In the previous module 6.1 we listed several of the desirable properties of estimates estimators. These are extremely fundamental properties one should always look for whenever they are trying to do statistical estimation because if you show your result to your statistician they will immediately challenge you by asking this question is it unbiased how efficient is this it is a constant estimate and so you have to be aware of these potential questions that an educated person who is well versed in statistics could ask because if you project data simulation as one of estimation of unknowns you cannot escape analyzing properties of estimates that one has to consider as a fundamental base. So, in this module we are going to turn to a particular method of estimation least square estimation we have already talked about deterministic least squares so far now we are going to be talking about statistically squares. So, the whole lectures are centered around these square methods until now we talked about deterministic least square principles now because we considered static deterministic problem dynamic deterministic problem everything was deterministic up until now now we have ventured into the world of stochastic or statistical estimation with respect to model being stochastic with respect to observations are always stochastic. So, this theory essentially calls for some nodding understanding of the fundamental underpinnings of stochastic or statistical estimation. One of the workhorse of the statistical estimation methodology is again statistically squares. So, we are going to not develop the grand theory of statistically squares but we are going to illustrate the fundamentals of statistically squares using simple examples that are relevant to data simulation. Again let us start with let z be an observation let z be a linear function of the unknown x v be a noise v be such that its mean is 0 its covariance is r is known r is spd r is a covariance matrix in statistics the inverse of the covariance matrix is called information matrix for example let us let us let us consider the following let sigma square be the variance if the sigma square is large what does it mean the random variable can find itself in a large domain that means the uncertainty is large when sigma square is small means what the variance is very small if the variance is very small there is a very small variation that the underlying random variable encounters. So, we know pretty much the values much more confidently than when the variance is large so the inverse of the variance if sigma square is large 1 over sigma square is very small your sigma square is small 1 over sigma square is large therefore the inverse of variance is called information they are inversely related to each other one is the reciprocal of the other so larger variance means less information and smaller variance means more information the covariance matrix inverse of a covariance matrix we simply do not say inverse of a covariance matrix another name for it is information matrix so if r is spd information matrix exists x is an unknown x could be random for example as we have already said the unknown could itself be a random variable for example a climate variable as a natural variability so climate variables are random variables I am going to make observations of this underlying climate variables v is the observation noise so x is a random variable v is another add on noise coming from observations so there are 2 components in here I am going to assume that the observation noise and the underlying intrinsic variation in x are uncorrelated that is a very standard assumption it is also a valid assumption the climate does not change because I am trying to measure the nature so v depends on the instruments I use but the climate itself has an underlying variability for example El Nino comes once in several years in some form of a cycle while we cannot pin down the exact period of the cycle we know roughly what happens in here so we know the overall behavior we cannot pin down the value of the period so the period with which El Nino occurs is a random variable so El Nino phenomena has a natural variation if El Nino varies naturally the associated climate also varies naturally so x is unknown x is endowed with a natural variability x is a random variable v is an additional randomness that I am introducing into by virtue of measuring about gaining information about x so when you have multiple random variables in statistics one of the things that we have to worry about what is the relation between these 2 random phenomena simplest possible assumption is also very valid assumption x and v are uncorrelated now define the residual a this sounds very familiar to us that is what we have been in deterministic squares so I can now consider the weighted sum of square residuals the weighted sum of square residuals ffx is equal to rx transpose r inverse rx r is the noise covariance which is also called the energy norm square of rx where the weight is r inverse the explicit expression is given by z-hx transpose r inverse z-hx I have stuck in a factor of half I have already argued that many of the arguments do not change the only convenience for half is that it simplifies a little bit of an algebra we can compute the we can compute the gradient of that we already know we have we have we should be already masters of this you may have by now we have seen this term several times over so this is the expression for the variance this is the expression for the Hessian if I set the variance to 0 and solve the equation 1 I get the estimate least square estimate like this if I assume z to be to be to be deterministic xl x hat ls the least square estimate is also be deterministic but in our case z is random z is random because x is random v is random z inherits randomness from 2 sources the 2 sources are uncorrelated therefore x hat ls is random so this estimate is a random the vector it has its own underlying sampling distribution the estimator the estimate has a distribution so I would like to be able to ask myself the following question under what condition this least square estimate is unbiased and so on we would like to be able to analyze the properties of this estimate so first question is this unbiased is this estimate unbiased so I am going to discuss unbiasedness please and from the previous slide this is the expression for the estimate but please remember x is equal z is equal to hx plus v so if I substitute this z in here and simplify simple algebra leads you to this so if I took the expected sides left hand side is this right hand side is this the expected value v is 0 so the expected value of the estimate is equal to the unknown and hence the estimate is unbiased so what is the mean the least square estimate is unbiased in this case I would like to be able to compute the covariance of the estimate the covariance of the estimate z of x ls being a vector the covariance consists of the outer product this is the column that is the row you have to consider a matrix expected value of the matrix expected value of the matrix as the expected value of the every element of the matrix but in this particular case from this equation I do know the expression for x hat ls so expression for x hat ls is equal to x plus this so using this relation in here as well as in here we can readily see these two factors relate to this I am going to talk about the expected value is very easy for me to draw these things and tell you but I would like you I would like to emphasize that you should do all the verify all these algebra that is very fundamental now look at this now this term is not random this term is not random this term consists of the covariance of the noise vector and we all know the covariance of the noise vector is R this R will get cancelled with one R inverse once it gets cancelled with R inverse the other term also gets cancelled leaving behind H transpose R inverse H inverse A that is the expression for the least square least covariance of the least square estimate now I would like to go back to the previous slide if you look at equation 2 the Hessian I want to emphasize the Hessian of the FFX is equal to H transpose R inverse H now let us come back in here the covariance of the estimate is H transpose R inverse H inverse so you can readily see the covariance of the least square estimate sorry the covariance of the least square estimate is simply inverse of the Hessian of the objective function it is a beautiful property is one of the one of the most one of the most beautiful property that relates the gradient the Hessian the estimate the variance of the estimate and so on therefore what is the conclusion the conclusion is the least square estimate in this setup is unbiased I can also compute its variance the variance is related to the inverse of the Hessian of the objective function that the least square estimate tries to minimize please remember we have already talked about the relation between projections and least square estimates so relation to projections I want I would like to bring to our attention so what is Z hat so let us we can think of it like this now this is the span of H we have already talked about it in our iterative deterministic methods this is Z if I project this. This is then orthogonal projection if I project this that is an oblique projection so I in the case of orthogonal projections Z hat in the case of oblique projection there should be this I could also get an oblique projection like this there is only one orthogonal projection all the other projections are oblique oblique projection occurs when there is a weighted least squares orthogonal projection comes in only when you do ordinary least squares these are wait at least squares. So we are going to get oblique projections in here therefore Z hat whichever projections you have Z hat is always is equal to H of XLS XLS the formula for that is already known you substitute that formula this is the expression for the projection of Z on to the span of H. I am going to concoct a matrix that matrix is a projection matrix P it is given by this expression we have already seen this in the case of deterministic case I am trying to redo them within the context of statistically squares to see the relation between least squares whether it is statistical or deterministic it can be verified this matrix P is idempotent what does it mean P square is equal to P please verify that this matrix is not symmetric P is equal not equal to P transpose please again verify that. So any idempotent matrix that is not symmetric gives rise to an oblique projection that is the basic theory we have not proved that theory we have not proved that theorem I would like to if you are a person who wants to know more and more of fundamentals what is the fundamental result in here any orthogonal projection matrix must be idempotent and symmetric any oblique projection matrix as an operator must be idempotent but not symmetric what is the intrinsic relation between projection operators and matrices with these properties these are produced in an advanced book on matrix analysis matrix theory there are number of books we have alluded to especially the book by horn and johnson is one of my favorites it is very nice proofs of these basic facts from operator theory as well as projection theory when R inverse is I P is equal to P transpose it becomes symmetric it becomes orthogonal projection. So this is something we have already come across within the context of deterministic I wanted simply inform ourselves of the fact that the same properties also carry over to the statistical counterpart. Now I am going to make some further simplification until now we assumed the covariance of V the covariance of V is equal to R in general R is a speedy so R is a symmetric matrix in general it can have nonzero off diagonal elements now I am going to come to a very special case I am going to consider the noise component V are uncorrelated. So what does it mean V has V1 V2 Vm the covariance of VI with respect to Vj if it is equal to 0 for I not equal to J what does it mean all the elements here are 0 all the elements on here are 0 further I am going to assume the variance of each of the is equal to sigma square that means they all have the common variance in this case my R becomes sigma square times I sigma square times I is an identity matrix multiplied by sigma square that means it is a matrix where off diagonal elements are 0 all the diagonal elements are same and this makes sense why is that I am going to measure observations at different places and collate them to a vector the observational errors of different places where I am measuring from they are uncorrelated I am using the same instrument at every place so they have a common variance so that is the physical import of this assumption so in this case R is equal to sigma square I in this case my expression for XLS if I substitute this becomes this now please remember now in this case there is no sigma square there is no R because it magically goes away that is the nature of the expressions in here the covariance of this is simply the inverse of the Hessian which is given by this H transpose H you remember that that is the H there is a Gramian H transpose H is symmetric if I do an Eigen decomposition of H transpose H I get this Q is the matrix of columns of the Eigen vectors H transpose H is equal to Q lambda Q transpose Q Q transpose Q transpose I is I lambda is the diagonal element with n Eigen values I know I am going a little faster but I am sure we have come across this several times so I do not want to overly repeat what we have already done we have come across these things in the context of the discussion of the value decomposition and the same ideas come over here because this is the Gramian so given this so if H transpose H is equal to this its inverse is equal to the inverse of the right hand side we already know inverse of the product is the product of the inverses in the case of Q Q is an orthogonal matrix transpose is the inverse I am combining all those results to get this this is a very special formula for the for the for the for the H transpose H inverse why am I interested in the H transpose H inverse because H transpose H inverse is a matrix that relates to the covariance of the the least square estimate least square estimate now if I consider a covariance matrix the diagonal elements of the covariance matrix are all variances of the different components so I am now going to compute the trace of the covariance of the estimate please remember we have already talked about trace the trace of a matrix is equal to sum of the diagonal elements when the matrix is the covariance matrix diagonal elements are all related to variance so the trace of the covariance matrix gives you the total variance in the components of the estimate so trace of the covariance of X hat LS is equal to the total variance in all the elements of the estimated vector X hat L of X hat LS now we already know that is equal to sigma square this this is a constant that multiplies the trace so that comes out from the definition of trace so now I am going to utilize this formula to to to substitute for this so this becomes this there are lots of matrix algebra in here and and we also have seen trace of a b c is equal to trace of c a b is equal to trace of b c a that trace is invariant under the circular shift of the product so because of that I get this q q transpose is I therefore this quantity becomes a trace of lambda inverse trained trace of lambda inverse is simply sum of 1 over lambda i times sigma square beautiful beautiful so that the total variance in the estimate is simply sigma square times sum of the reciprocals of the Eigen values of the Gramian H transpose H it is it is it is absolutely beautiful result so what does it tell you if H transpose H is nearly singular one of the Eigen values is going to be close to 0 when one of the Eigen values is going to be close to 0 1 over the smallest Eigen value that is explored on your face that means the covariance estimate is large so what does it tell you you remember the condition number we talked about is something similar to that when is the condition number goes goes large the condition number please remember is the ratio of the largest Eigen value to the smallest Eigen value if the smallest Eigen value while remaining positive is very close to 0 that is going to explode on your face that is exactly what is happening in here that is exactly what is happening in here the condition number the Gramian H transpose H is very large another interpretation in the context of statistical least squares is that the covariance of the least square estimate can explode on your face if the matrix H is ill condition or nearly ill condition that is that is exactly the conclusion coming from the last line of the slide now I am going to talk about A I can do all the estimates I am assuming my covariance noise covariance is sigma square I until now I assume my I know my sigma square what happens if sigma square is not known so I could estimate that statistics are very clever people they will relax every possible assumption what if that is known how to how to estimate this what if that is not known how to estimate this so they will talk about every possible combinations of knowns and unknowns in the context of estimation problem as we have seen in the previous exercise so the whole question is this can I use this framework to be able to estimate the observation noise covariance let me let me let me pose the problem like this we have already seen if we use satellite observations we generally do not know what is the covariance of satellite observations so is that a way to formulate your estimation problem such that using which I can at least hope to estimate the variance of statistical of the satellite observations or radar observations that is one way to think of this problem as a motivational motivational motivational question so I would like to be able to estimate I would like to be able to estimate sigma square sigma square is related or by this now to be able to do this estimation I am going to go back to my residual residue E which is the error in the estimate E is equal to I sometimes I call it or sometimes I call it E depending on the context I do not think that should bother you I can call the same quantity by different names at different places depending on convenience I hope it does not throw you out of the boat so Z-HFLX so this is the model counterpart this is the actual observation the difference is the error I can express this error as I-P times Z where P is the projection matrix which we have already talked about in the last slide I can substitute for Z as HX plus V and if I do the multiplication it turns out that this error is equal to I-P times V and why this is I-P times V because of the algebra that is given in here so because of this algebra it turns out that this product while it should have 4 terms essentially the product of I-P the HX becomes 0 because of the argument in here I am going to leave it to you to enjoy the argument so the expected value of this error is 0 that means this error is an unbiased it is the covariance of this error I am interested in estimating therefore I want to be able to estimate sigma square sorry I want to be able to estimate sigma square in order to estimate sigma square E is the error vector E transpose E what is that that is the sum of the squared errors I want you to remember two things now E E transpose is a matrix because E is a column vector but E transpose E is a scalar I want you to distinguish two things E E transpose E is simply sum of squared errors so expected value of the sum of squared errors is given by this because we have already derived the expression for E in the previous slide E is equal to I-P times V I plug in that value in here I am sorry I plug in that value in here sorry yeah this is the right place I plug in that value in here I-P P is the projection operator is idempotent so I-P times I-P is I-P I want you to verify I-P is idempotent because P is idempotent V transpose I-P V is a scalar therefore a scalar is a quadratic form in V that is equal to its own trace because trace of a scalar is itself that is a mathematical trickery that we bring in to make the analysis more meaningful and the trace of a product by rule by the cyclic rule the trace is essentially given by this this is essentially so I can now trace of V V transpose is E of I want you to remember E of V V transpose is equal to sigma I sigma square I so if you use that particular property this becomes this which is essentially trace of sigma square times trace of I-P trace of a sum is the sum of the traces you can see so many facts are used trace of I is M because I am concerned with the M by M matrix trace of P trace of the projection matrix is N that is proved in here I have attached the proof for every one of these sigma so the expected value of sum of squared errors is sigma square times M-N I am concerned with an over determined system where M is greater than N that is an underlying assumption we have made all through therefore what could be the structure of a estimator for sigma square this is the resulting structure for the estimator so E is the error please remember E is the error as a residual in the estimate so E is computable E transpose E is a scalar that is computable you divide that by M-N that tends to be an unbiased estimate that tends to that that is a very good unbiased estimate for sigma square so look at this now using this framework we not only can assume sigma square is known and estimate X which is unknown after having estimated X we can also use the same framework to be able to come to estimating sigma square so we can kill two birds in one straw we can kill two birds in one straw is a very powerful device is a very powerful device I hope you do go over this very leisurely and enjoy the moment to understand the beauty and the structure the arguments and the power of this example so now I am going to summarize everything we have seen so I am talking about the world of estimators U is the world of unbiased estimate that is given in green L is the world of linear estimates some estimates are linear some estimates are non-linear some estimates are biased some estimates are unbiased so outside the green is biased outside the red is non-linear the intersection of unbiased and linear is called linear unbiased among all the linear unbiased estimate LUE L for linear U or unbiased E for estimate I am interested in the best linear unbiased estimate best in what sense best in the sense of minimum variance A that is what Kalman filter rest on Kalman filter next estimations which are blue Kalman filter estimates are blue that is the reason why we use them repeatedly and what gives us the power to use Kalman filters and make predictions because it is a very beautiful underlying property called blue blue is the best linear unbiased estimate we have talked about unbiased we have talked about linear estimate we have talked about also best best relation efficiency linear relative structure of the estimator unbiased is also related to the property of the estimate so far we have seen via an example properties of least square statistical least square estimate a reader can very easily recognize the parallelism between deterministic least squares and statistical least squares we not only are able to estimate the unknown we are able to show that is unbiased we are able to compute its variance we are also able to test the properties of the resulting estimate against the standards unbiasedness and consistency and being able to estimate the unknown covariance and so on that gave raise to the notion of the class of all unbiased estimators the class of all linear estimators the why we are going to be concerned with linear estimators because computationally linear estimators are easy to compute nonlinear estimators requires more time if I can get very good results with linear estimators why not unbiasedness is very fundamental to any and every estimator whenever wherever possible so I am interested in the intersection of these two properties namely linear unbiased estimates LUE once you confine yourself to LUE you also want to be able to next talk about the other dimension namely variance of the estimate if the total variance of the estimate can be minimized then it is going to be optimal in some sense such optimal estimates are called best linear unbiased estimate so we are interested in blue blue is a key characterization of estimations and as we will see later in when we do the Kalman filtering this notion of blue plays a very fundamental role in the definition of filtering equations known as Kalman filter equations now I am going to bring to your attention you have very fundamental theorem called gas Markov theorem gas Markov theorem relates to an inherent optimality of the least square estimates thus far we talked about some of the routine properties this gas Markov theorem takes the least square estimates one step further and puts it on a pedestal so what does gas Markov theorem says among all the linear unbiased estimate then the least square estimate is the best so it tries to attest the importance of the least square estimate within the framework of estimation theory so that is what is called optimality of least square estimates least square estimates are optimal in a very natural sense the discovery of this natural property of least squares is called gas Markov theorem gas live in 1700s Markov lived in early 1900s Markov is a Russian mathematician gas is a famous German mathematician so in that honor I believe it is called gas Markov theorem so let us try to formulate the basic aspects of the claim of the gas Markov theorem let X be an unknown to be estimated let us pick a mu a vector a vector of the same size as X and let us concoct a function phi of X which is mu transpose X mu is a vector of fixed X is a variable so mu transpose X is a linear function is an inner product so phi of X is a linear function so this is a linear estimator phi of X is a linear function is a linear estimator now what is that we want to do even though mu is known even though mu is something I picked X is not known components of X is not known so mu of mu transpose X is essentially sum of mu i X i so instead of estimating X I would like to be able to estimate this linear combinations of the components of X which is sum of mu i X i so the problem is to consider estimating phi of X I hope that is clear now ultimately I am interested in estimating X but I am considering a special case of estimating phi of X so you can think of phi of X as a functional so phi is a mapping from Rn to R phi is a function to which if I give X it gives me mu transpose X and phi is defined by mu so that is the idea here so what is that we are seeking we are seeking a linear unbiased estimate for phi of X now how is this related to estimating X suppose so now you can see mu is a vector which is mu 1 mu 2 mu n suppose if I set mu 1 is 1 all the other mu are 0 mu transpose X becomes X 1 suppose I picked mu 2 as 1 everybody else is 0 I pick X 2 and so on so if I can if I know how to estimate mu transpose X I know how to estimate every component of X if I can estimate every component of X I can estimate X therefore the problem of estimating X is embedded in this problem of estimating a functional of X so without loss of generality let us consider the problem of estimating a linear functional of X so we all know linear function functional is a mapping from a vectors from Rn to R and so on so let Z so we talked about the structure of what we want to do let Z be the data or the observation that has information about X let A transpose Z be the estimator for phi of X so phi of X is the is the so what is that now we want to do we do not want to we do not want so that is X that is phi of X we want to be able to estimate phi of X to estimate phi of X I need an estimator estimator is a function of the observation so I am going to concoct that A transpose Z where Z is the observation be an estimator for phi of X phi of X is a scalar X is a vector let A be another vector I would like to be able to consider A transpose X to be a potential candidate for estimating the value of phi of X I hope that is all clear now A is Rn Z is Rn mu is Rn X is Rn so you can see all the relations all the players in this game if A transpose Z is my estimator I am sorry is my is the structure of my estimate I can compute the expected value my Z has a standard formula Z is the linear function of X so A transpose HX plus V I am substituting for Z in here the standard one expectation of the sum is the sum of the expectations expectation of V is 0 expectation of E of X if the unbiased estimate is X therefore E transpose A Z is A transpose HX so that comes very nicely that comes very nicely now A transpose Z is an unbiased estimate only if phi X is equal to mu transpose X is equal to expected value of A transpose Z is equal to A transpose HX please remember for the unbiased estimate it must be equal to the value I am seeking therefore if I were to relate these two quantities that essentially tells you mu transpose must be equal to A transpose H or H transpose A must be equal to mu so that provides the relation between the A vector that I use in the estimator and the mu vector I use in the definition of the functional to be estimator so the A and mu are cannot be two distinct they must be related through H H is a function that relates the state to the observation I hope that is clear since it is an unbiased estimator we already know the mean square value is equal to the variance please remember in one of the earlier lecture we have already said that if the estimate is unbiased mean square error is equal to the variance so minimizing the mean square error is equal to minimizing the variance therefore I am now going to compute the variance of the estimate A transpose Z the variance of the estimate is equal to expected value of the error square I am again going to substitute for Z which is given by this we already know A transpose Z expected value is A transpose H X so if you simplify this you get this A is a constant V is a random vector A transpose Z so we can write like this A transpose V square is equal to A transpose V transpose I am sorry A transpose once again times A transpose V here A transpose V times V transpose A because the symmetric the inner product is symmetric therefore this can be written as A transpose V V transpose A if I took the expectations I get what is this so to go from here to here this is the simple algebra so like this I would like you to do all the basic algebra to see which steps come from the previous one why and how and the variance V is R so this is A transpose R A A transpose R A R is a matrix A transpose R A is a quadratic form so A transpose is this this is the matrix that is the vector so this is a scalar so the variance of a scalar is a scalar everything matches so I have computed an expression for the variance of the estimate which is A transpose A the variance of the estimate is A transpose R A yes I know I am going a little faster but what I am trying to tell is nothing unknown to you except for the algebra so I would like you to be able to go through these algebra and convince yourself in my regular class where I teach a student I also do not go over this algebra in the class because it is a part of the folklore of the course where you need to struggle learn to do many of the matrix manipulations and these exercises these simplifications are very educative I hope you will pursue that now what is that what you want to do we want to be able to get the best estimate best estimate is in the sense it minimizes the variance of the estimate so what is the estimate the estimate is is is is is mu transpose x I am sorry the estimate is A transpose z what is the functional being estimated mu transpose x what is the variance of the estimate the variance of the estimate we have already seen is given by A transpose R A now I would like to minimize this variance with respect to A A is the variable you remember A is the vector I picked to design my estimator which is A transpose z z is given A is something I picked now I am going to take responsibility in how to pick A what are the conditions for the choice of A that is where we are coming to so I need to be able to minimize this quadratic function A with this quadratic function with respect to A but A is not a free variable please remember the unbiasedness requires A and mu to be related H transpose A must be mu therefore I am not interested in minimizing A for any A I am interested in minimizing A transpose R A under the condition that H transpose A is mu so this is the constraint this is an equality constraint so I am interested in minimizing something with the equality constraints now how many times we have seen this constraint minimization with the equality constraint what is the rule the grand general multiplier now you know the importance of multivariate calculus optimization theory so formulate this as a Lagrangian problem A is a free variable with respect to which I am going to do the minimization lambda is a vector with respect to which I am going to build my constraint into my Lagrangian function so this is the Lagrangian function compute the gradient of the Lagrangian function with respect to A compute the gradient of the Lagrangian function with respect to lambda equate them to 0 solve them simultaneously the solution process is given here ultimately the minimizing A is given by A is equal to R inverse H H transpose R inverse H inverse mu wow look at this now I want to go back I know there is lot of lot of side steps there is involved Z is equal to H X plus V I do not want to estimate X by I want to estimate a functional of X which is mu transpose X to estimate this I picked an estimator to be A transpose Z this estimator is unbiased this estimator has a variance which is A transpose R A I am interested so this estimator is linear I am sorry this estimate is linear it is also unbiased therefore it is already LUE I want to be able to introduce a B to that I want to make it a blue to make it a blue I have to minimize this with respect to A but the unbiasedness requires A and mu to be related H transpose A must be equal to mu therefore this is the constraint this is the function to be minimized so I combine the two I minimize this with this as a constraint using Lagrangian multiplier technique with a little bit of algebra and simplification I have now found a formula for the optimal A what is this value the A that I should use in my estimator is not coming out of the blue sky but it is going to be decided by this structure which is given by star now let us look at the structure star it depends on R the covariance matrix of the noise it depends on H which is the linear map between the model space and the observation space it also depends on mu what is mu mu is the coefficient of the functional that we originally started that we originally started with so we have solved the problem if I pick my A to be this and use that A in my A transpose Z the resulting estimator is not only is not only is not only linear it is unbiased it is also minimum variance it is also minimum variance. So linear unbiased minimum variance estimate of fx mu transpose x is equal to is given by is given by is given by the by this expression so you can readily see this is the least square estimate the H transpose R inverse H inverse H transpose R inverse Z that the least square estimate so it is mu transpose times the least square estimate so mu transpose Z sorry mu transpose Z that is the that is the structure of the that is the structure the estimate that comes out. So if I pick mu to be 1, 1, 0, 0, 0 I get the best estimate of x1. If I pick 0, 1, 0, 0, 0 I pick x2 and so on so I can by this formulation can estimate any component of x I can estimate all components of x each of the component of x will be it will be I can estimate use and the properties of the estimate is blue. So that is the basic idea of the that is the basic idea of the setup that is the basic idea of the setup. So I would like to take you through some of the slides so in the gas markup theorem we have established linearity we have established unbiasedness we have also now established it being the best being the best. Now I am going to further embellish this with some other claims some extensions if the noise so until okay good until now I only assume the noise v has a mean 0 and the covariance r we did not explicitly invoke to the Gaussian nature of v. So until now all the analysis depends on the fact that v has a mean 0 and the covariance r it need not necessarily be Gaussian. So in addition if somebody tells you v is indeed Gaussian as in this case there is a further extension of this gas markup theorem and that gas markup theorem is the extension is called Raub-Blackwell theorem the Raub-Blackwell theorem essentially tells you the least square estimate with a Gaussian noise is the best among all estimates linear and nonlinear. If v is not Gaussian there exist that there exist nonlinear estimates whose variance is smaller than the linear estimate these are all deeper results from statistical estimation theory these results can be gleaned from the book by Dr. Raub we refer to in the previous talk. So I believe I had given you a reasonably good picture of the properties of statistically squares as opposed to deterministically squares that we saw within the context of within the context of deterministic static estimation theory. My favourite book on this topic is again Melso and Cohen is the same book that I have already referred to earlier Decision and Estimation Theory by McGraw-Hill is a small little book no more than 200 pages but it is beautifully written it has an engineering flavour and whenever I have trouble with these I always fall back on on on Melso and Cohen or Melso and Sagan Melso these are the two books and we also refer to many of these things in our chapter 14 of our book on data simulation. So with these two you should be able to get a rather complete picture of the basic elements and properties of statistically square estimation and it is underlying properties. Thank you.


Happy Friday. This is Kaui Lukas here on Hawaii is my mainland today. I have Randy Bartlett who is the See if I can get it right and Interagency coordinator for the Hawaii invasive species Council. Yes, and that's his official title but What he really is is this amazing photographer of Hawaii's native birds Although he's a local boy from Oahu around he spent 30 years on Maui and There are just every native bird. I think there is you that's alive. Have you gotten them all Randy? Oh, no nowhere close No, no, no, not all the species. I'm still working on it. Okay. Well, let me just tell you if you're on Instagram check out Randy's feed because he's RTB 808 picks RTB 808 picks Extraordinary that's how I got hooked and Then and here you are so thank you. Thank you for having me on your show This is part of my ramp up to the IUCN and taking a look at what we're doing here in Hawaii And what I love is not only is the science side of you going out and finding these birds in their native habitat But they're gorgeous Photographs, did you thank you? How did you how did you get there? Did you just wing it or? We're gonna prayer. Yeah. No, I finally finally Invested in some better equipment better camera equipment a few years ago, and so that that really helped So what kind of you want to talk about that? Um, yeah, I've got a Canon EOS 6d, which is like the cheapest full-frame camera that they have that they sell and I got a decent some my decent zoom lens so I telephoto lens so I could Get close as I could to to the wildlife especially birds are kind of tricky to photograph Yeah, so how far away from them are you and let's let's start Let's let's have a bird to look at and then you can kind of tell the story about when you are out Hunting okay for the perfect shot, which you get Okay, this is this is the this is actually a lie Keokia or like here the white Koot Hawaiian Koot and one of our native endemic species and this one is actually taken on a wahoo at Kauai Nui or Hamakua Marsh complex on the windward side in Kailua So it's fairly close, you know, I'm up on the bank and I'm shooting down at the birds in the water So maybe yeah, here's another shot of the same bird a few yards few yards So if you go by behind down to earth and oh, and here's the other one. Here's the ally oola. Mm-hmm They're the red Red which means red forehead ally oola. So this is the one that in the Hawaiian legends that the demigod Maui Squeezed lice throat so to give up the secret of fire so so Yeah, so they could wow their couple and that's that's how they've got it's red Is Maui squeezed its throat the I think this the story goes and it could be wrong is that It got read from a burnt from an ember in the fire Well, I love how the water has sort of the same quality as the feathers and these photos around it It's just exquisite. Yeah, this one was actually taken at Keava pond or wetlands out in Hawaii Kai Right at the or the awful club is and there's also a hail. Yeah, I heard they had some chicks Born there of the ally oola Two months ago. Yeah, that wouldn't be surprising that was taken maybe a couple years ago Okay, so how are they doing the ulas? Yeah, and they seem to be doing fairly well. I mean there do have their biggest threats are probably predation from Maybe feral cats or dogs because they are kind of ground nesters So they could be vulnerable on the ground while they're nesting Yeah, that's why they had to put that that fence out there at Keava in Hawaii guy, right? Because they have those nasty feral cat colonies. Okay, we won't go there All right more birds, please See oh, so this is a Maui creeper or aloha heel. This one's a female And it's I like the shot because it kind of shows how she blends in with the lichens and almost the same color And this is a houseburg Grove on Maui in Haleakala National Park And it's just a little closer version and it might be moving Some of these are our gifts, but and should be moving but we're just experimenting with the the that we're experimenting with that with the technology here as always it's a Think tech science experiment. Oh, this is what apapani. Whoo. See that's so cool. Randy How do you do that and they're up on the oh here on the top of an oh he a tree Not too many flowers at this point But they are looking around probably for insects too because they not only go after nectar But they can go after the insects too along with the amakihi And there's another apapani there just gorgeous. So how are they doing? They're doing pretty well. You know the apapani are one of the more common of the native forest birds And they seem to have some Resistance a little more resistance to the avian pox or malaria that a lot of the native birds are Threatened with and those are all diseases that are spread by mosquitoes. Ah, which we learn no native mosquitoes and so those things just like people can get disease from Tropical diseases from mosquitoes like Zika or dengue or malaria or chicken guña The birds also get avian pox and malaria spread by mosquitoes and so as Is the earth warms up from global climate change then their habitat is gets warmer and so They are more threatened because the mosquitoes can go up higher to higher elevations On the in the forest where the birds are and so then they're more exposed to that to those disease vectors As time goes on so then their populations are more and more endangered Okay, who's this? This is another alohi. Oh a Maui creeper.  They've got that yellow kind of chest or breast So this one you can if you look closely you can see in his beak He's got a little bug a larvae that he picked off the branch of that of that bush right there I was wondering whether that was beak design or dinner. Yeah, that's his breakfast actually This was taken kind of early in the morning What if we had and this is an amakihi Hawaii amakihi Again on just a pine branch, but with the lichens in the background This one is probably a male also because of he's a little more brightly colored than the female females get Oh Love this shot Yeah, this one just this amakihi as well the Hawaii amakihi Just taking off from the Eliahi, which is a sandalwood, which is a holly. This is a haleakala soundalwood They're on the left so it just took off the branch and I was just Lucky to kind of get it So the the native forest up there where these are taken on Maui. Is it pretty pristine as far as I mean You've got Eliahi. You've got these birds. How how how was the forest? Yeah, where I where I was taking a lot of these Maui forest birds are we're at as a haasburg Grove and Haleakala National Park is just past the park entrance So it's just a little bit above 7,000 feet and it's just at the upper edge of the native forest and right below The park boundary there is The nature conservancies Waikamoe preserve and you get some really nice wet native forest down there But in the haasburg Grove, you're just right at the top end of it And so the birds come up out of the forest and they kind of forage around there So it's the probably one of the most accessible places in in all the islands to see more of the common native birds to see the really rare endangered birds like the Kivikiu the Maui parrotbill or the a kohe kohe the crusted honeycreeper Which are so endangered. I mean, I don't even have those yet and those are still on my bucket list to get But those you have to go down into the really really nice wetter forest down in Waikamoe preserve And those you need permission from the nature conservancy to do that Isn't that part of your job though? You could Used to be used to be when I was managing the East Maui Watership Partnership program The nature conservancy was part of that partnership and so was the national park Yeah, so for um ornithology fans here um we can ornithologists um the Going to Hawaii uh Maui up to this haasburg Grove sounds like a good spot to hang out Yeah, if you're really into birds and bird watching or we just want to see more That's that's really a really good spot on Maui. It's probably the best place to get to On the big island you'd want to go to either probably to volcano national park and bird park there They've got some really good places there just all around the volcano center area. You'll see native birds Or else also up on the saddle road They just opened up a monarchy on monarchy a palila kind of viewing trail I was just hoping that up a month or so ago and I haven't been there yet, but I hope to go soon Let's see some more. Oh, there we go. There's another. Yeah, this is a apapani again on the uh sandalwood or the iliyahi Do they exist on a while?  Um helping with some snail work and um We saw apapani flying around in the daytime, but you can see them in the koalas as well But you kind of have to hike a little farther. Um, maybe on the ia loop trail It would be a kind of fairly accessible place for most people to go to you can even probably see ellipio there The oahu ellipio Which are endangered. I I have seen um, there's the ellipio in aina. Oh, what do we have here? This is the alkuu. Yeah our black crown night heron This is um, this is a native species, but it's not endemic. We're Native just to hawaii. It's a pan tropical species. So it's throughout the tropics all around the world. Um, they have They don't call them alkuu, but they're black black crown night herons But we have our own. Um, yeah, they're not a Endemic subspecies So night heron they they you caught them in the daytime, but they they do they're fishing at night they um They can they do a lot of I just catch them in the daytime. I don't know why they're called night herons Okay, I've always taken pictures during the daytime So and here you are, um Where are you? This uh, was a couple jobs ago. Um, when I was managing, you know, the uh, Pukui watershed preserve at Maui Land and Pineapple Company on West Maui So on the Pukui West Maui, and this is up on the boardwalk up in the up in the bogs About a mile high Very wet and this is some uh, this is a lobilia gloria montus um, the glory of the mountain Um on kawaii they call lobilias pu'e. That's the only hawaiian word that that I know of That survives for the name of those flowers Well, um, we're gonna take a little break and then come back and look at some more of your gorgeous photos and talk about birds Okay, great Aloha, my name is Justina Spiritu and I'm the co-host of hawaii farmer series This is my co-host Matthew Johnson And you can catch us every thursday at 4 p.m at thinktech hawaii dot com What do we talk about matt? So on hawaii farmer series, we're going to be bringing on the farmers and also supporter of farmers Including restaurants cuters as well as government supporters and nonprofits to hear their background stories And understanding our local ag community just a little bit better Yeah, essentially there's a lot more that goes into Farming and the local food community beyond just producing the food And we want to feature and get the background story on all these folks and see how we all work together as a community So join us every thursday Aloha Welcome back to hawaii is my mainland on thinktech hawaii every friday at 3 p.m This week. I have randall t Bartlett who is not only um a phenomenal Photographer of the native birds, but he is um the interagency Coordinator for the invasive species council of hawaii. Did I get that right hawaii invasive species council? All right. Okay. Well next time I'll get it right So let's look at some more birds What do we have? Oh, we're going to talk about the eevs a little bit and where where do you find them usually? Um eevs are still pretty common on most of the islands um big island maui, especially kawaii also um They're also on moloka e and they're also on oahu though They're kind of the population is going down They seem to be kind of susceptible to to avian pox in malaria that are spread these diseases that are spread by mosquitoes as we mentioned earlier um So Yeah, you perfect. It'd be kind of rare to see one on oahu But if you went to kawaii and went up to the alakai or even kokei you'd probably see them um on Maui if you went up to haleak la national park around hawsmer grove you'd probably see them these are all taken um in hawsmer grove Either on the trail or sometimes right there in the parking lot um, so yeah, this was right in the parking lot and um in the parking lot This is a great shot because you can see why they have those curved Curved beaks right just to get right in after that momani nectar Uh, you know in the momani flower Just teasing it out of there This is a uh a great show that about um, you know your photographs often have them Where their food is naturally and to see why it is so important to take care of of the ohia And the momani and all those the native trees because they're the food for the birds Yeah, I'm affected for like the palila on the big island on monachea You know, that's they're almost their exclusive food sources are the seed pods of momani And so they really need those this is a juvenile eevee or eevee polena or eevee um Let's see Popole, I think as um sam ohugan has told me From the nature conservancy, but yeah, this is when they're still young They're they're kind of uh, they don't get their red feathers yet until they're adults And so just yellow black and white and sometimes they get a little green Uh and red so they kind of almost look rastafarian This one so it was a little more yellow their teenage rebel years Yeah, and you can see he's got a little bit of pollen on his beak on the tip of his beak So as as they're going from flower to flower they're moving pollen around and pollinating pollinating the flowers and Yeah, I need some of those they're apart for my My my fruit i'm needing pollinators. Maybe I can get some of those Beaked birds to fly over to my house. Yeah, we have to get rid of the mosquitoes first Unfortunately the mosquitoes. Okay, so here's this is this dapple. This is amazing shot With the sky behind but there those feathers are this is an even younger one and he's still got that's why he's still got his gray Downy feathers on his breast that you can see there and even on his thigh So he's really young. He's probably this could be the could have been the first time out of the nest Um because he was coming down and looking at me and both his parents were up above in the branch above You know above him like squawking at him madly and probably telling him stay away from that thing down there Which is me So are you are you generally that close like you're down on the ground Right by the tree and and the birds are okay with you being that close If you're lucky enough. Yeah, if you're lucky enough and you and you're patient enough now this one I hadn't seen before um your photos the kiawaya Um, yeah, this is the bristle thigh curlew and they're not year-round residents To hua ii. So you just see them seasonally Now summertime they're they've they're flown here from the mainland or alaska wherever it is they're coming from Um, kind of like the coleas, you know, which is why away. Where do these ones live? Um, well again the summertime they're here in the rest of the year. I mean they're Up. I'm not exactly sure on the bio. Where would I find one in hua ii on oahu? Um, this was taken that one was taken at james camville national wildlife refuge out in kahuku I see so and then during this season and that was taken Um, that was actually taken at the end of the season there That was in december end of the year end of last year And so it was like they opened it up on the weekends When it's not nesting season so it's closed now So the but you can go on the weekends when it's when it's open from like october to the end of the year and You can go on a little tour You had some some pictures of that on your um your instagram feed. So I haven't been there I want to check that out. So the um, so this bird comes um, see the name again The uh, kiauea kiauea. Okay. I looked at that and there were no Uh glottals or uh anything and I just couldn't figure out how that was going to come up kiauea Okay, so what did they eat?  Yeah, and this is another seabird.  Um, at least on a wahoo their main population is out near um Between coco head and coco crater and sandy beach out there near halona blowhole along the coast Yeah, and they just um, they're just kind of finishing up their breeding season now. Um, so then Probably the rest of the year they might be out to sea and you won't see them much Okay, so if one goes out to vava malo now You could see there might still be a few um with chicks that haven't fully fledged yet But yeah, and do they nest are they ground nesters also? Yeah, so they're also The um that a lot of our most of our seabirds are and so um that makes them really susceptible to things like rats or cats or dogs You know feral cats or even you know people go like what happened out at a point where people Walked out there with their dogs and they just Right decimated 15 or more or laser and albatross because They're defenseless against those kind of things Um, up next we have the um the The ducks. Yes, our uh Koloa maoli our hawaiian duck are endangered Um, most of the ducks on on all the islands except kawaii Um, are kind of are you are mostly hybrid hybridized with uh introduced mallard ducks Um, these were taken at um again at kawainui here on oahu and kailua kawainui marsh Um, so they're probably hybridized to some degree kawaii again has the most number of native ducks that are still Pure or mostly pure and how how are we controlling that? um The the only way to control it really is to um is to well Yeah, because there's been ducks that have been introduced Uh mallards and other species. It's it's kind of tough. They're already out there in the wild Um, so can they fly in rylan can mallards fly from oahu to kawaii for instance? Um, they probably can yeah, and they probably have um, but Um, but most of those um were the ones we have were probably introduced for Either pet trader or whatever we we had we raised them as a kid. I had no idea Yeah, so don't bring mallards in bring the white ones in because they don't mix or probably not But they they might they might just just the mallards somehow they're close enough to the mallards Well, the other species might be able to i'm not sure on the biology after that's a real expert So how how much time do you get to spend out in the out in the wild with these beautiful things? Um, not as much as i'd like probably not as much as i used to on maui But uh, because that was more part of my day job But i do um try to go out on the weekends in fact And these are the these are ones that are close to your day job. These are manuoku. Yeah, these are manuoku chick or white fairy turns This is a indigenous seabird native Again, it's all over the tropics, but um Here in honolulu. We're actually kind of lucky on oahu. We have the only Major colony on on the main islands hawaii islands the rest of them are all out on the northwest hawaii islands on places like midway or or leisanne But yeah, these are actually right in the state capital grounds or eolani palace grounds Um, they're capi eolani park uh manua the population is kind of growing has been growing since the early 60s And so, um, it's actually a growing population. So we'll see more of them I have uh Two two pairs in where I live at pia This year what is so extraordinary and I never I couldn't tell because I can't see up close like you can Um that that that that little branch that I mean somehow the the egg balanced on that branch and It hatched and this little chicklet is able to hand stay on that narrow little branch Yeah, they're they're that's the same bird that was in this previous shot It's the same chick and it's just a little younger here But they don't build nests these birds don't build build nests and they just lay the egg Right on a flat spot or a depression on the branch or on a building or on a rock or On the ground even on on the northwest hawaii islands But here they they tend to do it up in the trees where they're a little safer And yeah, the the chick will hatch right there at that spot where the egg was laid and stay there Pretty much until it's fledged Once it gets a little older, it'll start to move around up and down the branch a little bit, but It pretty much just hangs out right there and Until it gets fed and it gets its flight feathers Look at that. It's practicing already so adorable. It's just ridiculously cute So they are such a great um Mystery to me that they that that little chick can hang out on that narrow branch for For would she say six to eight weeks? Yep, about there a month and a half or so or a little bit more Wow um And they again are ones that are So the the big Pressures on on these birds seem seem to be cats and rats Cats and rats primarily and and mosquitoes and right and also yeah diseases for the for the forest birds Especially it's um avian diseases bird diseases spread by mosquitoes So we're going to wrap up with the Pueo Which has made the evening news last night. How's that? That's pretty neat. Yeah, I guess so where were these um these again? We're also um at haleakala national park uh, hawsburg grove This is the the headwaters of wikemoi gulch. They're right near the end of there the little nature loop trail they have Um, and it was just a pair of them us just flying around in circles around each other for a little while And I was managed to get a few shots That's that's pretty good. I mean I've seen Pueo from time to time on on different islands, but I've never seen A few of them at once Yeah, usually they're singly, but sometimes, you know, you'll have the pairs flying around each other Uh big island you can you can probably see a lot of them a lot of shots from the big island photographers Pueo and um, how's their habitat doing? They're doing a little better. I think they're they're raptors. Um, but they also the the main problem is their ground nesters also So they're not building a nest in a tree. They're on the ground. And so then they're also, uh, susceptible to predation from cats or dogs Thank you so much randy for coming and showing your gorgeous photos again your instagram fade is rtb 808 pics pix And um, you're um, you've started a a new venture which is the The Hawaii endangered Hawaii photography, right? That's what I'm kind of um, that's going to be a little bit of a side side job for me Thanks, randy. See you again soon. Thank you. Aloha. Aloha


Hi everybody, Joe for jazzy's casebreaks.com still new release day new release Wednesday at least for us here in Los Angeles 2022 Panini luminance football picker team one coming at you remember picker team three sold out first then four or five six sold out And then we circled back to one Thanks everyone for the fill. However, we a little roundabout way, but we got there big. Thanks everybody here for making it happen You got a little rooftop next to your name that means you won that spot in the Luminance team random that we did and Jose ended up with double last-pot mojo with the buccaneers and Titans got those last before For these teams were pulled now remember Jason did break number three and It's a 12 box case and we're doing four boxes So one of these is gonna be one or two right one of these stacks gonna be one or two So let's grab a die And we'll go one two three for the left side four five six for the right side and it's one For break number one. So this Will now say for break two All right, and that break two is coming up after this and we'll just be in a separate video. All right. Good luck everybody three autographs and one memorabilia card per box on average great photography great inserts and I feel like Compared to previous years. I feel like there's been some more If we seem some more year one on card autographs don't mind that I dig that Got nick chubb to 149 and some points will the winner take all on the points That's for the browns. That'll be for Chad Like that insert set right there. That looks really sharp It's great photography here as well Dak Prescott for the Cowboys to 249 That'll be for Joe and the boys And then we got Romeo Dobbs year one on card autograph. I have a checklist.  I suppose there's not too many Pass catcher catching options out there. So we'll see how that works out You got sky more to 149 for the chiefs I'm gonna go to sam And we got to the moon or in moon Vikings chris for the Vikings Patrick Mahomes 58 out of 100 for the chiefs We've got a bangle coming up here We got uh roberson to 249 for the cowboys and it's daxton hill and uh He is a bangle that'll be for patrick davis I like these sheesh insert. Oh, I thought that was numbered, but I like those inserts. This is numbered dynamic Are these cards bigger than standard? No It is the standard size Do they look different? Do they do they look bigger? Maybe my hands look smaller today. So the the cards look bigger More points winter take. Oh, wow 1500 points winter take all on the points It's a lot of points though Uh, there is Derek card 149 Raiders Stefan digs 249 and we've got Jerome for year one on card autograph Yeah, that's actually a a decent amount of points Uh, Jerome Ford is a brown cleavage. This is for you That's for chat. Oh got the browns straight up and a one of one davante Adams. Wow That's a sharp-looking card Davante Adams in his raiders gear one of one Jeremy port with my raiders Hoping for big things for him Jeremy All aboard the big hit express A raiders Nice and we got kyle pits behind him And that's going to go to brandon and the atl Aaron jones to 100 Givante williams to 99 and we've got uh, leo churnal That's 84 out of 349 That's what i'm hoping jack. It'll be nice all season long Chiefs that's going to be for sam. That's right rex Uh, I'll stretch out a train whistle for a raider Some more players in your post notes here all right Halfway through this four box break and then we'll have another one of these coming up Picker team two will come up in a separate video after this It's like a thicker card there Maybe a lower number perhaps With damien harris to 249 patriots that'll be for karen And we got a diante johnson autograph 10 out of 25 Light speed ink that looks really cool That will be for the steelers. Han picked up pittsburgh straight up There's out of 99 Calvin awesom the third and we got calvin awesom the third I feel like they had that happened in a couple breaks ago Calvin awesom third is a pittsburgh steeler Oliver is a steelers fan. Oliver tell me about calvin awesom the third I'm gonna meet malik willis rookie card as well for the titans for jose Good that's good news. Oliver saying he made the team should be the fifth or sixth receiver. All right, fine. All right. See you guys It's your own ford to 249 Nice jack good good trivia there his dad at calvin awesom the second Good one. There's givante williams piece of his jersey going to patrick and the broncos Yeah to the two the two uh Chronicles luminance and the baseball got it. I got it. Thank you Out of 50 sheesh cd lamb Love these uh, love those inserts george pickens Nice for the steelers. That's to 149 and We got a kyle phillips 10 out of 10 autograph for jose and the titans nice low number there Did he make the team? who so we have a picket and two Pickets and phillips's and pickens is Sorry kyle phillips. I'm losing my mind here kyle phillips kyle phillips Not sure, but he was a uc le guy wikipedia says roster status active Maybe he made the team All right fourth and final box here and pick your team one Thanks all pick your team two coming up After this And that's looking like that's going to be our last break of the night That's right reps. I'm not hourly Do staying late for the people It was nicer than to fill a break Thanks peoples All right An all-car ship ladies and gentlemen, that's that's a good thing out of 25. We got Tyler Goodson for the Packers Then we've got um Kind of hard to read there charles cross To 149 and a titans up nope and a patriot autograph tyquan thorton to 349 that's going to be for uh Karen and her patrons Brandon cooks to 249 and we have We've got a ikem ikwanwu Big dude That is for carolina jeremy port and the panthers Wow more points another 600 added to that pile That's the biggest i've seen that points pile in a long time long time We've got a justin herbert three out of ten Chargers chris Got a trend mcduffey the 99 and a van jefferson intensity Relic Yeah, is that is that his number as a professional or or in college? All right, and that is that A lot of points to give away So let's flip back over here Let's flip back to Grab some blank lists here All right, so let's go back to random.org. There's some new dice right there one dice rule for both lists The first list is winner take all on all those points. That's a lot So that's going to be a nice consolation prize or a nice cherry on top for somebody now for break credit You have to buy at least an even number of teams and half of those become entries Aaron bought two teams. That's an entry brandon bought two teams. That's an entry chad two teams an entry chris four teams two entries Han two teams an entry Jeremy two teams an entry jerry two teams an entry jose Still got two teams. That's an entry one a spot though patrick two teams entry Richard you won two teams that still counts as an entry And then shon healy the two teams in an entry all right, so shon richard patrick jerry jerry All right, these are some good odds too this I think the lowest total entry list I've ever seen 12 So two out of the 12 spots All right, let's roll it. Let's randomize names and uh names six and a one both lists seven times points first one two three four five six And a one seventh and final time After seven congrats to Oh jopey so close He wanted those points too. Congrats to Richard with all those points after seven times I got a green 600 a purple 1500 and a blue 400 nice congrats and seven and a one six and a one seven times top two after seven one two three four five six and Seven than final time After seven congrats to shon and jose 125 dollars a break credit going your way I'm joe that was picker team one from jaspercasekates.com. I'll see you in the next video for picker team two


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Man, welcome to the Trap Spot. I don't know if you heard. I don't know if you know. I don't know if you're privy to the information, but right now, the black market is up. We got somebody real dope and cool and hit with us, man. You know what I'm sayin'? Just to throw a few titles out. You know, this is a digital media content creator. You know, very well-established journalist, writer, documentary, filmmaker, cool-ass nigga who always on the scene. He always know what's goin' on.   I be like, what this nigga got goin' on? Hey, ladies and gentlemen, welcome to the Trap. My man, Torren Walker. Good, man. How you doing? I'm good, bro. I'm good. My turn. Good, man. Welcome to the Trap, once again, formally. How you been? I've been good, man. Out of the world, starting to open up, you know, and gettin' back out here and breathe a little bit more. What you been up to? Oh, man. Really just gettin' back into doing my journalism again. I had to do a lot of things remotely. Yeah. And that was gettin' on my nerves, and now I can go out and see people face-to-face so I'm doin' that again. That's good. Yeah, what type of journalism you been up to doin' this pandemic? Doing a lot of interviews. I managed to connect with some people who had been rockin' with the things I do, and so I set up, since I couldn't go out to do interviews anymore, I set up a Context Media website channel and a YouTube channel, so I just started doin' zooms and talkin' to people about some of the things that we were cared about.  Well, yeah. I really saw, you know, a lot of press conferences where, you know, the old president, he just ignored the black-lady questions most of the time. Yeah, but you know what? You gotta expect that from somebody like that. Right. I think what people have to really pay attention to is be a warrior of people who pretend to be your friend and will invite you in, and still don't answer your question, but they'll smile at you. Damn, bro, I never even knew that black journalism was that hard. Maybe all the black journalists should just focus on investigative reports then.  It's just that the ones that tend to get more shine are the ones who do more sensationalism. Like, you know, celebrity media's always gonna sell. You're never gonna get tired of that. Shit, I did. Word? I'm tired of it.  New people regular as hell if you look at it. Yeah, but you know, most people like that stuff because you know it's sensational, and it's just like watching reality TV.  But it's so prevalent that you can't really move anywhere else on that. That's real.   Well, you know, it's an escape for some people. And you know, it's nothing wrong with going on and taking a look at what's going on with somebody for a while. You know, for a couple of minutes and then moving on or something else, but because there's so much of it, and they keep pumping that stuff out on a daily basis 24-7, you can't really, it doesn't rule from nothing else.  I think it's real to the point where you gotta understand, there's a whole generation that grew up online and they don't know anything else. And there's a lot of people who don't really have any sort of outside stimulus, outside of being online all the time, literally.  They don't know anything else. So if your world is in there and all of a sudden you say something that somebody doesn't like and then they create this mob that attacks you, then yeah, cyberbullying is real at that point. Now if you're 18 and up, yeah, I don't really believe in it that much. Right. You can always turn the, you can always cut your phone off. I think I've been bullied by every demographic of people out there on social media. I think I probably have individually upset, except the indigenous people. They let everything slide. You know what? If you have any kind of presence on social media, anything you say is gonna make somebody mad at some point. I know. So you can either end up being so wishy-washy that you don't stand on anything or you can just be who you are and just keep it moving. They'll find something else to be mad about in a day or so. That's crazy.  Some people do. You know what I mean? Some people are so tied in to social media and they're so tied in to being in that cyber world that they really do believe in it. Some people don't, but they pretend that they do because it looks good. Just like some people pretend to care about certain causes that they really don't, you know? Yeah. I ain't one of them people.  You know, that's a controversial opinion, depending on who you ask.   We gotta save the black people, man.  I don't even know what the count is right now. I know it's a lot of us left, but leave us alone.  We gotta save us, man. Everybody else save themselves. They keep saying that that's the way to go, so we just go ahead to save ourselves. But you know what?  Everybody else knows that, except us. Yeah. We're not supposed to be everybody's life preserve and we're not supposed to come save everybody else, but when it's time for us to get something, we get nothing.  We need a lot.  We still just need, we probably got about 30 or 40 more silver rights we gotta get.  They cheated us out a lot of them. Well, America's gonna always be America. That's not gonna change. I feel you, I feel you.  I hope we just owe it enough to see it. That's the cold shit, like, I don't know. You know, there's no way to determine that, you know? There's people, you know, 200 years ago on the field who thought they would never get any further than that, you know what I mean? They were fighting, they would run away, but they only was trying to run away for their own freedom and try to take care of their individual needs.  We may not see them. I don't even think we got enough of them yet. Some of them people are so disappointed. Because they get on TV and you be like, oh, they those kind of black people. So it's kind of like they not even that. A lot of misrepresentation out there. Just, I don't think we should just have to accept people just because of, like, on our behalf, just because of that. If they don't have our interests, you know, on the forefront, then they just, they just as bad as the motherfuckers who's trying to replace. We got a real bad habit of doing that, though. We get caught up in the idea of representation.  I think we just have been having a whole lot of the wrong kind of black people put in the positions that we wanted to see real black people in.  Unless you hide who you are until you get to a position where you can do what you wanna do. Exactly, man. But it's a certain level of realness that gotta be in anybody and in profession. You just can't go along and get along with everything. Yeah, you can. That's not healthy, though. That doesn't, that doesn't, that's not progress. If you just accept anything and let anything be, then what are you doing? Taking care of the Wallace. I mean, that's, you know, that's what I mean. People gotta get out of this idea that just because somebody looks like you're there for you, you know what I mean? Exactly. A lot of people say everything that, people say stuff that sounds good, and they say things that they think people wanna hear, but they're out for the bottom line. That's just a...  Gotta look out for that bottom line. Journalism is some of the worst of that. Hmm. Like, what are some of the things that you have to, that you have to experience on that side of the journalism? They know that the public may not be privy to. All right. I started out in journalism working for mainstream news. I worked for Huffington Post. I worked for a couple other mainstream outlets. And right about... What was that experience like?   Hell no, it wasn't me. But you know exactly what I'm talking about.   Y'all met that one. I'll tell you about my story, my experience was, I started writing for Huffington Post right about the time when Mike Brown got killed in Ferguson. So the beat I was on was like, basically for lack of a better term, the race beat. It was like, cause you know, like every other week somebody was getting killed. You know, and there was a story and there was a protest and there was an uprising. So I was on a lot of those stories. While that was a hot story, nobody really bothered me that much. But then once you started seeing those court cases start coming to pass, I wanted to start trying to cover those. And that's when I started getting pushed back because it was like, it wasn't a hot story to talk about what was going on in the courts.  They loved that. But nobody really wants to talk about what happens after that.  They don't want to talk about that. So I had a lot of pushback with editors for that sort of thing. That and I had to deal with a lot of editors who weren't really from the culture who were trying to tell me how to talk about black people. Like what were some of their pointers? Whew. There was one time I wanted to go into the city. I'm not gonna say the city, but there was some brothers in the city who had been doin' a neighborhood watch. A lot of them were like fresh out of prison. A lot of them were ex-gang members. They didn't have any pedigrees. They weren't from college and all the stuff. They were just street cats. But they would make sure these kids got to school and keep out of like, from the dealers and the hustlers knew not to mess with them because these dudes were on the street. I wanted to interview them and they told me straight up what was the interesting story. Basically told me nobody wants to hear that. Really? What happened? Why are these guys now? We gotta get them on the show. That's interesting as hell, bro. Yeah, but it's like, damn, that's... But what is the goal of mainstream media and news from your experience knowing that they think that that type of shit is not important? What is important? Um, ratings. That's what they care about, ratings. However way they can get them. Even in news? Especially in news. News or what you want to call mainstream news or hard news is a loss leader for most corporations. You gotta understand like, most news that you get comes from maybe two, three, maybe four corporations right now in the United States. Four major corporations control 98% of the news that you get. So they're concerned about turning over money for their shareholders and they're concerned about getting ratings and whatever. They don't really care. I mean, there's a certain level of fact checking that has to go on, but they're more concerned about getting to a story first and getting it right. That's just the truth. If they can get paid from a story about somebody falling out in the street, they'll sell that. If they can get paid from somebody throwing a rock through a building, they'll sell that. Okay, so what is it really then? It's entertainment. Carlos Miller here and I get DMs all the time saying, Los, man, how can I make sure I leave my girl satisfied? If you were paying attention, I'd been giving you the game the whole time.  Bluetooth is a unique online service that delivers the same active ingredients as Viagra and Cialis. The first step is simple. Visit bluetooth.com, then consult with one of the licensed medical providers and once you approve, you will receive your prescription within day. You can take them anytime, day or night so you can plan ahead or be ready whenever an opportunity arises. So if you can benefit from extra confidence when it's time to perform, Bluetooth can help. And we've got a special deal for our listeners. Try Bluetooth for free when you use promo code blackblack at checkout. Just pay $5 shipping. That's bluetooth.com, promo code BLACK to receive your first month free. Visit bluetooth.com for more details and important safety information and we thank Bluetooth for sponsoring this podcast. So all the people who are watching the news who think they are getting legit information is just simply being entertained. Yeah, for the most part. There's exceptions, but for the most part, you're just looking at the same thing running over and over and over on the loop. And then when you get into black people and how people report on black people, that's a whole nother animal. Like how do they report on black people or like they're a... They report on black people like the circus. Basically, if something happens- Like the circus? Yeah. You ever notice like every time something happens on the block or there's a story that happens in a neighborhood that's not upper middle class, they find the wildest person they can find on the street and they put a camera in front of them and they let them talk? Yeah. You know what I mean? Not to say that that person doesn't have a valid point, but you know, they get somebody who's gonna act the fool so they can go viral and they just- I think they be trying to find those people because though, I don't know. That's a good thing and a bad thing. The bad thing is embarrassing.  I was going to get some gas. And that man came out of the car. And that man was like, is that a car that came out of the car? Came out of the car then. He's out of the car then. Right in front of me. I'm in the car, I'm in the car, came out. Everything happened right now. I seen everything. Yep.  I mean, sometimes that's funny. Like remember the guy who rescued the girls around that house a couple of years ago?   See, people love the dead giveaway. The golden part of that interview was when that man said, I used to eat ribs, man. Saucer Dan. Yeah. I used to eat ribs. Just as, man, where is that footage? Charles Rams over there with the slick bag of the Saucer Dancer.  Man, come on. He dropped a cigarette out of there one time I picked it up. Come on, man. Come on.  Now he still ain't get everything he was supposed to get for that. Yeah, and see that's the other part of it too. When you black people who move like that, get to know the writing, they're not able to capitalize on stuff like the way somebody who's white or whatever does that. Cause you see white people like in the food too and they walk in the record, they walk on the contracts, they walk into endorsement deals. I don't know what Charles Rams have got.  Well, I don't know. I know he ain't really get the real, you know, the real like respect that he deserve. He's people's lives right here. I have a fun in the interview is like, he still did a great thing.  So that's why I moved out of mainstream media and started doing my own thing. Because I felt like, you know, after being on the grind and watching how people were able to tell stories, I'm like, you know what? The technology is here. I can do this myself. I don't really need anybody to help me with this. And I can say what I want to say. That's where technology is pushing everybody to go independent. Like have you found more success on your own terms as opposed to going in and having to try to figure out what these people in the mood for the day? Well, yeah, in a way. I mean, I found success whereas I have freedom to be able to say what I want to say. And I can fact check it myself and I can put it out and I don't have to go through a lot of filters. But it's hard to do that by yourself because you're not plugged into the apparatus with like, you know, I can toss CNN or I can call MSNBC and say, hey, put this out, put this out. So what kind of advice are you giving right now to the young up and coming black journalists? I don't normally give advice but what I would say is everybody has a story and everybody's story is valid. And you have a right to tell that story.  I mean, everybody has one. Everybody has a right to. You don't have to, you have 15 degrees to be able to tell somebody's story.  As long as you don't break that, you're good. What kind of literature would you point them to? Books that may help? One book that I'm a huge fan of and it's helped me out is Gordon Parks's autobiography.  Gordon Parks was doing stuff that it's wild because some of the ground he was breaking in the 40s, in the 1930s, being one of the few black photographers and journalists working in mainstream media. Yeah, his photography is so dope, just like even without the context of his writing, it's just, it got all these books and shit you could just see rare images. Yeah, and he was really instrumental in teaching me how about the power of an image. You can talk and you can write, but that's only gonna catch a certain amount of people. If you have video and you have technology buried together and you put these images together and you can show people what's really going on, that's extremely powerful and that says more than just what you can write. Well shit, man, let them know where they can find you at, where we can get some links to some of your print media and things of that nature. You can find me at the site, Context Media Group, that's the website. All my social media is Context Media and you can find me, Torren Walker, mostly on Twitter. I'm on that way more than I need to be, but that's what I'm gonna do. Man, you be on that, bro. You see everything. Yeah, I see a lot, I see a lot too, but you know, better for worse. That's where you can find my stuff. That's what's up, man. Anything you wanna leave me with before we wrap it up? You got a question? Yeah, after the last election and the way that it transpired through the media, where do you think the ethical code lies now for journalists? Journalists gotta stop, a lot of people not gonna like this, but journalists gotta stop being groupies and start being journalists again. There's too much cheerleading on both sides. If you're there to get a story, don't go there to get a story and press these people. A lot of these people, I know what you're talking about, a lot of these people ran on, I'm black, ski-wee and all this type of stuff. If you're gonna go and beat in these people's face, you need to hit them with hard questions that don't take selfies. Be a journalist. I'm probably not gonna get invited to the White House dinner now, but I mean it is. You are though. You are. You gotta quote him. All right. You gonna be in there, man. Don't worry about that, bro. White House dinner. You gonna be straight, man. Hey, the black market is open. My man, Torrin Walker, 85 South Shore. We out of here. Get your journalism game together. Groupies. Thank you, bro. There we are. My food is here.


The National Chairman of the APC, Abdul-Ai Ademo, has expressed concern over the wave of the factions of federal lawmakers and the party to opposition parties. This was today after a close-down meeting with party members at the National Assembly. Ademo admitted that while the party is worried about the gale of the factions, nothing can be done about it. According to him, the factions are normal during an election year, as those who feel dissatisfied with the outcome of party primaries are free to test their popularities in other political parties. Well, it's an unfortunate development when it happens. But this is a season for all manner of behaviour in the political space in the country. And this is not Nigeria's own exception. In every election year, this kind of sub-step, you know, give course for people to take orders to sneeze. This is what we're experiencing. Nigeria has no exception, and the APC has no exception, you know.  I care what's happening in our party. But you and me know that it's not the APC that's having this experience, this situation. Is it? If we are the ruling party, yes, our problems are more prominent in the public glare. No responsible leader will not worry losing one member, not talk of two or three. At the moment, we know we are faced with the stark reality of their problems. And we have committed with my colleagues in the National Working Committee to face the problems quietly and see the problems that are solvable. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.