시계열 예측의 자동 회귀 모델 이해

자동 회귀 모델은 과거 값을 기반으로 예측하며, 고정성과 긴 기억력을 목표로 합니다.

### 자동 회귀(AR) 모델: 과거 값을 사용한 예측
* 📊 자동 회귀 모델은 **과거 값 또는 지연을 기반**으로 시리즈를 예측하는 고정 모델이며, **고정성 개념에 의존**합니다.
* 🔮 차수 1의 간단한 AR 모델은 **계열의 이전 값을 사용**하여 대상 변수를 예측하며, **절편, 계수 및 오차 항**을 사용합니다.
* 📈 예측에 지연된 값을 사용하는 개념은 데이터 집합의 모든 관측값에 이 개념을 적용하여 **전주 매출을 기반으로 주간 매출을 예측**하는 예시를 통해 설명할 수 있습니다.
* AR 모델은 **현재 값이 과거 값에 의존하는 장기기억 모델**이며, 이러한 의존성은 수학 방정식을 통해 반복되어 **과거의 관측이 현재에 지속적인 영향**을 미친다는 것을 보여줍니다.
* 🕰 과거 관측의 효과는 시간이 지남에 따라 사라지는데, 이는 고정성의 개념을 반영하며, AR 모델에는 **다음 동영상에서 설명할 ARMA 모델**이라고 하는 과거에 여러 개의 시차가 포함될 수 있습니다.

finally, our series on time series gets back to the idea of modeling. Specifically, today, what are Autoregressive, also known as AR, models? Well, first of all, let's remember stationarity. We can't have independence in time series, but we still want consistency. And that means that we want the distribution to depend only on differences in time, not on locations in time. Of course, if you're not quite sure about stationarity, you can always go back and watch my video on stationarity. The reason stationarity is so important is there are entire classes of models that depend on stationarity, and these models are well called stationary models. I know, very complicated. And one of the most popular of these stationary models are called Autoregressive models. Autoregressive models try to forecast a series based solely on the past values of the series. We call these lags. A model that only depends on the previous value of the previous lag, or one lag in the past, is called an AR model of order 1, or an AR 1 model, as you can see here. So we

have the target variable Y that we're still interested in, but we're essentially using previous values of Y to predict this target variable. Y, you can imagine, is like a lagged target. And then, of course, it's not going to be a perfect prediction, so we have some error as well. Now, there's an intercept, there's a coefficient. We'll deal with the coefficient in a little bit, but let's talk a little bit more about this lagged target. Maybe you're having a hard time conceptualizing this, so let's zoom in a little bit more on my sales chart on the right-hand side. Again, let's say I want to forecast my sales on a weekly basis. Well, how can I use the previous week to be able to help me out? Well, let's take a look at one observation. This one observation would essentially be looking back at the previous week sales to try and predict it. So again, I'm using a previous week to predict a current week. Now, of course, I'm not going to do this for just one observation. I'm doing this

for all of the observations in my dataset every observation is looking back at the week before it. Of course, except for the very first observation. It's lonely. It doesn't have one before it. But don't worry, first observation, you'll be important. Just wait. And so if you keep doing this whole idea of recursion where you have that today depends on the time period before, and that depends on the time period before that, and that depends on the time period before that, you can sort of recursively plug this in in a mathematical way. Which is why these are called long memory models. Let me show you what I mean mathematically. So let's say that today YT depends on YT minus 1. Well, that means that YT minus 1 depends on YT minus 2. And so I can actually plug these equations into each other and show in a roundabout way that YT depends on YT minus 2 with a little bit extra error and a little bit different of an intercept. I still have that YT minus 2 is somewhat influential on YT. Ok, well if

I know that YT is dependent on YT minus 2. Let's think about YT minus 2. Well, YT minus 2 depends on YT minus 3. Again, time to repeat the process. I can plug these two equations into each other and, again in a roundabout way, show that YT minus 3 has some minor effect on YT today. Again, with a little bit extra error and a little bit different of an intercept. Of course, I can repeat this over and over and over and over again. In fact, I can sort of do this all the way till we get back to the very very first observation. Essentially, plug in a bunch of math and you can get what you have here at the very bottom of the screen. And if you take a look, the first observation does matter. The first observation, in a roundabout way, still has a small effect on what's going on today. Now, the whole point of stationarity is that this effect goes away. So, the effect of these shocks that happened long ago will have only a little effect on today, if that.

coefficient in absolute value is less than 1. Again, this is why these are called long memory models. The effect doesn't ever fully go away, but the effect does dissipate over time, hence the idea of stationarity. Now, of course, why just stop with only having one like you could easily have two lags where if looking at today's weeks in sales. I can look back at the previous two weeks to try and predict this week in sales. Of course, again, I'm not doing this for one observation. I'm doing this for all of the observations. But if you try and draw this out, it gets a little bit complicated. And again, why stop with two? Why not have as many different lags in the model that you'd like? So again, you can go up to P lags into the past. These are called ARP models. And these models can get rather large. If only they had a counterpart to be able to help them out. Oh, you'll have to turn in to our next video to figure that one out. But for

right now, what are auto-regressive models? those are auto-regressive models in under five minutes.

What are Autoregressive (AR) Models

Data is impacting large portions or our lives! Data science and analytics try to help sift through this data to help drive decisions and measure impacts. This channel tries to explain techniques for data science in simple terms so everyone can understand what is happening as well as help discuss overall impacts data has on our lives.

Aric LaBarr

시계열의 자동 회귀(AR) 모델 이해하기

자동 회귀(AR) 모델은 과거 계열의 값을 사용하여 미래 값을 예측하는 일종의 고정 모델입니다.

AR 모델이라고도 하는 이러한 모델은 이전 값에만 의존하는 1 차수의 AR 모델과 함께 지연에 의존합니다.

기본적으로 AR 모델은 대상 변수의 이전 값을 사용하여 예측을 수행하므로 시계열 예측에서 중요한 역할을 합니다.

시계열 데이터에서 장기기억 모델의 개념

시계열 데이터는 종종 각 관측이 이전 기간에 의존하는 재귀 패턴을 따르기 때문에 긴 메모리 모델로 이어집니다.

즉, 첫 번째 관측치라도 추가 오차가 있더라도 현재 기간에 미치는 영향은 미미합니다.

그러나 고정성의 개념은 과거의 충격이 현재에 미치는 영향이 시간이 지남에 따라 줄어든다는 것을 시사합니다.

매출 예측에서 장기기억 모델의 개념

장기기억 모델은 절대값이 1보다 작은 계수를 가지므로 시간이 지남에 따라 소멸 효과가 발생하여 고정성이라는 개념에 기여합니다.

이러한 모델에는 여러 지연이 포함될 수 있으며, ARP 모델에는 많은 수의 과거 지연이 있을 수 있습니다.

이러한 모델의 복잡성은 매출 예측의 중요한 측면이며 다음 동영상에서 자세히 살펴볼 것입니다.

자동 회귀 모델은 이전 데이터 포인트를 기반으로 미래 값을 추정하는 시계열 분석의 한 유형입니다.

이러한 모델은 경제 동향, 주가, 날씨 패턴을 예측하는 데 널리 사용됩니다.

자동 회귀 모델은 과거 데이터를 기반으로 예측하는 데 유용하며, 많은 분야에서 필수적인 도구입니다.