Topics: Statistics & Experimental Design The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

Design of experiments Why is it important? We wish to draw meaningful conclusions from data collected Statistical methodology is the only objective approach to analysis

Design of experiments Recognize the problem Select factor to be varied, levels and ranges over which factors will be varied Select the response variable Choose experimental design: Sample size? Blocking? Randomization? Perform the experiment Statistical analysis Conclusions and recommendations

Let’s start easy We would like to compare the output of two systems. Design a testing protocol and run it several times RunSystem A System B 1y 1A y 1B 2y 2A y 2B 3y 3A y 3B ………

Visualize data For small data sets: Scatter plot

Visualize data For larger data sets: Histogram Divide horizontal axis into intervals (bins) Construct rectangle over interval with area proportional to number (frequency) of observations frequency, n i

Statistical inference Draw conclusions about a population using a sample from that population. Imagine hypothetical population containing a large number N of observations. Denote measure of location of population as 

Statistical inference Denote spread of population as variance 

Statistical inference A small group of observations is known as a sample. A statistic like the average is calculated from a set of data considered to be a sample from a population RunSystem A System B 1y 1A y 1B 2y 2A y 2B 3y 3A y 3B ………

Statistical inference Sample variance supplies a measure of the spread of the sample

Probability distribution functions P(a  x  b)

Probability distribution functions P(x i ) xixi P(x = x i ) = p(x i )

Mean, variance of pdf Mean is a measure of central tendency or location Variance measures the spread or dispersion

Normal distribution  = standard deviation = √    mean

Normal distribution, From previous examples we can see that mean =  and variance =  2 completely characterize the distribution. Knowing the pdf of the population from which sample is draw  determine pdf of particular statistic.

Normal distribution Probability that a positive deviation from the mean exceeds one standard deviation  is  1/6 = percentage of the total area under the curve. (Same as negative deviation) Probability that a deviation in either direction will exceed one standard deviation  is 2 x = Chance that a positive deviation from the mean will exceed two  =  1/40

Normal distribution Sample runs differ as a result of experimental error Often can be described by normal distribution

Standard Normal distribution, N(0,1) Values for N(0,1) are found in tables.

Standard Normal distribution, N(0,1)

Example: Suppose the outcome of a given experiment is approximately normally distributed with a  = 4.0 and  = 0.3. What is the probability that the outcome may be 4.4? Look in table in previous page, to find that the probability is 9%.

   distribution Another sampling distribution that can be defined in terms of normal random variables. Suppose z 1, z 2, …, z k are normally and independently distributed random variables with mean  = 0 and variance  2 = 1 (NID(0,1)), then let’s define Where  follows the chi-square distribution with k degrees of freedom.

   distribution k = 1 k = 5 k = 10 k = 15

Student’s t Distribution In practice we don’t know the theoretical parameter  This means we can’t really use and refer to the result of the table of standard normal distribution Assume that experimental standard deviation s can be used as an estimate of 

Student’s t Distribution Define a new variable It turns out that t has a known distribution. It was deduced by Gosset in 1908

Student’s t Distribution k=1 k=10 k=100 Probability points are given in tables. The form depends on the degree of uncertainty in s 2, measured by the number of degrees of freedom, k.

Inferences about differences in means Statistical hypothesis: Statement about the parameters of a probability distribution. Let’s go back to the example we started with, i.e., comparison of two imaging systems. We may think that the performance measurement of the two systems are equal.

Hypothesis testing First statement is the Null hypothesis, second statement is the Alternative hypothesis. In this case it is a two-sided alternative hypothesis. How to test hypothesis? Take a random sample, compute an appropriate test statistic and reject, or fail to reject the null hypothesis H 0. We need to specify a set of values for the test statistic that leads to rejection of H 0. This is the critical region.

Hypothesis testing Two errors can be made: Type I error: Reject null hypothesis when it is true Type II error: Null hypothesis is not rejected when it is not true In terms of probabilities:

Hypothesis testing We need to specify a value of the probability of type I error . This is known as significance level of the test. The test statistic for comparing the two systems is: Where

Hypothesis testing To determine whether to reject H 0, we would compare t 0 to the t distribution with k A +k B -2 degrees of freedom. If we reject H 0 and conclude that means are different. We have: System ASystem B

Hypothesis testing We have k A + k B – 2 = 18 Choose  = 0.05 We would reject H 0 if

Hypothesis testing

Since t 0 = < -t 0.025,18 = then we reject H 0 and conclude that the means are different. Hypothesis testing doesn’t always tell the whole story. It’s better to provide an interval within which the value of the parameter is expected to lie. Confidence interval. In other words, it’s better to find a confidence interval on the difference  A -  B

Confidence interval Using data from previous example: So the 95 percent confidence interval estimate on the difference in means extends from to Note that since  A –  B = 0 is not included in this interval, the data do not support the hypothesis that  A =  B at the 5% level of significance.