Atypical Physicist

Today a student asked Edward:

Can you please briefly give me a more general notion of convexity?

I’ve studied that in 2D a function is convex if, considering 2 points on it, the function stays under the segment that connects them but is there any more general formulation for this concept ?

Edward:

A more general definition could be formulated considering a $ L $ Vector Space and $ V \subset L $ a subspace. 

Given 2 any points $ x_{1}, x_{2} \in V $ it is said that $ V $ is convex if the linear parametrization 

$$ s(t) = t x_{1} + (1 - t) x_{2} \qquad t \in (0,1) $$ 

brings to $ s(t) \in V \quad \forall t \in (0,1) $ so the path stays alwyas inside the Space. 

Hence a convex space is closed wrt any linear path connecting any 2 elements in it. 

In this post I'm going to briefly sketch the different perspectives regarding the technical knowledge in ancient greek times, with Aristotle, and in modern times, with Galilei and the Scientific Method. 

The Téchne in Aristotle 

First, Aristotle observed that Man and Animals discover the World in different ways: while animals seem to rely most on instrinct and sometimes on memories, Man seem to rely much more on experiences, it is to say conscious knowledge of specific facts. 

These two categories of beings seem to live at two different levels of fusionality with the Nature : the animals, staying at higher level, rely mainly on their instinctual part thus existing as a part of Nature instead of something different that interacts with Nature. 

Regarding the Man, it seems the opposite: he lives at a lower level of fusionality and so he perceives Nature as something different from him that needs to be discovered by means of experience. 

Aristotle defines the experience as knowledge of specific facts but observes that the knowledge building process does not consists solely of the creation of collection of facts: a generalization process is always carried out on such a list, to try extracting some general knowledge out of the specific one it is to say with Aristotle words, to develop a technique. 

So in Aristotle opinion, technique is the result of a generalization process carried out over a collection of specific facts, obtained by the experience. 

Aristotle points out that the téchne (the original greek term for technique) involves disposing of a Power towards the World. 

By means of this Power the Man can 

act on the World to transform it, to ultimately give his body different conditions for existence 

act on the World to improve his knowledge of it 

Hence Aristotle notices the dualism that the téchne, and so the power, involves: transforming the World to improve existing conditions, thus bringing utility to the body, and transforming the World to improve the knowledge of it and ultimately of ourselves, thus bringing utility to the soul. 

The Scientific Method 

As it is widely known, the Scientific Method (developed by Galileo Galielei) relies on the possibility to carry out Experiments. 

Scientific Experiments are realized by means of a tèchne that permits a proper manipulation of the World. 

It is important to observe that in Aristotle the experiments coincide with the experience thus they are a fundamental part of the knowledge bringing process, proceeding from spefici to general. 

Ultimately knowledge discovery, at least in the Western part of the World, has always been (since the beginning of civilization) very interested in the Comprehension of Prescriptive Rules behind Reality that were assumed to exist, because of its coherence. 

From this point of view, the téchne is the quintessential product of this process. 

The Scientific Methods also aims at the abovementioned kind of general knowledge but with a completely different paradigm: while Aristotle proposed a knowledge building process (one of the many possible) the Scientific Method is a knowledge validation process. 

In order to start using the Scientific Method, a Theory is needed, so it means General Knowledge, that shall have been produced someway. 

The Scientifc Method could then provide a sort of Confidence Level for the Theory, combining it with Experiments and proceeding according to Bayes Theorem 

T = Theory and E = Experiments 

Let's think about the communication of a message. 

To perform the communication, an alphabet of $ M $ basic symbols is used to create a message of $ k $ of these symbols. 

The communication is performed on a channel with a maximum bandwidth that depends on the minimum amount of time $ T $ needed to transmit one basic symbol. 

The function representing the whole message is the following one 

Function representing a message of $ k $ symbols each of which is transmitted in an amount of time equal to $ T $ 

Let's define a set of eigen-functions that have with the property of being orthogonal it is to say 

Set of Orthogonal Eigen-Functions used to describe each symbol

Each signal representing a single symbol, could be expressed in terms of this set of orthogonal functions, it is to say 

Expressing a single symbol in the message chain as a linear combination of the orthogonal eigen-function set

Immediate candidates for this eigen-functions set are the sine and cosine given their orthogonality. 

It's also important to bandlimit them, in order not to exceed the channel bandwidth, and it could be performed using the sinc function, properly tuned on the minimum amount of time $ T $ available for the symbol transmission 

Sinc Functions is used to bandlimit the eigen-functions accordin to the $ T $ symbol transmission time

So a simple eigen-functions set could be the following one 

Simple Eigen-Function set. The $ T $ is the minimum amount of time allowed for symbol transmission.  The Parameters $ C_{1}, C_{2} $ will be tuned to achieve normalization. 

A desiderable property for this Eigen-Functions Set regards Normalization it is to say 

Normalization of the Eigen-Functions Set 

The Coefficients $ C_{1}, C_{2} $ are then simply computed 

Computation of $ C_{1} $ imposing Normalization 

Computation of $ C_{2} $ imposing Normalization 

Finally the complete Eigen-Functions Set is computed 

Complete Eigen-Functions Set properly bandlimited and with Normalization Coefficient 

Introduction to Recursive Bayesian Filters 

Let's introduce the general formula of Recursive Bayesian Filters for Hidden Markov Processes 

General Recursive Bayesian Filter 

The elements of the formula are the following ones 

$ P(x_{t} | y_{1:t}) $ State PDF given a set of Observations in different times 

$ \alpha $ Normalization Term 

$ P(y_{t} | x_{t}) $ Observation PDF given a certain State, also called Likelihood 

$ P(x_{t} | x_{t-1}) $ Process Dynamic, it is to say the PDF of the State at a certain time, given the State at the previous instant 

$ P(x_{t-1} | y_{t-1}) $ State PDF calculated at the previous iteration (it explains why this is called a Recursive Filter) 

Numerical Estimation 

In general the integral in the abovementioned formula has no closed form solution (except in very specific situations) so a numerical approximation is needed 

The Particle Filter is a Monte Carlo Method to compute a numerical approximation for this integral relying on the concept of particle 

The term Particle designate a Possible System State. 

In order to be able to adopt this solution, it is necessary to know 

the System Dynamic 

the System Likelihood 

The Particle Filter strategy so is simply as follows 

at the beginning of the iteration cycle (recursive filter) a certain amount of Particles is generated, for example by sampling a certain PDF chosen in some way 

at every iteration, the particles are evolved according to the System Dynamic thus to be able to numerically estimate the integral 

Let's consider a $ N $ Dimensional Phase Space for a System like a Robot Manipulator or a Vehicle. 

The whole System State is made of $ N $ States each related to a different couple of variables 

Position and Speed 

The evolution of each State could be forced imposing a certain law, for example, in this case, a Cubic Spline, it is to say a 3rd order polynomial. 

This leads us to 

Cubic Spline Evolution Law imposed to a certain state variable 

This polynomial presents 4 coefficients thus 4 equations are needed to solve it. 

Definition - Control Points 

A Control Point is the specification of the System State (for the specific variable) in a certain time. 

Every control point brings 2 equations thus with 2 control points it's possible to fully define the Cubic Spline. 

Conventionally the control points are 

the Starting State 

the Ending State 

Assumption 

Let's now assume without loss of generality that the motion start at $ t = 0 $ so the time interval is

  Let's also assume without loss of generality that the Motion starts at the center of the Reference System, it is to say that 

$$ q(0) = 0 $$ 

Let's also consider a simple motion where the System is at rest at the beginning and at the end of its motion. 

These assumptions lead us to 

Starting Conditions 

Ending Conditions 

Computation 

The initial conditions lead us to compute the value of 2 coefficients 

so the equations become 

and the remaining coefficients values are 

Considerations 

The terms $ a_{0}, a_{1} $ are the linear ones and they are fully defined by the Initial Conditions. 

The terms $ a_{2}, a_{3} $ are the non linear ones and they depend on the difference between the Initial and the Final Condition 

From a Physical Point of View this is absolutely logical: the Initial Conditions define a state of rest or of motion at constant velocity (see Newton's First Law) 

If no forces will act on the System, the evolution is fully determined and the future states are placed on a line defined by the first order spline with just $ a_{0}, a_{1} $ coefficients. 

Symmetry Breaking is an important topic appearing in different fields of Physics. 

It has become "widely" famous when it has been proposed as an explanation for a common phenomenon like Ferromagnetism. 

Let's think about a Ising Model, in first instance without specifying the number of dimensions, and so it's possible to formalize it this way 

A N particles system with 2 spin values possible 

Then let's consider the Hamiltonian of the System 

Standard Hamiltonian taking into account both

the endogenous interaction that's a pairwise nearest neighbour interaction $ \sigma_{i} \sigma_{j} $ 

the exogenous interaction that's single spin interaction with an external field 

Now let's consider just the endogenous interaction, assuming that the exogenous one is negligible compared to the first one 

This assumption transforms the Hamiltonian this way 

Approximated Hamiltonian under the assumption that the exogenous term is negligible compared to the endogenous one

  Let's now define a Transformation called Spin Flip Transformation this way 

Spin Flip Transformation just invertes the value of each spin in the System 

  Now it's time for a couple of important observation 

Observation 1: Approximated Hamiltonian is invariant to Spin Flip Transformation 

It is to say that 

The Energy Value computed using the approximated Hamiltonian is the same for the Normal System and the Spin Flipped System

The reason for this invariance stays in the exclusive presence of the pairwise interaction term: flipping the spin value doesn't change the spin product value. 

Observation 2:  Initial Hamiltonian is not invariant to Spin Flip Transformation 

The reason for the absence of invariance stays in the fact that in addition to the pairwise interaction term (that's invariant to spin flip transformation) there's a spin-external field interaction term that's not invariant to spin flip. 

Let's now consider an observable for the system like the system magnetization defined as follows 

  The System Magnetization is just the mean magnetization given by the orientation of every spin in the system 

Let's now observe that the system magnetization presents odd symmetry with respect to the spin flip transformation 

It means that the average system magnetization (defined as the average of all the possible system magnetization values for a given energy value) under the assumption that the external field is neglectible, thus to use the Approximated Hamiltonian, is expected to vanish.

This dynamic is indeed observed for 

finite systems (no Thermodynamic Limit) 

in the Thermodynamic Limit for $ D = 1 $ Monodimensional Systems 

in the Thermodynamic Limit for $ D \ge 2 $ only if the Temperature of the System is high enough 

It is indeed known that $ D \ge 2 $ Ising Systems present a Paramagnetic Phase and a Ferromagnetic Phase. 

In the Paramagnetic Phase the abovementioned Symmetry holds and thus the measured value of the system magnetization is 0 like it is expected to be (according to the model). 

In the Ferromagnetic Phase the measured value for the system magnetization in known to be all but 0 and this implies that the Symmetry has been broken. 

According to the Model, the case for no Symmetry is the one where the Assumption that allowed us to use the Approximated Hamiltonian is not holding anymore and it implies necessarily that some sort of external field is acting in a non negligible way. 

Conclusion 

The abovementioned observations tell us that some sort of external field (or something acting as an external field) is always present 

It's very interesting to observe that this sort of external field is not just an unerasable element of real world experiments, because of the observation that in every Monodimensional System the Symmetry is never broken and thus they present just the Paramagnetic Phase. 

This should suggest that there is a correlation of some kind between this sort of external field and the number of dimensions of the system. 

It could be shown that this correlation is due to the presence of Long Range Interactions that are responsible for this sort of external field. 

Introduction 

In this post we'll try to explain the way the Expected Value of a Random Variable is formally defined. 

It is assumed that notions like Random Variable, Sigma Algebra, Probability Measure, ... are known in a formal way. 

Definition of the problem 

A Random Variable is defined in a Measurable Probability Space $ (\Omega, F, P) $ where 

$ \Omega $ = Universe Space 

$ F $ = Sigma Algebra defined over $ \Omega $ 

$ P $ = Probability Measure 

Its Expected Value is defined as follow 

so now the problem becomes to find an operative definition for this integral. 

It's important to notice that, at this point, the only well defined Integral we have is this one 

This way we are able to associate a Probability Measure to each possible event in the Universe Space but we still don't know which way to integrate a Random Variable. 

As a start we can make an assumption to make the job easier: let's assume that $ X $ is a Random Variable with finite cardinality, it is to say that the possible outcomes of $ X $ are finite or more formally 

It implies that independently of the structure of $ \Omega $ it's possible to identify $ N $ Regions, each connected to a possible outcome. 

More formally this concept could be expressed exploiting the fact that the Random Variable has been defined as an invertible function 

This means that 

So we can express the Random Variable just as a linear combination of indicator funcions related to each of the segments, multiplied by the real value associated to each region thus 

At each of the regions in which the $ \Omega $ Universe Space has been segmented is associated a Measure of Probability as follows 

With $ 1_{A_{k}} $ Indicator Function 

It is a well defined integral. 

This leads us to a well defined integral associated to the initial one 

Introducation to ZCBF 

In the previous Post the concept of ZCB has been introduced 

The key concepts are that 

a Portfolio containing just one ZCB has a Cashflow Vector of this type 

it is possible to compute indicators like Implicit Simple Capitalization Rate and Implicit Exponential Capitalization Rate for the ZCB 

Now let's present another Financial Tool called ZCBF Zero Coupon Bond Forward appunto 

The ZCBF is different from the ZCB because the negative cashflow caused by buying it does not happen in $ t $ when the price is formed on the market but in $ S > t $ and $ S < T $ it is to say in a future time but before the maturity 

In our convention we introduce $ p(t, S, T) $ the indicates the Price of a ZCBF with the cash outflow happening in $ S $ 

The ZCBF has this kind of Cashflow Vector 

Hypothesis of Non Arbitrage 

The term Arbitrage identifies a situation in which it's possible to get an out of the market return with no risk. 

Actually it is not a proper definition but it should convey the idea that our ideal market is extremely efficient in forming prices and that it's possible to get a return only in 2 ways 

risk free returns originate from depriving of a resource, typically money, for a certaim amount of time 

risky returns originate from taking risks 

Until now we've just met a couple of financial tools, ZCB and ZCBF, that are related to risk free investments 

Under the hypothesis that no risk is present, and obviously no arbitrage situations are present too, profit coming from investments should depend only on the amount of time. 

This reasonment should immediately make us think that a relation should exist between ZCB and ZCBF prices 

Let's consider a ZCB and a ZCBF both with maturity in $ T $ and with a nominal return of $ C = 1 $ 

If an investor wants to start an investment in time $ t $ it should make no difference buying the ZCB or the ZCBF abovementioned but it does not mean that their prices are equal actually they shall be different for a very simple reason: the cash inflow happens at the same time for both the tools, it is to say at maturity in $ T $ but the cash outflow happens in different moment, it is to say for the ZCB in $ t $ and for ZCBF in $ S > t $ 

Logically we should expect that ZCBF should be more expensive than ZCB because it requires cash outflow later with respect to the latter. 

To compute quantitatively this relation it is necessary to build 2 equivalent Portfolio, one containing a ZCB and the other one the ZCBF. 

The former is trivial but to build the latter a further element is needed: a ZCB with maturity in $ S $ 

This way we get 

Obviously by convention 

but this term has not been omitted for a dimensional issue. 

The unknown $ x $ is the parameter we have to tune in order to make the Portfolio $ P_{2} $ equivalent to $ P_{1} $ 

It represents the quantity of ZCB with maturity in $ S $ to buy in order to make the 2 Portfolio equivalent. 

In $ t = S $ the $ P_{1} $ Portfolio has $ C = 0 $ so it should be the same for $ P_{2} $ Portfolio thus 

From a dimensional point of view this result is correct because it is a pure number as we were expecting it to be, according to the fact that we are looking for a quantity. 

Making a substitution in the relation and stating an equivalence in $ t $ we get that 

This leads us to the relation between the price of ZCBF and ZCB thus 

Conclusion 

In this post the concept of ZCB has been reclaimed and new concepts have been presented: ZCBF, No Arbitrage Hypothesis. 

In this series of posts the basic tools and the basic strategies of Financial Engineering are presented 

Capitalization Laws 

Capitalization Laws are equations used to compute the value of an amount money back and forth in time 

Those Laws depend on a fundamental parameter that's the interest rate: it defines a quantitative relation between time and money. 

The Interest Rate has always the dimension of a percentage, it is to say a pure number, over time. 

There 3 kind of Capitalization Laws 

Simple Capitalization 

Composite Capitalization 

Exponential Capitalization 

the difference between them resides in the time dependence 

By convention the term $ C_{0} $ identifies the amount of money in a certain time $ t = 0 $ 

Simple Capitalization Law 

with $ i $ = Simple Interest Rate 

Composite Capitalization Law 

with $ i $ = Composite Interest Rate 

It's interesting to observe that for $ t \in [0,1] $ the Simple Capitalization Law gives higher return while for $ t \in (+1, +\infty) $ the Composite Capitalization gives higher ones 

Simple Simulation with 

$ C_{0} = 10 $ Initial Capital 

$ i = 1.8 $ Interest Rate 

The Red Line identifies the Simple Capitalization while the Blue Line identifies the Composite Capitalization 

Observation on the Growth Rating 

The Simple Capitalization Law exposes a linear dependence on time because only the initial amount is involved in the dynamic. 

The Composite Capitalization Law exposes a more than linear dependence on time because the interests cumulate to the initial amount and produce other interests. 

Composite Capitalization is characterized by 2 regions 

for $ t \in [0,1] $ it unerperforms the Simple Capitalization Law 

for $ t \in (1, +\infty) $ it overperforms the Simple Capitalization law 

The value dividing the 2 regions is the Measurement Unit of Time for the Interest Rate 

Let's explain it with an example: if a the Interest Rate is 2% / month it means that 

unless the first month is passed the Composite Capitalization still underperforms 

after the first month is passed the Composite Capitalization overperforms 

Exponential Capitalization Law 

The Exponential Capitalization Law is the Composite Capitalization Law with Interest Rate expressed on a infinitely small amount of time or better it is related to the Instant Interest Rate 

Obviously this kind of Capitalization Law always overperforms tha Simple Capitalization Law 

Cashflow Vector 

First the notion of Cashflow Vector for a certain Portfolio is introduced: it's just a list of pairs $ (t, C) $ where 

$ t $ has the dimension of time and identifies a certain moment 

$ C $ has the dimension of a currency and identify a certain amout of cash flowing inside (conventionally by the plus sign) or outside (conventionally by the minus sign) the Portfolio 

ZCB 

ZCB (Zero Coupon Bonds) is the simpliest Financial Tool. 

It just pays back a certain amout of cash at maturity. 

The following convention is adopted 

$ T $ = Maturity 

$ C = 1 $ the amount cash payed back  Within this convention it has the dimension of a pure number but obviously all the results will be valid for any currency 

ZCB are traded on a market and this process establish their price in a certain moment, that is represented by $ p(t, T) $ where 

$ t $ = Time which the price is related to 

$ T $ = Bond Maturity 

Please notice that the cash payed back at bond maturity has been omitted because we've assumed it's always $ C = 1 $ 

So a Portfolio containing just one ZCB we'll have the following Cashflow Vector 

because it has been payed an amout of $ p(t, T) $ to buy the ZCB in $ t $ and the latter has payed back an amount of $ C = 1 $ at maturity in $ T $ 

This kind of tool is thus an investment because the investor chooses to renounce to a certain amount of money in a certain moment, to buy the right to get a higher amount of money in the future 

This observation leads us back to Capitalization Laws: knowing the way a certain amount of money changes over time allows us to compute the implicit interest rate in this kind of investment. 

As abovementioned, it's possible to compute different interest rates according to the Capitalization Law applied 

Note 

The notation will be slightly modified in order to be able to distinguish the 

$ i_{s} $ Simple Interest Rate 

$ i_{e} $ Exponent Interest Rate 

Simple Interest Rate in ZCB 

Let's now compute the Simple Interest Rate implicit in the ZCB investment 

The way the Capital gets transformed over time is given by the abovementioned Cashflow Vector 

So applying this information to the Simple Capitalization Law we get 

Solving for the unknown $ i $ Implicit Simple Interest Rate we get 

Exponential Interest Rate in ZCB 

In order to compute the Exponential Interest Rate implicit in ZCB investment let's simply apply the abovementioned Cashflow Vector 

to the Exponential Capitalization Law thus to get 

Solving for the unknown $ i $ Implicit Exponential Interest Rate we get 

Conclusion 

As stated in the previous post of this series, the Gaussian Distribution is not the only distribution that should be considered for modeling the error random variable 

Let's assume that the Error PDF follows a Cauchy Lorentz Distribution that has the following form 

The main difference between this kind of Distribution and the Gaussian one concerns that fat tails of the formers wrt to the latter, it means that it's more probable to find bigger values 

The assumption regarding the Independence still holds so the Likelihood is still factorized and we are going to work on the Log-Likelihood that in this case has the following form 

Like the previous case we get 

Also in this case the function to be minimized 

is convex 

Assuming $ \gamma = 1 $ but this term doesn't affect the convexity 

so it could be minimized using a classical descent method 

M-Estimators 

We can now introduce a generalization regarding the estimator, it is to say the function to minimize in order to estimate the model that best fits the data under certain assumptions, observing both the cases presented until now, it is to say 

The Minimum Least Square Estimator regarding the Gaussian Distribution 

The Cauchy-Lorentz Distribution Estimator 

The more general form of this kind of estimator is the following one 

With 

and 

$ \vec \alpha $ indicates possible different params like $ \sigma, \gamma $ according to the distribution chosen 

An estimator with this form is called M Estimator 

In the previous post of this series an important assumption has been made: 

The Errors, it is to say the Point-Line Distance, are IID Variables following a Gaussian Distribution 

So the corresponding Normalized PDF is the following one 

Setting the Normalization Constant properly to 

and the variable 

the proper PDF is obtained 

Let's remember that the Likelihood has been defined as 

and the factorization has been obtained supposing independence among all of the random variables 

When dealing with factorized Likelihood Function it's usually more convenient to define a Log-Likelihood this way 

Thus leading to 

Obviously maximizing the Log-Likelihood is equivalent to minimizing the negative term 

but it's now obvious that $ \frac{1}{2 \sigma^2} $ is just a constant, even if it's unknown, and so it doesn't influence the minimum that depends only on the squre of the errors so it follows that 

The function to be minimized is convex so it could be solved with descent direction method 

So it has been formally demonstrated the Optimality for the Least Square Fitting Method under certain assumption

Now it should be more clear the assumptions under which this method provides an optimum, it is to say 

No A Priori knowledge requires the Maximum Likelihood 

Errors are assumed to be Random Variables IID following a Gaussian Distribution  

The first assumption is quite clear and no further comment is needed. 

The second assumption needs a couple of final observations: 

in real world problem, the Independence Assumption could be a rather strong one 

Introduction to Data and Model 

In Model Fitting Problem the goal is to find the best fit for a hypothetical model on real data (data affected by errors, ...) 

At the beginning let's consider data as points on a 2D surface (the results will anyway easily generalize to euclidean space in higher dimension) 

With 

The model we want to fit is a line that in $ \mathbb{R}^{2} $ could be fully expressed this way 

This parameters leads to the equation in implicit form of the line in $ \mathbb{R}^{2} $ that is 

Cost Function 

The Cost Function plays a very central role in model fitting problems. 

In this situation, the most natural cost function is a distance function between the Model and each of the Data Point: ideally all the Data Points should stay on the Model and so the Total Cost would be zero. 

The first cost function presented is the Orthogonal Distance between each Data Point and the Line 

Observation 

This kind of Cost Function allows both positive and negative values, according to the position of the Data Point with respect to the Line: the sign tells if the Data Point stays on one side or the other side of the Line. 

Assumption on the Error Distribution 

The Point-Line distance abovementioned could be seen as an error: ideally everypoint should stay on the line (assuming that the Model is correct ! :) ) but practically Data is measured with added Noise. 

So a classical assumption on Error Distribution could be now made: let's suppose that Errors arre IID Random Variables following a Gaussian Distribution with zero mean and an unknown variance. 

With a high enough number of Data Points it could be a reasonable assumption because of the Central Limit Theorem (but how much is enough in real world applications?) 

Now let's jump back to our previous work 

Foundations of Bayesian Decision Theory

because we want to use Bayesian Theory to estimate the proper Probability that Data Points fit the Model 

Without any further information or assumption, we should use the more general approach relying on the concept of Likelihood that is defined as follows 

More generally the Likelihood shoule have been defined using the Joint PDF 

but because of the initial assumption of IID Variables, the Joint PDF could be factorized leading to the previous formulation. 

So this brings us to 

The problem here is to be able to estimate properly the $ \sigma^2 $ Variance of the Noise. 

Let's consider a Random Walk in one dimension, modeled by a Random Walker that moves in a discrete way both in time and in space 

$ \tau $ is the atomic time-step 

$ h $ is the atomic space-step 

The term space has a broad meaning: it could mean a real space,  measuredd in meters, or a price if we're in the context of Financial Modeling. 

The walker moves taking, at each time-step, the following space-step 

$$ X_{i} = h \xi_{i} $$ 

with $ \xi_{i} $ = Random Variable following a certain Distribution. 

All the $ \xi_{i} \qquad \forall i = 1,...,N $ are assumed to be IID. 

So after $ N $ steps the Random Walker is at the following position 

Being $ X_{N} $ the sum of $ N $ IID Variables it's possible to apply the CLT (Central Limit Theorem) stating that 

Defining the following quantities 

it holds that 

It's very important to observe that 

Usually the first kind of RW studied relies on the following distribution 

This implies that 

This leads us to 

This kind of process is as follows 

Being $ \xi_{i} \qquad \forall i=1,...,N $ IID it follows that 

This leads us to 

This equation brings us to the final relation 

Now it's time to consider an observation that has been made on the Real World Random Walk and states that 

So this is obviously a desiderable property for our Mathematical Model of the Real World Random Walk and so we can write that 

With $ C \in \mathbb{R} $ a certain constant 

This leads us to 

Now for convenience regarding reasons that will be explained forth we make a simple substitution that does not affect the result 

and this brings us to the final statement 

with $ D $ = Diffusion Constant of the Random Walk 

Final Considerations 

As lons as the $ \xi_{i} $ distribution remains at zero mean and only the Variance changes, this will result in a Random Walk of the same type of the previous one, only with a scaled Variance and so with a different $ D $ Diffusion Constant. 

Bayes Theorem for Bayesian Decision Theory 

Let's consider a Stochastic Process where 

$ x $ denotes the hidden state (supposed not to change in time)

$ y_{t} $ denotes the observation at time $ t $ that depends only on the state 

Usually the Observation is just the State plus some sort of Noise. 

There are 2 Random Variables 

$ (Y_{t} | X) $ the probablity that a certain state produces a certain set observations 

$ (X | Y_{t}) $ the probability that a certain set of observations is generated by a certain a state 

In the Bayesian Framework those Random Variables are correlated by the Bayes Theorem 

Once we know $ P(X | Y_{t}) $ it is to say the Probability Density Function for $ (X | Y_{t}) $ it is necessary to establish a criterion to select a specific value for that Random Value, it is to say a specific State 

Usually the criterion is the mimization of a well defined cost function, the most common one is the Mean Square Cost Function given by 

So the Minimization Problem is formally written this way 

Assuming that the Cost Function is differentiable, the problem could be easily solved calculating the minimum with the First Order Derivative thus leading to 

The Solution formally obtained this way is also the intuitive one, it is to say the Expected Value of the initial Random Variable 

Real World Applications for Bayesian Decision Theory 

In Real World Applications it's possible to do some experiments to estimate $ P(Y_{t} | X) $ but it's rather common not knowing values like 

$ P(X) $ A Priori Probability (prior to any observation) that a certain state will happen 

$ P(Y_{t}) $ Absolute Probablity that a certain Set of Observations will happen 

This make it unpossible to apply plain Bayes Theorem as previously explained. 

A Priori known 

In case the $ P(X) $ A Priori Probability is known and just the $ P(Y_{t}) $ is unknown, it's still possible to apply the Bayes Theorem working just with the Numerator of the Fraction, the only difference with respect to the full version will be the missing scaling factor, the Denominator, necessary for normalization purpose, but this won't affect the maximum. 

This way we can define the Pseudoprobability as a Non Normalized Probability 

The Estimated State in this approach is called Maximum A Priori or MAP and the maximization problem is 

A Priori unknown 

This kind of situations are treated defining a new kind of function, it is to say the Likelihood that relies on the assumption that all of the observation outcomes are independent 

The Estimated State in this approach is called Maximum Likelyhood or ML and the maximization problem is 

Usually $ P(y_{i} | x) $ is unknown but it is usually assumed to be Gaussian Distribution centered on $ y_{i} $ with unknown Variance. 

This way the Maximum Likelihood State is simply the average among all of the measurements 

Molecular Hydrogen Synthesis 

The word Hydrogenase identifies a class of enzymes that perform oxydation and redcution reactions on Hydrogen Molecule 

Oxydation Reaction 

$ H_2\text{aseAcceptor} $ identifies a Hydrogenase Enzyme ready to accept electrons 

Reduction Reaction 

$ H_2\text{aseDonor} $ identifies a Hydrogenase Enzyme ready to give electrons 

  Possible Applications 

This class of enzymes attract a lot of attention because of their ability to produce molecular hydrogen reducing protons that could be generate as a photosynthesis byproduct. 

Presenlty, the most common method to produce molecular hydrogen requires Platinum as a catalyzer but the abovementioned class of enzymes is interesting because of their higher efficiency in addition to the fact that the metals they're made of are Nickel and Iron that are much more earth-abundant.  

A possibile application could be found here 

Generation of hydrogen from NADPH using an [FeFe] hydrogenase

Source 

The most interesting enzymes are [NiFe]-hydrogenase and [FeFe]-hydrogenase: the former shows a higher molecular hydrogen synthesis rate wrt to the latter but it is irreversibly damaged by the reaction with molecular oxygen while the latter is only temporarly inhibited. 

The objective is thus to be able to design and synthesize a [FeFe]-hydrogenase variant O2 tolerant. 

  [FeFe]-hydrogenase analysis 

[FeFe]-hydrogenase structure 

Courtesy of Cell

$ H_{ox} $ identifies the enzyme in its oxidative state while  $ H_{red} $ identifies the enzyme in its reductive state. 

Sintesi dell'Idrogeno Molecolare 

Il termine Idrogenasi identifica una Classe di Enzimi che realizza reazioni di Ossidazione e Riduzione dell’Idrogeno descritte dalle seguenti reazioni 

Ossidazione Idrogeno Molecolare

$ H_2\text{aseAcceptor} $ Indica un Enzima Idrogenasi accettore di elettroni e quindi in grado di ossidare la molecola di Idrogeno

Riduzione Protoni in Idrogeno Molecolare

$ H_2\text{aseDonor} $ Indica un Enzima Idrogenasi donatore di elettroni e quindi in grado di ridurre protoni e produrre una molecola di Idrogeno

  Importanza applicativa dell’enzima idrogenasi

Il grande interesse per questo genere di enzimi deriva dalla loro capacità di produrre Idrogeno molecolare riducendo  singoli protoni i quali si potrebbero ottenere come risultato della fotosintesi.

Il metodo attualmente più utilizzato si basa sull’impiego del Platino come catalizzatore per la reazione di riduzione protonica sopra citata, ma alcuni enzimi idrogenasi si stanno rivelando dei promettenti candidati alla sostituzione del platino per via della loro maggiore efficienza oltre al fatto che i metalli che compongono questi enzimi sono Nichel e Ferro i quali sono molto più abbondanti, sulla Terra, del Platino.

  Si veda ad esempio 

Generation of hydrogen from NADPH using an [FeFe] hydrogenase

Source 

Gli enzimi di maggiore interesse sono [NiFe]Idrogenasi e [FeFe]Idrogenasi: il [FeFe]Idrogenasi ha un rendimento nella sintesi dell'idrogeno molecolare di molto superiore rispetto al [NiFe]Idrogenasi ma viene irrimediabilmente danneggiato dalla reazione con ossigeno molecolare mentre la funzionalità del secondo è solo parzialmente inibita.

L’obiettivo sarebbe quindi riuscire ad ottenere una variante del [FeFe]Idrogenasi tollerante al O2.

Analisi del [FeFe]Idrogenasi

Struttura della [FeFe]Idrogenasi

Courtesy of Cell

Il $ H_{ox} $ rappresenta (presumibilmente) l’enzima nel suo Stato Ossidante mentre  il $ H_{red} $ rappresenta (presumibilmente) l’enzima nel suo Stato Riducente.

A Pinhole Camera has 2 kind of parameters 

extrinsic parameters 

intrinsic parameters 

Extrinsic Parameters 

Extrinsic Parameters identify the camera coordinate reference system (C-CRS) in the world with respect to (wrt) a specific world coordinate reference system (W-CRS) and consist of 

a Rotation Matrix 3x3

a Translation Matrix 3x1 

The equation that applies this rototranslation is the following 

It could be expressed in a more a synthetic way 

With 

and 

$ P_{w} $ = Identifies a point in the W-CRS 

$ P_{c} $ = Identifies a point in the C-CRS 

Intrinsic Parameters 

Focal Length 

Intrinsic Paramters define the way the image forms on the Image Plane, it is to say by a projection 

$ f_{x}, f_{y} $ = Focal Length expressed in Pixel 

$ p_{ix}, p_{iy} $ = Image Coordinates in Pixel 

This kind of transformation erases the information related to depth it is to day $ P_{cz} $ 

The reprojection from 2D to 3D is simply got by the inversion of the previos equation, adding the lost information about depth it is to day $ P_{cz} $ 

Principal Point Coordinates 

The previous equations are correct under the assumption that the coordinates of the pinhole projection on the Image Plane are $ (0,0) $ in pixel, otherwise a correction is needed 

This way the abovementioned equations become 

and 

Putting it alotogether 

It's possibile to express abovementioned equations with a simple Matrix Relation