Chapter 4. Probability and Statistics. Probability and Statistics

Transcription

1 Chapter 4 Probability and Statistics Figliola and Beasley, (999) Probability and Statistics Engineering measurements taken repeatedly under seemingly ideal conditions will normally show variability. Measurement system Resolution Repeatability Measurement procedure and technique Repeatability Probability and Statistics Measured variable Temporal variation Spatial variation We want:. A single representative value that best characterizes the average of the data set.. A measure of the variation in a measured data set.

2 True Value True value is represented by X' = X ± ux X = most probable estimate of X u x = confidence interval at probability P% which is based on estimates of precision and bias error. Probability Density Function Central Tendency says that there is one central value about which all other values tend to be scattered. Probability Density the frequency with which the measured variable takes on a value within a given interval. The region where observations tend to gather is around the central value. Plotting Histograms The abscissa is divided in K small intervals between the minimum and maximum values. The abscissa will be divided between the maximum and minimum measured values of x into K small intervals. Let the number of times, nj, that a measured value assumes a value within an interval defined by x-δx x < x+δx be plotted on the ordinate. For small, K should be chosen so that n j 5 for at least one interval.

3 Plotting Histograms For >40 ; K=.87(-) The histogram displays the tendency and density. If the y-axis is normalized by dividing n j /, a frequency distribution results. K=.87(-) =7 Cell range: min = 0.68 ; max =.34 δ x = ( )/7 = 0.0 x-δx x < x+δx Cell = *δx = 0.0 ote: total frequency equals 00% ote: total occurrence equals Probability Density Function P(x) comes from the frequency distribution Px ( ) = lim ( nj) / ( ( δx)), δ x 0 - Defines the probability that a measured variable might assume a particular value on any given observation, and also provides the central tendency - The shape depends on the variable in consideration and its natural circumstances/processes. - Plot histograms compare to common distribution and then fit the parameter. - Unifit is a good PC-based distribution fitting software. 3

4 Infinite Statistics The most common distribution is the normal or Gaussian distribution. This predicts that the distribution will be evenly distributed around the central tendency. ( bell curve ) The operating conditions that are held fixed are length, temperature, pressure, and velocity. Gaussian Distribution Pdf for Gaussian: / px ( ) = / ( v( π) exp [( / )(( x x') / σ )] x = true mean of x ; σ = true variance of x To find/predict the probability that a future measurement falls within some interval. Probability P(x) is the area under the curve between X ±δx on p(x) 4

5 Gaussian Distribution Simplification of integration: Px ( ' x x x x x ' x ) ' + δ δ + δ = pxdx ( ) x' δx P(x)=probability p(x)=pdf if z =(x -x )/σ and β=(x-x )/σ dx=σdβ and P ( z z ) /( ) z / z e β / β = π d β β=standardized normal variant ; z =interval on p(x) Table 4.3 Solutions given for this in table 4.3 For z=, 68.7% of observations within standard deviation of x For z=, 95.45% For z=3, 99.73% The values of z in table 4.3 can be used to predict probability of a unique value occurring in an infinite data set. The Z Table 5

6 Finite Statistics When <<, we do not have true representation of the population. We have finite statistics which describe the sample and estimate the population. Sample mean (probable estimate of true mean) x = / xi i= Sample variance (measurement of precision) x i= s = /( ) ( xi x) Finite Statistics con t: Deviation of x i : x x Sample standard deviation: sx = s i Degree of freedom: the number of samples less the central tendency measurement (-) x t-estimate For a finite data set, we use the t-estimate instead of z, which was used in the infinite. One can state that ± tvps xi = x ± tv, psx (P%), x represents the precision interval P% = probability v = degrees of freedom x = sample mean Table 4.4 gives t distribution 6

7 Standard Deviation of Means If we measure a variable times under fixed conditions, and replicate this M times, we will end up with slightly different means for each replication. It can be shown that regardless of the form of the pdf of the individualized replication, the mean values themselves will be normally distributed. Variation depends on. The sample variance s x. The sample size sx = sx/ / Difference increases as s x increases and decreases with / The expected variance of means can be estimated by a single finite data set Pooled Statistics Samples that are grouped in a manner so as to determine a common set of statistics are called pooled. If we have M replicates of variable x, with repeated measurements producing data set x i,j ; i = to ; j = to M 7

8 Pooled mean of x: x = / M xij j= i= Pooled standard deviation of x: (with v=m(-) degrees of freedom) Pooled standard deviation of means: M sx = [ /( M( )) ( xij xj) ] = M /M sxj j= M j= i= sx = sx /( M) / Chi Squared Distribution Estimates the precision by which s x predicts σ. s x = sample variance ; σ = population variance If we plotted s x for many sets having samples each, we would generate the pdf for P(χ ) (chi-squared) For a normal distribution chi-squared χ = ν(s / σ ) x ν =- 8

9 Precision Interval in Sample Variance The precision interval for the sample variance can be formulated by the probability statement: P( χ χ α χ α ) = α / / with a probability of P(χ)= - α ; α = level of significance Combining: [ x / α / x / α / ] For 95% precision interval by which s x estimates σ : νs x χ σ νs /. 05 < < x / χ. 975 P νs χ σ νs χ = α Precision Interval in Sample Variance The χ distribution estimates the discrepancy expected as a result of random chance. Values for χ α are tabulated in Table 4.5 as a function of the degrees of freedom. The P(χ ) value equals the area under p(χ ) as measured from the left, and the α value is the area as measured from the right. The total area under p(χ ) is equal to unity. 9

10 Values for χ α Goodness of Fit Test We can use a chi-squared test to determine how good of a fit our selected pdf represents the actual distribution of data. The χ test gives us the measure of error between the variation in the data set and variation predicted by the assumed pdf. Construct histogram of data and histogram from predicted pdf, where n j is actual and n j is predicted number of occurrences per cell. Then: k χ = ( nj n' j) / nj j= For given d.o.f. the better fit gives lower χ χ α Goodness of Fit Test The table, given in the previous slide, can be interpreted as a measure of the discrepancy expected as a result of random chance. For example, a value for α of 0.95 implies that 95% of the discrepancy between the histogram and the assumed distribution is due to random variation only. 0

11 Goodness of Fit Test With P(χ ) = - α, this leaves only 5% of the discrepancy caused by a systematic tendency, such as different distribution. In general, a P(χ ) < 0.05 confers a very strong measure of a good fit to the assumed distribution, an unequivocal result. Regression Analysis Regression analysis is used to establish a functional relationship between the independent and dependent variables. It is assumed that the relationship can be described by either a polynomial or a fourier series. It is also assumed that the variation in the dependent variable is normally distributed about a fixed value of the independent variable. Regression Analysis Such behavior is illustrated in the previous slide by considering the dependent variable y i,j consisting of measurements, i =,,,, of y at each of n values of independent variable, x j, j =,,, n. This behavior is most common during calibrations, where the input to the measuring system, x, is held nominally fixed while the measurement of y occurs.

12 Regression Analysis Repeated measurements of y will yield a normal distribution with variance S y( xj), about some mean value, y( xj). mth Order Polynomials (least squares regression) y i =a 0 +a x+a x + +a m x m gives the estimate of y i at independent variable value χ I -x. The standard deviation of the deviation of each point is given by: syx = (( ( yi yci) / ν) / i= d.o.f: ν= - (m+) ; = observations, m = order

13 Linear Polynomials For linear polynomials, a correlation coefficient, r, can be found by: r = syx ( ) s y Linear Polynomials The correlation coefficient represents a quantitative measure of the linear association between x and y. It is bounded by ±, which represents prefect correlation; the sign indicates that y increases or decreases with x. For ±0.9 < r < ±, a linear regression can be considered as a reliable relation between y and x. Variance Sy = /( ) ( yi y) i= For ±0.4 < r ± is considered a good fit between x and y. r is often used, but these are not true measures of the precision of y i. S yx should be used for precision. 3

14 Data Outlier Detection It is not uncommon for a data set to contain one or more spurious data points that appear to be out of range of expected trends. Data that lie outside the probability of normal variation can bias the sample mean and variance. Statistical methods and histograms are helpful to identify outliers. If removed, data needs to be re-evaluated for stats. Method Method (Three Sigma Test) Use area under Pdf to find basis for outlier detection Calculate data set statistics Discard all points outside x± tν99. 8Sx Good for n > 0, easy to program Method Method (modified three sigma test for large data sets) Estimate the sample mean and standard deviation Compute modified z variable for each data point: z0 = ( xi x)/ sx Probability that x lies outside one-sided range from 0 to z 0 is 0.5-P(z 0 ), where P(z 0 ) is found from the one-sided z chart. For data points, if [0.5-P(z 0 )] 0. data is outlier. Example 4. 4

15 umber of Measurements Required How many observations are required to estimate the true mean x, with acceptable precision? Earlier we saw that precision interval is a quantified measure of precision error in the estimate of true mean. / x' = x± τ v, psx sx = sx/ Based on sample mean and its precision interval: / x' = x± tv, 95sx/ umber of Measurements Required: 95% confidence interval: CI =± tv, 95sx/ / (95%) To evaluate, we must have an estimate of S x, based on previous test data, experience, or manufacturer specs. sx = standard deviation of means sx = standard deviation of samples umber of Measurements Required: Precision interval is two-sided about the mean. / / x tv, 95sx/ x' < x+ tv, 95sx/ One-sided precision value: / d = CI/ = tv, 95sx/ Then required measurements: ( tv,95sx/ d) P= (95%) -approximation due to S x 5

16 umber of Measurements Required: The approximation serves as a reminder that this expression is based in an assumed value for S x. The accuracy of equation 4.43 will depend on how well the assumed value for S x approximates σ. The obvious deficiency in this method is that an estimate for the sample variance is needed. One way around this is to make a preliminary small number of measurements,, to obtain an estimate of the sample variance, S, to be expected. umber of Measurements Required: The total number of measurements, T, will be estimated by: t T (. 95 d S ) (95%) This establishes that T additional measurements will be required. 6