Data analysis from the perspective of control theory

Data analysis from the perspective of control theory

As the title , Talk about my cognition of mathematical and physical models . This article is also for 1024 medal , An article deliberately rushed , No manuscript saved , Just briefly express your ideas . If you have any questions, welcome to exchange and discuss .

My professional direction and control are inseparable , After junior year , Learn modern control theory 、 Testing Technology 、 After the principle of computer control , I really realized that the mathematical model should be raised to a dynamic system . Data analysis is actually the estimation of system state . This is the new understanding I have had during this period of time .

When doing robot control , There will inevitably be a problem —— Navigation . Then what is navigation ？ The task of navigation covers three parts ：

1. I （ robot ） Where is the ？
2. I （ robot ） Where are you going? ？
3. I （ robot ） How to get there ？

Our work revolves around these three issues . First, the first question , location . How can we achieve positioning —— Where is the robot ？

The positioning of robot is based on its perception . Perception comes from detectors , It usually includes optical sensors such as cameras 、 Laser radar 、 Ultrasonic ranging and so on . I temporarily call the data obtained by these sensors observation data . You've seen me Statistical pattern recognition series learning notes My friends must know Bayesian prior distribution and posterior distribution . That is, how to get the current posture of the robot in the world coordinate system from the observation data ？ Now I quote directly from my Statistical pattern recognition learning notes （ Two ） In the words of .

How do we observe data $x$ To estimate the current state of the robot ？
In short , We hope that through the observation data $x$ To infer the state （ And their probability distributions ）. therefore , We say the estimation of robot state , Is known observation data $x$ Under the condition of , Calculate the conditional probability distribution of the state ：
$$p({\varpi }_i|x)$$
In order to have a better connection with the previous article , The expression used is ${\varpi }_i$ and $x$ . And the above formula is also called Posterior probability . Using Bayesian formula , A posteriori probability can also be expressed as ：
$$p({\varpi }_i|x)=\frac{p(x|{\varpi }_i)p({\varpi }_i)}{p(x)}$$
$p(x|{\varpi }_i)$ It's called likelihood , $p({\varpi }_i)$ It's called a priori . Solving the maximum a posteriori probability is equivalent to the product of maximum likelihood and a priori .

The significance of a priori probability and a posteriori probability is discussed too much in the notes , No more details here . actually , If state estimation is strongly combined with classification problems , Then each state corresponds to a category . This is the source of my inspiration .

In the chapter of modern control theory about system controllability and observability , There is a sentence that inspires me a lot ,“ Input affects the internal state quantity of the system , The state quantity determines the output of the system .”

In computer control theory , We often study discrete systems . Because computers process digital signals , It is discrete in time and amplitude . We know that the physical meaning of differential equations is actually the law of motion of the system . Differential equations are mathematical models of continuous signals , The difference equation is the mathematical model of discrete signal . Discrete signals can be sampled from continuous systems , Problems related to sampling points , I can easily rise to the application problem . For example, a small shop needs to purchase goods every month , Input is the purchase quantity , Output is profit . And user preferences , Regional factors are often intermediate variables in this input and output process . I try to find a way to describe quantitatively , Although there is no further study this week , But I believe this problem can be solved next Monday and Tuesday .

I am now , Yes 、 Complex domain 、 frequency domain 、 State space has a new understanding , I believe my control system will go a long way . Thank you for meeting these lovely teachers this semester , I also found that my efforts gradually began to pay off .