{\text{converted} \rightarrow \infty}(t) = \frac{1}{1 + \exp(-(a-b)^Tz)} + +This is the `sigmoid function `_. If you set :math:`\beta = a - b` then it turns into ordinary logistic regression where :math:`\beta` is the unknown feature weights that we are trying to learn. This shows that the regression method in convoys turns into logistic regression in the limit where :math:`t \rightarrow \infty`. + +Weibull, gamma, and generalized gamma distributions +--------------------------------------------------- + +Moving on from exponential distributions, there are some good reasons we would want a bit more flexibility with the conversion rates. The `Weibull distribution `_ adds one single parameter and is widely used in time-to-event analysis. Empirically, the Weibull model seems to fit a large range of applications, where the common pattern is that conversions start immediately at :math:`t=0`. + +Another class of processes model the behavior where there might be some internal states between "undecided" and "converted" that causes conversions not to start immediately. The sum of multiple exponential distributions is a `gamma distribution `_. It also requires one more parameter than the exponential distribution. + +.. image:: images/gamma-markov-chain.png + :align: center + :height: 300px + +Finally, the generalized gamma distribution unifies the Weibull and the gamma distribution, and requires two more parameters than the exponential distribution. The relationship between all four distributions can be summarized in this chart: -This is the `sigmoid function `_ which means that we are doing logistic regression in the limit where :math:`t \rightarrow \infty`. +.. image:: images/probability-distributions.png + :align: center + :height: 150px From 83c87e7a3d09654c9bc62a2059395a419bc89f64 Mon Sep 17 00:00:00 2001 From: Erik Bernhardsson Date: Tue, 4 Dec 2018 12:44:41 -0500 Subject: [PATCH 6/6] typo --- docs/motivation.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/motivation.rst b/docs/motivation.rst index bc71898..c4af44c 100644 --- a/docs/motivation.rst +++ b/docs/motivation.rst @@ -19,7 +19,7 @@ A simple toy problem also demonstrates why we would expect to get a time-depende :align: center :height: 200px -Everyone starts out "undecided" but either converts or dies with rates :math:`\lambda_1` and :math:`\lambda_2. However, we *only observe the conversions,* not the deaths. We can solve for the distribution by thinking of this as a partial differential equation: +Everyone starts out "undecided" but either converts or dies with rates :math:`\lambda_1` and :math:`\lambda_2`. However, we *only observe the conversions,* not the deaths. We can solve for the distribution by thinking of this as a partial differential equation: .. math:: \frac{\partial P_{\text{converted}}(t)}{\partial t} = \lambda_1 P_{\text{undecided}}(t) \\

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