Data Analysis


Home Up Background knowledge Operation on GPS receiver Site Survey Data Analysis

Technique 1 - Calculation of sample mean, and sample standard deviation.

After finishing the experiment, you know the position or time parameters reported by GPS surely exhibit certain distribution, and your business is to search out what it is. Given the statistical distribution, you wish to extract some useful informations about such distribution. The most straight forward approach is to do experiment. Through sampling, you can estimate the some parameters such as the expectation value, variance of such distribution. Imagining you throw a die, a discrete distribution (integer 1 - 6) concerning the number on the upper face of die can be obtained. What is the expection value on the upper face of die, you know it is 3.5 ((1+2+3+4+5+6)/6). The variance is summation of 1/6*(i-3.5)2. The standard deviation is simply defined as square root of variance. The physical meaning of standard deviation is to measure how far for the distribution spread out from mean. Suppose you do not know how to calculation the mean and standard deviation, you can still do experiment to figure them out. In fact, the mean and standard deviation cannot be calculated in many practical problems. How can we estimate mean and standard deviation by experiment? In the case of throwing die, you will take the measurement many times and calculate the sample mean and standard deviation. Your result of sample mean and sample standard deviation are the estimation of mean and standard deviation of the original distribution. If you take N measurements (X1, X2, ... XN), the formula for sample mean is (summation Xi) / N and that for sample standard deviation is sqrt ((summation( Xi-Xo)2)/(N-1)) where Xo is the sample mean. Notice that dominator in expression is N-1 instead of N. Consult your mathematics teacher or us if you can't see the reason. The idea of proof is not diffcult. (Hints : Remember the expression is a estimator of standard deviation, so the expectation of the expression should be exactly the same as the standard deviation of the distribution. Physically, when we calculate the sample deviation by such formula for many time, we expect the average value of sample deviation converges to standard deviation of distribution.)

In part 3 for experiment 2, GPS height, latitude, longitude are treated as independent variables. You can manipulate their sample mean and sample deviation separately.

Technique 2 - Convertion from angular position to metric position

In WinOncore, standard deviation are represented by length scale not by angle. Here a simplified version for calculation. The idea is to treat earth as a perfect ellipsoid. (Don't blame us lazy and simple minded. There is actually a huge improvement from treating earth as perfect sphere to perfect ellipsoid. Unfortunately, error due to deviation from sphere to ellipsoid is comparable to systematic error of radius of equator. Can you suggest the reason why earth is flatten hence radius of equator is largest) In a circle, from a slightly angular defection, you know arc-length is different by  radius multiplied by angular defection (in radian). In ellipsoidal, the situaion is approximately the same. Once you calculate the distance from center of earth to sea level of you site, from angular deviation, you can calculate length deviation in plane coordinate by simply multiply radius to angle defection provided that angle defection is not too large. To calculate the distance from center to site, we need 3 parameters (latitude(l in radian) of your site, semi-major axis (a in meter) and eccentricity of earth(e dimensionless)). Semi major axis and eccentricity are important parameter to describe elliptic orbit, remember ephemeris on satellites system carries the last two informations. Enter the following link to check http://www.colorado.edu/geography/gcraft/notes/gps/ephclock.html. For earth, semi major axis is the radius of equator (Equator is treated as perfect circle). Eccentricity measures the deviation for tangential plane from perfect circle. (The distance from pole to center of earth is a(1-e2)1/2. Now you can see earth is flatten when e does not equal to zero.) 

Take those values for such parameters if you need them in experiment

  • eccentricity of earth (e) = 0.0167
  • radius of equator (a) = 6378km

The distance from earth center to your site with altitude (l) is given by a*(1-(e*sin(l))2)1/2. You can prove it youselt. If you thick the correction is meaningless, just skip this part.

Technique 3 - Analyse the problem by EXCEL

Making use of the integrated development environment of EXCEL, lots of messy tasks in calculation can be simplified. The screen shots show you how to analyse the problem by EXCEL.

Firstly, import report files generated by gpscomm.exe to EXCEL. EXCEL able to dilimit your text in report file and output as a database table ready to be analysed. (See the figure below) Here list some skills will be helpful to you if you don't familiar to EXCEL.

  • You can calculate sample mean and standard deviation by command AVERAGE and STDEV. For example,  =AVERAGE(A2:A100) calculates the mean of number from column A, row 2 to row 100. Similarly, =STDEV(A2:A100) calulates the sample standard deviation.
  • You may need to combine hour, minute, second to single field time. Use =TIME (A:A,B:B,C:C) as taught in data analysis part of experiment 1.
  • You may need to sort you datum as you want to select datum with smaller DOP, more tracked number. This can be accomplished by clicking the icon with A, Z and down arrow. But remember EXCEL do not know all columns in table are correlated, so it is useless to sort a particular column. Sorting several columns with a key field in EXCEL is a little bit tricky. See the description on the right and practise youself.

Secondly, you may need to perform graphical analysis. In Excel, generating graph from data table is easy. You should follow the steps of graph wizard. Here is a example for ploting GPS height against time. Hints : To activate Graph wizard, click the icon with 3-D bar chart. There are several options about graph style, choose line in this application. Select suitable column (Time and GPS height in this case) and add custom legends, scale and so the graph is ready.

The following figure demostrates the Angular position plot on observatory in CU by using Excel. Exactly the skills you have been learnt to plot height against time except you choose XY scatter in graph wizard. Notice that longitude and latitude are represented in milli arc sec unit. One milli arc sec equals to 1/3600000 degree. You may not satisfy with the plot of angular position since it does not give you a intuitive feeling. Read further to perform more complicate analysis by Excel.

 

Do you think rectangular coordinate is better? The idea of converting angular position to plane coordinate is to treat the position for mean latitude and mean longitude as (0,0) in plane coordinate. Next, making use of the radius of earth, transform angle deviated from mean to distance deviated from (0,0). In this part, you may use the skills taught in Technique 2. Practically, you will enter a formula in the cell, generate X, Y pos by copy and paste. (see the figure shown below)

At last, a similar 2-D position plot can be obtained but in plane coordinate. We have over 1000 data points captured within 2 hours. The resolution are within 10m. (Now, you know how to obtain standard deviation of position in length scale) By using WinOncore, the results are

  • Latitude: N22o 25.3029'   STD 5.986m
  • Longitude: E114o 12.5172'  STD 1.401m
  • Height : 171.39m  STD 6.150m

Compare to the results of graphical analysis.