Model Fitting Jean Yves Le Boudec 0
Contents 1. What is model fitting? 2. Linear Regression 3. Linear regression with norm minimization 4. Choosing a distribution 5. Heavy Tail 1
Virus Infection Data We would like to capture the growth of infected hosts (explanatory model) An exponential model seems appropriate How can we fit the model, in particular, what is the value of? 2
Least Square Fit of Virus Infection Data = 0.5173 Mean doubling time 1.34 hours Prediction at +6 hours: 100 000 hosts Least square fit 3
Least Square Fit of Virus Infection Data In Log Scale = 0.39 Mean doubling time 1.77 hours Prediction at +6 hours: 39 000 hosts Least square fit 4
Compare the Two LS fit in natural scale LS fit in log scale 5
Which Fitting Method should I use? Which optimization criterion should I use? The answer is in a statistical model. Model not only the interesting part, but also the noise For example = 0.5173 6
= 0.39 How can I tell which is correct? 7
Look at Residuals = validate model 8
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Least Square Fit = Gaussian iid Noise Assume model (homoscedasticity) The theorem says: minimize least squares = compute MLE for this model This is how we computed the estimates for the virus example 10
Least Square and Projection Data point Predicted response Manifold Where the data point would lie if there would be no noise Estimated parameter Skrivañ war an daol petra zo: data point, predicted response and estimated parameter for virus example 11
Confidence Intervals 12
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Robustness to «Outliers» 14
A Simple Example Least Square Model: noise L1 Norm Minimization Model : noise What is m? What is m? Confidence interval? Confidence interval? 15
Mean Versus Median 16
Also called «ANOVA» (Analysis of Variance») 2. Linear Regression = least square + linear dependence on parameter A special case where computations are easy 17
Example 4.3 What is the parameter? Is it a linear model? How many degrees of freedom? What do we assume on i? What is the matrix X? 18
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Does this model have full rank? 20
Some Terminology x i are called explanatory variable Assumed fixed and known y i are called response variables They are «the data» Assumed to be one sample output of the model 21
Least Square and Projection Data point Predicted response Manifold Where the data point would lie if there would be no noise Estimated parameter 22
Solution of the Linear Regression Model 23
Least Square and Projection The theorem gives H and K residuals data Predicted response Manifold Where the data point would lie if there would be no noise Estimated parameter 24
The Theorem Gives with Confidence Interval 25
SSR Confidence Intervals use the quantity s s 2 is called «Sum of Squared Residuals» residuals data Predicted response 26
Validate the Assumptions with Residuals 27
Residuals are given by the theorem Residuals residuals data Predicted response 28
Standardized Residuals The residuals e i are an estimate of the noise terms i They are not (exactly) normal iid The variance of e i is???? A: 1 H i,i Standardized residuals are not exactly normal iid either but their variance is 1 29
Which of these two models could be a linear regression model? A: both Linear regression does not mean that y i is a linear function of x i Achtung: There isa hidden assumption Noise is iid gaussian > homoscedasticity 30
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3. Linear Regression with L1 norm = L1 norm minimization + linear dependency on parameter More robust Less traditional minimization 32
This is convex programming 33
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Confidence Intervals No closed form Compare to median! Boostrap: How? 35
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4. Choosing a Distribution Know a catalog of distributions, guess a fit Shape Kurtosis, Skewness Power laws Hazard Rate Fit Verify the fit visually or with a test (see later) 37
Distribution Shape Distributions have a shape By definition: the shape is what remains the same when we Shift Rescale Example: normal distribution: what is the shape parameter? Example: exponential distribution: what is the shape parameter? 38
Standard Distributions In a given catalog of distributions, we give only the distributions with different shapes. For each shape, we pick one particular distribution, which we call standard. Standard normal: N(0,1) Standard exponential: Exp(1) Standard Uniform: U(0,1) 39
Log Normal Distribution 40
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Skewness and Curtosis 42
Power Laws and Pareto Distribution 43
Complementary Distribution Functions Log log Scales Lognormal Pareto Normal 44
Zipf s Law 45
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Hazard Rate Interpretation: probability that a flow dies in next dt seconds given still alive Used to classify distribs Aging Memoriless Fat tail Ex: normal? Exponential? Pareto? Log Normal? 47
The Weibull Distribution Standard Weibull CDF: Aging for c > 1 Memoriless for c = 1 Fat tailed for c <1 48
Fitting A Distribution Assume iid Use maximum likelihood Ex: assume gaussian; what are parameters? Frequent issues Censoring Combinations 49
Censored Data We want to fit a log normal distrib, but we have only data samples with values less than some max Idea: use the model Lognormal is fat tailed so we cannot ignore the tail and estimate F0 and a (truncation threshold) 50
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We want to fit a log normal distrib to the body and pareto to the tail Combinations Model: MLE satisfies 52
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5. Heavy Tails Recall what fat tail is Heavier than fat: 54
Heavy Tail means Central Limit does not hold Central limit theorem: a sum of n independent random variables with finite second moment tends to have a normal distribution, when n is large explains why we can often use normal assumption But it does not always hold. It does not hold if random variables have infinite second moment. 55
Central Limit Theorem for Heavy Tails normal qqplot histogram complementary d.f. log-log One Sample of 10000 points Pareto p = 1 56
p=1 1 sample, 10000 points average of 1000 samples p=1.5 p=2 p=2.5 p=3 57
Convergence for heavy tailed distributions 58
Importance of Second Moment 59
RWP with Heavy Tail Stationary? 60
Evidence of Heavy Tail 61
Testing Heavy Tail Assume you have very large data set Else no statement can be made One can look at empirical cdf in log scale 62
Taqqu s method A better method (numerically safer is as follows). Aggregate data multiple times 63
We should have and If log ( m 2 / m 1 ) then measure p = / p est = average of all p s 64
Example log ( 2) log ( 2) / p 65
Evidence of Heavy Tail p = 1.08 ± 0.1 66
A Load Generator: Surge Designed to create load for a web server Used in next lab Sophisticated load model It is an example of a benchmark, there are many others see lecture 67
User Equivalent Model Idea: find a stochastice model that represents user well User modelled as sequence of downloads, followed by think time Tool can implement several user equivalents Used to generate real work over TCP connections 68
Characterization of UE Weibull dsitributions 69
Successive file requests are not independent Q: What would be the distribution if they were independent? A: geometric 70
Fitting the distributions Done by Surge authors with aest tool + ad hoc (least quare fit of histogram) What other method could one use? A: maximum likelihood with numerical optimization issue is non iid ness 71