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+---
+title: "Location Based Services"
+author: "Erik Neller"
+date: "`r Sys.Date()`"
+output: pdf_document
+---
+
+```{r setup, include=FALSE}
+knitr::opts_chunk$set(echo = TRUE)
+rm(list=ls())
+```
+# Data Info
+- GPST: Timestamp
+- latitude/longitude/height: Position estimates.
+- Q: Quality indicator (e.g., 5 = single-point positioning).
+- ns: Number of satellites used
+- sdn/sde/sdu: Standard deviations (North, East, Up)
+- sdne/sdeu/sdun: Combined standard deviations
+
+
+
+```{r echo=FALSE}
+library(DataExplorer)
+library(sf)
+library(lubridate)
+earthrad <- 6378137
+d <- read.csv('rec.csv',header = FALSE, sep='',skip=15)
+
+doy <- yday(d$V1)
+res1 <- ymd_hms(paste(d$V1,d$V2, sep=' '))
+hour <- hour(res1)
+min <- minute(res1)
+sec <- second(res1)
+
+timer <- doy + hour/24 + min/(24*60) + sec/(24*3600)
+# calculate the error of lat lon elevation
+latitude <- (d$V3 + 12.843697222222223)*earthrad*pi/180
+longitude <- (d$V4 - 131.13274444444443)*earthrad*pi/180
+height <- d$V5 - 125.1
+df <- as.data.frame(cbind(longitude, latitude, height))
+```
+# Error Plot
+```{r echo=FALSE}
+plot(timer, latitude, type='l', col = 'red', 
+     ylim = c(min(latitude, longitude,height),
+              max(latitude, longitude,height)),
+     xlab='time [day in 2015]',ylab='errors [m]')
+lines(timer, longitude, col= 'dark green')
+lines(timer, height, col= 'blue')
+legend('topright', legend=c('latitude', 'longitude','height'),
+       col=c('red','dark green', 'blue'), lty=1:1, cex=0.8,
+       box.lty=0)
+
+```
+# Error Summary
+```{r}
+summary(latitude)
+summary(longitude)
+summary(height)
+```
+
+# Tests
+```{r}
+shapiro.test(latitude)
+ks.test(latitude, 'pnorm')
+
+shapiro.test(longitude)
+ks.test(longitude, 'pnorm')
+
+shapiro.test(height)
+ks.test(height, 'pnorm')
+
+ks.test(latitude, 'pnorm')
+```
+Both SW and KS Test suggest that the errors do not follow a normal distribution which is supported by visual inspection:
+
+```{r visualerr}
+library(ggplot2)
+  ggplot(df, aes(x = latitude)) +
+  geom_histogram(bins = 30, fill = "blue", alpha = 0.7) +
+  ggtitle("Distribution of Latitude Error") +
+  xlab("Error (m)") +
+  ylab("Frequency")
+
+ggplot(df, aes(x = longitude)) +
+  geom_histogram(bins = 30, fill = "green", alpha = 0.7) +
+  ggtitle("Distribution of Longitude Error") +
+  xlab("Error (m)") +
+  ylab("Frequency")
+
+ggplot(df, aes(x = height)) +
+  geom_histogram(bins = 30, fill = "red", alpha = 0.7) +
+  ggtitle("Distribution of Height Error") +
+  xlab("Error (m)") +
+  ylab("Frequency")
+```
+```{r}
+var.test(latitude, longitude)
+cor.test(latitude, longitude)
+
+var.test(height, longitude)
+cor.test(height, longitude)
+
+var.test(latitude, height)
+cor.test(latitude, height)
+
+
+
+
+
+```
+Variances of the samples are not equal and not similar.
+However, the correlation of latitude and height is highly correlated, suggesting a linear relationship.
+# Grouping by numsatellites
+```{r numsatellites}
+d_split <- split(d,d$V7)
+summary(d_split)
+```
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