In [5]:
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
In [6]:
#Onesided normal percentile point function
#alpha = 0.05    ==>  1-alpha/2 = 0.975
print stats.norm.ppf(0.95)
print stats.t.ppf(0.95,1000)

1.64485362695
1.64637881729

Laboratory Assignment 1

Exercise A

Question 1: How much time will the download require if all the seeders are used together, given that their individual speeds do not change?

In [3]:
#<your calculations here>
1.0 / (1.0/4.0 + 1.0/7.0 + 1.0/6.0)
Out[3]:
1.7872340425531914
Let M denotes the size of Linux distribution to be downloaded. The individual speeds of seeders are given by M/4, M/7, and M/6 and these values are constant. Let h denotes total hours requires to download the distribution if all the seeders are used together. The portion of the distribution will be downloaded from each seeder are h*M/4, h*M/7, and h*M/6, respectively. The sum of these values is M. Then we have h*M/4 + h*M/7 + h*M/6 = M => h = 1/(1/4 + 1/7 + 1/6) = 1.787 hours Answer: ~ 1.79 hours

Question 2: How large was the distribution (in Mbytes)?

In [4]:
#<your calculations here>

# the ammount is downloaded in the first hour with speed of 290Kbps  
S1 = 290*3600*1000
# the ammount is downloaded in the first hour with speed of 253Kbps  
S2 = 253*3600*1000
# the ammount is downloaded in the first hour with speed of 269Kbps
S3 = 269*3600*1000
# the ammount is downloaded in the first hour with speed of 1008Kbps
S4 = 1008*3600*1000

#Total ammount in bit
S = S1 + S2 + S3 + S4
#Total ammount in Mbytes
SM = S/(8.0*pow(2,20))
print SM

781.059265137
The size of the distribution in Mbytes is Answer: ~781.06 MB

Exercise B

In [8]:
dataset = list()
with open("labdata.dat") as datafile:
    for line in datafile:
        dataset.append(int(line))
    
    
#print dataset
print np.mean(dataset)
onewayRTT = np.mean(dataset)/2.0
print onewayRTT

31.527
15.7635
In [13]:
data_col1 = list()
data_col2 = list()
with open("labdata2.dat") as datafile:
    for line in datafile:
        data_col1.append(int(line.split()[0]))
        data_col2.append(int(line.split()[1]))
print data_col1

print data_col2

[428, 423, 426, 404, 399, 403, 440, 439, 443, 354, 353, 357, 324, 328, 325, 328, 325, 328, 325, 325, 324, 326, 329, 325, 329, 325, 329, 337, 342, 337, 351, 349, 349]
[541, 525, 525, 440, 440, 441, 444, 445, 440, 451, 451, 455, 404, 399, 399, 436, 437, 437, 446, 445, 448, 378, 377, 376, 390, 390, 388, 405, 404, 407, 371, 372, 373]

Question 1: How much time, approximately, is required for a message to go from A to B?

In [16]:
print np.mean(dataset)
onewayRTT = np.mean(dataset)/2.0
print onewayRTT

31.527
15.7635
The approximate time which is required for a message to go from A to B is calculated as a half of average RTT between A and B Answer: 15.7635 ms

Question 2: Considering all values, what seems to be the typical RTT?

In [6]:
RTT = np.mean(dataset)
print RTT

31.527
The typical RTT is calculated by averaging of all 1000 RTT samples Answer: 31.527 ms

Question 3: Approximately, how much does the RTT vary on average?

In [10]:
stdev = np.std(dataset)
print stdev

19.354463852
How much the RTT varies on average is represented by the standard deviation variable. It is given below Answer: ~ 19.35 ms

Question 4: How does the distribution of RTT samples look like?

In [48]:
plt.hist(dataset)
plt.show()
Answer: By ploting histogram of RTT samples from dataset, we get the shape of the distribution of RTT samples in the above figure in which x-basis represents RTT value in milisecond and y-basis represents frequency of appearance of each value

Question 5: How would you like to describe the shape of the distribution, and why do you think it has this shape?

In [6]:
#<your calculations here>
Answer: From the above histogram, it looks like the shape of exponential distribution.

Exercise C

Question 1: Calculate [c1,c2] for a confidence level of 90%?

In [24]:
def conf_func(X, clevel):
    avg = np.mean(X)
    stdev = np.std(X)
    alpha = 1 - clevel
    n = len(X)
    temp = stats.norm.ppf(1 - alpha/2)*stdev/(pow(n,1/2.0)) #number of measurement = 1000 >> 30

    c1, c2 = avg - temp, avg + temp
    
    return c1, c2

print conf_func(dataset, 0.90)
    
    

(30.520280682916461, 32.533719317083538)
Calculate [c1, c2] for a confidence level of 90% Answer: [c1, c2] = [30.520280682916461, 32.533719317083538]

Question 2: Calculate [c1,c2] for a confidence level of 95%?

In [12]:
print conf_func(dataset, 0.95)

(30.327419945158617, 32.726580054841385)
Calculate [c1, c2] for a confidence level of 95% Answer: [c1, c2] = [30.327419945158617, 32.726580054841385]

Question 3: Why is the interval larger for 95% than for a 90% interval? Should not a 95% interval be a more precise measure of the interval containing the "real" value?

In [25]:
print "Confidence Intervals for confidence level of 90%: [c1,c2] = ",conf_func(dataset, 0.90)
print "Confidence Intervals for confidence level of 95%: [c3,c4] = ",conf_func(dataset, 0.95)

Confidence Intervals for confidence level of 90%: [c1,c2] =  (30.520280682916461, 32.533719317083538)
Confidence Intervals for confidence level of 95%: [c3,c4] =  (30.327419945158617, 32.726580054841385)
Answer 01: Confidence Interval is the range of values so defined that is a specified probability that the value of a parameter lies within it or probability of including the actual value, the mean value in this case. The size of this range affects the precison of measurement. 90% confidence means that there is a 90% chance that the actual mean is within that interval. Increasing the confidence level to 95% means that we are increasing the probability that the actual mean is in the resulting interval. That's the reason why the interval is larger for 95% than for a 90% interval. Answer 02: Increasing the confidence level means we are inscreasing size of precision domain. The wider interval implies that we know less about the actual mean. The higher our confidence about the mean, the less precise is the information we seem to have.

Question 4: Between what values can we expect the "real" transmission time to be? Choose a 95% level of confidence.

In [14]:
print conf_func(data_col1, 0.95)

(351.85107458580217, 365.05801632328871)
The range of values that we can expect the real transmission time to be is represented by confidence intervals Answer: (351.85107458580217, 365.05801632328871)

Question 5: How many repetitions of the experiment are required to be sure that the average transmission time is within 3% of the average?

In [ ]:
def num_rep(X, err_rang, clevel):
    avg = np.mean(X)
    alpha = 1.0 - clevel
    err = err_rang/2.0
    stdev = np.std(X)
    temp = (stats.norm.ppf(1 - alpha/2)*stdev)/(avg*err)
    n = temp**2
    return n
print num_rep(data_col1, 0.03, 0.95)
The number of repetitions of the experiment are required to be sure that the average transmission time is within 3% of the average is given as below Answer: 228

Question 5: Compare the achieved results to the unmodified TCP stack. What can you say about the difference?

In [20]:
# Values for original TCP stack
print "Original TCP Stack"
print "Mean = ", np.mean(data_col1)
print "Standard Deviation = ",np.std(data_col1)
# Values for modified TCP stack
print "Modified TCP Stack"
print "Mean = ", np.mean(data_col2)
print "Standard Deviation = ",np.std(data_col2)

Original TCP Stack
Mean =  358.454545455
Standard Deviation =  41.3990857088
Modified TCP Stack
Mean =  426.666666667
Standard Deviation =  43.0670137898
To compare two versions of TCP stack before and after modifying, we compute mean and standard deviatoin Answer: After modifying, new TCP stack increases the average of transmission time compared to the original one, from 358.45 to 426.67 (time units)

Question 6: Do one set of calculations according to the Means of Difference approach, and one set of calculation according to the Difference of Means approach, both using the same confidence level. Do they show the same results? Why/Why not?

In [19]:
# Means of Difference Approach
d = list()
d = list(np.array(data_col2) - np.array(data_col1))
print "List d =",d
print "Average of d=",np.mean(d)
print "Standard Deviation of d=", np.std(d)
print "Number of degrees of freedom =",len(d)
print "Confidence Intervals [c1,c2] = ",conf_func(d,0.95)

# Difference of Mean Approach
avg_dif = np.mean(data_col2) - np.mean(data_col1)
print "Difference of Average=", avg_dif
n1 = len(data_col1)
n2 = len(data_col2)
s1 = np.std(data_col1)
s2 = np.std(data_col2)
temp1 = s1**2/n1 + s2**2/n2
std_comb = pow(temp1,1/2.0)
print "Combined standard deviation = ",std_comb
temp2 = pow(s1**2/n1,2)/(n1-1) + pow(s2**2/n2,2)/(n2-1)
n_dof = temp1**2/temp2
print "Number of degrees of freedom", n_dof
temp3 = stats.norm.ppf(0.975)*std_comb/pow(n_dof,1/2.0)
c1, c2 = avg_dif - temp3,avg_dif + temp3
print c1, c2, 

List d = [113, 102, 99, 36, 41, 38, 4, 6, -3, 97, 98, 98, 80, 71, 74, 108, 112, 109, 121, 120, 124, 52, 48, 51, 61, 65, 59, 68, 62, 70, 20, 23, 24]
Average of d= 68.2121212121
Standard Deviation of d= 36.4564975925
Number of degrees of freedom = 33
Confidence Intervals [c1,c2] =  (61.608650343377938, 74.815592080864491)
Difference of Average= 68.2121212121
Combined standard deviation =  10.3990850039
Number of degrees of freedom 63.9004102198
65.6624076375 70.7618347867
Answer: The confidence intervals for means of difference approach are [c1,c2] = [61.608650343377938, 74.815592080864491] and the confidence intervals for difference of means approach are [c3,c4] = [65.6624076375, 70.7618347867]. These values are different. Although, two mean values in both approaches are the same but the differences between the standard deviation values and number of degrees of freedom when calculating distribution function (n # n_dof) are reasons why these values are different.

Exercise D

Question 1: Are there any statistically significant difference between the network providers i.e.: what can you say about the difference with 95% confidence?

In [134]:
#<your calculations here>
#net1, net2, net3 = list(), list(), list()
net1 = [129,139,142,138,133]
net2 = [144,132,135,145,139]
net3 = [142,147,144,149,138]
#print net1[:], net2[:], net3[:]
matrix = [net1,net2,net3]
print matrix
# First way
print "Pre-built Function:",stats.f_oneway(net1,net2,net3)

# Second way
mean_list = list()
n = 5
k = 3
for j in range(k):
    mean_list.append(np.mean(matrix[:][j]))

SSA = 0
SSE = 0 
SST = 0
# Calculate variation due to effects of alternatives
for j in range(k):
    SSA += n*(mean_list[j] - np.mean(mean_list))**2
print "SSA =",SSA

# Calculate variation due to errors in measurements
for j in range(k):
    for i in range(n):
        SSE += (matrix[j][i] - mean_list[j])**2
print "SSE =",SSE

# Calculate difference between each measurement and overall mean
for j in range(k):
    for i in range(n):
        SST += (matrix[j][i] - np.mean(mean_list))**2
#print SSA + SSE, STT = SSA + SSE
print "STT =",SST 

# Degrees of freedom
df_SSA = k-1
df_SSE = k*(n-1)
df_SST = k*n-1

# Mean square values
sa_sq = SSA/df_SSA
se_sq = SSE/df_SSE

# Comparing variances
F_cacluated = sa_sq/se_sq
F_calculated2 = (SSA/SSE)*k*(n-1)/(k-1)
#print "F_calculated2 =",F_calculated2
print "F_calcuated =",F_cacluated 

[[129, 139, 142, 138, 133], [144, 132, 135, 145, 139], [142, 147, 144, 149, 138]]
Pre-built Function: F_onewayResult(statistic=3.0534550195573615, pvalue=0.084726918790441252)
SSA = 156.133333333
SSE = 306.8
STT = 462.933333333
F_calcuated = 3.05345501956
Answer: Comparing the F value calculated from the set of computation above to F value got from the F-distribution table, we see that F_calculated = 3.0535 < F_tabulated[0.95;2;12] = 3.89. Therefore, with 95% confidence that differences among the alternatives are not statistically significant

Question 2: What can you say with 90% confidence?

In [6]:
Answer: Using F-distribution table, we can get the F value with 90% confidence as F_tabulated_90 = 2.81 which is smaller than F_calculated = 3.0535. Therefore, it can be concluded that with 90% confidence, the differences among the alternatives are statistically significant.

Question 3: Is it possible to provide any useful comments about the experimental design based on the information provided above?

In [6]:
Answer: In the experimental design, we need to know if our new design can have better performce than other alternatives. In order to do that we need a technique for determining whether any changes we see are due to random fluctuations in the measurements or they are statistically significant. As shown above, ANOVA can help us but we need to carefully choice the confidence level because this value can affect comparision results. In previous computation for example, with confidence level of 95% we cannot see any difference or improvement between three alternatives but confidence level of 90% does. However, ANOVA doesn't tell us where the differences occur. Therefore, we may need to use contrast method to find more detailed comparisions between those alternatives.
In [ ]: