Curso de Geoestadistica en Ingles

MATH RECALL / NOTATIONPROGRAMS / FACT SHEETS

Content

Double Bend Sign- For your information only- Not to be remembered!

Math Recall- Sum sign

- Integral Area- Derivative Slope- Minima / Maxima

Notation- Statistics- Geostatistics

Fact Sheets

1

MATH RECALL - Conventions

Computation order- Priority 1: power- Priority 2: multiplication, division- Priority 3: addition, subtraction- Left to right- Parenthesis have priority, inside before outside- Examples:

- 1 + 2 3 = 1 + (2 3) = 1 + 6 = 7

- 1 4/2 = 4/2 = 2

- (1 + 2) 3 = 3 3 = 9

- 1 + (2 3) = 1 + 6 = 7

- ((1 + 3) + (2 3)) = (4 + 6) = 10

Summation sign:

4i=1

ai = a1 + a2 + a3 + a4

Product sign: 4i=1

ai = a1 a2 a3 a4

Factorial: n!4! = 4 3 2 1

2

MATH RECALL - Conventions

More about

ijij = 1111 + 1212 + 1313 + 2121 + 2222 + 2323

i

i = 1 + 2 i=1

2 1

2

ij = 11 + 12 + 13 + 21 + 22 + 23 i=1 j=1

2 3

1,1 1,2 1,3

2,1 2,2 2,3i

j = 1 + 2 + 3 j=1

31 2 3

j

i=1 j=1

2 3

j

i j

f.150

3

MATH RECALL: Exercise 1

Let 3 Au values: z1 = 1, z2 = 2, z3 = 3 g/t

Compute:

z1 + z2 + z3

(z1 + z2) z3

z1 + z2 z33

i=1 zi

3i=1 2 zi

12

3i=1(zi 2)

2

4

CALCULUS - Definitions

Variable: x- An expression, the value of which is unknown orsubject to change.

Domain of definition of x- Set of values that can be taken by x.

- [a, b], i.e. a x b, ]a, b], i.e. a < x b

- [a, b[, i.e. a x < b, ]a, b[, i.e. a < x < bwhere a & b are two constants.

Function- An ordered set of pairs (x, y) such that for each x,there is one and only one y. Usual notation is:

y = f(x)

- Defined by:

- domain of definition of x

- the condition that must be satisfied by x & y

5

CALCULUS - Functions

Parabola: y = x2, 2 x 2

3 2 1 0 1 2 3 x

y = f(x)4

2

f. 66

3

1

Normal Distribution: f(x) = 12pi2

exp[ (x)

2

22

],

x +

f(x)

: Mean

2: Variance

x

f. 1a

6

CALCULUS - Integral

Graphical representation

y = f(x)

a b

y

x

dx 0

f. 68aa

y = f(x)

a b

y

x

dx >>> 0

dxx1

f(x1+dx) + f(x1) dx 2

y = f(x)

a b

y

x

dx >> 0

dxx x1 2

2 f(xi+dx) + f(xi) dxi=1 2

y = f(x)

a b

y

x

dx > 0

dx

x x1 2

3 f(xi+dx) + f(xi) dxi=1 2

b

f(x)dxa

x3

7

CALCULUS - Integral

DefinitionThe integral of the function f(x) on the interval [a,b] isthe limit of the sum

ni=1

f(xi + dx) + f(xi)

2dx

when limdx 0. Note that n + in [a, b].

Notation

ba

f(x)dx = limdx0,n

ni=1

f(xi + dx) + f(xi)

2dx


y = f(x)

a b

y

x

f. 68

b

f(x)dxa

8

CALCULUS - Derivative

DefinitionThe derivative y = f (x) is, when it exists, the limitof the quotient of the increment of the variables x & y,when the increment of x tends towards 0.

f (x) = limdx0

dy

dx= lim

dx0

f(x + dx) f(x)

dx

The function f(x) is derivable at x = x0 if f(x0) exists.


y = f(x)

x0

f(x0)

y = f(x)

x

f(x0+dx)

x0+dx

dx > 0

y = f(x)

x0

f(x0)

y = f(x)

x

f(x0+dx)

x0+dx

dx >> 0

f. 67

f (x) = tg(), i.e. slope of tangent at x0.

9

MINIMA / MAXIMA: 1 variable

y = f(x)

x

f(x)

x2x1

f(x1)=0

f(x2)=0

3 2 1 0 1 2 3 x

y = f(x)

4

2

f.123

3

1

f(0)=0

f(x+dx) f(x) dxf(x) = lim

dx > 0

(x+dx)2 x2

dx= lim

dx > 0

x2 + 2xdx + (dx)2 x2

dx= lim

dx > 0

= lim (2x + dx)dx > 0

= 2x + 0 = 0

f(x) = 2x = 0 iff x = 0

Example: y = x2

Slope of tangent = 0 = Minima or Maxima (local orglobal).

10

MINIMA / MAXIMA: Several variables

f(x,b)f(a,y)

dzdy

fy(a,y) = 0

y = b

dzdx

fx(x,b) = 0

x = a

x

y

z=f(x,y)

ab

0

(a,b,0)

y

x

z

f(x,y)=x2 + y2

x=0

y=0

f(x+dx, y) f(x, y)

dxfx(x, y) = lim

dx>0

(x+dx)2+y2 (x2+y2)

dx= lim

dx>0

x2 + 2xdx + dx2 x2

dx = lim

dx>0

= lim (2x + dx)dx>0

fx(x,y) = 2x = 0 iff x = 0

fy(x,y) = 2y = 0 iff y = 0

Example:

f.124

11

MINIMA / MAXIMA: Under Constraint

y

x

z

f(x,y)=x2 + y2

Free Minimum

Constrained Minimum

Min f(x, y) = x2 + y2

Min f(x, y) = x2 + y2

s.t.: g(x, y) = x 1 = 0

Min h(x, y, ) = f(x, y) g(x, y)

= x2 + y2 + x

dh(x, y, )

dx = 2x = 0

dh(x, y, )

dy = 2y = 0

dh(x, y, )

d = x 1 = 0

(x, y, ) = (1, 1, 2)

{

f(x,y) = x2 + y2 such that:

g(x, y) = x 1 = 0

df(x, y)

dx = 2x = 0

df(x, y)

dy = 2y = 0

(x, y) = (0, 0) {{

f.125

12

CALCULUS: Summary

Variable: x

Function: y = f(x)

Integral:

ba

f(x)dx = limdx0,n

ni=1

f(xi + dx) + f(xi)

2dx

Derivative:

f (x) = limdx0

dy

dx= lim

dx0

f(x + dx) f(x)

dx

f(x0) = tg

y = f(x)

a bx0

y = f(x)

x

a

bf(x)dx

x1

f(x1) = 0

f.68b

Minimum/Maximum if derivative is zero.13

FACT SHEET

Integral = area Derivative = slope Integrals and derivatives are linear operators. Minimum/Maximum when derivative is zero.

14

15 16

NOTATION (1)

Summation sign:

4i=1

ai = a1 + a2 + a3 + a4

Random Variable (RV)- X, Y : random variables- Z(x): random variable Z at location x- x, y, z(x): realizations of X, Y , Z(x)- P , Prob: probability

Random Function- Z(x): random function- Z(x), x D: set of RV Z(x), x within thedeposit D (or geological domain)

Descriptive Statistics:- Roman letters- m, mX , x: mean, average- s2, s2X : variance- cvX : coefficient of variation- xi: ith quantile; x50: median- Cov(X, Y ): covariance- Cor(X, Y ): correlation coefficient

17

NOTATION (2)

Model Parameters:- Greek letters- , X , E(X): mean, average, expectation- 2, 2X , V ar(X): variance- X,Y : covariance- X,Y : correlation coefficient

Distributions- pdf : probability density function (probabilitydistribution)

- cdf : cumulative density function- fX , fXY : univariate, bivariate pdfs- FX , FXY : univariate, bivariate cdfs- X N(X ,

2X): normal distribution, mean X ,

variance 2X

Variogram- (h): variogram for distance h- (V, V ): average variogram value within block V- (V, i) = (i, V ): average variogram valuebetween sample i and block V

Block variance:- V ar(ZV ), D

2(V |), (V = Volume)- V ar(ZB), D

2(B|), (B = Block)

18

NOTATION (3)

Miscellaneous:- Greek letters

- , , , , , , - AFC : affine correction- ILC : indirect lognormal correction- IK : indicator kriging- OK : ordinary kriging- QQ : quantile/quantile plot- RV : random variable- RF : random function- SMU : selective mining unit

- AVE, Ave: average- VAR, Var : variance- RLVAR, RlVar : relative variance- STDV, StDv : standard deviation- RLSTDV, RlStDv : relative standard deviation- CV : coefficient of variation

- RLP : relative pairwise (variogram)

19 20

FACT SHEET (1)

Recall Calculus

Integral = area Derivative = slope

Minimum/Maximum when derivative is zero.

Univariate Statistics (Description)

mX = (1/N)

xi

s2X = (1/N)(xi mX)

2 = (1/N)

x2i m2X

m(aX+bY+c) = a mX + b mY + c

s2aX+b = a2s2X

Univariate Statistics (Model)

E(X) = X

V ar(X) = 2X = E(X X)2 = E(X2) 2X

21

FACT SHEET (2)


E[aX + bY + c] = aE[X] + bE[Y ] + c

V ar[aX + b] = a2V ar[X]

Bivariate Statistics (Description)

sX,Y =1N

[xi mX ][yi mY ] =

1N

[xiyi]mXmY

V ar(aX + bY + c) = a2V ar(X) + b2V ar(Y )+2abCov(X, Y )

Bivariate Statistics (Model)

Cov[X, Y ] = X,Y = E[(X X)(Y Y )

]= E(XY ) XY

22

FACT SHEET (3)

Variogram, Covariance Function

(h) = (1/2Nh)[z(xi) z(xi + h)]

2

(h) = (1/2)E[(Z(x) Z(x + h)

)2]

C(h) = E[(Z(x)Z(x+ h)] Z(x)Z(x+h)

(h) = C(0) C(h)

Block Variance

V ar[ZV (x)

]= V ar

[Z(x)

] (V, V )

Estimation Variance

V ar(Er) = 2N

i=1 i(V, i)

N

i=1

Nj=1 ijij (V, V )

23

FACT SHEET (4)

Ordinary Kriging

2OK =N

i=1 i(V, i) + (V, V )

Indicator Kriging

No new formulas

Cross-Validation/Reconciliation

No new formulas

24

UNIVARIATE STATISTICS

Content

Concepts, notation:- Random variable, random function- Probability- Stationarity

Univariate description:- Maps: values, contours,...- Graphs: histogram, boxplot,...- Statistics: mean, variance,...

Univariate model:- Probability distribution- Parameters: mean (expectation), variance,...- Distributions: normal, lognormal,...

Applications:- EDA envelope- Geology model- Declustering- Compositing

1

BASIC STATISTICS: Random Variable

Cast of a die:

P (X = i) = 1/6, i = 1, . . . , 6,

- X is the random variable (RV) value shown onthe face after casting the die.

Random variable (RV):- Value cannot be predicted, but- Possible results are outcomes or realizations- Outcomes occur according to probability law

Many geological attributes, CU grade, lithotype, etc.,have random characteristics:

- Are not known, unless sampled- Possible values are generally known- Some values are more frequent than others

Attributes can be interpreted as realizations RVs:- Discrete: lithotype, ...- Continuous: Au, Cu, density ...

Will try to infer relevant characteristics of these RVs.

2

BASIC STATISTICS: Probability

Cast of a die:- X: RV result from casting a die- The observed relative frequency of getting 2 is:

p(2) =Number of 2s obtained

Number of casts

- p(2) tends towards the probability of having 2when the No. of casts is large (infinite):

p(2) = Prob(X = 2), if No. of casts is infinite

Probability of an event: its relative frequency when theNo. of trials is large.

Properties:- 0 probability 1- Sum of probabilities of all disjoint outcomes is 1

Cast of a die, X:

P (X = i) = 1/6, i = 1, . . . , 6

P (X = 1) + + P (X = 6) = 1

Z: CU value at a given location (%):

P (Z = 4.4673829) = 0,

0 P(Z [0.1, 1.5]

) 1

P(Z 100) = 1

3 4

BASIC STATISTICS: Notation

z(x) is grade at location x and is interpreted as arealization of a random variable (RV) Z(x).

Attention!- x is a location here- Sometimes use X for a random variable- X(x) is RV X located in x ...

UPPER CASE: Random Variables (RV) Z(x)lower case: Realizations z(x)

Random Variables- Point grades at locations:

x : Z(x) x0: Z(x0), or Z0 xi: Z(xi), or Zi

- Block/Volume grades at location x:

ZB(x), or ZV (x)

Realizations- z(x), z(x0), z0, z(xi), zi, zB(x), zV (x)

Important:- Do not mix different supports:

- e.g. sample & block grades- They have different statistical characteristics.

- Do not mix apples and oranges.

5

BASIC STATISTICS: Stationarity

Roughly means that the characteristics of the RV Z(x)do not depend on the location x.

Concept intuitively used by geologists when:- Defining geological domains- Computing statistics

Important:- Split the deposit within geological domains thathave different mineralization characteristics.

- Do not mix apples and oranges.

Geology first!

Mineralization within a geology domain is usually con-sidered +/- stationary, i.e. with similar characteristics.

6

7 8

UNIVARIATE DESCRIPTION: Maps

List of values and coordinates

East 134.2 628.2 330.5 333.8 ...

North 253.2 431.7 320.3 382.8 ...

Au (g/t) 8.5 3.2 6.2 8.1 ...

Maps

0.8

1.5

1.2

8.5

7.5

13

6.2

8.0

3.23.9

8.1

0.8

1.5

1.2

8.5

7.5

13

6.2

8.0

3.23.9

8.18

6

24

0.8

1.5

1.2

8.5

7.513

6.2

8.0

3.23.9

8.1

Locations & Values Symbols

Contours Colour Scale

f. 175

9

UNIVARIATE DESCRIPTION: Histogram

A graph showing the (relative) frequencies of occurrenceof values within classes of equal amplitudes

0

10

20

30

Freq

(%)

Au (g/t)0 4 8 12 16

Histogram of Au grades

f. 3a

P(2. < Au < 4.)

Remarks:- Class interval: 2.0 g/t- Class interval includes the upper bound, but notthe lower bound

- 20% of Au values are in the ]2.0, 4.0]% class- Total area defined by classes is 100% or 1.- Different scales are sometimes needed to observelow and high values.

10

UNIVARIATE DESCRIPTION: Examples

HISTOGRAM - Logarithmic Scale

Fre

qu

en

cy

BH Au (g/t)

.1 1. 10. 100.

0.000

0.050

0.100

0.150

0.200 Number of Data 99688Number trimmed 2295

mean 2.170std. dev 4.949

coef. of var 2.281

maximum 330.000minimum 0.100

f. 62

HISTOGRAM - Arithmetic Scale

Fre

qu

en

cy

BH Au (g/t)

0.0 4.0 8.0 12.0 16.0 20.0

0.000

0.100

0.200

0.300 Number of Data 99688Number trimmed 2295

mean 2.170std. dev 4.949

coef. of var 2.281


11

UNIV. DESCRIPTION: Cumulative Histogram

A graph showing the (relative) cumulative frequenciesof occurrence of values below a cutoff

0

25

50

75

100

Cum. freq.

(%)Cumulative histogram

of AU grades

f. 4a

22%

Au (g/t)0 4 8 12 16

Histogram

Remarks- 22% of AU values are 4 g/t, or- Prop(AU 4) = 0.22,- Cumulative frequency of last class is 100% or 1

12

UNIV. DESC.: Location vs. Dispersion Statistics

0

10

20

30

Freq(%)

Location

Au (g/t)

f. 3f

Dispersion

0 4 8 12 16

Location Statistics

- Value around which grades are distributed

- Ex.: mean grade

Dispersion (Spread) Statistics

- Statistics that indicates the spread of values

- Ex.: variance

13

UNIV. DESCRIPTION: Location Statistics

Let N grade values:-[z(xi), i = 1, . . . , N

], xis stand for the

geographical coordinates.

Location statistics:- Mean:

mZ =z(x1) + z(x2) + ...+ z(xN )

N

mZ =1

N

Ni=1

z(xi)

- Mode: most likely value

- Quantile or percentile:

zq , such that Prop[z(xi) zq

]= q%

- Median: z50, or 50th percentile

14

UNIV. DESCRIPTION: Dispersion Statistics

Dispersion statistics:- Variance:

s2Z =1

N

Ni=1

[z(xi)mZ ]

2 0

- Standard deviation:

sZ = +s2Z .

- Coefficient of variation:

coef. var. = sZ/mZ .

- Interquartile range:

z75 z25

An easy way to compute the variance (Exercise 2):

s2Z =( 1N

Ni=1

z2i

)m2Z

s2Z = Mean of Squares - Square of Mean

15

UNIV. DESCRIPTION: Statistics

0

10

20

30

freq

(%)

Mean

Au (g/t)

f. 3e

Variance

0 4 8 12 16

Mode

25% 25%

Au25 Au75

Interquartile RangeAu75 Au25

16

UNIV. DESCRIPTIONCoefficient of Variation

What is it?- Another measure of spread

Coef.Var. =Standard Deviation

Mean

Why?- Unit-less

Example (Unit = oz/t)- Mean grade = 1 oz/t 30 g/t- Variance = 1.00 (oz/t)2

- Coef.Var. = 1.00

Unit Grade Variance Coef.Var.oz/t 1.00 1.00 1.00dwt 20.0 400. 1.00mopt 1000. 106 1.00g/t 30.0. 900. 1.00

Rule of thumb- Coef.Var. < 1.5 = No problems- Coef.Var. > 3.0 = Problems

17

UNIV. DESC.: Weighted Mean and Variance

Let N (Grade, Weight) values:

-[zi, wi, i = 1, . . . , N

],N

i=1 wi = 1

Weighted mean

mZ =

Ni=1

wi zi

Weighted variance

s2Z =

Ni=1

[wi (zi mZ)

2]=( Ni=1

wi z2i

)m2Z

= Example: Declustered mean and variance (to beseen later)

18

UNIV. DESC.: Declustered Histogram

AU Naive Histogram

Au (g/t)

Fre

quency

0.0 2.0 4.0 6.0 8.0 10.0

0.000

0.040

0.080

0.120

Nb. of data 4296

mean 2.059std. dev. 2.179coef. var 1.058


AU Declustered Histogram

Au (g/t)

Fre

quency

0.0 2.0 4.0 6.0 8.0 10.0

0.000

0.040

0.080

0.120

Nb. of data 4296



Note 14% drop in grade due to declustering.19

UNIV. DESCRIPTION: Various Means

Some software list several means when computingstatistics.

Arithmetic mean:

maZ =1

N

Ni=1

z(xi)

Geometric mean:

mgZ =N

Ni=1

z(xi)

Harmonic mean:

mhZ =

(1

N

Ni=1

1

z(xi)

)1

Relation:mhZ mgZ maZ

20

UNIVARIATE DESCRIPTION: Exercise 3

Let 11 Au values

.1 .2 .7 .8 .9

1.2 2.0 2.4 3.5 5.7 18.0

Draw the histogram, choosing a class interval so thatthe shape of the distribution is reproduced.

Compute the:- mean, median;- variance, standard deviation;- coefficient of variation.

Comment you results.

HINTS for variance:

- Variance = Mean of Squares - Square of Mean, e.g:

s2Z =(

1N

Ni=1 z

2i

)m2Z

- Compute the means of zi and z2i

- Then compute the variance

21

UNIV. DESCRIPTION: Properties of m

Demonstration given in Exercise 4

Let two RVs X and Y , sampled at N locations.We know:

- The means: mX and mY

- Three constants: a, b, c

Properties of the mean:

- m(X+Y ) = mX +mY

- m(aX) = a mX

- m(X+a) = mX + a

More generally

- m(aX+bY +c) = a mX + b mY + c

= The mean (average) is a linear operator.

22

UNIV. DESCRIPTION: Properties of s2

Demonstration given in Exercise 4

Let two RVs X and Y , sampled at N locations.We know:

- The means: mX and mY

- The variances: s2X and s2Y

- Two constants: a, b

Properties of the variance:

- s2(X+Y ) =? (see bivariate statistics)

- s2(aX) = a2s2X

- s2(X+a) = s2X

More generally

- s2(aX+b) = a2s2X

The variance is not a linear operator.

23 24

UNIV. DESC.: Boxplot

A graph summarizing a distribution essential statistics

0

Boxplot of AU grades

f. 61

Au (g/t)

4

8

12

16

Minimum

Lower Quartile

Median

Mean

Upper Quartile

Maximum

Outliers}

Multiple boxplots can be displayed on the same page

Great display!

25

UNIV. DESC.: Boxplots

f. 63

AU BOXPLOTS

DOM-03 DOM-04 DOM-05 DOM-06 DOM-07

0.1 0.1

1.0 1.0

10.0 10.0

100.0 100.0

1000.0 1000.0

4859Number of data Number of data4.7667Mean Mean21.9575Std. Dev. Std. Dev.4.6065Coef. of Var. Coef. of Var.680.0Maximum Maximum

2.9Upper quartile Upper quartile1.0Median Median0.3Lower quartile Lower quartile0.01Minimum Minimum


0.9Upper quartile Upper quartile0.23Median Median0.1Lower quartile Lower quartile

0.01Minimum Minimum

20902Number of data Number of data0.7509Mean Mean3.7238Std. Dev. Std. Dev.4.9593Coef. of Var. Coef. of Var.398.0Maximum Maximum0.5Upper quartile Upper quartile0.11Median Median0.1Lower quartile Lower quartile0.01Minimum Minimum

13793Number of data Number of data6.1199Mean Mean

36.3533Std. Dev. Std. Dev.5.9402Coef. of Var. Coef. of Var.972.0Maximum Maximum4.5Upper quartile Upper quartile1.2Median Median0.2Lower quartile Lower quartile0.01Minimum Minimum

19117Number of data Number of data10.6783Mean Mean45.2865Std. Dev. Std. Dev.

4.241Coef. of Var. Coef. of Var.1000.0Maximum Maximum


26

UNIV. DESC.: Piecharts

f. 64

AU PIECHARTS

6%DOM-03

32%

DOM-04

24%DOM-05

16%

DOM-06

22%

DOM-07

By number of samples(Total = 86721 samples)

2%DOM-03

29%

DOM-04

56%DOM-05

6%

DOM-06

7%DOM-07

By sample weight

6%DOM-03

17%

DOM-04

21%

DOM-05

18%DOM-06

38%

DOM-07

By sample weight x grade

Useful to show geological domain relative importances.

27

UNIV. DESC.: Orientation Plot

Orientations of Consecutive Pairs in Same Hole

< 10 pairs

10 to 100 pairs

100 to 1,000 pairs

1,000 to 10,000 pairs

> 10,000 pairs

fig192

E

80

70

60

50

40

30

20

10N350340

330

320

310

300

290

280

W

260

250

240

230

220

210

200

190S

170

160

150

140

130

120

110

100

-10 -20 -30 -40 -50 -60 -70 -80

28

UNIV. DESC.: Standardized Variable

Standardized random variables are often used in statis-tics. For example:

- Affine correction (e.g. sample to block grade dis-tribution).

Standardizing consists in subtracting the mean from thevariable, and then dividing by the standard deviation.

Standardizing X mX

sX

f.176

m

0.0

0.0

1.0

X mX ORIGINAL DISTRIBUTION

X

X mX X

1

2

STANDARDIZED DISTRIBUTION

Mean and variance of a standardized random variable(Exercise 5) :

- Mean: 0- Variance: 1

29

UNIV. DESC.: Standardized Variable

Example

Let 2 AU values (X): 1 g/t, 9 g/t

The mean, variance, and std. deviation are:

mX = 5

s2X =( 2i=1

x2i

)m2X = (1 + 81)/2 25 = 16

sX = 4

The standardized values (Y) are:

(1 5)/4 = 1, (9 5)/4 = 1

The standardized variable mean, variance, andstd. dev. are:

mY = 0

s2Y =( 2i=1

y2i

)m2Y = (1 + 1)/2 0 = 1

sY = 1

30

UNIVARIATE STATISTICS: Summary 1/3

Random Variable (RV) Z; Realization z.

z(x), grade at location x, is interpreted as a realizationof the RV Z(x).

Stationarity roughly means that characteristics of RVZ(x) do not depend on location x.

Univariate Description:- histogram, boxplot, piechart;- mean, variance, standard deviation;- coefficient of variation;- quantiles, median.

Properties

- Mean

- m(aX+bY +c) = amX + bmY + c

- Variance:

- Mean of Squares Square of Mean

- s2(aX+b) = a2s2X

31

FACT SHEET

Univariate Statistics (Description)

mX = (1/N)

xi

s2X = (1/N)(xi mX)

2 = (1/N)

x2i m2X

m(aX+bY +c) = a mX + b mY + c

s2aX+b = a2s2X

32

UNIVARIATE MODEL: Introduction

Models are needed to go beyond description. Forexample:

- To build estimators (prediction)- To simplify problems (e.g. using normal/lognormaldistribution)

- To solve problems (e.g. conditions needed tobuild unbiased estimator)

Examples of models:- Assumption that grade at location x is a realiza-tion random variable at that location

- z(x) is realization of RV Z(x)- Assumption of stationarity

33

UNIV. MODEL: Probability Density Function (PDF)

Description: histogram

0

10

20

30

Freq

(%)

Au (g/t)0 4 8 12 16

Histogram of Au grades

f. 3b

P(2. < Au < 4.)

P (2 < AU 4) = 20%

Model: probability density function (pdf)- Recall: integral

0 8 160

fZ(z)

.1

.2

.3

z

f. 5a

2 4

P(2

UNIV. MODEL: Parameters

Let Z(x) a RV. Histogram has Nc classes.f(z) is the pdf of Z(x).

- Recall: weighted mean and variance

frq Histogram

f. 165

f(z) pdf

z zz1 , z2 , z3 ,...

f1 , f2 ,...

Description Model

fi=1

Mean: mZ Z

mZ =

Nci=1

fizi Z =

+

zf(z)dz

Quantile: zq zq

P[zi zq

]= q

i,zizq

fi

zq

f(z)dz = q

Variance: s2Z 2Z

s2Z =

Nci=1

fi(zi mZ)2 2Z =

+

(z Z)2f(z)dz

37

UNIV. MODEL: Parameters

frq Histogram

f. 165

f(z) pdf

z zz1 , z2 , z3 ,...

f1 , f2 ,...

Description Model

fi=1

Std. Dev.: sZ Z

sZ =s2Z Z =

2Z

Coef. Var.: cvZ cvZ

sZ/mZ Z/Z

Remarks- Greek instead of Roman letters are used for modelparameters: e.g. and instead of m and s.

38

UNIV. MODEL: Properties of and 2

Properties of and 2

- Are similar to properties of m and s2.

- Demonstration is given in Exercise 6.

Let 2 RVs, X and Y , and 3 constants, a, b, and c.

Properties of the mean:

- (aX+bY+c) = a X + b Y + c

= The mean is a linear operator

Properties of the variance:

- 2(aX+b) = a22X

- 2(X+Y ) =?

39

UNIV. MODEL: Notation

Mean: () E()

- E() stands for Expectation

- Ex.: (X+Y ) = E(X + Y )

Variance: 2() V AR()

- Ex.: 2(aX) = V AR(aX)

Standard Deviation: () STDV ()

40

UNIV. MODEL: Expectations - Moments

Suppose:- RV Z with density f ;- A function g(Z).

Then:

E[g(Z)] =

+

g(z)f(z)dz

where E[] stands for Expectation.

Examples:

g(Z) E[g(Z)]

1 1Z Z (Moment of order 1)(Z Z)

2 2Z (Moment of order 2)(Z Z)

3 Moment of order 3(Z Z)

4 Moment of order 4(Z Z)

3/3 Skewness(Z Z)

4/4 Kurtosis (peakedness)

Some software list the skewness and kurtosis whencomputing statistics.

41

UNIV. MODEL: Skewness - Kurtosis

Skewness: E(Z- mZ)3

s3Z

Symmetry

= 0

< 0

Modez50mZ

> 0

Modez50

mZ

Kurtosis: E(Z- mZ)4

s4Z = 3 (Normal)> 3

< 3

s2Z

Same for all

42

43 44

UNIV. MODEL: Normal Distribution

Let a RV X with mean X and variance 2X

X is normally distributed if its pdf gX(x) is:

gX(x) =12pi2X

exp

[(x X)

2

22X

].

X

X

gX(x)

X: MeanX: Variance

x

f. 1

22

The notation is:

X N(X , 2X),

A normal distribution is fully defined by its mean, X ,and variance, 2X

45

UNIV. MODEL: Normal Distribution

Important probabilities:

P (X X X X + X) = 68%

P (X 2X X X + 2X) = 95%

X

gX(x)

68%95%

f. 2

13.5% 13.5%

X-2X X-X X+2X X+XX

34% 34%

= Used for estimation confidence intervals.Cf. Section Estimation Variance.

46

U. MODEL: Normal Cumulative Distribution

X N(X , 2X)

The cdf of X, GX(x) is such that:

GX(b) = P (X b)

0

.5

1.GX(x)

b x

P(X

UNIV. MODEL: Lognormal Distribution

If X is lognormally distributed, then ln(X) is normallydistributed

= ln(x)50

fX(x)

x50

X ~ Lognormal ln(x) ~ Normal

f 42a

x ln(x)

X ~ LN(,2) ln(X) ~ N(,2)

gln(X)

(ln(x))

X lognormal distribution parameters: = e+

2

2

2 = 2(e2

1)

x50 = e

Y = ln(X) normal distribution parameters:

= ln() 2

2

2 = ln(1 +2

2)

y50 = = ln(x50) 6= ln()

49

UNIV. MODEL: Probability Plots

Probability of exceeding grade

Gra

de

PROBABILITY PLOT - DISTRIBUTION IS NOT NORMAL

0

10

20

30

40

50

60

70

80

90

100

f. 65

Gra

de

LOG PROBABILITY PLOT - DISTRIBUTION IS APPROX. LOGNORMAL

0.1

0.2

0.3 0.4 0.5

1

2

3 4 5

10

20

30 40 50

100

200

300 400 500

1000

99.99 99.9 99.8 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.01

Probability of exceeding grade

99.99 99.9 99.8 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.01

50

UNIV. MODEL: Poisson Distribution

Occurs for discrete variable randomly distributed withinvolume. Ex: No of AU grains within a 50g pulp sample.

= 6 = 7

0 1 2 3 4

= 1

= 2 = 3

= 0.5 = 0.75

0

0.1

0.2

0.3

0.4

0.5

0.6

= 4 = 5

f. 142

Poisson

Distributions

Prob (X=k)

k(Salamd, Ingamells, 1974)

Parameters ( =mean = variance)):

Prob(x = k) =ek

k!

= Sample size matters!51 52

UNIV. STATISTICS: Summary 2/3

Random Variable (X)

Probability: P (X < a)

Probability distribution function (pdf), cdf- Expectation or mean: E(X) = X- Variance: 2X- Quantiles: x25, x50, x75- Coef. variation: X/X- Skewness, kurtosis

Normal distribution- Bell shape- Fully defined by its mean and variance- Notation: X N(X ,

2X)

- P (X X < X < X + X) = 68%- Straight line on probability paper

Lognormal distribution- Straight line on log-probability paper

Poisson distribution

53

FACT SHEET


E(X) = X

V ar(X) = 2X = E(X X)2 = E(X2) 2X

E[aX + bY + c] = aE[X] + bE[Y ] + c

V ar[aX + b] = a2V ar[X]

54

55 56

Geology Model

A good geology model is most important.

The geologymodel should tell something about the min-eralization, the structures, etc.

A geology model consists of several geology domains.Each domain has its own statistical characteristics thatdo not depend on location (homogeneity station-arity).

Boundaries between geological domains can be hard,soft, or in-between.

Statistics to be computed per geology domain andwithin the EDA envelope.

Useful statistics to check differences between domains:- Multiple boxplots, histograms- Multiple cumulative distribution- Contact plots- Variograms (different directions)

57

Overb

urd

en

Ox. S

apro

lite

Sul. S

apro

lite

Sapro

ck

Crb

Lch B

rk

Crb

Stb

l B

rk

Dyke

Wst

0.0

2.0

Au

0.5

4 g

/t

DD

H

21400

21400

21600

21600

21800

21800

22000

22000

22200

22200

22400

22400

-400

-400

-300

-300

-200

-200

-100

-100

0

0

100

100

200

200

PA

RA

DIS

E IN

C.

SE

CT

ION

: 9200 N

Dep

osit

: P

ara

dis

eG

EO

LO

GY

+ D

DH

SA

MP

LE

S

las

cri

s_

01

2

58

EDAPurpose

Data familiarization

Detecting possible errors

Identifying/confirming different mineralizations

Answering questions such as:

- Ordinary or indicator kriging ?

- What trimming values ?

- Mean and variance ?

Providing information for:

- Model validation

- Reconciliation

59 60

EDA Envelope (1/4)

An EDA envelope is a 3D envelope within which statis-tics are computed.

Which statistics?- Declustered mean, variance.- Choice of trimming values.- Choice of indicator cut-off grades.- Variogram.- Resource block model validation.

Why an EDA envelope?

- To restrict statistics to where it matters.

- Fairly tight around sample locations

- No extensive waste areas

- Well covered with samples

- To reduce the impact of fringes whendeclustering and computing statistics.

- To make sure that comparisons made duringvalidation (e.g. sample vs. kriged averagegrades) correspond to the same material (i.e.material within the EDA envelope for both samplesand kriged estimates).

61

EDA Envelope (2/4)

Project Area

EDA Envelope

Very low grades"Other" grades

f.185

How to define and EDA envelope?- Dont be too precise.- Fairly tight around reasonably well sampled zone.- Around material that matters. Significant wastezone well below cut-off can be ignored.

- Digitize on a series of benches, then wireframe andcreate a 0/1 indicator grid.

- Generally, geology can be ignored when definingthe EDA envelope.

62

EDA Envelope (3/4)

431600

431600

431800

431800

432000

432000

432200

432200

432400

432400

432600

432600

432800

432800

80

78

00

80

78

00

80

80

00

80

80

00

80

82

00

80

82

00

80

84

00

80

84

00

80

86

00

80

86

00

80

88

00

80

88

00

80

90

00

80

90

00

RL = 302.5 +/- 2.5 -- 5 ft Bench Toe = 300

Paradise - 10m Composite Au values

1.0

10m cmp AU (g/t)

0.2

2.0

EDA Envelope

pdi-10

63

EDA Envelope (4/4)

Some remarks:- The EDA envelope is used to compute statistics.- All data within and outside the envelope can beused at the estimation step.

- There can be estimates outside the EDA envelope.- A geology model has to be considered in additionto the envelope.

64

DECLUSTERING: Introduction

Clusters of samples are common in the mining industry.

f.116

CLUSTERS

N

Potential problem:- Clusters are often located within high grade zones.Their impact can be a serious overestimation ofthe average grade and variability if not accountedfor.

Solution:- Declustering.- Objective is to reduce the weight of each clus-tered data +/ proportionally to the cluster sam-pling density.

65

DECLUSTERING: Methods

Cell Declustering

- Superimpose a grid of cells on the data;- Cell size roughly the average sample spacing, ignor-ing clusters. There is on average 1 data per cell,where clustered. The cell can be rectangular.

- The declustered weight of a given sample is 1/Ncwhere Nc is the number of samples located in thecorresponding cell.

1

2

86

2

1

f.184b

2 per cell ==> w=0.50 per sample

1 per cell ==> w=1.0 per sample

Average

Naive: 1 + 2 + 8 + 6 + 1 + 26

= 3.33 g/t Au

Declustered:

8 + 62

1 + 2 + + 1 + 2

5= 2.60 g/t Au

66


Polygonal declustering

- In 2D, the declustered weights are proportional tothe polygons of influence of the corresponding data.

- In 3D, the same principle is applied on a benchbasis.

1

2

86 2

1

f.184C

Average

Naive: 1 + 2 + 8 + 6 + 1 + 26

= 3.33 g/t Au

Declustered:

= 2.54 g/t Au

37*1 + 28*2 + 17*8 + 21*6 + 47*2 + 44*137 + 28 + 17 + 21 + 47 + 44

Area : 37 17 44

Area : 28 21 47

67


Kriging

- Kriging is a good declustering tool (see later).- A regular grid of cells is superimposed on the data.The size of the cell does not matter too much.

- The cells are kriged using the samples. The sam-ple kriging weights are kept in memory.

- The declustered weight of a given sample is thesum of the corresponding kriging weights kept inmemory.

- Note:

- Declustering depends very much on the vari-ogram model (and nugget value).

68


Nearest Neighbour Model

- Data is used to estimate a regular cell/block model.Closest data is used to estimate each block.

- Resulting distribution is the distribution ofestimated blocks.

- The shape of the resulting distribution is verysimilar to the shape of the polygonally declustereddistribution.

- Advantage:

- Can be done with commercial software.- Disadvantage:

- Does not attach declustered weights to sam-ples.

- Notes:

- Usual AMEC procedure.

- NN model block size should be small, other-wise many samples may not be considered.

69


Automatic cell declustering

- Based on assumption that clusters are always inhigh grade zone. The naive average is thereforeoverestimated.

- Several cell size are automatically used for declus-tering.

- The selected cell size is that one that gives the low-est average.

DeclusteredMean

OptimumCell Size

f.177

0

Tiny HugeCell Size

Note: tiny or huge cell size No declustering

- Nice in theory, but often inconclusive in practice. Not recommended.

70


Note 1:- Some declustered weights can be very large due to:

- huge polygonal area (on the fringe)- special sample location (start/end of hole)

- The solution consists in:- setting a maximum value when declustering- declustering within an EDA envelope

Note 2:- Polygonal declustering incomplete due to declus-tering radius smaller than small polygons.

- Solution consists in:- Checking maps of declustered areas- Increasing declustering radius and using EDAenvelope to control the fringes

- Check declustering weights and eventuallytrim them.

Useful displays- Histogram of weights- Maps of weight values- Maps of declustered areas

71

DECLUSTERING: Statistics

Let N AU values[z(xi), i = 1, . . . , N

]and[

wi, i = 1, . . . , N]rescaled declustered weights such

that:Ni=1

wi = 1

Declustered statistics are:- Mean:

mZ =Ni=1

wiz(xi)

- Variance:

s2Z =

Ni=1

wi[z(xi)mZ

]2

=

Ni=1

wiz2(xi) m

2Z

- Standard deviation: sZ =s2Z .

- Median: z50 such that the sum of the declusteredweights of the values less than z50 is 0.5.

Note: if pairs of values are available, the covariance (seebivariate statistics) can also be declustered.

72

DECLUSTERING: Example (1/3)

AU Naive Histogram

Au (g/t)

Fre

quency

0.0 2.0 4.0 6.0 8.0 10.0

0.000

0.040

0.080

0.120

Nb. of data 4296



AU Declustered Histogram

Au (g/t)

Fre

quency

0.0 2.0 4.0 6.0 8.0 10.0

0.000

0.040

0.080

0.120

Nb. of data 4296



-14% change in grade73


Histogram of Declustered Weights

Decl. Weight

Fre

qu

en

cy

.01 .1 1. 10. 100.

0.000

0.040

0.080

0.120

Nb. of data 4296



Very few excessive weights. Keep as is, or trim to 40.

74


Polygons

1.

5200.

Geology

75600

75600

75800

75800

76000

76000

76200

76200

76400

76400

94

40

0

94

40

0

94

60

0

94

60

0

94

80

0

94

80

0

95

00

0

95

00

0

95

20

0

95

20

0

95

40

0

95

40

0

Declustering polygons - Elev: 202.5

Incomplete declustering in the Northern portion of themap and in elongated domain in the SW.

75

DECLUSTERING: Exercise 7

Let the following sampling situation:

f.118

N

100m

What would be a reasonable cell declustering size?

76

COMPOSITING 1/3

Support size (point, 2 m sample, block, etc.) is im-portant.

- Different support sizes in different variabilities.- Blocks are less variable than samples.

In theory, samples must be representative of the pop-ulation. 5m samples are not representative of a 1 msample population.

(Most) estimation algorithms do not account for samplesize, e.g. do not make the difference between a 10 anda 1 m sample.

Solution: composite samples so that resulting com-posite lengths are identical.

"Hard" Geological Boundary

f. 178

OriginalSamples

RegularComposites

Rock A

Rock B

77

COMPOSITING 2/3

Compositing may be required if:- Sample lengths are much different: average lengthof 1.5 m, many 50 cm long samples centered onhigh grade veins.

Before compositing:- Histogram of sample lengths.- Histograms of sample grades per interval of lengths.- Eventually trim or cut very high grade (outliers)to avoid smearing them over much longer lengths(More on outliers in Section Bivariate Statistics).

Composite length should be such that:- Enough variability is retained when estimating.- No geological boundary crossing.- Do not exceed block size:

- 5m benches: 2/3 m composites OK; 5m ismaximum.

If possible, composite only what is needed, i.e. leaveuntouched composites if in specified Min/Max limits.

78

COMPOSITING 3/3

Impact of compositing:- Loose original samples;- Grade variability reduced;- Number of samples reduced;- Geological contacts can be smeared out.

If an original sample length is very long, compositingwill split it in many regular smaller lengths.

- OK if the original grade is very low.- Problem if original grade is very high, because thelocation of the high grade is unknown.

After compositing:- Check for smallest composites

- Eventually discard some of them.- Check pre/post length weighted histograms.

- Means should be the same.- Variance should decreases after compositing.

Visual check: display drill holes with the compositedgrade histogram on one side and the original grade his-togram on the other side.

79 80

OTHER EDA TOOLS: Checking Trends

2.5ft Composites, Declustered - All Domains

Grade Profiles by Coordinate Axes

Number of Data by Coordinate Axes

Composites

Composites

0.0

0.0 0.443 0.886 1.329 1.772 2.215 2.658 3.101 3.544 3.987

39300.0

39607.0

39914.0

40221.0

40528.0

40835.0

41142.0

41449.0

41756.0

42063.0

42370.0

bh_met1_tr (g/t)

Easting

(m)

0.0 0.443 0.886 1.329 1.772 2.215 2.658 3.101 3.544 3.987

77820.0

78050.0

78280.0

78510.0

78740.0

78970.0

79200.0

79430.0

79660.0

79890.0

80120.0

bh_met1_tr (g/t)

Nort

hin

g(m

)

0.0 0.443 0.886 1.329 1.772 2.215 2.658 3.101 3.544 3.987

-320.0

-245.5

-171.0

-96.5

-22.0

52.5

127.0

201.5

276.0

350.5

425.0

bh_met1_tr (g/t)

Ele

vation m

(m)

0.0 266.2 532.4 798.6 1064.8 1331.0 1597.2 1863.4 2129.6 2395.8 2662.0

39300.0

39607.0

39914.0

40221.0

40528.0

40835.0

41142.0

41449.0

41756.0

42063.0

42370.0

Number of Data

Easting

(m)

0.0 266.2 532.4 798.6 1064.8 1331.0 1597.2 1863.4 2129.6 2395.8 2662.0

77820.0

78050.0

78280.0

78510.0

78740.0

78970.0

79200.0

79430.0

79660.0

79890.0

80120.0

Number of Data

Nort

hin

g(m

)

0.0 266.2 532.4 798.6 1064.8 1331.0 1597.2 1863.4 2129.6 2395.8 2662.0

-320.0

-245.5

-171.0

-96.5

-22.0

52.5

127.0

201.5

276.0

350.5

425.0

Number of Data

Ele

vation

(m)

f197a

f197a

81

OTHER EDA TOOLS: Grade profiles

Comparison of Au,Ag,Cu,Zn,C and S for DDH161

Depth Depth

0.01

0.01

0.1

0.1

1.0

1.0

10.0

10.0

100

100.

Gold + Silver (in g/t)

0.01

0.01

0.1

0.1

1.0

1.0

10.0

10.0

100

100

Copper + Zinc (in %)

0.01

0.01

0.1

0.1

1.0

1.0

10.0

10.0

100

100

Carbon + Sulphur (in %)

0

20

40

60

80

100

120

140

160

180

200

220

0

20

40

60

80

100

120

140

160

180

200

220

82

OTHER EDA TOOLS Boxplots

f. 63

AU BOXPLOTS

DOM-03 DOM-04 DOM-05 DOM-06 DOM-07

0.1 0.1

1.0 1.0

10.0 10.0

100.0 100.0

1000.0 1000.0




0.9Upper quartile Upper quartile0.23Median Median0.1Lower quartile Lower quartile

0.01Minimum Minimum

20902Number of data Number of data0.7509Mean Mean3.7238Std. Dev. Std. Dev.4.9593Coef. of Var. Coef. of Var.398.0Maximum Maximum0.5Upper quartile Upper quartile0.11Median Median0.1Lower quartile Lower quartile0.01Minimum Minimum

13793Number of data Number of data6.1199Mean Mean

36.3533Std. Dev. Std. Dev.5.9402Coef. of Var. Coef. of Var.972.0Maximum Maximum4.5Upper quartile Upper quartile1.2Median Median0.2Lower quartile Lower quartile0.01Minimum Minimum

19117Number of data Number of data10.6783Mean Mean45.2865Std. Dev. Std. Dev.

4.241Coef. of Var. Coef. of Var.1000.0Maximum Maximum


83

OTHER EDA TOOLS Multiple Probability Plots

Different Mineralizations

PROBALITY OFEXCEEDING GRADE

GR

AD

E

PROBALITY Plot - Domain 04 - 07 - Au Dwt

0.0100

0.0200 0.0300 0.0500 0.1000

0.2 0.3 0.4 0.5 1.0

2.0 3.0 5.0 10.0

20.0 30.0 50.0 100.0

200.0 300.0 500.0

99.99 99.9 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.1 0.01


GR

AD

E

0.0100

0.0200 0.0300 0.0500 0.1000

0.2 0.3 0.4 0.5 1.0

2.0 3.0 5.0 10.0

20.0 30.0 50.0 100.0

200.0 300.0 500.0 900.0

99.99 99.9 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.1 0.01


GR

AD

E

0.0100

0.0200 0.0300 0.0500 0.1000

0.2 0.3 0.4 0.5 1.0

2.0 3.0 5.0 10.0

20.0 30.0 50.0 100.0

200.0 300.0 500.0 900.0

99.99 99.9 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.1 0.01


GR

AD

E

0.0100

0.0200 0.0300 0.0500 0.1000

0.2 0.3 0.4 0.5 1.0

2.0 3.0 5.0 10.0

20.0 30.0 50.0 100.0

200.0 300.0 500.0 900.0

99.99 99.9 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.1 0.01


GR

AD

E

0.0100

0.0200 0.0300 0.0500 0.1000

0.2 0.3 0.4 0.5 1.0

2.0 3.0 5.0 10.0

20.0 30.0 50.0 100.0

200.0 300.0 500.0 900.0

99.99 99.9 99.8 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.1 0.01

pdi_12

84

UNIV. STATISTICS: Summary 3/3

Geology model- Geology must be relevant to mineralization.

EDA envelope

Clusters:- Overestimation possible- Various declustering techniques:

- Cell, polygonal, kriging, NN- Declustered mean, variance, and covariance.

Compositing

Other EDA tools:- Assay above cut-off statistics- Checking spatial anomalies- Checking trends- Checking grade profiles

85

FACT SHEET

No New Formulas

86

BIVARIATE STATISTICS

Content

Bivariate description:- Graphs: scattergrams,...- Statistics: covariance, coefficient of correlation,...

Bivariate model:- Intuitive introduction

Marginal and conditional statistics

Applications:- Regression- Trimming, cutting outliers- Checking pairs of values- Checking geological boundaries

1

BIVARIATE DESCRIPTION

Scattergram Bivariate Histogram

List of pairs of values:

X (Zn) 4.61 6.07 4.60 7.89 ...

Y (Pb) 3.2 4.9 3.9 5.3 ...

Scattergram:

x

y

Scattergram

f. 89

Bivariate histogram:

1

2

2

1

2

1 3

2

1

1

1 1

1 1

3

2

1

1

x

yBiv. histogram

f.90

2

BIV. DESCRIPTION: Examples

cu_new

(%

)

cu_pre (%)

Arithmetic ScalePast VS Current Cu Sample Values (

BIV. DESCRIPTION: Joint Statistics

X and Y considered jointly:

Covariance:

Cov(X, Y ) =1

N

Ni=1

([xi mX ][yi mY ]

),

=

(1

N

Ni=1

[xiyi]

)mXmY .

Cov.= Mean of Products Product of Means

Coefficient of correlation:

Cor(X, Y ) =Cov(X, Y )

sXsY

1 Cor(X, Y ) 1

Notes:

- Cov(X,X) = V ar(X)

- Cor(X,X) = 1

5

BIV. DESCRIPTION: Examples of Correlation

Various coefficients of correlation (Davis, 1973):

Cor = 0.98 Cor = 0.54

Cor = 0.16Cor = 0.80

Cor = undefined

Cor = 0

y

x

f. 33

y

x

y

x

y

x

y

x

y

x

The covariance and coefficient of correlation are mea-sures of linear correlation.

6

BIV. DESCRIPTION: Induced Correlation

Pebble example (Davis, 1973):

A

A A

A B

B

B

B

C C

C C

AXIS "A" > AXIS "B" > AXIS "C"

AXES "A" , "B" , "C", PICKED RANDOMLY

(Davis, 1976)

f.95

Two elements A and B such that A+ B = 100%.

A

B

f.96

100

100

Similar but less obvious results with more than two el-ements that add to a constant.

7

BIV. DESCRIPTION: Induced Correlation

X versus X2 :

VAR 1: V1 ~ N(0., 1.)

Fre

qu

en

cy

V1

-3.0 -2.0 -1.0 0.0 1.0 2.0

0.00

0.02

0.04

0.06

0.08

0.10 No. of Data 1000

Mean -0.04Std. Dev 1.00

Max. 4.00Min. -3.34

VAR 2: V2 = V1 ** 2

Fre

qu

en

cy

V2

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

0.00

0.10

0.20

0.30

v2

v1

v1/v2 scattergram

-3.36 -2.36 -1.36 -0.36 0.64 1.64 2.64 3.64

0.0

4.0

8.0

12.0

16.0

Correlation -0.06Rank correlation -0.05

No. of Data 1000

Mean 0.99Std. Dev 1.44

Max. 16.07Min. 0.00

No. of Data 1000

Max. 16.07Min. 0.00

cor_induced.eps

8

BIV. DESCRIPTION: Exercise 8

Let 11 pairs of AuS values

X (Au) .1 .2 .7 .8 .9 1.2

Y (S) 1.01 2.54 0.60 1.72 2.63 1.82

X (Au) 2.0 2.4 3.5 5.7 18.0

Y (S) 3.25 2.34 3.68 4.89 2.24

Draw the scattergram

Compute the:- covariance;- coefficient of correlation.

Comment you results.

HINTS

- The stats ofX (mx, s2x, sx)have already been computed.

- For the covariance

- Cov = Mean of Products - Product of Means

Cov(X, Y ) =(

1N

Ni=1 xiyi

)mXmY

- Compute the mean of xiyi, then mx and my .

- For the correlation coefficient, use the fact that:

- Variance of X = s2x = 24.3 (computed earlier)

- Variance of Y = s2y = 1.32 (graciously provided).9

10

11 12

BIV. DESC.: Rank Correlation Coefficient

Previous Exercise 8 Au and S values are:

X (Au) .1 .2 .7 .8 .9 1.2

Y (S) 1.01 2.54 0.60 1.72 2.63 1.82

X (Au) 2.0 2.4 3.5 5.7 18.0

Y (S) 3.25 2.34 3.68 4.89 2.24

The ranks RAu and RS are:

Au 1 2 3 4 5 6 7 8 9 10 11

S 2 7 1 3 8 4 9 6 10 11 5

The rank correlation coefficient is computed from theranks of the data values, not the values themselves.

The rank correlation coefficient is:- Robust with respect to outliers- Often a better measure of correlation for preciousmetal deposits.

Example with previous Exercise 8 data:- Correlation coefficient = 0.22 (cf. solution)- Rank correlation coefficient = 0.85 (cf. below)

13

BIV. DESC.: Rank Correlation Coefficient

Compute intermediate quantities:

- N = 11: number of data

-

RAu,i =

RS,i = 66

-

R2Au,i =

R2S,i = 506

-

RAu,iRS,i = 472

Compute covariance of the ranks:

- mRau =1N

RAu,i = 55/11 = 5

- mRs = mRau = 5

- Cov(RAu, RS) =(1N

AuiSi

)mRaumRs

= 472/11 5 5 = 17.91

Compute rank correlation coefficient:

- s2Rau =(1N

R2Au,i

)m2Rau

= 506/11 52 = 21

- s2Rs = s2Rau = 21

- Cor(RAu, RS) = Cov(RAu, RS)/sRausRs = 0.85

The rank correlation coefficient is not much affected bythe 18 g/t Au outlier.

14

BIV. DESCRIPTION: Properties of m and s2

Demonstration given in Exercises 4 and 9.

Let two RVs X and Y . We have:- The means: mX and my- The variances: s2X and s

2y

- The covariance: Cov(X,Y)- Three constants: a, b, c

Properties of 1 RV (X; recall):

maX = a mX mX+b = mX + b

s2aX = a2s2X s

2X+b = s

2X

Properties of 2 RVs (X & Y; new):

mX+Y = mX +mY

s2X+Y = s2X + s

2Y + 2Cov(X, Y ) (NEW!)

More generally

maX+bY +c = a mX + b mY + c

s2aX+bY +c = a2s2X + b

2s2Y + 2abCov(X, Y ) (NEW!)

The average is a linear operator15

BIV. DESCRIPTION: Conditional Statistics

Conditional statistics:

1

2

2

1

2

1 3

2

1

1

1 1

1 1

3

2

1

1

x

yy|xBiv. histogram

f.92

x|y

Conditional

histograms

- Univariate statistics computed on X, given that Ytakes some value(s), and vice versa.

- Conditional Means: mX|Y=y and mY |X=x- Conditional Variances: s2X|Y=y and s

2Y |X=x

16

BIVARIATE MODEL: Introduction

As in the univariate case, models are needed to go be-yond description to:

- Build estimators (regression, prediction)- Solve problems (minimizing spread or errors whenestimating)

- Demonstrate properties

In fact, multivariate models are sometimes needed:- Simulation (sequential gaussian)

Following section is just a quick glance at the bivariatemodel. Purpose is to see parallel between descriptionand modeling when two variables are involved.

17

BIV. MODEL: Bivariate pdf fXY (x, y)

Description: bivariate histogram:

1

2

2

1

2

1 3

2

1

1

1 1

1 1

3

2

1

1

x

yBiv. histogram

f.90

P (4 < X 5 & 4 < Y 5) = 2/26 = 7.7%.

Model: biv. probability distribution function (pdf):

x

y

a b

cd

f.97

fXY(x,y)

- P (a < X b & c < Y d) = Double Integral.

18

BIV. MODEL: Marginal pdf s

Description: marginal histograms:

1

2

2

1

2

1 3

2

1

1

1 1

1 1

3

2

1

1

x

yBiv. histogram

Y

histog.

X

histog. f.91

Model: marginal probability density functions (pdfs):

x

y

fXY

(x,y)

f.98

fY

(y)

fX

(x)

Marginal PDFS fX

(x) & fY

(y)

19

BIV. MODEL: Joint Statistics

Description (recall)- Covariance:

Cov(X, Y ) =1

N

Ni=1

([xi mX ][yi mY ]

)

=

(1

N

Ni=1

[xiyi]

)mXmY .

- Correlation coefficient:

1 Cor(X, Y ) =Cov(X, Y )

sXsY 1

Model (new)- Covariance:

X,Y = E[(X X)(Y Y )]

= E(XY ) XY

- Correlation coefficient:

1 XY =XYXY

1

20

BIV. MODEL: Parameter Properties

Properties of and 2 similar to properties ofm amd s2.

We have:- Two RVs (e.g. Au & Cu): X, Y- Three constants: a, b, c

Properties of 1 RV (X; recall):

aX = aX X+b = X + b

2aX = a22X

2X+b =

2X

Properties of 2 RVs (X & Y; new):

X+Y = X + Y

2X+Y = 2X +

2Y + 2X,Y

More generally:

aX+bY+c = aX + bY + c

2aX+bY+c = a22X + b

22Y + 2abX,Y

21

BIV. MODEL: Conditional pdf s

Description: conditional histogram of X given Y = y0:

y

1

2

2

1

2

1 3

2

1

1

1 1

1 1

3

2

1

1

x

f.92a

x/yConditionalhistogramof X given Y

Biv. histogram

Conditionon Y

Model: conditional pdf of X given y = y0:

x

y

fXY(x,y)

yo

fX|Y=yo(x)

Conditional pdf

f.100a

- Parameters: X|Y=yo, X|Y=yo, . . .

Same for Y given x.

22

23 24

REGRESSION: Introduction

Function summarizing some conditional statistics:- mean of a variable Y given another variable X orvice versa;

- median of Y given X;- etc.

Used for prediction, or to study trend between 2 vari-ables.

Obtained by minimizing deviations between theexperimental values and the regression line.

Example from Davis (1973):

X

Y

A

B

C

f.102

Mo

istu

re

Depth

(Davis, 1976)

- A: minimize deviations in Y (Y |X: Y given X);- C: minimize deviations in X (X|Y : X given Y );- B: minimize joint deviations;

= 3 possible regressions (of the mean).25

LINEAR REGRESSION

X

Y

f.103a

Y|X

X|Y

Binormal Cloud

of points

Two regression lines: Y |X and X|Y

Optimal if samples are (bi)normally distributed and notclustered.

Regression of Y |X = x (binormal case):Y |X=x = Y +

YX

XY (x X)

2Y |X=x = 2Y (1

2XY )

Main problem is lack of normality. Solution:- Log transform prior to linear regression;- Polynomial regression;- Smoothed regression.

26

LINEAR REGRESSION ON LOGS

Recall on lognormal distribution.

= ln(x)50

gln(X)

(ln(X))

fX(x)

x50

X ~ Lognormal ln(x) ~ Normal

X ~ LN (, 2) ln(X) ~ N(, 2)

f 42a

x ln(x)

x50 = e

= e + 2

2

If (X,Y) are bi-lognormally distributed,then (ln(X), ln(Y)) are bi-normally distributed.

- (X, Y ) LN(X , Y , 2X ,

2Y , XY );

- (lnX, lnY ) N(X , Y , 2X ,

2Y , XY );

Linear regression on the logs is optimal:

Y |X=x = Y +YX

XY (ln(x) X)

27

LINEAR REGRESSION ON LOGS

Linear regression on the logs is optimal but:

- The regression is optimal for the logs.

Taking the anti-log of Y |X=x provides the conditionalmedian of Y

y50Y |X = exp(Y |X)

The conditional mean of Y is obtained by:

Y |X = exp(Y |X +2Y2)

X

Y

Y|X

Bilognormal Cloud

of points

y50Y|X

ln(X)

ln(Y)

f.104

Y|X

Binormal Cloud

of points

28

POLYNOMIAL REGRESSION

Polynomial of order 2 (Davis, 1973):

X

Y

f.105

Mois

ture

Depth

(Davis, 1976)

Dont overdo it!

X

Y

f.106

29

PIECEWISE LINEAR REGRESSION

2 linear segments:

X

Y

f.105a

Mois

ture

Depth

30

SMOOTHED REGRESSION

Average behaviour of Y is computed withinmoving win-dow along X.

Does not make any assumption about the (X, Y ) dis-tribution.

More difficult to use as predictor, but good enough inmost cases, when we want to look at the general trendbetween 2 variables.

Example:

RE

JE

CT

AU

ORIGINAL AU

ORIGINAL AU VERSUS REJECTLC - TRENCHES

1. 10. 100.

1.

10.

100.NB. OF DATA 477

X VAR: MEAN 5.814STD. DEV. 6.110

Y VAR: MEAN 6.021STD. DEV. 7.264

CORRELATION 0.935

f107

31 32

EDA: Checking Twin Holes

Comparison of Drill Holes D-1 and D-2

Depth

D-2 D-1

0.01 0.1 1.0 10.0 100.0

Au (in g/t)

0

10

20

30

40

50

60

70

80

90

100

Number of pairs: 66

D-1 mean: 1.779

D-2 mean: 1.854

D-1 std. dev.: 1.487

D-2 std. dev: 0.927Linear correlation: 0.553

Rank correlation: 0.591

0.01 0.1 1.0 10.0 100.00.01

0.1

1.0

10.0

100.0

RS

2 A

u (

in g

/t)

DDH161 Au (in g/t)

0.01 0.1 1.0 10.0 100.0

33

HARD / SOFT GEOLOGICAL BOUNDARIES

Waste

Ore{{ {Waste / Waste Waste Ore

Ore / Ore

f. 144

COMPARISON OF CONSECUTIVE DOWN HOLE AU ASSAYS

WASTE

0.01

0.1

1.0

10.0

100.0

0.01 0.1 1.0 10.0 100.0

Au in A

dj. W

aste

Au in Waste

Number of pairs: 89

X Mean: 1.517

Y Mean: 1.558

X Std.Dev.: 2.238

Y Std.Dev.: 2.239

Correlation (on logs): 0.214

WASTE / ORE

0.01

0.1

1.0

10.0

100.0

0.01 0.1 1.0 10.0 100.0

Au in O

re

Au in Waste

Number of pairs: 129

X Mean: 1.852

Y Mean: 7.363

X Std.Dev.: 2.521

Y Std.Dev.: 6.328


ORE

0.01

0.1

1.0

10.0

100.0

0.01 0.1 1.0 10.0 100.0

Au in A

dj. O

re

Au in Ore

Number of pairs: 102

X Mean: 8.882

Y Mean: 7.142

X Std.Dev.: 7.264

Y Std.Dev.: 6.296


f.110

Waste / Waste Waste Ore Ore / Ore

Musselwhite Comparison of Consecutive Down Hole Assays

Note: sometimes, mineralization occurs at contact.

34

HARD / SOFT GEOLOGICAL BOUNDARIES

Distance From Contact, m

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

DOMAIN: In_Stope

N = 2146 Mean = 2.72

DOMAIN: Out_Stope

N = 16721 Mean = 0.79

486

351

196

164104

90

74

52

640426

219

188

125

120

94

56

Contact Plot

f144a

35

BIV. DESC.: QQ Plot

Useful to compare 2 populations, say A and B.

The quantiles of A and B,(a1, b1), (a2, b2), , (a100, b100),

are plotted on a X/Y graph.

B B B

- Same shape- "A" more variable than "B"

- Same shape- "A" less variable than "B"

- Different shape

B

- Same distribution

0 100

10

0 100

10

0 100

10

0 100

10

f.108

A A AA

36

BIV. DESC.: Relative Difference Plot

Useful to investigate conditional bias between twopopulations, say A and B

- X axis: mean of pair of values (A+B)2

- Y axis: relative difference between values(AB)(A+B)/2 100%

f.109

A + B2

A -

B

(A +

B

) / 2

100%

0 1 2 3 4 5 6 7 8 9

0

-10

-2 0

-3 0

+10

+20

+30

RELATIVE DIFFERENCEPLOT

Notes on graph:- Low values of A < B- High values of A > B- A few outliers (analytical errors?).

37

BIV. DESC.: Checking Pairs of Values

ASSAY A: NORMAL ORIGINAL ASSAY B: NORMAL REJECT

LINEAR CORRELATION: 0.699RANK CORRELATION: 0.799

0.01 0.1 1.00.01

0.1

1.0

AS

SA

Y B

(O

Z/T

)

ASSAY A (OZ/T)

SCATTERPLOT (log scaling)

0 10

1

1

AS

SA

Y B

(O

Z/T

)

ASSAY A (OZ/T)

SCATTERPLOT

0 10

1

AS

SA

Y B

QU

AN

TIL

ES

(O

Z/T

)

ASSAY A QUANTILES (OZ/T)

Q-Q PLOT

0.01 0.1 1.00.01

0.1

1.0

AS

SA

Y B

QU

AN

TIL

ES

(O

Z/T

)

ASSAY A QUANTILES (OZ/T)

Q-Q PLOT (log scaling)

0.0 1.0-100

-75

-50

-25

0

25

50

75

100

RE

LD

IFF

[A

-B]/

AV

G (

%)

AVERAGE [A+B]/2 (OZ/T)

RELDIFF PLOT

0.01 0.1 1.0-100

-75

-50

-25

0

25

50

75

100

RE

LD

IFF

[A

-B]/

AV

G (

%)

AVERAGE [A+B]/2 (OZ/T)

RELDIFF PLOT (log scali ng)

SIDE-BY-SIDE BOXPLOT

ASSAY AASSAY B

0.01

0.1

1.0

10.0

189NUMBER0.143MEAN0.136STDEV1.207MAXIMUM0.17775TH %-ILE0.094MEDIAN0.06425TH %-ILE0.001MINIMUM

189NUMBER0.146MEAN0.162STDEV1.285MAXIMUM0.17575TH %-ILE0.091MEDIAN0.05525TH %-ILE0.001MINIMUM

AU

(O

Z/T

)

pdi_0014.eps

38

39 40

OUTLIER TRIMMING / TOPCUT (1)

When computing statistics within a geological domain,we make the assumption that there is only one popula-tion and that all samples belong to that population.

Outliers or extreme values are often observed. Theirimpact can be a serious overestimation of the blockmodel grade and variability.

There is also considerable uncertainty as to thegrade and tonnage represented by these very high val-ues.

Various solutions:- Outliers are erroneous: delete or correct them;- Outliers are from different population:

- define new geology domain;

- trim them down prior to computing statistics;

- restrict their influence during estimation (1/d2

or kriging);

- use indicator kriging.- High values are from same population:

- trim them down to reduce the risk.

In this section, trimming = topcut.

41


The main questions are:- Is trimming/cutting warranted?- If yes, which value(s) to choose?

The answers are subjective.

Useful graph to assess outliers:- Actual versus smoothed grade profiles- Histogram, cumulative probability plots- Decile analysis- Indicator correlation plot- Coefficient of variation plot- Quantity of metal plot

Method to assess the risk- Metal at risk

Other useful information:- Number of trimmed/cut data- Quantity of metal reduction after trimming

42


Grade profiles along holes

0 50 100

1

10

100

25 12575

DHxxxx

f.171

Geological

ContactsOutliers Sample

Grades

Smoothed

Sample Grades

- Outliers stand out with respect to smoothed gradeprofile.

Similar techniques can be applied in 2 and 3D:- 2D: sample values & contours; map of residuals.- 3D: sample values & 3D estimates; list of residuals.

Advantage: detect local outliers.

43


Histogram

f114_a

HISTOGRAM

Fre

quency

Au

.1 1. 10. 100. 1000.

0.000

0.020

0.040

0.060

0.080

0.100 NUMBER OF DATA 455NB CUT-OUT 93

CUT VALUE (MIN) 0.110

MEAN 10.528STD. DEV 18.818

COEF. OF VAR 1.787

MAXIMUM 201.000MINIMUM 0.110

80 g/t

- A possible trimming/cutting value is where thehistogram classes start to be isolated on thehorizontal axis.

- Possible trimming value from graph: 80 g/t.

44


Cumulative (log)probability plots.

f114_b

CU

MU

LA

TIV

E P

RO

BA

BIL

ITY

VARIABLE

CUM. DISTRIBUTION

0.01

0.10.2

12

5

10

20304050607080

90

95

9899

99.899.9

99.99

0.100 1.00 10.0 100.

70 g/t

1000.

- A single population would be represented on the plotby a gradually increasing line.

- A kink or a break in the curve might indicate twopopulations or the presence outliers.

- A possible trimming/cutting value is around thekink/break where the second population (outliers) getspredominant.

- Possible trimming value from graph: 70 g/t.

45


Decile Analysis% of Contained Metal Decile # of Samples Average (g/t) Minimum (g/t) Maximum (g/t) Contained Metal (g)

Percentile (of last decile)

Suggestion: cutting may be warranted

0-1010-2020-3030-4040-5050-6060-7070-8080-90

90-100Total

90-9191-9292-9393-9494-9595-9696-9797-9898-99

99-100

*

38.018.0

12.39.3

6.95.34.23.01.91.0

9.87.4

2.92.72.63.62.22.12.91.7

1.35 0.8 1.9 36

32.35 32.0 32.7 64

2.72 2.0 3.4 73

36.67 35.0 39.0 110

4.22 3.4 5.25 114

40.35 39.7 41.0 80

5.91 5.28 6.65 159

41.5 41.0 42.0 83

7.44 6.69 8.4 200

45.5 42.0 47.5 136

9.66 8.5 11.0 260

49.25 49.0 49.5 98

13.04 11.1 15.0 352

51.5 50.0 53.0 103

17.24 15.0 21.0 465

55.5 54.0 57.0 111

25.26 21.0 31.5 681

93.33 61.0 110.0 280

62.54 32.0 201.0 1438

185.5 170.0 201.0 371

27

2

27

3

27

2

27

2

27

3

27

2

27

2

27

2

27

3

23

2

266 14.22 0.8 201.0 3783

Decile Analysis

- Introduced by I.S. Parrish, Min. Eng., Apr. 97.

- See next page for 40/10 rule of thumb

- 40/10 rule to be reduced if last decile / percentiledo no contain a full complement of samples.

46


Decile Analysis (Contd.)

- If Top decile contains:- More than 40% of metal, or- More than twice the metal of previous decile

Split it in 10 percentiles

- If top percentile contains:- More than 10% of metal

Trimming is warranted

- Suggested trimming value is then:- Highest value of previous percentile

Possible trimming value from graph- Note that last decile / percentile not full.- Trimming may be warranted.- Previous percentiles 3 values: 61, 109, 110 g/t- Possible trimming value: 100 g/t

47


Indicator correlation plot.

f114_c

0.1 1.0 10.0 100.0 1000.0

Indicator Threshold (g/t Au)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Indic

ato

r C

orr

ela

tion for

Lag 1

60 g/t

- This plot shows the correlation coefficient of two adja-cent down-hole sample indicators for increasing cut-offs.

- Indicator: ic(x) =

{1, if the grade z(x) zc0, otherwise

where zc is the cut-off (indicator threshold).

- As the cut-off zc increases, the correlation decreases.

- A possible trimming/cutting value is when the correla-tion is or is getting close to 0.

- Possible trimming value from graph: 60 g/t.48


Coefficient of variation plot.

f114_d

0.1 1.0 10.0 100.0 1000.0

Cutting Limit (g/t Au)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0C

oeffic

ient of V

ariation

- This plot shows the coefficient of variation of the cutgrades for increasing cutting limits.

- As the cutting limit increases, coefficient of variationincreases.

- Indicates the impact of cutting on the CV.

49


Quantity of metal plot.

f114_e

0.1 1.0 10.0 100.0 1000.0

Trimming Value (g/t Au)

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

% o

f C

onta

ined M

eta

l in

Sam

ple

s

- This plot shows the relative quantity of metal containedwithin the trimmed-down samples for various trimmingvalues.

- Useful to know the quantity of metal discarded bytrimming.

- 93% of metal corresponding to 70g/t trimming value.

= 7% of metal loss if larger than 80 g/t Au samplesare trimmed down to 80 g/t.

50


D6:

HW

Sh

ear

--

AU

Decl, T

rim

1500 g

/t, E

nv=

2, In

sid

e T

rust.

Cu

ttin

g S

tati

sti

cs

0.1

1.0

10.0

100.0

1000.0

Ind

ica

tor

Th

resh

old

(g

/t A

u)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Indicator Correlation for Lag 1

0.1

1.0

10.0

100.0

1000.0

Trim

min

g L

imit (

g/t

Au

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Coefficient of Variation

0.1

1.0

10.0

100.0

1000.0

Cu

ttin

g V

alu

e (

g/t

Au

)

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

% of Contained Metal in Samples

Variable

: au (

weig

hte

d b

y w

tpoly

) fr

om

0.1

to 9

999.0

fig114_f

51


Metal at Risk

Objective:

- Assess the risk associated to high grade material

- Trim down high grade values to reduce the risk

What is the metal at risk?

- Consider:- Ore only, not waste- Yearly production increments

- Distribution of the yearly metal production:- Average = estimated yearly production- 20th percentile is such that:

- 4 times out of 5, production will be higher- 1 time out of 5, production will be lower

Yearly Metal Production

f.200Metal

ProductionAverage20th

Percentile

Metal

at risk

52


What is the metal at risk? (Contd)

- Metal at risk is:- Average - 20th percentile

What is the trimming value?

- Such that the trimmed value average is close to the20th percentile.

Procedure: Monte Carlo simulation

1. Get distribution of ore samples

2. Assess numberN of ore samples mined out per year

3. Draw N samples out of the distribution

- Calculate metal content

4. Repeat No 3 many times

- Get distribution of metal production

5. Pick 20th percentile from distribution.Compute corresponding topcut

53


Metal at Risk vs. Other Methods

Metal at risk:

- Objective

- Trimming value depends on production- The higher the production, the lesser the risk,the higher the trimming value

- Topcut model on the conservative side- Actual production assumed to exceed predic-tion 4 times out of 5.

- Metal loss exists but we do not know where itis.

Other methods

- Sometimes subjective

- Trimming does not depend on production

- Topcut model assumed to be middle of the roadmodel

- i.e. metal loss does not exist

54


Trimming/Cutting Summary Table

Histogram 80 g/tProbability Plot 70 g/t

Decile Analysis 100 g/tIndicator Correlation 60 g/t

Metal at risk (1)Final Choice 80 g/t

Coefficient of Variation 1.4Number of Data Trimmed 4 of 455

Metal Loss 7%

(1) Notes:- Metal at risk topcut yet to be included inthis example.

- Amec policy yet to be formalized.

55 56

BIVARIATE STATISTICS: Summary

Description- Scattergram, bivariate histogram- Marginal distributions and statistics- Conditional distributions and statistics- Covariance, correlation of coefficient,rank correlation coefficient

Model- Bivariate PDF of (X, Y )- Marginal and conditional PDFs- Covariance, correlation coefficient

Regression- Linear, non linear, smoothed

Special Problems- Checking twinned holes- Checking hard/soft geological boundaries- Checking pairs of values- Choose cutting/trimming values

- Several graphs- Metal at risk

57

FACT SHEET

Bivariate Statistics

sX,Y =1N

[xi mX ][yi mY ] =

1N

[xiyi]mXmY

Cov[X, Y ] = X,Y = E[(X X)(Y Y )

]= E(XY ) XY

V ar(aX+bY +c) = a2V ar(X)+b2V ar(Y )+2abCov(X, Y )

58

VARIOGRAM

Content

Theory- Definition- Nugget, sill, range- Anisotropy- Models: nugget, spherical, exponential- Variogram versus covariance

Practice- Proportional effect- Alternative variograms:

- Relative pairwise variogram- Correlogram

- Computing variograms- Variogram cross- Variogram maps

- Modeling variograms

1

VARIOGRAM: Exercise 10

Suppose a mineralized zone as following:

A B C

D

6m

3m

3m

Ni Saprolite

Zone

f. 71a

How would you compare the similarity of B, C, and Dsample grades with respect to the grade of sample A?

2

3 4

VARIOGRAM Introduction

A B C

D

6m

3m

3m

Ni Saprolite

Zone

f. 71a

The correlation between sample grades:- increases with decreasing distance betweensamples;

- can vary with direction;- varies with sample size;- depends on the continuity.

The variogram is a function that quantifies the notionof geological continuity or spatial correlation.

5

VARIOGRAM

2 % Cu

3 % Cu

Example with 3 samples

Distance

(300 Azim)

Diff.

0 10 20 30

3

2

1

0

1 % Cu

Direction: 300 AzimDistance: 10m 20mDifference 1: 2-1 = 1 3-1=2Difference 2: 3-2 = 1Avg Diff: 1 2

Distance Between

Sample Locations

along one direction

Range

Nugget

f. 12b

Average

Difference

Between

Sample

Values

Variogram

N

10m

Sill

00

Model

In fact, variogram is 1/2 average of squared differences.6

7 8

VARIOGRAM: Definition

The variogram quantifies spatial correlation by lookingat the average square difference between two values adistance (and angle) h apart.

Experimental variogram:

(h) =1

2Nh

Nhi=1

[z(xi) z(xi + h)]2

where Nh is the number of couples (z(xi), z(xi + h))separated by h (distance + direction).

Experimental variogram and model

Distance (h)

RangeNugget

SillTotal

Sill

(h)

f. 12

Theoretical expression(h) =

1

2E[(Z(x) Z(x+ h)

)2]9

VARIOGRAM: Zone of influence

Horizontal Range

Range

Range

Range

Vertical

Up to 65m verticallyin iron ore formation

No more than 2m in a bauxite deposit

Can be different indifferent directions

f. 72

VA

RIO

GR

AM

DISTANCE

IRON ORE FORMATION (DAVID, 1977)

0. 20. 40. 60. 80. 100.

0.00

1.00

2.00

3.00

4.00

5.00

VA

RIO

GR

AM

DISTANCE

BAUXITE (DAVID, 1977)

0.00 1.00 2.00 3.00 4.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

VA

RIO

GR

AM

DISTANCE

ANISOTROPY (DAVID, 1977)

0.0 10.0 20.0 30.0

0.00

1.00

2.00

3.00

4.00

5.00

6.00

10

VARIOGRAM: Examples

There is one variogram for every deposit and forevery spatially correlated variable.

VA

RIO

GR

AM

DISTANCE

(DAVID, 1979)U3O8 - WYOMING ROLL FRONT

0. 40. 80. 120.

0.00

1.00

2.00

3.00

VA

RIO

GR

AM

DISTANCE

(DAVID, 1979)U3O8 - NEW MEXICO

0. 50. 100. 150. 200. 250. 300.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

VA

RIO

GR

AM

DISTANCE

(DAVID, 1977)OIL IN TAR SAND

0. 200. 400. 600. 800. 1000.

0.00

1.00

2.00

3.00

4.00

VA

RIO

GR

AM

DISTANCE

(DAVID, 1977)WESTERN US COAL(SO2/BTU)

0. 1000. 2000. 3000. 4000. 5000. 6000.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

VA

RIO

GR

AM

DISTANCE

(DAVID, 1977)AG - MEXICO

0.0 10.0 20.0 30.0 40.0 50.0 60.0

0.00

1.00

2.00

3.00

4.00

5.00

6.00

VA

RIO

GR

AM

DISTANCE

(DAVID, 1977)CU - EXOTICA

0.0 10.0 20.0 30.0 40.0 50.0 60.0

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

VA

RIO

GR

AM

DISTANCE

(J & H, 1978)OIL GRADES

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0

0.0

5.0

10.0

15.0

20.0

25.0

30.0

VA

RIO

GR

AM

DISTANCE

(J & H, 1978)TOPOGRAPHIC HEIGHTS

0. 50. 100. 150. 200.

0.0

4.0

8.0

12.0

VA

RIO

GR

AM

DISTANCE

(J & H, 1978)CU - LOS BRONCES

0.0 10.0 20.0 30.0 40.0 50.0 60.0

0.00

0.20

0.40

0.60

0.80

f. 73

11 12

VARIOGRAM: Models

Variances are computed using the variogram model.The model therefore must ensure that all computedvariances are positive.

Example of models are:

(h)

h

Gaussian

Spherical

Exponential

Range

Nugget

Sill

Linear

Power

f. 16

Models with a sill- Nugget: zero range- Spherical- Exponential: practical range- Gaussian: practical range

Models with no sill- Linear- Power

Any linear combination of the above Models with several structures

13

VARIOGRAM: Examples

h

h

h

h

h

Dist.

Dist.

Dist.

Dist.

Dist.

Nugget

Spherical

Power

Gaussian

Hole effect

f. 34

Likely variograms Grade transects

Grade

Grade

Grade

Grade

Grade

(h)

(h)

(h)

(h)

(h)

(+ small nugget)

(+ small nugget)

(+ small nugget)

14

15 16

VARIOGRAM: Geometrical Anisotropy

Occurs when isotropy can be obtained by stretching/squashing the deposit along 1 or 2 main directions.

hR90R45

R0

0 045 090

0

f.17

N

Ellipse of Ranges

Long Range Azimuth

R90R45

R0

Variogram

Notes:- Variogram model covers all directions.

17

VARIOGRAM: Anisotropy

Azim

= 90 0

Azim=45 0

Azim=0 0

f.17a

Ellipse of RangesDist

Dist

Dis

t

Azim

= 90 0

Azim=45 0Azim=0 0

f.17b

Variogram Map

Dist

Dist

Dis

t

0 1/3 x Sill

1/3 2/3 x Sill

2/3 1 x Sill

18

VARIOGRAM: Variogram Map

-218.

-118.

-18.

83.

183.

-218.

-118.

-18.

83.

183.

FA

1 T

rad

itio

na

l V

ari

og

ram

Ma

pG

eo

log

y 1

: 1

-NE W

est

-> E

ast

South -> North

0.3

5

0.4

4

0.5

2

0.6

1

0.7

0

0.7

8

0.8

7

0.9

5

1.0

1.1

1.2

-218.

-118.

-18.

83.

183.

-218.

-118.

-18.

83.

183.

FA

1 X

Y m

ap

West ->

East

South -> North

0.3

5

0.4

4

0.5

2

0.6

1

0.7

0

0.7

8

0.8

7

0.9

5

1.0

1.1

1.2

-218.

-118.

-18.

83.

Curso de Geoestadistica en Ingles

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