Seasonal ARIMA.Rmd

---
title: "STAT4181 HW5 Min Yang"
output:
  pdf_document: default
  word_document: default
  html_notebook: default
---

###A. Seasonal ARIMA

####1.
```{r}
library(zoo)
library(rdatamarket)
library(forecast)
library(astsa)
```

####3.
```{r}
data<-dmseries("https://datamarket.com/data/set/22pw/monthly-lake-erie-levels-1921-1970#!ds=22pw&display=line")
str(data)
```

####4.
```{r}
plot.ts(data)
```
```{r}
difference<-diff(data,12)
tsdisplay(difference,lag.max=500)
```

The acf plot suggests there that the data are highly correlated, and there are very strong autocorrelation. Both positive and negative autocorrelation occure, with a negative followed by a positive. A negative autocorrelation means the lake level decrease and positive autocorrelation means the lake level increase. So there is a seasonality. 


####5.
```{r}
auto.arima(data, stepwise = FALSE)
```

The best model is ARIMA(0,1,2)(2,0,0)[12].

####6.
```{r}
sarima(data,p = 0,d = 1,q = 2,P=2,D = 0,Q = 0,S = 12,details=FALSE)
```

The estimated generating equation:
$$
(1-0.3076B^{12}-0.3771B^{24}) (1-0.2599B-0.1074B^2) \nabla^1 x_t = 0.0006 + w_t,
$$


###B. Linear Regression
####1.
```{r}
forest<-read.csv("~/desktop/forestfires.csv",header=TRUE)
```

####2.
```{r}
lm_model<-lm(RH~temp,forest)
```

####3.
```{r}
plot(forest$temp,forest$RH,xlab="Temperature", ylab="Relative Humidity", 
     main="Relative Humidity vs. Temperature")
abline(lm_model,col=2)
```

####4.
```{r}
summary(lm_model)
```

The estimated equation is RH=72.2828-1.482temp.
With p-value <0.0001, the regression model is significant.But the r-square value shows the model only explains 27.81% of the data. Since this is a simple linear regression model, there should be other varaibles that affect the relative humidity.