Since Covariance matrix is symmetric it is Hermitian (self adjoint) and always diagonalizable. If the matrix has all non zero eigen-values its is a full rank matrix. But what is the importance of the covariance matrix being positive definite?  I am asking this question in context of principle component analysis. We often come across the statement that covariance matrix is positive semi-definite and also the proof for it. The proof and information are nice to know. But, by not being able to figure out its importance in given context i feel I am missing some important point.  For example the importance of positive definiteness of a kernel is often stated clearly that a positive definite kernel corresponds to a RKHS. But what is the importance of positive definiteness of covariance matrix.
I have been researching the meaning of positive semi-definite property of correlation or covariance matrices.   I am looking for any information on   Definition of positive semi-definiteness; Its important properties, practical implications; The consequence of having negative determinant, impact on multivariate analysis or simulation results etc.
Get total amount of time between records when changed state I have a table, showing the status (healthy, broken) for each device at a given timestamp. I need to get the total amount of time of a whole breakdown.  So the data is stored like this:  device_owner   device_id    timestamp          status owner1         device_1     2001-01-01 09:00   0 owner1         device_2     2001-01-01 09:15   0 owner1         device_1     2001-01-01 09:30   1 owner1         device_2     2001-01-01 09:45   1   and so on.  The example above shows    device_1 was down from 09:00 to 09:30 (total 30 mins),   device_2 was down from 09:15 to 09:45 (total 30 mins)   What I need  Need to calculate the total amount of time for the whole breakdown for owner_1 will be 45 mins (from 09:00 to 09:45) instead of 60 mins (30 + 30).  device_owner   total_breakdown_min owner_1        45   What I've done  Currently I can define a breakdown time for each device separately (device_1 = 30mins, device_2 = 30mins). For this I transformed the table using LEAD over partition by device_owner, device_id and date part from timestamp:  device_owner  device_id     timestamp        status lead_timestamp   lead_status owner1        device_1     2001-01-01 09:00   0     2001-01-01 09:30     1 owner1        device_1     2001-01-01 09:30   1     NULL                 NULL owner1        device_2     2001-01-01 09:15   0     2001-01-01 09:45     1 owner1        device_2     2001-01-01 09:45   1     NULL                 NULL   So the datetime difference between timestamp and lead_timestamp in sum gives me the total breakdown time for a device.  ;WITH LeadStatus AS (     SELECT         D.*,         lead_status = LEAD(D.status) OVER (PARTITION BY D.device_owner, D.device_id, D.date ORDER BY D.datetime ASC),         lead_timestamp = LEAD(D.datetime) OVER (PARTITION BY D.device_owner, D.device_id, D.date ORDER BY D.datetime ASC)     FROM         #DeviceStatus AS D ) SELECT * FROM LeadStatus   It is the same device list for any owner. The data snapshot is to be taken between 2 dates (StartDate, EndDate). The records are written to a DB each n seconds and also when the device status changes. What registers to save in the ARM C calling convention? It's been a while since I last coded arm assembler and I'm a little rusty on the details. If I call a C function from arm,  I only have to worry about saving r0-r3 and lr, right?  If the C function uses any other registers, is it responsible for saving those on the stack and restoring them? In other words, the compiler would generate code to do this for C functions.   For example if I use r10 in an assembler function, I don't have to push its value on the stack, or to memory, and pop/restore it after a C call, do I?  This is for arm-eabi-gcc 4.3.0. Algorithms for Time Series Anomaly Detection I'm currently using Twitter's AnomalyDetection in R: .  This algorithm provides time series anomaly detection for data with seasonality.  Question: are there any other algorithms similar to this (controlling for seasonality doesn't matter)?  I'm trying to score as many time series algorithms as possible on my data so that I can pick the best one / ensemble. I have been facing hard time understanding meaning of "random sample" as well as "iid random variable". I tried to find out the meaning from several sources, but just got more and more confused. I am posting here what I tried and got to know:  Degroot's Probability &amp; Statistics says:     Random Samples / i.i.d. / Sample Size : Consider a given probability distribution on the real line that can be represented by either a p.f. or a p.d.f. $f$. It is said that $n$ random variables $X_1 , . . . , X_n$ form a random sample from this distribution if these random variables are independent and the marginal p.f. or p.d.f. of each of them is $f$. Such random variables are also said to be independent and identically distributed, abbreviated i.i.d. We refer to the number n of random variables as the sample size.   But one of the other statistics book I have says:       In a Random Sampling, we guarantee that every individual unit in the population gets an equal chance(probability) of being selected.   So, I have a feeling that i.i.d.s are elements that construct random sample, and the procedure to have random sample is random sampling. Am I right?   P.S.: I am very confused about this topic, so I will appreciate elaborate reply. Thanks. Safest way to transport a laptop during flight I flew a long range flight carrying my laptop, a quite heavy one, inside a neoprene briefcase in the passenger cabin. The recommendation by the aircrew was to place it in the overhead compartment or under the seat in front of me. However, during take-off and landing, the vibrations on the plane are quite strong and, as a result, it received minor damage.   Placing them in the overhead compartment might be equivalent to placing it on the floor under the seat, with the added issue of additional luggage in the compartment shifting.  I considered the alternative of holding the laptop by the handle while sitting, so it won't be any contact with the aircraft's floor. This is impracticable during cruise, but I thought it could be better during the take-off and landing segments. Another option I considered is to wrap it with completely with bubble wrap before boarding the plane.   Which is the safest way to transport a laptop during flight? /ɪ/ sound when not stressed I've seen that some words in English are pronounced with the /ɪ/ sound when the vowel is not stressed.  Some examples include: pocket /ˈpɒkɪt/, comet /ˈkɒmɪt/. But hundred /ˈhʌndrəd/.  Is there any pronunciation rule here when the words end in /t/? I can prove the converse of it, but I cannot do this one. Here is the problem:     Prove that the implicit function theorem implies the inverse function theorem. How to find a global minimum for $\frac{1}{2}x^tQx-b^tx$ I am an Indian passport holder travelling from China to India via Kuala Lumpur, on Air Asia tickets bought separately. The layover is 4 hours and 30 minutes.  Do I need a transit visa? I'm not a native English speaker. I am having difficulty with some words. Could anyone please tell me if there's any difference between the above mentioned short sentences, as I'm a little confused. I am trying to encrypt an entire USB drive with Truecrypt 7.1a. But as long as there is a partition it will only encrypt that. This will make it appear like the USB is full instead of completely empty, which isnt what Im aiming for.   I've tried with DISKPART, USB Partition Tool, and the built-in storage manager. When I delete all partitions, Windows still assigns the drive a letter and assumes that it has a partition covering the whole space.  I know that it is possible as I have done it with another USB, but I just don't remember how I did it.  Ideas? While I was reading the book Exploratory examples for real analysis, I came across this:     to show that : $$\lim_{x\rightarrow x_o}\frac{1}{h(x)} = \frac{1}{\lim_{x\rightarrow x_o}{h(x)}} = \frac{1}{M}$$      Discussion:Let $\epsilon \gt 0$.we want to find $\delta \gt 0$ such that       if  $0 \lt |x-x_0| \lt \delta $,then $\Big|\frac{1}{h(x)}-\frac{1}{M}\Big|\lt \epsilon$.      In order to get expression involving $|h(x)-M|$,we first find the common denominator:        $\Big|\frac{1}{h(x)}-\frac{1}{M}\Big|=\Big|\frac{M-h(x)}{M~h(x)}\Big|$      $~=\Big|\frac{h(x)-M}{|M|~|h(x)|}\Big|$.      As we have to deal with variable term $|h(x)|$ in denominator.We can construct bound .Let $\epsilon =\frac{|M|}{2}$ ,since lim$_{x\to x_o}h(x)=M$,there exists $\delta_1\gt 0~~$ s.t.       if $0 \lt |x-x_0| \lt \delta_1 $,then $|h(x)-{M}|\lt \frac{|M|}{2}$.      As you study steps below,identify that step where $h(x)\neq 0$ for all $0 \lt |x-x_0| \lt \delta_1 $ plays a critical role:      $|M|=|M-h(x)+h(x)|$      $~~~~~~\leq |M-h(x)|+|h(x)|$      $~~~~~~= |h(x)-M|+|h(x)|$      $~~~~~~\lt \frac{M}{2}+|h(x)| \implies$      $\frac{|M|}{2} \lt |h(x)| \implies$      $\frac{1}{h(x)}\lt \frac{2}{|M|}$,where $0 \lt |x-x_0| \lt \delta_1 $.      Now that we've bound for $1/h(x)$,we can apply the definition to term $|h(x)-M|$.      Since     lim$_{x\rightarrow x_o}h(x)=M$,we can find $\delta_2 \gt 0$ such that,       $~~~~~~~~~~~~~~~~~~~~~~~~~~~~$if  $0 \lt |x-x_0| \lt \delta_2$, then $|h(x)-M|\lt \frac{|M|^2 \epsilon }{2}$   The thing I can't understand is for what purpose did the author work with such complicated expression involving $\epsilon $ in the end of the discussion? Using a comma before "but" Is short-circuiting logical operators mandated? And evaluation order? I have 2-3 different column names that I want to look up in the entire DB and list out all tables which have those columns. Any easy script? $M + N$ and $M \cap N$ are finitely generated $A$-modules implies $M$ and $N$ finitely generated using exact sequences Let $0 \to M_1 \to M \to M_2 \to 0$ be an exact sequence of $A$-modules.  i) Prove: If $M_1$ and $M_2$ are finitely generated, then $M$ is too.  ii) Let $M$ and $N$ be sub-modules of an $A$-module $L$. Assume that $M+N$ and $M\cap N$  are finitely generated $A$-modules. Prove that $M$ and $N$ are finitely generated $A$-modules.  If have proved the first part, but i don't see how to do the second part. I tried to use the first part, so to find an exact sequence with $M$ in the middle and finitely generated modules in place of $M_1$ and $M_2$, but I can't find a surjective map $M \to P$ (for some f.g. $A$-module) in the sequence:  $$0 \to M\cap N \to M \to P \to 0$$  any hints/help would be appreciated. Tensor products of simple modules over algebras Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules.   We can regard $M$ and $N$ as $A\oplus B$-modules in natural way, namely, $AN=0$ and $BM=0$. Let $M\otimes_{\mathbb{C}} N$ be the tensor product of $A\oplus B$-modules $A$ and $B$.  ${\bf My\ Question:}$ Is $M\otimes_{\mathbb{C}} N$ still a simple $A\oplus B$-module? What conditions on $A$ and $B$ will make this result true? Thanks very much in advance! How will Ubuntu automatically allocate the disk partition sizes when we select Erase disk and install Ubuntu during installation?    Can the settings for automatic space allocation be changed? Prove, in this figure, that $EFGH$ is a parallelogram How should I phrase a question that must be answered with an ordinal number (e.g., the third prime)?