Modeling Obesity and S&P500 Using Normal Inverse Gaussian

Transcription

1 Modeling Obesity and S&P500 Using Normal Inverse Gaussian Presented by Keith Resendes and Jorge Fernandes University of Massachusetts, Dartmouth August 16, 2012

2 Diabetes and Obesity Data Data obtained from CDC Data is in % diabetes and % obesity (BMI >30) by U.S. county (3,143 counties) Data is collected for % diabetic data is approximately normally distributed

3 S&P500 Stock Return Standard&Poor s 500 stock Return 2009 Stock market index based on the common stock price of 500 top publicly traded American company Log of the Returns

4 Diabetes, Obesity, and S&P500 Data Normal distributions takes the form of 1 mean µ and standard deviation σ σ 2π e 1 2 (x µ) 2 σ with

5 Diabetes Data Figure: Diabetes 2004 Q-Q Plot

6 % obese data is not normally distributed Figure: Obesity 2004 Q-Q Plot and Histogram with Normal Distribution

7 % obesity data has a high kurtosis; this is what shows as being peaky Figure: Obesity 2004 Histogram High kurtosis!

8 SP500 Figure: Normal Distribution

9 Kurtosis What is a kurtosis? it is a measure of the peakedness of the probability distribution of a real-valued random variable. µ 4 σ 4 µ 4 = average (x µ) 4 (known as the fourth moment about the mean) σ = standard deviation A normal distribution has a kurtosis of 3

10 Obesity So we took an informed guess: % obesity has an normal inverse Gaussian High kurtosis and relative symmetry suggest this may be true

11 Normal Inverse Gaussian What is NIG? it is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. it consists of five parameters that need to be solved

12 Normal Inverse Gaussian f NIG (x; α, β, µ, δ) = α π exp(δ α 2 β 2 βµ) k 1(αδ 1+( x µ δ )2 ) 1+( x µ δ )2 exp(βx)

13 Normal Inverse Gaussian Figure: Ole Barndoff-Nielsen Ref: Barndorff-Nielsen invented the Normal Inverse Gaussian his uses with NIG were for data on turbulence data in wind tunnels and in log returns in finance.

14 Normal Inverse Gaussian f NIG (x; α, β, µ, δ) = α π exp(δ α 2 β 2 βµ) k 1(αδ µ = location α = tail heaviness β = asymmetry parameter δ = scale parameter γ = α 2 β 2 1+( x µ δ )2 ) 1+( x µ δ )2 exp(βx)

15 Normal Inverse Gaussian Moments mean = µ + δβ γ variance = δα2 γ 3 skewness = kurtosis = 3β α δγ 3+12β 2 α 2 δγ

16 How? Moment Estimation Maximum Likelihood x, s 2 = sample mean and variance γ 1 = µ 3 µ γ 2 = µ 4 µ 2 2

17 Normal Inverse Gaussian How do you fit a theoretical NIG distribution pdf to the % obesity data? we use the moment estimates of the parameters Not good enough: Why?

18 Normal Inverse Gaussian Figure: Histogram vs. NIG moment estimates Bad fit

19 Normal Inverse Gaussian Figure: Histogram vs. NIG moment estimates Bad fit Any better ideas?

20 Likelihood Function Yes! Maximize the likelihood function L = NIG(x i, α, β, µ, δ) the probability of getting the data we have, if the data, county by county, is independent is likely a decent approximation. Method of estimating the parameters of a statistical model Estimation provides estimates for the model s parameters

21 Normal Inverse Gaussian maximizing L is the same as maximizing log(l), which is easier because it s a sum; not a product. Likelihood functioin is a product harder to take the derivative Log of Likelihood function is a sum easier to take the derivative L=-nln(π) + nln(α) + n(δγ βµ) 1 2 n K 1 (δαφ(x i ) 1 2 ) i=1 n n φ(x i ) + β x i + i=1 i=1

22 Normal Inverse Gaussian Use random optimization to maximize log-likelihood (as suggested by Sidafa Conde) Good fit? Figure: Histogram vs. NIG moment estimates vs. Random Optimization red curve is moment estimates blue curve is random opt.

23 Normal Inverse Gaussian Using random optimization values P-P Plot Figure: P-P Plot 2004

24 χ 2 Test n (expected observed) 2 i=1 expected is the value small? What is small? degrees of freedom needed

25 χ 2 Test Figure: χ 2 Test Plot χ 2 test didn t work. We don t know why!

26 χ 2 Test To determine df, simply subtract: (# of bins - # parameters - 1) 9-4-1=4 df

27 Effect Size Effect Size What is an Effect Size? an effect size is a measure of the strength of the relationship between two variables in a statistical population Example: Reduced class size in Florida classroom

28 Degree of Overlap What is the degree of overlap?

29 Degree of Overlap What is the degree of overlap? Figure: 2004 and 2009 NIG pdf s Red curve is 2004 Blue curve is 2009

30 Degree of Overlap Figure: 2004 and 2009 NIG pdf s % overlap = area of min. of curves = b a 0.4 min(nig 2004 (x), NIG 2009 (x))dx

31 Degree of Overlap So, there is a 60% non-overlap between

32 Degree of Overlap Figure: NIG pdf s PDF plots are successive and suggests people are getting more obese.

33 Degree of Overlap Figure: Histogram and Smoothed Histogram

34 Brief Summary Fitting the Normal Inverse Gaussian(NIG) Distribution to our Data

35 Brief Summary Normal-Inverse Gaussian Four parameter distribution Estimation of its parameters is not easy due to complicated quantities involved Maximizing the log of the likelihood function Random optmization FindMaximum command

36 Results Figure: After using the random optimization as initial value for Mathematica built in FindMaximum.

37 PP-Plot Figure: PP-Plot to test the goodness of fit

38 Questions?