# Accuracy Analysis

```{r include=FALSE}
library(data.table)
library(forecast)
require(data.table)
require(TSA)
require(gridExtra)
require(ggplot2)

#Get Data
pathToFiles <- '../input/'
numberOfRowsToRead <- -1
set.seed(55446)
dt.train <- read.table(paste0(pathToFiles, "train.csv"),  header=T, sep=",",nrows=numberOfRowsToRead)
dt.train <- as.data.table(dt.train)

dt.train <- dt.train[order(time)]
dt.train[,accuracy:=as.numeric(accuracy)]
dt.train.subset.accuracy <- dt.train[,list(accuracy=median(accuracy)),by=time]
rm(dt.train)
```
The **accuracy** itself is an interesting variable that oscillates in several frequencies. Its values are extremely centered even if its outliers magnitude is quite high.

```{r echo=FALSE,warning=FALSE}
acLineplot <- ggplot(data=dt.train.subset.accuracy[1:3000], aes(x=time, y=accuracy, colour="orange")) +
  geom_line()+
  geom_point()+
  theme(legend.position="none")

acBoxPlot <-  ggplot(data=dt.train.subset.accuracy, aes(x=time, y=accuracy)) + geom_boxplot()
 
acDensityPlot <-  ggplot(dt.train.subset.accuracy, aes(x=accuracy)) + geom_density(fill="white")

acLineplot
```

The graphs verify how is the data distributed.
```{r,echo=FALSE,warning=FALSE}
grid.arrange(acBoxPlot,acDensityPlot, ncol=2)
```

In order to find periodicity, the spectral analysis shows this:

```{r, warning=FALSE,echo=FALSE}
movingAverage<-60
hour.coef = 60
four.hour.coef<-60*4
day.coef = hour.coef*24
week.coef = day.coef*7
year.coef = day.coef*365

accuracy.sp.pgram <- spec.pgram(scale(dt.train.subset.accuracy$accuracy), spans=(movingAverage*2+1), taper=0, log="no")
accuracy.frequencies <- data.table(freq=accuracy.sp.pgram$freq,spec=accuracy.sp.pgram$spec)[order(-spec)]

#Calculating period cycles.
accuracy.frequencies[,cyclePeriod:=round(1/freq)]

#accuracy.frequencies[,cycleHours:=round(cyclePeriod/hour.coef)]
accuracy.frequencies[,cycleDays:=round(cyclePeriod/day.coef)]
accuracy.frequencies[,cycleWeeks:=round(cyclePeriod/week.coef)]
accuracy.frequencies[,cycleMonths:=round(cyclePeriod/(day.coef*30))]
#accuracy.frequencies[,cycleYears:=round(cyclePeriod/year.coef,1)]

#Selecting frequencies up to 1/500 of max density power.
accuracy.frequencies <- accuracy.frequencies[
  accuracy.frequencies$spec>max(accuracy.frequencies$spec)/250][1:25]

data.frame(accuracy.frequencies[1:nrow(accuracy.frequencies)])
```

There is a monthly component that defines the strongest frequencies around the following amplitudes in **months**:

```{r, echo=FALSE}

bpData<-accuracy.frequencies[,list(power=max(round(spec))),by=list(factor(cycleMonths))][order(factor)]

colnames(bpData) <- c("Names","Heights")

barplot(height = as.vector(bpData$Heights),names.arg = bpData$Names
        ,ylim = c(0,500),ylab = "Power Spectral Density",xlab="Frequency in Months"
        ,main="Monthly Accuracy Spectral Analysis"
        )
```

As per the graph, it seems that the frequency is highly related to monthly, quarterly and biannual periodicities. There is also a notable density of sub-monthly frequency. Let's take a look to the **weekly** densities.

```{r, echo=FALSE}

bpData<-accuracy.frequencies[,list(power=max(round(spec))),by=list(factor(cycleWeeks))][order(factor)]

colnames(bpData) <- c("Names","Heights")

barplot(height = as.vector(bpData$Heights),names.arg = bpData$Names
        ,ylim = c(0,500),ylab = "Weekly Power Spectral Density",xlab="Frequency in Weeks"
        ,main="Accuracy Spectral Analysis"
        )
```

Where the correspondence in **days** are:
```{r, echo=FALSE}
bpData<-accuracy.frequencies[,list(power=max(round(spec))),by=list(factor(cycleDays))][order(factor)]

colnames(bpData) <- c("Period","Heights")

data.frame(bpData)
```

In conclusion, the spectral analysis shows that accuracy is intimately related to monthly and weekly cycles.