Semi-implicit Euler method: Difference between revisions

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. In probability theory and statistics, the definition of variance is either the expected value (when considering a theoretical distribution), or average value (for actual experimental data), of squared deviations from the mean. Computations for analysis of variance involve the partitioning of a sum of squared deviations. An understanding of the complex computations involved is greatly enhanced by a detailed study of the statistical value:

E (X^{2}) .

It is well known that for a random variable $X$ with mean $μ$ and variance $σ^{2}$ :

σ^{2} = E (X^{2}) - μ^{2}

^[1]

Therefore

E (X^{2}) = σ^{2} + μ^{2} .

From the above, the following are easily derived:

E (\sum (X^{2})) = n σ^{2} + n μ^{2}

E ({(\sum X)}^{2}) = n σ^{2} + n^{2} μ^{2}

If $\hat{Y}$ is a vector of n predictions, and $Y$ is the vector of the true values, then the SSE of the predictor is: $S S E = \frac{1}{2} \sum_{i = 1}^{n} (\hat{Y_{i}} - Y_{i})^{2}$

Sample variance

The sum of squared deviations needed to calculate variance (before deciding whether to divide by n or n − 1) is most easily calculated as

S = \sum x^{2} - \frac{{(\sum x)}^{2}}{n}

From the two derived expectations above the expected value of this sum is

E (S) = n σ^{2} + n μ^{2} - \frac{n σ^{2} + n^{2} μ^{2}}{n}

which implies

E (S) = (n - 1) σ^{2} .

This effectively proves the use of the divisor n − 1 in the calculation of an unbiased sample estimate of σ².

Partition — analysis of variance

In the situation where data is available for k different treatment groups having size n_i where i varies from 1 to k, then it is assumed that the expected mean of each group is

E (μ_{i}) = μ + T_{i}

and the variance of each treatment group is unchanged from the population variance $σ^{2}$ .

Under the Null Hypothesis that the treatments have no effect, then each of the $T_{i}$ will be zero.

It is now possible to calculate three sums of squares:

Individual

I = \sum x^{2}

E (I) = n σ^{2} + n μ^{2}

Treatments

T = \sum_{i = 1}^{k} ({(\sum x)}^{2} / n_{i})

E (T) = k σ^{2} + \sum_{i = 1}^{k} n_{i} (μ + T_{i})^{2}

E (T) = k σ^{2} + n μ^{2} + 2 μ \sum_{i = 1}^{k} (n_{i} T_{i}) + \sum_{i = 1}^{k} n_{i} (T_{i})^{2}

Under the null hypothesis that the treatments cause no differences and all the $T_{i}$ are zero, the expectation simplifies to

E (T) = k σ^{2} + n μ^{2} .

Combination

C = {(\sum x)}^{2} / n

E (C) = σ^{2} + n μ^{2}

Sums of squared deviations

Under the null hypothesis, the difference of any pair of I, T, and C does not contain any dependency on $μ$ , only $σ^{2}$ .

E (I - C) = (n - 1) σ^{2}

total squared deviations aka total sum of squares

E (T - C) = (k - 1) σ^{2}

treatment squared deviations aka explained sum of squares

E (I - T) = (n - k) σ^{2}

residual squared deviations aka residual sum of squares

The constants (n − 1), (k − 1), and (n − k) are normally referred to as the number of degrees of freedom.

Example

In a very simple example, 5 observations arise from two treatments. The first treatment gives three values 1, 2, and 3, and the second treatment gives two values 4, and 6.

I = \frac{1^{2}}{1} + \frac{2^{2}}{1} + \frac{3^{2}}{1} + \frac{4^{2}}{1} + \frac{6^{2}}{1} = 66

T = \frac{(1 + 2 + 3)^{2}}{3} + \frac{(4 + 6)^{2}}{2} = 12 + 50 = 62

C = \frac{(1 + 2 + 3 + 4 + 6)^{2}}{5} = 256 / 5 = 51.2

Giving

Total squared deviations = 66 − 51.2 = 14.8 with 4 degrees of freedom.

Treatment squared deviations = 62 − 51.2 = 10.8 with 1 degree of freedom.

Residual squared deviations = 66 − 62 = 4 with 3 degrees of freedom.

Two-way analysis of variance

The following hypothetical example gives the yields of 15 plants subject to two different environmental variations, and three different fertilisers.

	Extra CO₂	Extra humidity
No fertiliser	7, 2, 1	7, 6
Nitrate	11, 6	10, 7, 3
Phosphate	5, 3, 4	11, 4

Five sums of squares are calculated:

Factor	Calculation	Sum	$σ^{2}$
Individual	$7^{2} + 2^{2} + 1^{2} + 7^{2} + 6^{2} + 1 1^{2} + 6^{2} + 1 0^{2} + 7^{2} + 3^{2} + 5^{2} + 3^{2} + 4^{2} + 1 1^{2} + 4^{2}$	641	15
Fertiliser × Environment	$\frac{(7 + 2 + 1)^{2}}{3} + \frac{(7 + 6)^{2}}{2} + \frac{(11 + 6)^{2}}{2} + \frac{(10 + 7 + 3)^{2}}{3} + \frac{(5 + 3 + 4)^{2}}{3} + \frac{(11 + 4)^{2}}{2}$	556.1667	6
Fertiliser	$\frac{(7 + 2 + 1 + 7 + 6)^{2}}{5} + \frac{(11 + 6 + 10 + 7 + 3)^{2}}{5} + \frac{(5 + 3 + 4 + 11 + 4)^{2}}{5}$	525.4	3
Environment	$\frac{(7 + 2 + 1 + 11 + 6 + 5 + 3 + 4)^{2}}{8} + \frac{(7 + 6 + 10 + 7 + 3 + 11 + 4)^{2}}{7}$	519.2679	2
Composite	$\frac{(7 + 2 + 1 + 11 + 6 + 5 + 3 + 4 + 7 + 6 + 10 + 7 + 3 + 11 + 4)^{2}}{15}$	504.6	1

Finally, the sums of squared deviations required for the analysis of variance can be calculated.

Factor	Sum	$σ^{2}$	Total	Environment	Fertiliser	Fertiliser × Environment	Residual
Individual	641	15	1				1
Fertiliser × Environment	556.1667	6				1	−1
Fertiliser	525.4	3			1	−1
Environment	519.2679	2		1		−1
Composite	504.6	1	−1	−1	−1	1

Squared deviations			136.4	14.668	20.8	16.099	84.833
Degrees of freedom			14	1	2	2	9

References

↑ Mood & Graybill: An introduction to the Theory of Statistics (McGraw Hill)

[1] Mood & Graybill: An introduction to the Theory of Statistics (McGraw Hill)

[1]

Semi-implicit Euler method: Difference between revisions

Revision as of 23:23, 29 January 2014

Contents

Sample variance

Partition — analysis of variance

Sums of squared deviations

Example

Two-way analysis of variance

See also

References

Navigation menu

@@ Line 1: / Line 1: @@
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+{{See also|Least squares|Mean squared error|Partition of sums of squares|Residual sum of squares}}
+In [[probability theory]] and [[statistics]], the definition of '''[[variance]]''' is either the [[expected value]] (when considering a theoretical [[probability distribution|distribution]]), or average value (for actual experimental data), of '''squared deviations''' from the mean.  Computations for '''[[analysis of variance]]''' involve the partitioning of a sum of '''squared deviations'''.  An understanding of the complex computations involved is greatly enhanced by a detailed study of the statistical value:
+: <math>\operatorname{E}(  X ^ 2 ).</math>
+It is well known that for a [[random variable]] <math>X</math> with mean <math>\mu</math> and variance <math>\sigma^2</math>:
+: <math>\sigma^2 = \operatorname{E}(  X ^ 2 ) - \mu^2</math><ref>Mood & Graybill: ''An introduction to the Theory of Statistics'' (McGraw Hill)</ref>
+Therefore
+: <math>\operatorname{E}(  X ^ 2 ) = \sigma^2 + \mu^2.</math>
+From the above, the following are easily derived:
+: <math>\operatorname{E}\left( \sum\left( X ^ 2\right) \right) = n\sigma^2 + n\mu^2</math>
+: <math>\operatorname{E}\left( \left(\sum X \right)^ 2 \right) = n\sigma^2 + n^2\mu^2</math>
+If <math>\hat{Y}</math> is a vector of n predictions, and <math>Y</math> is the vector of the true values, then the SSE of the predictor is:
+<math>SSE=\frac{1}{2}\sum_{i=1}^n(\hat{Y_i} - Y_i)^2</math>
+== Sample variance ==
+The sum of squared deviations needed to calculate variance (before deciding whether to divide by ''n'' or ''n''&nbsp;&minus;&nbsp;1) is most easily calculated as
+: <math>S = \sum x ^ 2 - \frac{\left(\sum x\right)^2}{n}</math>
+From the two derived expectations above the expected value of this sum is
+: <math>\operatorname{E}(S) = n\sigma^2 + n\mu^2 - \frac{n\sigma^2 + n^2\mu^2}{n}</math>
+which implies
+: <math>\operatorname{E}(S) = (n - 1)\sigma^2. </math>
+This effectively proves the use of the divisor ''n''&nbsp;&minus;&nbsp;1 in the calculation of an '''unbiased''' sample estimate of&nbsp;''&sigma;''<sup>2</sup>.
+== Partition &mdash; analysis of variance ==
+In the situation where data is available for ''k'' different treatment groups having size ''n<sub>i</sub>''  where ''i'' varies from 1 to ''k'', then it is assumed that the expected mean of each group is
+: <math>\operatorname{E}(\mu_i) = \mu + T_i</math>
+and the variance of each treatment group is unchanged from the population variance <math>\sigma^2</math>.
+Under the Null Hypothesis that the treatments have no effect, then each of the <math>T_i</math> will be zero.
+It is now possible to calculate three sums of squares:
+;Individual
+:<math>I = \sum x^2 </math>
+:<math>\operatorname{E}(I) = n\sigma^2 + n\mu^2</math>
+;Treatments
+:<math>T = \sum_{i=1}^k \left(\left(\sum x\right)^2/n_i\right)</math>
+:<math>\operatorname{E}(T) = k\sigma^2 + \sum_{i=1}^k n_i(\mu + T_i)^2</math>
+:<math>\operatorname{E}(T) = k\sigma^2 + n\mu^2 + 2\mu \sum_{i=1}^k (n_iT_i) + \sum_{i=1}^k n_i(T_i)^2</math>
+Under the null hypothesis that the treatments cause no differences and all the <math>T_i</math> are zero, the expectation simplifies to
+:<math>\operatorname{E}(T) = k\sigma^2 + n\mu^2.</math>
+;Combination
+:<math>C = \left(\sum x\right)^2/n</math>
+:<math>\operatorname{E}(C) = \sigma^2 + n\mu^2</math>
+===Sums of squared deviations===
+Under the null hypothesis, the difference of any pair of ''I'', ''T'', and ''C'' does not contain any dependency on <math>\mu</math>, only <math>\sigma^2</math>.
+:<math>\operatorname{E}(I - C) = (n - 1)\sigma^2</math> total squared deviations aka ''[[total sum of squares]]''
+:<math>\operatorname{E}(T - C) = (k - 1)\sigma^2</math> treatment squared deviations aka ''[[explained sum of squares]]''
+:<math>\operatorname{E}(I - T) = (n - k)\sigma^2</math> residual squared deviations aka ''[[residual sum of squares]]''
+The constants (''n''&nbsp;&minus;&nbsp;1), (''k''&nbsp;&minus;&nbsp;1), and (''n''&nbsp;&minus;&nbsp;''k'') are normally referred to as the number of [[degrees of freedom (statistics)|degrees of freedom]].
+===Example===
+In a very simple example, 5 observations arise from two treatments.  The first treatment gives three values 1, 2, and 3, and the second treatment gives two values 4, and 6.
+:<math>I = \frac{1^2}{1} + \frac{2^2}{1} + \frac{3^2}{1} + \frac{4^2}{1} + \frac{6^2}{1} = 66</math>
+:<math>T = \frac{(1 + 2 + 3)^2}{3} + \frac{(4 + 6)^2}{2} = 12 + 50 = 62</math>
+:<math>C = \frac{(1 + 2 + 3 + 4 + 6)^2}{5} = 256/5 = 51.2</math>
+Giving
+: Total squared deviations = 66 &minus; 51.2 = 14.8 with 4 degrees of freedom.
+: Treatment squared deviations = 62 &minus; 51.2 = 10.8 with 1 degree of freedom.
+: Residual squared deviations = 66 &minus; 62 = 4 with 3 degrees of freedom.
+==Two-way analysis of variance==
+The following hypothetical example gives the yields of 15 plants subject to two different environmental variations, and three different fertilisers.
+{| class="wikitable"
+|-
+!
+! Extra CO<sub>2</sub>
+! Extra humidity
+|-
+| No fertiliser
+| 7, 2, 1
+| 7, 6
+|-
+| Nitrate
+| 11, 6
+| 10, 7, 3
+|-
+| Phosphate
+| 5, 3, 4
+| 11, 4
+|}
+Five sums of squares are calculated:
+{| class="wikitable"
+|-
+! Factor
+! Calculation
+! Sum
+! <math>\sigma^2</math>
+|-
+| Individual
+| <math>7^2+2^2+1^2 + 7^2+6^2 + 11^2+6^2 + 10^2+7^2+3^2 + 5^2+3^2+4^2 + 11^2+4^2</math>
+| 641
+| 15
+|-
+| Fertiliser &times; Environment
+| <math>\frac{(7+2+1)^2}{3} + \frac{(7+6)^2}{2} + \frac{(11+6)^2}{2} + \frac{(10+7+3)^2}{3} + \frac{(5+3+4)^2}{3} + \frac{(11+4)^2}{2}</math>
+| 556.1667
+| 6
+|-
+| Fertiliser
+| <math>\frac{(7+2+1+7+6)^2}{5} + \frac{(11+6+10+7+3)^2}{5} + \frac{(5+3+4+11+4)^2}{5}</math>
+| 525.4
+| 3
+|-
+| Environment
+| <math>\frac{(7+2+1+11+6+5+3+4)^2}{8} + \frac{(7+6+10+7+3+11+4)^2}{7} </math>
+| 519.2679
+| 2
+|-
+| Composite
+| <math>\frac{(7+2+1+11+6+5+3+4+7+6+10+7+3+11+4)^2}{15} </math>
+| 504.6
+| 1
+|}
+Finally, the sums of squared deviations required for the [[analysis of variance]] can be calculated.
+{| class="wikitable"
+|-
+! Factor
+! Sum
+! <math>\sigma^2</math>
+! Total
+! Environment
+! Fertiliser
+! Fertiliser &times; Environment
+! Residual
+|-
+| Individual
+| 641
+| 15
+| 1
+|
+|
+|
+| 1
+|-
+| Fertiliser &times; Environment
+| 556.1667
+| 6
+|
+|
+|
+| 1
+| &minus;1
+|-
+| Fertiliser
+| 525.4
+| 3
+|
+|
+| 1
+| &minus;1
+|
+|-
+| Environment
+| 519.2679
+| 2
+|
+| 1
+|
+| &minus;1
+|
+|-
+| Composite
+| 504.6
+| 1
+| &minus;1
+| &minus;1
+| &minus;1
+| 1
+|
+|-
+|
+|
+|
+|
+|
+|
+|
+|
+|-
+| Squared deviations
+|
+|
+| 136.4
+| 14.668
+| 20.8
+| 16.099
+| 84.833
+|-
+| Degrees of freedom
+|
+|
+| 14
+| 1
+| 2
+| 2
+| 9
+|}
+==See also==
+* [[Variance decomposition]]
+* [[Errors and residuals in statistics]]
+* [[Absolute deviation]]
+==References==
+<References/>
+[[Category:Statistical deviation and dispersion]]
+[[Category:Analysis of variance]]

Semi-implicit Euler method: Difference between revisions

Revision as of 23:23, 29 January 2014

Sample variance

Partition — analysis of variance

Sums of squared deviations

Example

Two-way analysis of variance

See also

References

Navigation menu

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