prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Why does Craig want to find his own place?

Read the following text and answer briefly.

Context 1: My Aspergers Child: COMMENTS & QUESTIONS [for Feb , 2017]
I emailed you a while back and you mentioned that I could email when I needed to  Thank you  I last wrote you in December that my son became involved in a dispute involving the local police  We have had 3 court dates  It keeps delaying due to not being able to come to an agreement  But the attorney, even though he was just vaguely familiar with Aspergers, has been very good with Craig  He has the compassion and excellence that is needed here  What started out very bad is turning into a good thing  It will probably take another 90 days or more 
But Craig is working hard  Too hard sometimes  He goes to therapy 3 times a week  Doing excellent  He's more focused and can calm down easier  He's got a lot on his plate but has support from his family  From his attorney  From therapy  And from his work 
He has been renting a room from a lady who has a son with ADHD  It is good for him  I'm a little worried though because since she smokes he wants to find his own place  With all the costs he has to balance it out financially  That is good  I can't help him more than I am which is good  He is stepping up and taking responsibility  He is listening much better 
He is going to have an evaluation today to get an accurate diagnosis  I understand that is a little difficult since he is an adult  Also the PTSD may cover it over  The attorney stated it would help to have the diagnosis 
Aware this is a long update, but thanks for reading  I am fighting much guilt still but I have a lot of peace now  My daughter and her 4 year old son also have Aspergers symptoms  So my life chapters may not close for a while  :-)
My name is Mac  I'm sure you're quite busy, so I'll get right to it I just wanted to pass on compliments on My Aspergers Child and your post, How to Implement the GFCF Diet: Tips for Parents of Autistic Children 
Me and my wife absolutely loved it!
I got a facebook message from him today begging to be able to come home saying he misses home and he will change  He says he will follow rules now  I stated to him the simple rules he has to follow which were - No weed in my house, or smoked in my house, coming home at curfew, going to school, no skipping, no drugs at school, and to drop the attitude of I am 17 I can do whatever I want 
I have made it very clear that if I see any drugs in my home I will be calling the police, as well as if I see signs of it being sold by him I will report him  (He has never had selling amounts in my house,     I believe it's being kept at his "friends" which of course I have no proof of    I just know it is not here 
I know my battle is not over by a long shot, I am sure we will have more consequences and possibly another being kicked out, but I am going to think positive and hope that he learned some form of a valuable lesson here 
Thank you so much for the guidance, never in a million years did I ever think I'd be on this side, (the one needing the help, as I am the one who helps )
I am going to go back to the start of the program like I said earlier and keep notes close by for reference 
Thanks for all you do, helping us all with ODD children/teens
I have a small company providing educational support services to a few families who have children with various disabilities in Ohio  One of the families has multiple adopted children of whom several have significant attachment disorders including RAD  As an experienced teacher and foster parent I have some experience in working with children who have extensive trauma backgrounds  However, I could use additional training  Also working with these children are two staff members with minimal background in attachment disorders who would also benefit from training primarily in behavior management  The primary caregiver to the children does a wonderful job managing their needs  In order to further develop team cohesion, I'm hoping to include her in any training as well 
Is it possible to schedule such a training session with you? If so, please let us know what will work for you including time, place, and cost  Thank you for your assistance 
I just listed to your tapes on dealing with an out of control, defiant teen  I'd like to ask your advice on a particular situation we have  Our 15 year old daughter is smoking pot almost every day at school  Because we had no way to control the situation, we told her, fine, go ahead and smoke weed  However, you will no longer receive the same support from us  You will not have your phone, lunch money to go off campus (she has an account at the school for the cafeteria she can use), and you will be grounded until you can pass a drug test  We will not be testing you except for when you tell us you are ready to be tested  She is now saying she's suicidal because she feels so isolated, yet she continues to smoke weed  In fact, she tried to sneak out last night but was foiled by our alarm system  For the particular drug test we have, I read it takes about 10 days of not smoking to pass the test  What would you do? Please advise 
I am having a problem with my 18 year old son, Danny, with high functioning autism  We finally had him diagnosed when he was 16 years old  I always knew something was going on with him but the doctors misdiagnosed him as bipolar  It's been 2 years now and he will not accept his diagnosis  He won't talk about it and when I try to bring it up he gets very angry  I've tried telling him that it's not a bad thing, that there's been many, many very successful people with Aspergers  He won't tell anyone and refuses to learn about managing life with it  He once shared with me that the other kids at school use it as an insult, like saying someone is so autistic when they do something they don't approve of  So he doesn't want anyone to know  He's turned down services that could help him  He has a girlfriend, going on 8 months  He won't tell her and they're having problems arguing a lot and I wonder if it would help for her to know 
I'm sad that he thinks it's a life sentence to something horrible instead of accepting, embracing it and learning about it more so he maybe can understand why he's struggling  I told him that he doesn't need to shout it out to the whole world but he won't even accept it himself 
I don't know how to help him with it and because he's almost 19 I have limited control now  It's made my life easier knowing what we're dealing with and I think his life would be easier is he accepted it 
Please help me help him 
I am a clinical psychologist in NYC who now has several (!!) children I see who have RAD  In 20 years of practice, I’d seen only one case  Now, I have at least three children with this  I have no training, per se, in working with this children though I know about setting structure, consistency, etc  I do a lot of work with parents about parenting  I work primarily within the school setting in a charter school whose mission is to educate children on the autism spectrum in a mainstream setting  We use Michelle Garcia Winner’s social thinking program with our ASD kids  I also work with gen ed kids in the school who are at-risk; the school is in the inner city from where the majority of our non-ASD kids live 
It would have been so much easier to mention to my adult son that I think (I know he does, but want to ease into the subject)
he has Asperger's when we were living together two years ago  He has since moved to Tennessee working in his field of interest
which is 3-D printing and software development  I am so happy for him that he has found his way into a job that he truly enjoys
even though he's socially isolated 
He's not diagnosed and does not know he has it  How I know is his classic symptoms being sensory issues (fabric feeling like sandpaper)
communication difficulties, meltdowns and much more  Throughout his childhood I just felt he was a bit different  Nothing major stood out and time
just passes, misdiagnosis of ADHD, low frustration, etc  We've talked about his ADHD numerous times (which I now know he doesn't have) 
It's so much easier to communicate with him now that I know he has Asperger's  I keep it "slow and low" in talking, with long moments
of silence and then we connect  It's really too bad that Asperger's got a diagnostic code back in the 90's, yet all the so called doctors,
physiologist's, etc, didn't know how to diagnose it  Too bad 
There seems to be no one answer to "should I tell my adult son he has Asperger's" from a few specialists I asked  He is typical Asperger,
complicated, highly intelligent (high IQ), anxiety at times, socially isolated, hard to make friends  Not knowing how he will react is the hard part 
How will he be better off knowing he has it? Do I wait to tell him in person, or ease into it with him over Skype? He likes direct, honest, concrete communication 
Why is this so hard for me? Maybe because no one know's if he is going to be better off knowing or not  Do you know if people are better off
knowing? I try to get up the courage to just let him know, then I back down 
I have been searching the web looking for advice and came upon your site  I am trying to read blogs, websites, books, and articles to help guide me  I was so happy when you said that I could ask you a question  My husband and I are struggling with my 27 year old son who lives with us 
Kyle is the youngest of 4 sons  He is a college graduate but never could find the "right" job  He has always been quiet and never had a lot of friends  Two years ago, his girlfriend broke up with him  Kyle had an online gambling addiction and was using pot all the time  After the breakup, Kyle was very depressed and started using heroin and finally told my husband he was using  He is now seeing a psychiatrist who has him on suboxone and antidepressants  He is also seeing a psychologist weekly for counseling but it does not seem to be helping 
Last October,, Kyle lost his job, got drunk, and was agitated and came home , fighting with us, damaging our home and being verbally abusive  My other son , age 32, who also lives with us called the police and Kyle got arrested  He is currently in the family court system  He went through an anger management course and now is in substance abuse classes  Kyle continues to verbally abusive to me and blame me for everything  He says he "hates me "and calls me terrible names  At times, he pushes my husband and intimidates me  My husband and I are so upset  We just hired an attorney for him because since he has been going to these classes, he is getting more depressed and not getting better  Kyle continues to drink while taking his meds prescribed by the psychiatrist and then he has his "moods " My husband and I have met once with the psychiatrist just to give him background information when Kyle started with him 
At this point, we do not know what to do  We never thought at this stage of our life, we would be supporting and spending our retirement money on adult children  I do not know why Kyle hates me, I could not have been a better mom  My husband and I have no life and just do not know what it the right path we should take  Kyle does not want anything to do with us  He spends all his time in his room playing football online We have tried tough love versus caring and love and understanding  Do you have any advice for me?
This whole ODD and ADHD is killing me as a parent  I work in the field of adult psych and addictions so I am well educated  I have been dealing with my teen being like this for almost 3 years and I totally lost my cool today with my 17-year-old son to the point I told him he is out of the house  He can never simple rules, comes and goes as he pleases sometimes doesn't come home, just recently back in school from several suspension for drug related    I am just so exhausted  He has made me hate life, hate being a parent and sometimes I just feel like not even being here  I bought your program in hopes to it would help, I am at week three and I feel things are getting worse    what am I doing wrong??
My partner hasn't been diagnosed yet but I know he has aspergers   day to day is a struggle   I feel I'm going crazy with how he makes me feel Feel let down constantly  He lies alot but I've been told they can't but I know he does I just feel trapped and unloved We have a 4yr old daughter together and my main worry with how he is that it will effect our daughter ; (his skills as a parent are so weak He can't disapline at all Feel so alone  he hides it well too I just wondered if things will get worse? He's angry so quick in arguments Scares me etc I can't leave as he's the main bread winner and our daughter loves him to bits Don't know why I'm writing this  Sorry if I'm going on and not making sense :(
I wanted to let you know about a research opportunity for children, teens, and young adults with autism  I am studying the effects of Brazilian Jiu Jitsu, and psychotherapy on helping people with autism develop subjective awareness of others 
I am writing you to see if this might help someone in your practice, or to see if you might know of someone with autism who may benefit from participating in this study  The requirements of the study will be:
1  A participant should be between 7-21 years of age and have a diagnosis of Autism Spectrum Disorder 
2  The participant should enroll in an approved Jiu Jitsu Academy and attend at least two sessions a week for a period of six months 
3  The participant should enroll in social skills groups, provided by my office or be in a steady psychotherapeutic relationship in your office, at least once a week, or minimally two to three times a month 
4  The participant will be given a SRS (Social Responsiveness Scale) test at the beginning of the study, at three months, and again at six months 
If you know of anyone who might benefit from this novel approach to helping to develop social awareness in autism, please do not hesitate to contact me for further information 
I have a 10 year old daughter who has outbursts with prolonged crying almost like tantrums that 2 year olds have when they cannot express themselves 
I had her in therapy from age 6-8 years old for the same thing but I feel that the sessions didn't really help much 
She has severe sensitivities to light, sound, vibration, frequencies which trigger irritability and crying 
We changed her diet and tried getting her involved with activities but she is anti-social and prefers reading than being social  She is terrified of change even in daily routine (even that will trigger prolonged crying) 
It frustrates me because I don't know what else to do with her behavior 
I've tried acupuncture (she refused at the first session); she refuses massage too 
She is an honor-roll student at school and has very minimal issues at school but if she has had a bad day it does result in a tantrum or crying and defiance 
How can I get her tested for Asperger's Syndrome?
Last night our 24 year old son with Aspergers told his dad and I that he is pulling out of the 4 college classes that he recetnly enrolled in because he has not been attending class or turning in his assignments  He paid $2800 (his own money) for tuition and I reminded him of this when he told us but it did not seem to bother him 
This is the 3rd time he has started college courses and has not completed them  (He also took some concurrent college classes while he was in high school that he failed)  This is a son who basically had a 4 0 grade point average through 10th grade and got a 34 on the ACT the first time he took it 
With the news that he was once again not sticking with college courses I did not sleep well  When I got up this mornning I began looking online for help in how to deal with his situation  I found your "Launching Adult Children With Aspergers" and purchased it  Most of what is included are things we have done or did with our son throughout his life  I was hoping for more help so I am emailing you now in hopes of more specific ideas 
We noticed some things with our son, Taylor, as a yound child but as we had not heard of Aspergers at that time we just did what we thought would help him  As a toddler and a child at pre-school he generally went off on his own to play  When I talked to his pre-school teacher about my concerns (that I was worried he would end up a hermit) she said she did not see him being a loner and that he seemed to interact fine with others in many situations  We worked with him on making eye contact when talking with others  We explained different emotions in people's faces and mannerisms to help him know how to interact with others  We discussed the fact that people would say things that did not mean what they souneded like - such as "I'm so hungry I could eat a horse"  As we did these things he worked hard to better understand communication with others 
During his 4th grade year I had a teacher from the gifted program ask me if I had ever heard of Aspergers  I told her that I had not heard of it  She proceeded to read me some of the charateristics and so many of them described my son  So we had him tested by the school district during the summer between 4th and 5th grade and they did find that he had Aspergers but that he was high functioning  We then set him up with and EIP which stayed with him until his sophomore year  We pulled him from it at that time because we had moved and the new district was requiring him to take one class a day that was a study class  This reduced the number of required classes he could take and he was doing fine with his studies at the time 
It was during the 2nd half of his Junior year that we noticed some of his grades going down  Then during his Senior year is when he started skipping classes and not doing assignments  We had not realized it before then but we soon became aware that he was addicted to gaming  He would go to the library or somewhere else on campus and play games on the computer rather than go to class  It was also at this time that he began lying about his actions (so as not to get in trouble) 
Based on his grades and his ACT score he received offers from colleges for full tuition scholarships  He chose the college where he had taken concurrent classes during his high school years  But he proceeded to skip class and not turn in assignments so he lost his scholarship and quit attending college  During this time he was only able to find employment through an employment agency where he was mostly sent to manuel labor type jobs (which is not something he enjoys but he did it anyway)  It was during this time that at one place had gone to on numerous occasions he was told if he came late one more time they would tell the emplyment agency they did not want him to come there anymore  (This seemed to make an impression on him because he has continued to be reliable and responsbile at his places of employment) 
At 19 1/2 he left to serve a 2 year full-time mission for our church  He completed his mission successfully  (I don't think it was without some struggle, stress and depression, but he was able to pick himself up and move on from those times) 
When he came home he started working for the employment agency again but began looking for employment elsewhere  He got a job at a local Chick Fil-A where he has worked for 3 years  He started college again shortly after he came home but as before it was short lived  He did finish out the semester but failed most of the classes due to his skipping class and not turning in assignments  When he skipped class he would usually sleep in his car 
Taylor's life consists of working (where to the best of our knowledge) he does well, he is reliable and his employer likes him  When he comes home from work he either sleeps or plays video games or other games - such as kakuro  He spendes most of his time in the basement where his bedroom is and this is where he games  Taylor owns his own car, bought his own laptop and very rarely spends money  He pays us $200 /month to still live at home, unloads the dishwasher on a regular basis and does the weekly garbage  However, his room is a mess and he only cleans his bathroom when I tell him he needs to clean it 
Taylor used to read quite a bit and loved to learn  It has just been in his adult years that he has not read as much - I think because of his gaming addiction  Taylor goes to church on a regular basis but sleeps through the main meeting  In Sunday class room settings he stays awake - I think because he is able to particpate in discussions 
Taylor has only had 2 real friends since entering Junior High school  And as of now he only keeps in contact with one of them who still lives in Georgia  We have lived in Utah since the summer of 2007 and he has never had a friend to do things with since we have lived here  He has two younger siblings, a brother 22 and a sister 20  They love Taylor and spend time with him when they are home  They are both at college and doing well 
Throughout Taylor's school years he has seen a counsleor on a fairly regular basis  One summer during junior high he attended a weekly class where he interacted with other kids with Aspergers  We did see a lot of change in him from this group  After he returned from his mission he went to see a counselor for a short period - this counselor tried to help him with some social skills  His dad and I went with him the first 3 or 4 times but we found out that after we quit going with him he only went a few more times and then scheduled appointments but did not show a couple of the times  We only found this out when a bill came for a "no show" appointment 
I don't know if this is too much information but were are in dire need of help for him  In the information that we purchased from you you mentioned that you do coaching for Aspergers adults  I don't know if you can help us but I thought I would check with you just in case 
Alas I think I have found your information too late to save my marriage but I am hoping to save myself 
I am currently going through a very very painful separation after a 27 year relationship with my husband whom I am convinced has aspergers syndrome  It is a long and painful story and I am desperately trying to process it all alongside dealing with a very conflictual separation  My partner is angry non communicative and totally dismissive of me and our long shared history 
He walked out last year after I discovered he had been visiting massage parlours and developed a relationship with an illegal Chinese escourt whom he subsequently moved in with  He had been seeing this woman behind my back for over 18 months  The pain of all this indescribable and his dismissal of my pain and very existence beyond belief 
Leading up to this I had been battling anxiety and depression which my husband found very hard to cope with 
Over the years of our relationship I knew something was off but I just could not put my finger on it  I often felt a complete lack of validation and empathy  Communication was also difficult as my husband was defensive and unwilling to look at issues in our marriage 
Please Mark could you help me validate some of this pain and try and make dense of 27 years of my life without drowning in fear guilt and despair about my future 
Thank you for listening and your site 
I have had problems with drunkenness, being late for school, not handing in school work, buying pot from a dealer etc  I chose to focus on the drinking and did the grounding then (grounding happened 3 times)  I also stopped sleep overs at friends 100%  I have stopped handing out money for no reason or even buying treats like chocolate 
I did lose it one evening (and didn't do the poker face) when I was trying to unplug the internet at midnight on a school night (she’s always late for school so I am trying to get her to sleep at a reasonable hour)  I was physically stopped and pushed around so I slapped my daughter (it was not hard)  This ended up with her saying she didn’t want to come home (the next day after school)  By this stage, I also had enough and didn’t go get her  I thought I am not begging  You will run out of money soon  It was quite a relief to have some peace  Daughter’s Dad was in town (from another country) and called a family meeting with the counsellor  To cut a long story short, daughter and her counsellor put it on the table that daughter wants to go live somewhere else (with her friends family) because of the stress at home with me (we live on our own) (i e  stricter rules and her bucking up against it) 
I didn’t really want this but made a compromise that daughter would go there Tues morning – Friday afternoon as the friend is an A student whereas my daughter is failing  They do the same subjects  I made the decision at the end of the day based on what is good for me – some time away from the daughter  I also thought of your book when the child went to live with the grandparents – daughter will dig her own hole over at the friend’s house  They have a week day no going out policy which made me think it is OK  I went and discussed with them the problems experienced (drinking, pot, late nights, not handing in work)
I am also trying to follow the let go of school thing per your book  I find it really difficult to remain calm when I can see daughter on her phone and watching series (when I have her on the weekends) when I know there are projects due  I hired her a private tutor once a week for help with a subject  The tutor has just fired my daughter for not handing in work and being not committed  It’s not the first time private tutoring has not been appreciated  The school give me a report back on a Friday as to whether everything is handed in  The deal is – if the work is not handed in – no pocket money and no Friday night out)  Her school is a "progressive" school and there are no repercussions for her being late or not handing in work  I would change schools if I could but there are only 8 months left of school (she turns 18 in August) 
We have just completed the first week and beginning week two of your material  We are agreeing with your take and see our son and ourselves in most of what you are saying  Prior to finding your material and starting your program we had been having extreme out of control behaviors and had to call the police because he was breaking things in our house and pushed my husband  This happened three weeks ago  After that incident we took away privileges ie  PS4, phone (which had already been taken for a few days), and friends  So, last week while doing your program he already didn’t have privileges and has continued with poor behavior – name calling, throwing things, slamming doors  We are not sure when to give privileges back  He has been given the privilege of playing with friends on occasion  His 13th birthday is tomorrow  This past weekend, for his birthday my husband and he went boar hunting  Of course we debated about it but decided to go ahead since it was his bday  We are cooking some of the meet on the grill tomorrow night for his bday and inviting a couple of his friends over for a cookout  No more gifts other than cards and balloons  We are wondering if we should go ahead and give him his privileges back and not sure how to do it  Last Friday morning we attempted to talk giving him a date to return privileges and that conversation ended with him getting angry but he gathered from our conversation that he is getting his stuff back on his bday  We are starting week 2 assignments today but not sure how to handle what was already in place  Of course, we aren’t seeing the respect and responsibility we are looking for but realize it has been a long time  We were wanting him to pay for his phone and thought it might be a good time to introduce that idea  Allowing him to earn his phone  We expect that he will be angry with this idea and not sure how to implement 
My son and myself are interested in a inpatient Aspergers program  We line in Calif which is preferable  My son is very high functioning and was diagnosed dry late  He was eight years old  He has never been in or attended a full day of class  Partially due to depression,anxiety, and trouble with his ADHD also his aversion and being bullied and of course his Aspergers  He will not attend his freshmen year due to surgery on both Achilles' tendons from walking on his toes  With physical therapy he should be ready by his sophomore year! We all feel he needs in patient therapy to give him the tools on how to work with his issues in a structured setting and a place that will give him tools for the rest of his life 
In my utter desperation to find a way to get some help for my daughter's increasingly challenging behaviour I trawled the internet to see if I could find some strategies that would provide specific methods on dealing with teenagers with Asperger's syndrome  When I came across your website, I couldn't believe that every statement you made was exactly what I have been going through with my daughter  She has just turned 14 last week, and was diagnosed with Asperger's/ Autism Spectrum Disorder 15 months ago  I have already been seeing a child psychologist for the past five months, however the methods she has been advising have not been very effective 
Our main difficulty with our daughter is her overwhelming obsession to use her cell phone (and to a lesser extent her laptop) constantly  Without any restriction, she will be on it every minute of the day, and will be awake until the early hours every day  We have tried to incorporate her input around rules as to when she has to give in her phone, but she is unwilling to compromise on a time that she should give it to us, believing that she should have unlimited use  I believe she is unable to do any adequate study or homework, as she is constantly having to look at the phone  We have tried to put rules in place that she has to give in her phone and laptop on school nights at 22:15  If she is able to do this then she is given rewards, and if she doesn't then she knows that there will be consequences  The consequence has been restricted use the following day  However, this is usually where we fail, because taking her phone away from her results in tantrums, screaming, and even threatening to harm herself  This behaviour is relentless to the point where the whole family becomes deeply distressed, and inevitably results in her getting the phone back 
This obsession is affecting her schoolwork, and more severely her eyesight  She has become very shortsighted, and her eyesight continues to deteriorate as a result of holding the phone or laptop very close, and mostly in the dark without any lights on  My husband and I have a constant battle on our hands daily, in all areas of discipline with our daughter, but our main concern is that we have been unable to find a way to minimise this obsessive behaviour centred around her phone and laptop  Please can you provide some strategies that can help us specifically with this problem 
First of all, I thank you for developing this program and I am only at the first stage of assignment 1  I have loads of books I have bought, attended psychiatrists for my son and myself, family therapy, occupational therapy, begged and prayed for change but have been dealing with behavioural issues for so long I am definitely exhausted and resentful 
I am a mum to a 15 yr old boy with ASD, dyslexia, OCD and ODD  Sorry to focus on the labels but just to give you an idea of what I am dealing with  I also have a 13 yr old son whom finds his brother’s behaviours difficult, embarassing and challenging  My husband whom is not in great health ( he had a cerebral aneurysm clamped two years ago and has two further aneurysms that are inoperable so endures fatigue, headaches and stress)  We have however a pet cat that is very social and a calming influence in the home! I was fortunate enough to have loving parents but I lost both my mum and dad in 2008 and 2015  My inlaws are elderly and quite directly say they are too old to help us so it feels we are alone in dealing with the issues we have 
I am desperate for change as the household is one of stress and anger and I feel all the control lies in my son Patrick’s hands  I am hopeful your programme can make life better for all of us but I wonder if it is too early to ask you two questions?
The first lies with what to do when Patrick goes into my other son Brendan’s room and will either turn on a light when he is sleeping, yell when he is on his phone or create some disturbance  He will not leave the room when asked to do so and the situation always escalates into yelling and Brendan attempting to physically remove him  This happens regularly and always ends badly with doors slamming, my husband being woken and myself in tears feeling the lack of control and also I admit I seem to think “Why me?” which rationally I know is of no help 
The second problem is leaving the house for school  Patrick refuses personal hygiene (either morning or night) and any request to even brush his teeth is fraught with swearing and abuse  If I can get him to shower, he will watch the water roll down the drain and turn up the water really high temp (mu husband has had to turn down the thermostat on the hot water service) without so much as getting wet  My husband leaves for work at 6am but I leave at 745 to work as a nurse in a busy outpatients department in the Alfred Hospital (Melbourne)  My work is my sanity as it is a paid break from home but most days I am late which is causing considerable stress and anxiety not to mention my responsibility to do my job  Patrick simply refuses to leave the house and as much as I am tempted to just walk out and leave I know the house would be left unlocked and wonder if Patrick would even attend school  The time I need to leave is not negotiable but Patrick uses this to his advantage and seems to delight in stressing me out and subsequently speeding to work in a frazzled mess 
The interesting and frustrating element in all of this is that although he is socially isolated at school (he has no friends) and academically challenged his behaviour at school is not a problem  He is quiet and his teachers report he does his best and is compliant and well mannered  It is like a Jekyll and Hyde situation where another side of him at home is so angry and abusive yet at school this behaviour does not happen 
I’m Jackie, I now work primarily as a freelance tech writer, after starting my career in software development and moving on to teach IT to young adults at a variety of colleges and schools 
My freelance work is pretty varied and looks at many aspects of the computer industry as a whole, and I’ve just recently completed a piece which gives help and advice to anyone wanting to become a game designer, which you can read here: http://www gamedesigning org/become-a-game-designer/  It highlights the hard work and effort it takes to get into such a role, and also how you can further your career and continue to learn and improve as you go  I hope you’ll agree it shows that starting work in the industry takes dedication and skill and that becoming a game designer isn’t just a fly-by-night job!
If you’d be interested in sharing a quick mention of my work on your blog that would be really wonderful and I’d appreciate the chance to get my work out there to a wider audience  Alternatively, I’d be happy to write a short blurb or paragraph or two (or a longer piece - just let me know) highlighting the key points because I think some of your readers might get a lot of value from it 
My son just turned 15 and is a freshman in high school  Although this is his first year in a general ed environment, he is struggling with behaviors in school  He has meltdowns and does not express why he would have them until much later  Once we all know what caused it, the school will accommodate him and try to "change up" things so as not to cause his meltdown  Once that is resolved, another issue comes up and causes him to melt down  He is a high functioning and academically does well, when he wants to do the work  We battle at home over homework  He does not care how it is done, as long as he hands it in  He thinks failing a test is ok, at least he took the test  Homework is never on his mind when he gets home from school  If I never prompt him, he would never open is backpack  He can be aggressive but is never intentionally trying to hurt anyone  He may push over a chair in school, but it is not directed at anyone  We know how that in itself could hurt someone who gets hit by it though  He is defiant in that he only wants to do what interests him  He does not go out by himself (still immature), or abuse alcohol or drugs and never curses  He is a very funny kid and very talented  His main problems are task avoidance and seeking attention  He can be disrespectful to adults in that he is "cheeky" with them, trying to be funny or cute  And he has no "filters" 
I’ve just finished reading your Living with an Aspergers Partner ebook  I found it so informative, thank you 
You offered some personal advise, and i wanted to run a situation past you and seek your input as to a strategy for what to do next 
I’ve been seeing a guy for about 7 months now who I believe has Aspergers  I came to this conclusion months ago and I don’t think he realizes, (or acknowledges) although he is aware he has some traits 
He’s highly intelligent and successful, a pattern seeker, has a tendency to focus on the project to hand to the total exclusion of all else for as long sit takes (work or home) socially awkward (has learned coping strategies), sensitive to loud noise, high anxiety with control strategies, black and white thinking etc  He’s currently not working and I’ve seen a slow withdrawal over the last 6 weeks, including the need to ‘escape’ and leave a situation at least once 
He also has a bipolar ex overseas who has primary custody one daughter where there has been ongoing patterns of drama which has recently increased 
Over the past couple of months (since stopping work and drama increase) I’ve gone from being ‘wonderful’ in his eyes to him now being sorry and not having the ‘urge’ to spend close/intimate time with me and offering friendship  Since he shared that with me in a message he’s stonewalled and has retreated to the safety of minimal messages and talks about not knowing what best to say and not being able to find the right words somehow 
He’s a good kind man who I feel is struggling  I’m concerned about his anxiety and possibly the risk of depression  I’m fairly resilient and whilst i’m disappointed he doesn’t want to pursue a relationship with me, i’m concerned for him and his well being  One of his very few close friends is also just leaving the country to live overseas 
The strategy I’ve used so far is simply to back off and give him space  I’ve asked to take him up on an original offer he made to talk but haven’t pushed it  I also haven’t been aggressive or accusatory in the few messages i’ve sent 
Any advise you could give would be greatly appreciated,
Carli who is 10 years old and has had behavioral issues her whole life  The other night she came home very upset after having a conflict with a friend  She was at her friend's house and her and her friend wanted to get on the computer and the older sister was using it  Carli made up a story that someone was at the door to get the older sister off the computer  Her friend didn't understand that she was making up a story to get the sister off the computer  She got excited that someone was at the door and ran downstairs to answer the door  In the process of getting the door, she fell and yelled at Carli  Carli became extremely upset  She was able to control her feelings at her friend's house, but when she came home, she proceeded to cry extremely loudly for over an hour  Her dad spent most of that time with her, talking to her and trying to calm her down  After an hour, I asked him if he could please tell her to be more quiet because the other members of the household were trying to go to sleep 
My question is    how do I as the girlfriend, handle this? He did not like that I asked her to be quiet  We have a rule that if she is having bad behavior, and can't calm down in 5 minutes, he takes her out of the house because her yelling doesn't stop for a long time and is very upsetting to everyone in the household  I would like to ask him to do this with this kind of situation as well  Is this a reasonable request? His thought was that she shouldn't be made to calm down, because everyone handles being upset in a different way  But, she was literally sobbing and wailing very loudly 
My other question is should she have been told that if she wouldn't have lied, this wouldn't have happened? She has a history of lying and of not accepting responsibility for her actions  My boyfriend became very upset with me when I brought this up  He was being very sympathetic and understanding to her  I feel like he was giving her negative attention, and being an over indulgent parent by not putting his foot gown and saying, "you can't carry on like this, even though you are upset"  Please let me know how we can handle these situations better 
I am contacting you for help with adult AS  I am taking initiative to pre screen potential therapists to help my current boyfriend get therapy and help with Adult AS 
He has seen many therapists, but it seems like they aren’t really helping him with his problems  They don’t seem to understand how his (undiagnosed) AS would affect therapy approaches  For example, he may not share enough in therapy session and I’m assuming an AS therapist would recognize that is part of the AS and employ strategies to get information from him that helps with treatment  Sometime he tunes out when he is processing something heavy or that he doesn’t want to hear necessarily, or he gets distracted and I’m hoping an As therapist would recognize that and get that he may need repeated something for example, if this is happening 
He is currently suffering from depression that appears clinical in nature as well as reoccurring negative thoughts about something specific that has been worrying him about our relationship  Today he told me these reoccurring thoughts happen during all waking hours unless he watches TV, he never gets a break from them and they make him feel like he is going crazy  As his girlfriend, I am extremely concerned that he cannot get relief from these thoughts and that the therapists he is seeing are unable to help him with his problems  Therefore, I am taking initiative to try and help him find better therapy options, because I want to see him someone who can better help him get to the bottom of things and help him with the challenges he is facing  He really needs an advocate that will help him go deep to figure things out and not just assume therapies are working well, without seeing changes or getting supporting feedback from him in that regard 
Here are some questions I am trying to ask in advance to find the right people to help us with this  As you may know, insurance for these therapies are not often available  We don’t have a lot of money to go from therapist to therapist to find the right person and are hoping prescreening will help 
I recently downloaded your e-book and listened to your talks and your information is by far the most helpful I have been able to find to date  It's very accurately describes my situation as an NT wife married to a very probable AS husband  I think you for taking the time to write this and sharing your insights as well as the experiences of many of your clients  It has really helped me understand the last 32 years of our marriage and get a grasp on how to move forward 
One area that is of primary concern to me, that I did not see addressed, is stimming  I believe that is the behavior my husband is showing through constant vocal singing, repetition of words, shouting out, as well as slapping himself in the chest and general nervous activity  It is very loud and disruptive to our household and it is often a relief when he is not at home  I think there may be a level of Tourette's syndrome as well 
I did some searches on the Internet and could not find anything that really describes his behavior  Most of what I found was flapping or children's behavior  I understand that it is a release of nervous tension but I am really trying to find some strategies to help him stop this behavior as it is extremely frustrating and builds my resentment in dealing with it daily  A lot of it is embarrassing as well and sounds childish to me 
He usually does this when close family members are around and will reign himself in if he is around other people besides us  When we are home it is constant  He also has a lot of anger, mostly at himself, and blows up at unimportant things, it is as if he has a ton of negative energy inside him that need to get out and stimming is one outlet 
I will try to build my acceptance of it, but I also would just like him to stop especially the loudest and most annoying portions  Would you have any resources you could point me to?
Context 2: Paper Info

Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation
Publish Date: 7 March 2023
Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS)

Figure

FIG  5  Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG  7  Comparison between predictions and reference solutions for a new initial condition fort = 1 25, 3 75, 7 5, 12 5, 20, 30  (from left to right and top to down) We note that with longer prediction time the uncertainty bounds increases Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12 5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering  However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial  We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data 
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space  It readily allows for the inclusion of sparse and irregularly sampled training data  The learned, latent dynamics are interpretable and reveal the different temporal scales involved 
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix  Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation 
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework 

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making  They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations   In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution 
In most cases, data-driven reduced-order models are not interpretable  In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics  A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure 
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data  Other frameworks are tailored to specific problems such as molecular dynamics  
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics  The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model  
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics  Therefore we do not have to enforce any parametrizations on the Koopman matrix  
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase  By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed  Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function 
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately   Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed   We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis  
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons  The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section II 
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics  In section III we present three numerical illustrations, i e  a system of linear ODEs, a hidden Markov Model and the discretized KS-equation  We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation 
We conclude with a summary and a short discussion about possible next steps  We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process  We consider data from high-dimensional time series x n ∈ R f with n = 1,    , T   We remark that the intervals between the different states do not need to be uniformly spaced 

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f   We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder  The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued  The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ  As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them 

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system  In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables  In addition the latent variables are treated independently  The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters 
This is in contrast to temporal propagators such as LSTMs   For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states 
The symbol is used to indicate a component-wise multiplication  We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue  The real part is associated with growth and decay whereas the imaginary part is representing the periodic component 
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition   In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space 

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data 
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator  At a designated time step we are using the decoder to reconstruct the high-dimensional solution  We applied our algorithm to three systems  First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues 
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified  As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation  Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z  The loss function was optimized using the Adam algorithm   As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm 
We observe in Figure that the algorithm was able to recover the correct values, i e  the eigenvalues 7 and 3 of the given linear ODE  The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value  Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution 
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components 

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process  In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W  
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component  dp 2 dt = (−0 9 + 1 5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points 
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0 0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder  The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3  
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure   We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process  With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i e  we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time 
Afterwards, we employ the decoder to reconstruct the full high-dimensional system  The results can be found in Figure and show very good agreement between predictions and reference data  This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different 
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged 

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]  For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data 
;   The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs  This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers 
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space  We obtained the λ's in Figure   Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying 
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data  We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data  The results for the phase space can be seen in Figure  
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution  All phase-spaces were obtained by using a finite-difference operator on the data or predictions  These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions 
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging  We believe that a probabilistic version of our algorithm could be advantageous here  This section contains a fully probabilistic formulation for the deterministic model discussed before 
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE  The loss function which we optimize is the Evidence-Lower Bound (ELBO) 

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent  Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems 
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i )  The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section)  These parameters along with the state variables z t have to be inferred from the data x t  
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1  As a consequence, a priori, the latent dynamics are stationary  A derivation and reasoning for this choice can be found in Appendix A  Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics 

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n   In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section 

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ  We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed 
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters  In the context of variational inference, we use the following factorization of the approximate posterior i e  we infer only the mean µ and variance σ for each state variable based on the given data points 
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before  It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B  We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm  

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation  We used the same settings as for the deterministic approach but considered up to 10 complex latent variables  The obtained λ's are in Figure   The probabilistic model allows us to quantify the uncertainty in predictions  In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition 
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon  Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds 
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution  The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model  As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar 
Context 3: \section*{Dynamical Behaviour of $O$ in Lattice Gases}

The dynamical behaviour of the anisotropic order parameter  $m$ [see Eq ~\eqref{eq:def-m} in the Letter] following a quench to the critical point is well described by
the Gaussian theory for all the three lattice gas models studied, $i e ,$ driven lattice gas with either constant (IDLG) or random (RDLG) infinite drive and equilibrium lattice gas (LG)  In other words, in the short-time regime, $m \sim t^{1/2}$ [see Eq  \eqref{eq:mt}] and the Binder cumulant $g$ of the lowest transverse mode [defined in Eq  \eqref{eq:binder}] is zero in this regime  The alternative order parameter $O,$ however, distinguishes between the driven (IDLG, RDLG) and the equilibrium (LG) lattice gases  

In order to understand  this, we first write the phenomenological scaling form for $O$,  analogous to Eq  \eqref{eq:scalingass} in  the Letter,
\begin{eqnarray}
O (t, L_{\parallel} ; S_\Delta) = L_{\parallel}^{-\beta/[\nu(1+\Delta)]} \tilde f_O (t/L_{\parallel}^{z/(1+\Delta)} ; S_\Delta) \quad
\label{eq:Oscalingass}
\end{eqnarray}
We already remarked that, in the LG, this scaling form is not compatible with the prediction $O \sim t^{1/8}  L_{\parallel}^{-1/2}$ of the Gaussian theory   However, following Ref  \cite{AS2002}, it can be argued that, at short times, the only dependence of $O$ on the system size $L_{\parallel}$ is of the form $O \sim L_\parallel^{-1/2}$ which is very well confirmed by numerical simulations  Accordingly,  the generic behaviour of $O$ can be assumed to be
\begin{eqnarray}
O \sim t^{\alpha} L_\parallel^{-1/2}, \label{eq:O}
\end{eqnarray}
where $\alpha$ is a phenomenological exponent to be determined  This, along with Eq  \eqref{eq:Oscalingass}, implies $\tilde f_O(x) \sim x^{\alpha} $ Comparing the finite-size behaviour in Eq ~\eqref{eq:O} with Eq ~\eqref{eq:Oscalingass} one actually infers,
\begin{eqnarray}
\alpha &=& \frac{1+ \Delta -2 \beta/\nu}{2 \, (4- \eta)}  \label{eq:alpha}
\end{eqnarray}
This equation, together with the hyperscaling relation $\Delta - 2 \beta/\nu= - \eta$ in two spatial dimensions, shows that the prediction $\alpha = 1/8$ of the Gaussian theory [see Eq  \eqref{eq:Ot}] can be obtained only when $\eta=0,$ which is the case for the IDLG (exactly) and the RDLG (approximately) but not for the LG  

On the other hand,  Eq ~\eqref{eq:alpha} predicts $\alpha = 1/10$ upon substituting the values of the critical exponents corresponding to the Ising  universality class (LG)  This is consistent with the numerical simulation results presented in the main text, see Fig  \ref{fig:ising}(b) therein 

\begin{figure}[th]
\vspace*{0 2 cm}
 \centering
 \includegraphics[width=10 cm]{ /compare_binder pdf}

\caption{Comparison between the temporal evolution of the Binder cumulants $g$ corresponding to the $12^{th}$ transverse mode, $i e ,$ with $n_\perp =12,$ in the LG (lowest curve), IDLG and RDLG (two upper curves) on a $32 \times 32$ lattice  \label{fig:b}}
 \label{fig:binder}
\end{figure}


The emergence of this new value $1/10$ of the exponent $\alpha$ must be traced back to the non-Gaussian nature of higher fluctuating modes in the LG  In fact, even though the lowest mode behaves identically in all the three models we considered,  characterized by the same behaviour of $m$, higher modes show a significant difference in the non-driven case   


To illustrate this, we measured the Binder cumulants of higher modes which is defined  analogously to Eq ~(11), using transverse modes other than the first, i e , with $\mu=\tilde \sigma(0,2 \pi n_\bot/L_\bot)$ and $n_\bot>1 $  
 Figure \ref{fig:b} compares the same for all the three lattice gases for the mode with $n_\perp =12$ on a $32 \times 32$ lattice  Clearly, the curve corresponding to the LG (lowest, blue) departs from Gaussian behaviour $g=0$ (in practice, $e g ,$ $|g| \lesssim 0 005,$ corresponding to the shaded gray area) much earlier than it does for the IDLG  or RDLG (two upper curves, red and green respectively) 

Accordingly, the different dynamical behaviour of $O$, which involves a sum over all modes, can be attributed to the non-Gaussian nature of the higher modes in the LG  
Such a departure is not entirely surprising  In fact, for higher modes, mesoscopic descriptions such as the ones in Eqs  \eqref{eq:L-DLG} or \eqref{eq:g_evol} are not expected to hold, while the anisotropy at the microscopic level could be the mechanism leading to the Gaussianity of higher modes in the driven models 





ground_truth: Because his roommate smokes.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the framework capture the reduced-order dynamics?

Read the following text and answer briefly.

Context 4: \section{Introduction}
\label{sec:introduction}

Probabilistic models have proven to be very useful in a lot of applications in signal processing where signal estimation is needed \cite{rabiner1989tutorial,arulampalam2002tutorial,ji2008bayesian}  Some of their advantages are that 1) they force the designer to specify all the assumptions of the model, 2) they provide a clear separation between the model and the algorithm used to solve it, and 3) they usually provide some measure of uncertainty about the estimation 

On the other hand, adaptive filtering is a standard approach in estimation problems when the input is received as a stream of data that is potentially non-stationary  This approach is widely understood and applied to several problems such as echo cancellation \cite{gilloire1992adaptive}, noise cancellation \cite{nelson1991active}, and channel equalization \cite{falconer2002frequency} 

Although these two approaches share some underlying relations, there are very few connections in the literature  The first important attempt in the signal processing community to relate these two fields was the connection between a linear Gaussian state-space model (i e  Kalman filter) and the RLS filter, by Sayed and Kailath \cite{sayed1994state} and then by Haykin \emph{et al } \cite{haykin1997adaptive}  The RLS adaptive filtering algorithm emerges naturally when one defines a particular state-space model (SSM) and then performs exact inference in that model  This approach was later exploited in \cite{van2012kernel} to design a kernel RLS algorithm based on Gaussian processes 

A first attempt to approximate the LMS filter from a probabilistic perspective was presented in \cite{park2014probabilistic}, focusing on a kernel-based implementation  The algorithm of \cite{park2014probabilistic} makes use of a Maximum a Posteriori (MAP) estimate as an approximation for the predictive step  However, this approximation does not preserve the estimate of the uncertainty in each step, therefore degrading the performance of the algorithm 

In this work, we provide a similar connection between state-space models and least-mean-squares (LMS)  Our approach is based on approximating the posterior distribution with an isotropic Gaussian distribution  We show how the computation of this approximated posterior leads to a linear-complexity algorithm, comparable to the standard LMS  Similar approaches have already been developed for a variety of problems such as channel equalization using recurrent RBF neural networks \cite{cid1994recurrent}, or Bayesian forecasting \cite{harrison1999bayesian}  Here, we show the usefulness of this probabilistic approach for adaptive filtering 

The probabilistic perspective we adopt throughout this work presents two main advantages  Firstly, a novel LMS algorithm with adaptable step size emerges naturally with this approach, making it suitable for both stationary and non-stationary environments  The proposed algorithm has less free parameters than previous LMS algorithms with variable step size \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}, and its parameters are easier to be tuned w r t  these algorithms and standard LMS  Secondly, the use of a probabilistic model provides us with an estimate of the error variance, which is useful in many applications 

Experiments with simulated and real data show the advantages of the presented approach with respect to previous works  However, we remark that the main contribution of this paper is that it opens the door to introduce more Bayesian machine learning techniques, such as variational inference and Monte Carlo sampling methods \cite{barber2012bayesian}, to adaptive filtering \\


\section{Probabilistic Model}

Throughout this work, we assume the observation model to be linear-Gaussian with the following distribution,

\begin{equation}
p(y_k|{\bf w}_k) = \mathcal{N}(y_k;{\bf x}_k^T {\bf w}_k , \sigma_n^2),
\label{eq:mess_eq}
\end{equation}
where  $\sigma_n^2$ is the variance of the observation noise, ${\bf x}_k$ is the regression vector and ${\bf w}_k$ is the parameter vector to be sequentially estimated, both $M$-dimensional column vectors 


In a non-stationary scenario, ${\bf w}_k$ follows a dynamic process  In particular, we consider a diffusion process (random-walk model) with variance $\sigma_d^2$ for this parameter vector:


\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;{\bf w}_{k-1}, \sigma_d^2 {\bf I}),
\label{eq:trans_eq}
\end{equation}
where $\bf I$ denotes the identity matrix  In order to initiate the recursion, we assume the following prior distribution on ${\bf w}_k$

\begin{equation}
p({\bf w}_0)= \mathcal{N}({\bf w}_0;0, \sigma_d^2{\bf I}) \nonumber
\end{equation}

\section{Exact inference in this model: Revisiting the RLS filter}

Given the described probabilistic SSM, we would like to infer the posterior probability distribution $p({\bf w}_k|y_{1:k})$ 
Since all involved distributions are Gaussian, one can perform exact inference, leveraging the probability rules in a straightforward manner  The resulting probability distribution is
\begin{equation}
p({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k}, \boldsymbol\Sigma_{k}), \nonumber
\end{equation}
in which the mean vector ${\bf\boldsymbol\mu}_{k}$ is given by
\begin{equation}
{\bf\boldsymbol\mu}_k = {\bf\boldsymbol\mu}_{k-1} + {\bf K}_k (y_k - {\bf x}_k^T {\bf\boldsymbol\mu}_{k-1}){\bf x}_k, \nonumber
\end{equation}
where we have introduced the auxiliary variable
\begin{equation}
{\bf K}_k = \frac{ \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)}{{\bf x}_k^T  \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)  {\bf x}_k + \sigma_n^2}, \nonumber
\end{equation}
and the covariance matrix $\boldsymbol\Sigma_k$ is obtained as
\begin{equation}
\boldsymbol\Sigma_k = \left( {\bf I} -  {\bf K}_k{\bf x}_k {\bf x}_k^T \right) ( \boldsymbol\Sigma_{k-1} +\sigma_d^2), \nonumber
\end{equation}
Note that the mode of $p({\bf w}_k|y_{1:k})$, i e  the maximum-a-posteriori estimate (MAP), coincides with the RLS adaptive rule
\begin{equation}
{{\bf w}}_k^{(RLS)} = {{\bf w}}_{k-1}^{(RLS)} + {\bf K}_k (y_k - {\bf x}_k^T {{\bf w}}_{k-1}^{(RLS)}){\bf x}_k  
\label{eq:prob_rls}
\end{equation}
This rule is similar to the one introduced in \cite{haykin1997adaptive} 

Finally, note that the covariance matrix $\boldsymbol\Sigma_k$ is a measure of the uncertainty of the estimate ${\bf w}_k$ conditioned on the observed data $y_{1:k}$  Nevertheless, for many applications a single scalar summarizing the variance of the estimate could prove to be sufficiently useful  In the next section, we show how such a scalar is obtained naturally when $p({\bf w}_k|y_{1:k})$ is approximated with an isotropic Gaussian distribution  We also show that this approximation leads to an LMS-like estimation 
 


\section{Approximating the posterior distribution: LMS filter }

The proposed approach consists in approximating the posterior distribution $p({\bf w}_k|y_{1:k})$, in general a multivariate Gaussian distribution with a full covariance matrix, by an isotropic spherical Gaussian distribution 

\begin{equation}
\label{eq:aprox_post}
\hat{p}({\bf w}_{k}|y_{1:k})=\mathcal{N}({\bf w}_{k};{\bf \hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_{k}^2 {\bf I} ) 
\end{equation}

In order to estimate the mean and covariance of the approximate distribution $\hat{p}({\bf w}_{k}|y_{1:k})$, we propose to select those that minimize the Kullback-Leibler divergence with respect to the original distribution, i e , 

\begin{equation}
\{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k\}=\arg \displaystyle{  \min_{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k}} \{ D_{KL}\left(p({\bf w}_{k}|y_{1:k}))\| \hat{p}({\bf w}_{k}|y_{1:k})\right) \}  \nonumber
\end{equation}

The derivation of the corresponding minimization problem can be found in Appendix A  In particular, the optimal mean and the covariance are found as
\begin{equation}
{\hat{\boldsymbol\mu}}_{k} = {\boldsymbol\mu}_{k};~~~~~~ \hat{\sigma}_{k}^2 = \frac{{\sf Tr}\{ \boldsymbol\Sigma_k\} }{M} 
\label{eq:sigma_hat}
\end{equation}


We now show that by using \eqref{eq:aprox_post} in the recursive predictive and filtering expressions we obtain an LMS-like adaptive rule  First, let us assume that we have an approximate posterior distribution at $k-1$, $\hat{p}({\bf w}_{k-1}|y_{1:k-1}) =  \mathcal{N}({\bf w}_{k-1};\hat{\bf\boldsymbol\mu}_{k-1}, \hat{\sigma}_{k-1}^2 {\bf I} )$  Since all involved distributions are Gaussian, the predictive distribution
is obtained as %
\begin{eqnarray}
\hat{p}({\bf w}_k|y_{1:k-1}) &=& \int p({\bf w}_k|{\bf w}_{k-1}) \hat{p}({\bf w}_{k-1}|y_{1:k-1}) d{\bf w}_{k-1} \nonumber\\
&=& \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k|k-1}, \boldsymbol\Sigma_{k|k-1}), 
\label{eq:approx_pred}
\end{eqnarray}
where the mean vector and covariance matrix are given by
\begin{eqnarray}
\hat{\bf\boldsymbol\mu}_{k|k-1} &=& \hat{\bf\boldsymbol\mu}_{k-1} \nonumber \\
\hat{\boldsymbol\Sigma}_{k|k-1} &=& (\hat{\sigma}_{k-1}^2 + \sigma_d^2 ){\bf I}\nonumber 
\end{eqnarray}

From \eqref{eq:approx_pred}, the posterior distribution at time $k$ can be computed using Bayes' Theorem and standard Gaussian manipulations (see for instance \cite[Ch  4]{murphy2012machine})  Then, we approximate the posterior $p({\bf w}_k|y_{1:k})$ with an isotropic Gaussian,
\begin{equation}
\hat{p}({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k ; {\hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_k^2 {\bf I} ),\nonumber
\end{equation}
where 
\begin{eqnarray}
{\hat{\boldsymbol\mu}}_{k} &= & {\hat{\boldsymbol\mu}}_{k-1}+ \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k  \nonumber  \\
&=& {\hat{\boldsymbol\mu}}_{k-1}+ \eta_k (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k   
\label{eq:prob_lms}
\end{eqnarray}
Note that, instead of a gain matrix ${\bf K}_k$ as in Eq ~\eqref{eq:prob_rls}, we now have a scalar gain $\eta_k$ that operates as a variable step size 


Finally, to obtain the posterior variance, which is our measure of uncertainty, we apply \eqref{eq:sigma_hat} and the trick ${\sf Tr}\{{\bf x}_k{\bf x}_k^T\}= {\bf x}_k^T{\bf x}_k= \|{\bf x}_k \|^2$,

\begin{eqnarray}
\hat{\sigma}_k^2 &=& \frac{{\sf Tr}(\boldsymbol\Sigma_k)}{M} \\
&=& \frac{1}{M}{\sf Tr}\left\{ \left( {\bf I} -  \eta_k {\bf x}_k {\bf x}_k^T \right) (\hat{\sigma}_{k-1}^2 +\sigma_d^2)\right\} \\
&=& \left(1 - \frac{\eta_k \|{\bf x}_k\|^2}{M}\right)(\hat{\sigma}_{k-1}^2 +\sigma_d^2) 
\label{eq:sig_k}
\end{eqnarray}
If MAP estimation is performed, we obtain  an adaptable step-size LMS estimation

\begin{equation}
{\bf w}_{k}^{(LMS)} = {\bf w}_{k-1}^{(LMS)} + \eta_k (y_k - {\bf x}_k^T {\bf w}_{k-1}^{(LMS)}){\bf x}_k, 	
\label{eq:lms}
\end{equation}
with
\begin{equation}
\eta_k = \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} \nonumber
\end{equation}
At this point, several interesting remarks can be made:

\begin{itemize}

\item The adaptive rule \eqref{eq:lms} has linear complexity since it does not require us to compute the full matrix $\boldsymbol\Sigma_k$ 

\item For a stationary model, we have $\sigma_d^2=0$ in \eqref{eq:prob_lms} and \eqref{eq:sig_k}  In this case, the algorithm remains valid and both the step size and the error variance, $\hat{\sigma}_{k}$, vanish over time $k$  

\item Finally, the proposed adaptable step-size LMS has only two parameters, $\sigma_d^2$ and $\sigma_n^2$, (and only one, $\sigma_n^2$, in stationary scenarios) in contrast to other variable step-size algorithms \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}  More interestingly, both $\sigma_d^2$ and $\sigma_n^2$ have a clear underlying physical meaning, and they can be estimated in many cases  We will comment more about this in the next section  
\end{itemize}



\section{Experiments}
\label{sec:experiments}

We evaluate the performance of the proposed algorithm in both stationary and tracking experiments  In the first experiment, we estimate a fixed vector ${\bf w}^{o}$ of dimension $M=50$  The entries of the vector are independently and uniformly chosen in the range $[-1,1]$  Then, the vector is normalized so that $\|{\bf w}^o\|=1$  Regressors $\boldsymbol{x}_{k}$ are zero-mean Gaussian vectors with identity covariance matrix  The additive noise variance is such that the SNR is $20$ dB



ground_truth: By using a propagator in the latent space.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the dynamical behavior of the anisotropic order parameter following a quench to the critical point?

Read the following text and answer briefly.

Context 4:  We compare our algorithm with standard RLS and three other LMS-based algorithms: LMS, NLMS \cite{sayed2008adaptive}, VSS-LMS \cite{shin2004variable} \footnote{The used parameters for each algorithm are: for RLS $\lambda=1$, $\epsilon^{-1}=0 01$; for LMS $\mu=0 01$; for NLMS $\mu=0 5$; and for VSS-LMS $\mu_{max}=1$, $\alpha=0 95$, $C=1e-4$ } The probabilistic LMS algorithm in \cite{park2014probabilistic} is not simulated because it is not suitable for stationary environments 

In stationary environments, the proposed algorithm has only one parameter, $\sigma^2_n$  We simulate both the scenario where we have perfectly knowledge of the amount of noise (probLMS1) and the case where the value $\sigma^2_n$ is $100$ times smaller than the actual value (probLMS2)  The Mean-Square Deviation (${\sf MSD} = {\mathbb E} \| {\bf w}_0 - {\bf w}_k \|^2$), averaged out over $50$ independent simulations, is presented in Fig  \ref{fig:msd_statationary} 



\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{results_stationary_MSD}}
\end{minipage}
\caption{Performance in terms of MSD of probabilistic LMS with both optimal (probLMS1) and suboptimal (probLMS2) compared to LMS, NLMS, VS-LMS, and RLS }
\label{fig:msd_statationary}
\end{figure}

The performance of probabilistic LMS is close to RLS (obviously at a much lower computational cost) and largely outperforms previous variable step-size LMS algorithms proposed in the literature  Note that, when the model is stationary, i e  $\sigma^2_d=0$ in \eqref{eq:trans_eq},  both the uncertainty $\hat{\sigma}^2_k$, and the adaptive step size $\eta_k$, vanish over time  This implies that the error tends to zero when $k$ goes to infinity  Fig  \ref{fig:msd_statationary} also shows that the proposed approach is not very sensitive to a bad choice of its only parameter, as demonstrated by the good results of probLMS2, which uses a $\sigma^2_n$ that is $100$ times smaller than the optimal value  


\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{fig2_final}}
\end{minipage}
\caption{Real part of one coefficient of the measured and estimated channel in experiment two  The shaded area represents two standard deviations from the prediction {(the mean of the posterior distribution)} }
\label{fig_2}
\end{figure}


\begin{table}[ht]
\begin{footnotesize}
\setlength{\tabcolsep}{2pt}
\def1 5mm{1 5mm}
\begin{center}
\begin{tabular}{|l@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|}
\hline
Method &  LMS &  NLMS & LMS-2013 & VSSNLMS & probLMS & RLS \\
\hline
\hline
MSD (dB) &-28 45 &-21 07 &-14 36 &-26 90 &-28 36 &-25 97\\
\hline                                                                     
\end{tabular}
\end{center}
\caption{Steady-state MSD of the different algorithms for the tracking of a real MISO channel }
\label{tab:table_MSD}
\end{footnotesize}

\end{table}
\newpage
In a second experiment, we test the tracking capabilities of the proposed algorithm with {real} data of a wireless MISO channel acquired in a realistic indoor scenario  More details on the setup can be found in \cite{gutierrez2011frequency}  Fig  \ref{fig_2} shows the real part of one of the channels, and the estimate of the proposed algorithm  The shaded area represents the estimated uncertainty for each prediction, i e  $\hat{\mu}_k\pm2\hat{\sigma}_k$  Since the experimental setup does not allow us to obtain the optimal values for the parameters, we fix these parameters to their values that optimize the steady-state mean square deviation (MSD)  \hbox{Table \ref{tab:table_MSD}} shows this steady-state MSD of the estimate of the MISO channel with different methods  As can be seen, the best tracking performance is obtained by standard LMS and the proposed method  





\section{Conclusions and Opened Extensions}
\label{sec:conclusions}

{We have presented a probabilistic interpretation of the least-mean-square filter  The resulting algorithm is an adaptable step-size LMS that performs well both in stationary and tracking scenarios  Moreover, it has fewer free parameters than previous approaches and these parameters have a clear physical meaning  Finally, as stated in the introduction, one of the advantages of having a probabilistic model is that it is easily extensible:}

\begin{itemize}
\item If, instead of using an isotropic Gaussian distribution in the approximation, we used a Gaussian with diagonal covariance matrix, we would obtain a similar algorithm with different step sizes and measures of uncertainty, for each component of ${\bf w}_k$  Although this model can be more descriptive, it needs more parameters to be tuned, and the parallelism with LMS vanishes 
\item Similarly, if we substitute the transition model of \eqref{eq:trans_eq} by an Ornstein-Uhlenbeck process, 

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;\lambda {\bf w}_{k-1}, \sigma_d^2), \nonumber
\label{eq:trans_eq_lambda}
\end{equation}
a similar algorithm is obtained but with a forgetting factor $\lambda$ multiplying ${\bf w}_{k-1}^{(LMS)}$ in \eqref{eq:lms}  This algorithm may have improved performance under such a kind of autoregresive dynamics of ${\bf w}_{k}$, though, again, the connection with standard LMS becomes dimmer 

\item As in \cite{park2014probabilistic}, the measurement model \eqref{eq:mess_eq} can be changed to obtain similar adaptive algorithms for classification, ordinal regression, and Dirichlet regression for compositional data  

\item A similar approximation technique could be applied to more complex dynamical models, i e  switching dynamical models \cite{barber2010graphical}  The derivation of efficient adaptive algorithms that explicitly take into account a switch in the dynamics of the parameters of interest is a non-trivial and open problem, though the proposed approach could be useful 

\item Finally, like standard LMS, this algorithm can be kernelized for its application in estimation under non-linear scenarios 

\end{itemize}


\begin{appendices}

\section{KL divergence between a general gaussian distribution and an isotropic gaussian}
\label{sec:kl}

 We want to approximate  $p_{{\bf x}_1}(x) = \mathcal{N}({\bf x}; \boldsymbol\mu_1,\boldsymbol\Sigma_1)$ by $p_{{\bf x}_2}({\bf x}) = \mathcal{N}({\bf x}; \boldsymbol\mu_2,\sigma_2^2 {\bf I})$  In order to do so, we have to compute the parameters of $p_{{\bf x}_2}({\bf x})$, $\boldsymbol\mu_2$ and $\sigma_2^2$, that minimize the following Kullback-Leibler divergence,

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) &=&\int_{-\infty}^{\infty} p_{{\bf x}_1}({\bf x}) \ln{\frac{p_{{\bf x}_1}({\bf x})}{p_{{\bf x}_2}({\bf x})}}d{\bf x} \nonumber  \\
&= &  \frac{1}{2} \{ -M + {\sf Tr}(\sigma_2^{-2} {\bf I}\cdot \boldsymbol\Sigma_1^{-1})  \nonumber \\
  & &  + (\boldsymbol\mu_2 - \boldsymbol\mu_1 )^T \sigma^{-2}_2{\bf I} (\boldsymbol\mu_2 - \boldsymbol\mu_1 )  \nonumber \\
 & &   +  \ln \frac{{\sigma_2^2}^M}{\det\boldsymbol\Sigma_1} \}   
\label{eq:divergence}
\end{eqnarray}
Using symmetry arguments, we obtain 
\begin{equation}
\boldsymbol\mu_2^{*} =\arg \displaystyle{  \min_{\boldsymbol\mu_2}} \{ D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) \} = \boldsymbol\mu_1 
\end{equation}
Then, \eqref{eq:divergence} gets simplified into 

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) = \frac{1}{2}\lbrace { -M + {\sf Tr}(\frac{\boldsymbol\Sigma_1}{\sigma_2^{2}}) + \ln \frac{\sigma_2^{2M}}{\det\boldsymbol\Sigma_1}}\rbrace 
\end{eqnarray}
The variance $\sigma_2^2$ is computed in order to minimize this Kullback-Leibler divergence as

\begin{eqnarray}
\sigma_2^{2*} &=& \arg\min_{\sigma_2^2} D_{KL}(P_{x_1}\| P_{x_2}) \nonumber \\
 &=& \arg\min_{\sigma_2^2}\{ \sigma_2^{-2}{\sf Tr}\{\boldsymbol\Sigma_1\} + M\ln \sigma_2^{2} \}  
\end{eqnarray}
Deriving and making it equal zero leads to

\begin{equation}
\frac{\partial}{\partial \sigma_2^2} \left[ \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{\sigma_2^{2}} + M \ln \sigma_2^{2} \right] = \left  {\frac{M}{\sigma_2^{2}}-\frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{(\sigma_2^{2})^2}}\right|_{\sigma_2^{2}=\sigma_2^{2*}}\left  =0 \right   
\nonumber
\end{equation}
Finally, since the divergence has a single extremum in $R_+$,
\begin{equation}
\sigma_2^{2*} = \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{M} 
\end{equation}




\end{appendices}

\vfill
\clearpage

\bibliographystyle{IEEEbib}




ground_truth: It is well described by the Gaussian theory.