Time-Delay State Space Embedding for Gender Identity Formation Dynamics

Copyright �2006: Gendercare.com



Gender identity formation is a complex multifactor process that starts with the egg inside the mother�s womb. As we described in a separate paper [1], the Gendercare Gender Clinic developed �unexpected gender tests� (which we call the MFX and FMX tests for MtF and FtM respectively). The results of these tests show there are different typical dynamic process structures and age relations during gender identity development. The mathematical approach we utilized was not derived from or based on any etiology hypothesis for any Gender Dysphoria condition ( the approach is independent of any nature vs nurture hypothesis). Due to the lack of enough data to use time-delay embedding to generate phase space from time-series, we developed a simpler method to generate phase space. Now we are proposing a method to generate a phase space utilizing time �delay embedding, also in a simple way (Return Map, with one single variable: gender as one single variable). These tests are performed online, so any person with a computer and an internet connection, from any place around the world, may access this resource for Gender Dysphoria and other possible gender variance evaluation assistance.



Let us consider a family of systems and processes that develops (as human diversity develops from human eggs) from an initial starting point or condition (a �source�).

Keep in mind that such systems are very complex, and some characteristics could naturally diverge or converge.

For example we know all humans naturally diverge from each other in terms of many different characteristics.

When they diverge, they may diverge following some divergence structures, depending on the system dynamics.

For example let us consider two human identical twins, from the same human egg: in some respects they will converge (general appearance for example, hair color, etc.), but in other respects they will diverge despite having the same initial condition.

A remarkable example of divergence among identical male twins is that they will always have different fingerprints. If one identical twin has a GID (Gender Dysphoria) and would like to have a physical gender transition, the other has a large probability of also desiring one (nearly a 55-60% probability according to some data from Dr. Milton Diamond from University of Hawaii - via private communication), but... also significantly, the brother has a large probability of not desiring it!

The consequence of these arguments & facts is: in complex issues, we have 3 main possible complex processes & systems:

  • A Deterministic Simple System when everything is always organized and predictable.
  • A Deterministic Chaotic System in which the process shows inner organization, but it is not predictable after a given time or after a given intensity or energy level or after any other specific factor.
  • A Stochastic or Random System in which the process shows no inner organization whatsoever and absolute unpredictability all the time.


Gender identity is the name we give to the core feeling of being a man (male & masculine) or a woman (female & feminine).

A human feeling is almost always a complex phenomenon, being based on some factors of which a person is consciously aware, but mostly on others of which he or she is not aware. Therefore we may surely say gender identity (GI) is the result of one or more of these complex human formation processes.

Gender identity formation starts with the development of the body during gestation, from one particular human egg. It continues through 3 physical �sexual differentiation� formation steps:

  • Chromosomal step (structured on the egg)
  • Gonad step (totally structured during gestation)
  • Genital or Sexual step (totally structured during gestation)

After some decades of intensive research, starting with guinea pigs (Phoenix et al., 1959) and later with rats (D�rner & Docke, 1964as well as Pfaff & Sakuma, 1979a,b, Ogawa et al, 1997, 1998a, 1998b, 1999), continuing with studies of non-human primates (mainly Bonsall & Michael since the 80�s at Atlanta and also Resko at the Oregon Primates Center), and more recently of humans (cadavers) as D�rner in Germany, Gorski at UCLA, Swaab in Amsterdam, among others and most recently living humans also (as Kawamura et al, 2001); we discovered there are two very important additional hard-wired, physical & biological gender differentiation steps about which we presently know very little:

  • Basal brain step (surely totally structured during gestation in non-human primates and possibly also in humans)
  • Cortical brain step (surely the structure differentiation starts during gestation and probably continues until maturity; we are not sure how long it continues for humans and non-human primates)

We may also surely say that environment is involved in gender identity formation... starting with the egg,mainly during gestation, and during childhood & adolescence also!


The 5 steps we talked about are hard-wired biological ones; but even among them, one of the later ones may be very sensitive to the mother�s emotional state during gestation (the emotional state of the mother affects her immune system, which interacts with and may inhibit the endocrine system of the fetus). In other words, even the hard-wired biological steps � mainly the �basal brain step� -- may be affected by environmental factors, including those that emotional disturbances of the mother may cause inside the womb.

In addition there are surely many other important less hard-wired variables such as:

  • Family (sex of rearing, maternal rejection, abuse...);
  • Traumas (child abuse, sexual abuse...);
  • Society (exclusion of those who are different, lack of opportunities...);
  • Local Culture (religion & cultural influences, religious �laws� & dogmas...);
  • Fears (due to any number of pressures, low self-esteem�);
  • Violence (of any kind and intensity that may trigger a post-traumatic stress disorder /PTSD...);
  • Etc ...


In summary, we do not know ALL of the POSSIBLE factors and their relative importance; only some of the probable important variables are available for studying the complex system of gender identity formation.


Therefore Gender Identity (GI) may be considered as the result of these processes: a multivariate system with known & unknown variables -- a complex system and developmental process that during gestation mainly has a hard-wired biological character, but that also during gestation can be influenced by the womb environment. Later, this process is affected by a lot of many environmental influences during childhood and adolescence; such effects may continue on into adulthood, and surely this process, that for some subjects may be marked by tension and discord, ends at maturity, leaving traces of all the traumas and social experiences of that complexity as scars that we may define mathematically as random instability.


But even with GI development being so complex and having so many unknown variables � we may still study it just as we may study the formation of a hurricane, even though we may not know all of its generating factors and controls.

We can employ the same reasoning for gender identity...we may not know all the factors that define it, but we can understand the complex dynamics of its formation, if we have adequate time series data (a series with enough data to use time-delay embedding to generate Return Maps).

Obviously to develop dynamic knowledge about a hurricane, we need an enormous amount of good data� and to develop understanding of gender identity we have only a few units of data available for each patient� but the reasoning is the same (and to study GI we may consider gender as one variable when to study hurricanes surely we have multivariable and much more complex systems).

By understanding the dynamic results and characteristics, we can have better insight into the causes and main factors important for gender identity formation.


Gendercare approach to the problem


The eternal nature vs. nurture debate is absolutely sterile for understanding gender identity formation, because both are important, although one may be more important in specific stages of life than the other. The crucial point is that gender identity is a complex result of multiple factors which we can study without needing to know its causes.

To study dynamic processes such as gender identity formation we have to follow one of three main procedures:

  1. If we have mathematical models (such as �theoretical maps� for example), we may use them;
  2. With no model, we need adequate time series data (hundreds to thousands of numerical data) to study the optimum embedding dimension and time-delay dimension to erect a consistent phase space where the process develops, through time delay embedding;
  3. If we have no model and inadequate time series data, we may try an alternative approach, utilizing a small data set for each time series and referencing a family of time series (although this is not the best mathematical approach to the problem, depending on the system, it may be the only practical approach).


We worked in the beginning with the most feasible idea given our situation: to discover a model or to generate a large time series would be almost impossible, but to generate some families of small time series for gender identity formation was possible using a new measurement instrument we developed: our Gendercare unexpected gender tests.



Gendercare MFX (for MtF�s) and FMX (for FtM�s)
Unexpected Gender Tests

What are these tests?

  • They are mainly two clusters of 100 questions specific to different periods of a patient�s life.
  • Based on the patient�s answers for each period of his/her life (ages 0-7, 7-10, 10-14, 14-18, >18) we determine a score for each scale (there are 4 scales: Unexpected Gender scale, GID & GIDNOS scale, Sexual Orientation scale and Sexual Action scale);
  • Therefore for each patient we could derive a specific time series trajectory for unexpected gender identity development (with a dozen numerical values for each time series);
  • Through these time series results, and analyzing some hundreds of patients, we could build and evaluate a global time series family (and some typical sub-families), to study GI (gender identity) in terms of dynamic concepts.
  • We developed special software that properly processes the results;
  • For each patient, we generate from those results a complete test report, with all the dynamic data for the unexpected gender scale as well as the other scales.
  • To answer all 100 questions is mandatory for a complete dynamic evaluation.
  • The patient answers the test online (after a complete e-mail anamnesis), allowing for a complete GID web evaluation and diagnosis. If the patient does not require a complete web evaluation, he/she may answer the test (with no previous e-mail anamnesis) and we can send the report to the local therapist (the patient also always receives a copy of the report, in PDF format) or directly to the patient.
  • For Gendercare patients, after the e-mail anamnesis and unexpected gender test we later perform an MMPI psychiatric screening online to complete the diagnosis.


Developing the Method



The gender identity trajectory (a time series) may show structures and typical patterns of gender identity development. We can study these possible structures and patterns using the theory of complex systems (allowing us to research whether inner organization exists or not).

To study gender identity using these theories we do not need to know its causes -- we only need to know the results of the complex process as a time series (in a family of time series with the identical or almost identical original source).From that analysis we may gain insight into the main factors that are important for gender identity formation and later its stabilization or disturbance.


The MFX/FMX trajectories generated a time-series with a few points for each patient.

We had two possible ways to interpret a time series :

--- We could utilize the few points only and try to generate dynamic knowledge from them, as we showed in our other work [1] or

--- We could utilize a mathematical model, (a curve fitting model), for the gender trajectories, and from that model generate a new time-series , and then use a simple time-delay embedding technique considering the time-series data generated by the model (that way generating a Return Map).

The latter is the new approach we present here.

Elimination of time: phase space diagrams & embedding.

By seeing only one trajectory of a result as a time series, we might not understand very well what has happened.

Traditional statistical analysis of time series shows us very little (we have very little data, trend analysis means nothing for us and auto-correlation factors also help very little ...).

A good evaluation method is to proceed using topology (the mathematics of continuous space and figures), eliminating time as a system variable (as for an ideal pendulum representing a family of different levels of energy � see Stewart, 1989 [2]; Torres, 2005 [3]). The elimination of time is referred to here as embedding and the result is that we obtain a Phase Space Diagram.

Unexpected Gender Phase Space Variables

The solution we proposed in our previous paper [1] was:

Using a simple embedding function for our unexpected gender data:

In our Phase Space diagrams obtained from MFX tests we define the Y axis as unexpected femininity. For our FMX test we define it as unexpected masculinity, and for both, we name it f(t), at time of life t.

On our diagram�s X axis, we show the embedded function we call gender gradient:

Gender Gradient={f[t(n+1)]-f[t(n)]}/[t(n+1)-t(n)]

  • For any simple system�s Phase Space diagram a simple GI attractor will immediately be evident, as a sink (i.e., a convergence point) or as a very small vortex; from the point of view of the etiology of the process, simple attractors mean absolutely hard-wired ones;
  • For a chaotic deterministic system (such as biological, genetic and endocrine-derived systems) the diagram will show a structure of a �strange� attractor (with a �fractal� dimension as an inner structure) ending as a �strange� limit cycle (the system develops with some stability but retains some structural instability);
  • For random systems we will see no structure, but a mixture of curves & points (due to random environmental disturbances by family, violence, culture warring against nature, �sex of rearing�, etc...).

Typical Phase Space Diagram for Gendercare Tests

Understanding the MFX test phase diagram:

All sources are male (0 unexpected femininity at the Y axis);

Near the source we see a bifurcation... normal men (the yellow family of curves near the male pole) soon split off from GID and GIDNOS ones (all other curves). Among the �normal men� tested are heterosexual, bisexual and homosexual men. Based on these men we may establish some divergence coefficients for the system, thereby using them as controls.

Later there is another bifurcation, nearage 7, between CD and GIDNOS (TG) � all mixed up in pink;and TS (almost all in blue). ...

The TS curves in light blue show a family with very consistent structure, while some curves (dark blue) diverge from that structure, mainly after age 7-10. As we can see in the figure, the closer the gender identity is to the female polarity (>70% unexpected femininity), the more organized or �strange� the attractor is, showing a chaotic and deterministic behavior that means hard-wired factors are the most important ones governing those TS situations.

After some years (at age > 10-14), mainly for GIDNOS, TG and CD (the pink curves but also some of the dark blue TS ones) the structure loses inner strength, and a lot of random factors, acting as late influences, may start interfering with gender identity formation (late instability due to early traumas and late external interferences).

The same happens for some TS subjects (light blue curves at late ages), but with much less intensity.

Our New Mathematical Approach to define a Phase Space for GI

We start from the same trajectory data from the same MFX/FMX tests.

But now, we begin by using Curve Fitting Software to try to generate new time-series data * �����

* Curve Expert 1.3 by Daniel Hyams..*

In this way, from the original data we can interpolate a fitting function of the type :

X= a + bY + c Y2 + d Y3 �����������������������[a, b, c and d are parameters and 2 and 3 are exponents]

(a cubic polynomial fitting curve)

From this curve we could generate a table of data, with regular time delay (we tried 10 weeks, 54 weeks & 108 weeks as time-delay = D).

With the data generated by these time-series we could use time-delay embedding, establishing a phase space (as Return Map) with the coordinates:

Taking Dt as 10, 54 and 108 weeks to plot the Return Map** (these weeks are considered since conception and not since birth)

For practical purposes we consider Dt=108 weeks

** Considering gender identity formation as a one variable only system (gender variation), the Return Map will be very effective for our purpose**

When we intend to study possible etiologies for the GI formation process we consider Dt=10 or even a lower value, to try to study possible details of the generated phase space curves.

We plot these curves for patients, and we obtain the following typical result:

Generating the Return Map for Gender Phase Space

From the identical source, the yellow curve shows the Phase Space Diagram (Return Map) for normal men or women (heterosexual, bisexual and homosexual) with Dt = 108.

The gray family of curves shows the typical TS family also with the same Dt value.

Sometimes, some patients show unique curves such as:

Or for example also:

For these two figures, in the same phase space (always considering a return map with Dt = 108 weeks), with the yellow curve for usual M or F people shown for reference, we see curves that are atypical for TS and we could classify better possibly as GID => GIDNOS

In the first figure, the patients identify themselves as TS but they do not show a typical TS development which means the therapist needs to evaluate them very carefully, because we must not make mistakes here. Some of them, through their anamnesis, may reveal what happened in their lives to determine their particular GI development, but for others only the MMPI may be able to explain some possible GIDNOS situations.

More detailed anamnesis, MMPI verification, careful and detailed transition, light HRT with strict supervision are always necessary for such patients, and sometimes, depending mainly on the MMPI , a local psychiatric/psychotherapeutic face-to-face verification will also be important.

The second curve shows some GIDNOS situations in which the patients are sometimes aware of their TG/CD condition, and sometimes plan to have SRS (sex reassignment surgery). This shows the therapist the same thing: these patients need special supervision and more detailed observation, and sometimes they may need a local psychiatric/psychotherapeutic face-to-face supervision.

Most of Gendercare TS MtF and FtM patients show typical TS patterns, and do not need supplemental local face-to-face supervision, but when in transition they always need special strategy, special FFS (from surgeons that know specifically FFS for MtF patients) and secondary surgeries and treatments and HRT-hormone-therapy supervision, and finally the SRS surgical correction, when that is the goal of the patient.

Advantages of that new embedding procedure (Return Map)

We work with more points for each curve plot, so we may also use time-delay with equidistant points.

With the older �gender gradient� approach, we used real measured values, but sometimes with an irregular time delay. For example, if the patient is 18 at the time of testing, we will need to compare him or her with another patient for whom the last point represents age 65 for example (we compare the curves, one that represents from conception to maturity and the other from conception to adolescence only!), and these time discrepancies may generate some distortions of the gender gradient calculations.

That old procedure generated some curve distortions, which we can avoid with the new procedure.


With both methods, we have the same curve aspects, but the new method is more practical than the older one.

For GID diagnostics both methods are very interesting, but the new one is more precise and practical (for gender therapy and GI diagnostics) and the old perhaps better to evaluate some etiology considerations.

With time we hope we will have more time-series data, to develop better typical curves and curve-fitting models for typical families of TS, CD, GIDNOS and TG peopleand to study in depth the same method for interxex gender evaluation.


We would like to thank Sonia John for her editing of our English text.


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Scientific Background

We did develop scientific knowledge as the background to develop a method to diagnose people with gender issues at a distance, through the web. See the links for details:

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