Integrating the given expansion equation gives $R^{3/2}=t/t_0$, with the integration constant fixed by $R(t_0)=1$. Thus the scale factor and Hubble parameter are $R=(t/t_0{)}^{2/3}$ and $H=2/(3t)$. The Newtonian derivation of the flat Friedmann equation gives $H^2=8\pi G\rho /3$, soUse the comoving coordinate $\mathbf{q}=\mathbf{x}/R$ and the density contrast $\delta ={\rho }_1/\rho$. Linearized mass conservation, the peculiar-velocity Euler equation for fluid motion and the Poisson equation for the peculiar gravitational potential areDifferentiate the first equation, substitute the second and use the third. A Fourier component of comoving wavenumber $k$ obeysSince $c^2{k}^2/R^2=k^2{c}^2(t_0)t_0^{2\gamma -2/3}{t}^{-2\gamma +2/3}$, the normalization is $\Lambda =kc(t_0)t_0^{\gamma -1/3}$, as printed in the PDF. The resulting linear cosmological density perturbation equation isTo reduce it to a Bessel equation, put $\nu =\gamma -4/3$, $z=\Lambda /(\nu t^{\nu })$ and $\epsilon =t^{a}y(z)$. As $\dot{z}=-\nu z/t$, multiplying the substituted equation by $t^{2-a}$ givesTaking $a=-1/6$ makes the first two coefficients proportional to those in the Bessel equation, and the remaining constant is $-25/36$. HenceFor noninteger $\lambda$, the Bessel density modes of a polytropic expanding universe areThere is an exceptional case within the allowed interval: if $\lambda =n$ is an integer, $J_{-n}=(-1{)}^n{J}_n$, so these are not two independent solutions. The full solution then uses $t^{-1/6}[A{Y}_n(z)+B{J}_n(z)]$, with the Bessel function of the second kind providing the missing branch. For example $\gamma =29/18$ gives $\lambda =3$. A basis using $J_{\lambda },Y_{\lambda }$ is valid for every allowed exponent.

For early times, $z\to \infty$. The large-argument asymptotic expansion of the Bessel function of the first kind has oscillatory phase $z-\pi a/2-\pi /4$ and envelope $(2/(\pi z){)}^{1/2}$ for order $a$; the Bessel function of the second kind has the same envelope. ThusThis decreases because $3\gamma <5$. At late times $z\to 0$, the leading powers are $J_{\pm \lambda }(z)\propto z^{\pm \lambda }$ for noninteger order, giving $t^{-1}$ and $t^{2/3}$. The $Y_n$ branch gives the same growing power in the integer case. Therefore a generic perturbation grows as $t^{2/3}$; an initial condition with zero growing-mode coefficient instead has the decaying $t^{-1}$ mode.

To find the growth scale including its dependence on $\nu$, set $\epsilon =t^{2/3}h(t)$. The equation becomesSubstitute $h=1+a{\Lambda }^2{t}^{-2\nu }+\cdots$. The leading coefficient equation is $-2\nu (5/3-2\nu )a+1=0$. The pressure correction to a growing polytropic cosmological mode is consequentlySince $ck/R=\Lambda t^{-\nu -1}$ and $t^{-2}=6\pi G\rho$, the correction becomes small when $(ck/R{)}^2{t}^2$ is small compared with $2\nu (5/3-2\nu )$. The factor $2(5/3-2\nu )$ is between two and $10/3$, giving the stated order-of-magnitude onset conditionFor a well-developed asymptotic growing mode, replace $\lesssim$ by $\ll$. This is a Jeans instability interpretation: the physical acoustic frequency must be sufficiently low compared with the gravitational expansion scale, or equivalently the physical wavelength must be sufficiently long for self-gravity to overcome pressure support. It is an approximate condition for the dust-like growing branch, not an exact initial-condition-independent instant at which any perturbation begins increasing. The instantaneous sign change of the coefficient in the original equation instead occurs at $c^2{k}^2/R^2=4\pi G\rho$.

A linear filter of a stationary time series with real coefficients $(a_j{)}_{j\in \mathbb{Z}}$ satisfying ${\sum }_j|a_j|<\infty$ is $Y_t={\sum }_j{a}_j{X}_{t-j}$. Its filter generating function is the Laurent series $A(z)={\sum }_j{a}_j{z}^j$, absolutely convergent on $|z|=1$, and its filter transfer function is $H_A(\lambda )=A(e^{-i\lambda })$. For a one-sided filter this is a power series; a general two-sided filter need not converge on a larger annulus.

The filter theorem states that if $X$ is second-order stationary, then $Y$ exists in mean square, is second-order stationary, and has mean and covarianceIts spectral measure of a stationary time series is $d{F}_Y(\lambda )=|H_A(\lambda ){|}^{2}d{F}_X(\lambda )$. In particular, if $X$ has a time-series spectral density in the convention ${\gamma }_X(h)={\int }_{-\pi }^{\pi }{e}^{ih\lambda }{f}_X(\lambda )d\lambda$, thenTo prove existence, the $L^2$ norm of a tail of the filter is at most $\Vert X_0{\Vert }_2$ times the sum of the omitted $|a_j|$, which tends to zero. Hence finite partial sums converge in $L^2$. Expectations and covariances pass to the limit, and the covariance double sum is absolutely convergent because $|{\gamma }_X(h)|\le {\gamma }_X(0)$. Its formula depends only on $h$, proving stationarity. Insert the spectral representation of ${\gamma }_X(h-j+k)$ and interchange sums and integration; the bound ${\sum }_{j,k}|a_j{a}_k|F_X([-\pi ,\pi ])<\infty$ justifies the interchange. The resulting factor is ${\sum }_j{a}_j{e}^{-ij\lambda }{\sum }_k{a}_k{e}^{ik\lambda }=|A(e^{-i\lambda }){|}^2$. This proves the spectral statement, including the spectral density transformation under a linear filter without assuming that the input autocovariances are absolutely summable.

For an autoregressive moving-average model, use real polynomials normalized by ${\varphi }_0={\theta }_0=1$ and first work with the reduced noise-driven representation. If $\varphi$ has no zero on the unit circle, the rational function $C(z)=\theta (z)/\varphi (z)$ has a Laurent expansion in an annulus containing that circle, with absolutely summable coefficients. Filtering the white noise with it gives a stationary solution. Its time-series spectral density isConversely, for coprime $\varphi ,\theta$, a unit-circle zero of $\varphi$ would make this ratio nonintegrable, since the numerator does not vanish there; thus there is no finite-variance stationary noise-driven solution. A stationary solution can use a bilateral filter when autoregressive roots lie inside the unit circle. The stronger causal condition, meaning dependence only on present and past noise, requires every zero of $\varphi$ strictly outside the unit disk. Common factors must be handled before applying these conditions; a cancelled unit-root factor can also allow additional deterministic stationary components, which are outside the minimal noise-driven convention.

Identifiability means that different admissible parameters cannot produce the same stationary process law. In this zero-mean Gaussian setting, its law is determined by its covariance, equivalently its spectrum. Unrestricted ARMA coefficients are not identifiable from that law: if a moving-average factor has root $r$ inside the disk, then on $|z|=1$,Reflecting the root outside and multiplying the innovation variance by $|r{|}^{-2}$ preserves the spectrum, and therefore the Gaussian law. Common polynomial factors and arbitrary overall scaling give further redundancy. A regular canonical identifiable parameterization requires constant coefficients one, coprime polynomials of their stated minimal orders, all zeros of both $\varphi$ and $\theta$ outside the closed unit disk, and ${\sigma }^2>0$.

Here is a proof of canonical identifiability of Gaussian ARMA models. Suppose two such models have equal spectra and write their innovation standard deviations as $\sigma ,\tau >0$. The ratio $R(z)=\sigma \theta (z)\tilde{\varphi }(z)/[\tau \tilde{\theta }(z)\varphi (z)]$ and its reciprocal are analytic on a neighbourhood of the closed unit disk, with modulus one on its boundary. The maximum modulus principle applied to both makes $|R|=1$ throughout the disk, so $R$ is constant. Its value at zero is the positive number $\sigma /\tau$, forcing $R=1$ and $\sigma =\tau$. Therefore $\theta \tilde{\varphi }=\tilde{\theta }\varphi$; coprimality and the normalization at zero force equality of both polynomials. This proves the claimed parameter uniqueness within that canonical class.

In this class, $D(z)=\varphi (z)/\theta (z)$ is analytic on a disk of radius greater than one. Its Taylor series coefficients are absolutely summable, soThis is an invertible time-series representation, proved by composition of the stable filters. Causality makes ${\epsilon }_t$ orthogonal to past observations; invertibility makes past noises recoverable from past observations. Their closed past linear spans agree, so this noise is the linear innovation and the one-sided expansion $X_t={\sum }_{j\ge 0}{c}_j{\epsilon }_{t-j}$ is the Wold representation. Mere stationarity without the stated canonical restrictions does not imply invertibility of the originally written noise representation.

For this one-sided canonical representation, $C(z)=\theta (z)/\varphi (z)={\sum }_{j\ge 0}{c}_j{z}^j$. Comparing coefficients gives the ARMA coefficient recursionsThus for exactly the range asked for,The characteristic rates are reciprocals of the zeros of $\varphi$. Partial fractions therefore express the tail as sums of $P_r(k)r^k$, where $|r|<1$ and the degree of $P_r$ is at most one less than the corresponding root multiplicity. Hence the coefficients decay exponentially, with possible polynomial and damped oscillatory factors. If $p=0$, they instead vanish after $q$. For a noncausal or noninvertible original specification, the Wold coefficients refer to the canonical spectral factor; using the original driving noise as the Wold innovation without checking these restrictions would be incorrect.

The autocovariance is $\gamma (h)={\sigma }^2{\sum }_{j\ge 0}{c}_j{c}_{j+|h|}$. Analyticity of $C$ on a disk of radius $R>1$ gives $|c_j|\le M{R}^{-j}$ for a slightly smaller radius than its first pole, and consequently $|\gamma (h)|\le {\sigma }^2{M}^2{R}^{-|h|}/(1-R^{-2})$. Moreover, multiply the ARMA equation at time $t$ by $X_{t-h}$ and take expectations. For $h>q$, all noises ${\epsilon }_{t-s}$ with $0\le s\le q$ are later than $X_{t-h}$ and orthogonal to it. For $h>\max (p-1,q)$ this givesThe covariance tail therefore has the same characteristic exponential rates and multiplicity patterns as the filter coefficients, subject to possible cancellation of amplitudes. Pure moving averages have an exact covariance cutoff.

For the final process, factor the polynomials:The common factor has its zero at $3/2$, so it can be cancelled as a stable filter. Explicitly, let $Y_t=(1+B/3)X_t-{\epsilon }_t$; the original equation gives $Y_t=(2/3)Y_{t-1}$. The second moments of $Y_t$ are uniformly bounded because those of $X_t$ and ${\epsilon }_t$ are. Iterating gives $Y_t=(2/3{)}^m{Y}_{t-m}$, whose $L^2$ norm tends to zero as $m\to \infty$, so $Y_t=0$ in mean square. ThusSumming the covariance series givesThe process is a causal stationary autoregressive process of order one, with alternating-sign geometric correlations, and is invertible. The displayed ARMA(2,1) description is nonminimal because of its common factor; its minimal identifiable class is ARMA(1,0).

For a weakly stationary process with finite second moments, let $\mu =\mathbb{E}{X}_t$ and $\gamma (h)=\mathrm{Cov}(X_{t+h},X_t)$, independent of $t$. If $\gamma (0)>0$, its autocorrelation function is $\rho (h)=\gamma (h)/\gamma (0)$. Given observations $x_1,\dots ,x_N$, one usual sample autocovariance function isThe correlogram plots $\hat{\rho }(h)$ against lag $h$. The partial autocorrelation function at lag $h$ is the correlation between $X_t$ and $X_{t-h}$ after linearly projecting each onto the intervening observations. Its sample version can be calculated by solving the Yule-Walker regression equationsthen taking the last coefficient ${\hat{\varphi }}_{h,h}$. Equivalently, use the corresponding residual correlation from the sample regressions, with a consistent covariance convention.

For a minimal causal autoregressive model of order $p$, the population partial autocorrelation function is zero after lag $p$, while the autocorrelation function follows an exponentially decaying or damped oscillatory tail. For a moving-average model of order $q$, its autocovariance and hence autocorrelation are zero after lag $q$, because observations farther apart share no driving-noise terms; its partial correlations usually tail off. These order identification by autocorrelation cutoffs patterns suggest AR or MA candidate orders. Sample cutoffs are approximate and should be judged with sampling uncertainty and residual diagnostics; the familiar white-noise bands near $\pm 1.96/\sqrt{N}$ are a null approximation, not universal confidence limits for a dependent series.

For the trend problem, write a symmetric estimator asReproducing arbitrary constant and quadratic trends requires $2a+2b+c=1$ and $8a+2b=0$; the linear-trend condition holds automatically by symmetry. These constraints follow by expanding $T_{t+j}=T_t+j(b_0+2c_{0}t)+c_0{j}^2$ for a trend $T_t=a_0+b_{0}t+c_0{t}^2$. Thus $b=-4a$, $c=1+6a$. Since the errors are uncorrelated and the estimator is unbiased, its mean squared error isThe strictly convex quadratic is minimized at $a=-3/35$, giving the five-point symmetric quadratic trend smootherThis weighted moving average allows negative coefficients; a convex average could not cancel the quadratic moment.

For three points, symmetry gives weights $(a,b,a)$. Unbiasedness for every quadratic requires $2a+b=1$ and $2a=0$, so the only solution is $(0,1,0)$. Thus there is no nontrivial three-point quadratic-preserving smoother. The literal claim that there is no unbiased estimator at all is false, since ${\hat{T}}_t=X_t$ is unbiased with error variance ${\sigma }^2$. This is the three-point quadratic reproduction forces the identity filter distinction.

For seasonal extraction by a centred moving average, use the additive decomposition $X_t=T_t+S_t+{\epsilon }_t$, where $S_t$ has period $m$ and its values over one period sum to zero. An average covering a full period removes the seasonal component and approximately retains a slowly changing trend. If $m$ is odd, take the centred length-$m$ average; if $m$ is even, centre it by averaging two adjacent length-$m$ averages, giving half weights at the two endpoints. Subtract this trend estimate from the data, average residuals within each seasonal phase over different cycles, and centre the resulting phase estimates to sum to zero. For a multiplicative decomposition, use ratios and normalize the phase estimates to average one.

Successive moving averages compose by convolution of their weights. Even an initially uncorrelated noise series becomes correlated: for weights $w_j$, the filtered noise has covariance ${\sigma }^2{\sum }_j{w}_j{w}_{j+h}$. Further smoothing changes this covariance again and can turn random variation into apparently persistent patterns while obscuring genuine short-scale features. Spectrally, repeated use of a transfer function $H$ multiplies the spectrum by $|H{|}^{2r}$ after $r$ applications, so repeated smoothing is not harmless repetition of the same estimate.

In simple exponential smoothing, choose $0<\alpha <1$ and update ${\ell }_t=\alpha X_t+(1-\alpha ){\ell }_{t-1}$. Iteration gives ${\ell }_t=\alpha {\sum }_{j=0}^{t-1}(1-\alpha {)}^j{X}_{t-j}+(1-\alpha {)}^t{\ell }_0$, so older observations receive exponentially decreasing weights. Forecast ${\hat{X}}_{t+h\mid t}={\ell }_t$ for every $h\ge 1$. This is appropriate for a locally stable level with no persistent trend or seasonality.

For a local linear trend, double exponential smoothing supplies level and slope. Holt's linear trend method usesAnother standard convention is Brown's double exponential smoothing: smooth twice with the same gain, $s_t^{(1)}=\alpha X_t+(1-\alpha )s_{t-1}^{(1)}$ and $s_t^{(2)}=\alpha s_t^{(1)}+(1-\alpha )s_{t-1}^{(2)}$, then put $a_t=2s_t^{(1)}-s_t^{(2)}$, $b_t=\alpha (s_t^{(1)}-s_t^{(2)})/(1-\alpha )$ and forecast $a_t+h{b}_t$. These variants handle a nonseasonal trend; seasonal effects need separate modelling, and a long-horizon linear extrapolation assumes the current local slope persists.

Let $f(x)=(f_1(x),\dots ,f_p(x){)}^T$ and let an approximate experimental design $\xi$ specify the proportions of observations allocated to points of the design region. Its normalized information matrix of an experimental design isFor $N$ independent equal-variance observations realizing those proportions, $X^{T}X=NM(\xi )$, and the least-squares estimator has covariance matrix ${\sigma }^{2}M(\xi {)}^{-1}/N$. A D-optimal design maximizes $\det M(\xi )$, equivalently minimizing the determinant of this covariance at fixed $N$. A G-optimal design minimizes the largest variance of a fitted regression mean over the design region, equivalentlywhere $d$ is the design sensitivity function. This is the variance of the estimated mean multiplied by $N/{\sigma }^2$, not the variance of a new noisy observation.

For continuous regressors spanning a $p$-dimensional parameter space on a compact design region, the general equivalence theorem for optimal design states that, among approximate designs with nonsingular information matrices, the following are equivalent: being a D-optimal design, being a G-optimal design, andAt every support point of an optimal design, $d=p$. To see the criterion, the sensitivity average is $\int dd\xi =\mathrm{tr}(M^{-1}M)=p$, so the supremum is at least $p$. Adding an infinitesimal allocation at $x$ has directional derivative $d(x,\xi )-p$ for $\log \det M$. At a D-optimum each derivative is nonpositive. Conversely, if all sensitivities are at most $p$, the tangent bound for the concave function $\log \det M$ shows that no competing design increases it. A nonsingular D-optimum exists by the spanning assumption and compactness of the information-matrix set, so the smallest possible G-value is also $p$. Equality at support points follows from the average. The theorem refers to the full approximate-design class; constrained integer allocations or mandated augmentations require separate comparison.

For the linear mean ${\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$, take $f(x)=(1,x_1,x_2{)}^T$. Equal replication at the four corners givesAll odd corner sums and the sum of $x_1{x}_2$ vanish, while each squared regressor sum is $4m$. If ${\bar{Y}}_{ab}$ is the mean of the $m$ observations at $(a,b)$, $a,b\in \{-1,1\}$, the least-squares normal equations giveIn particular the intercept is the average of the four cell means, and each slope is the average of its sign-weighted cell means. Independence and equal error variance suffice for this covariance formula; normal errors are not needed for it.

The normalized corner design has $M=I_3$, so its sensitivity iswith equality at all four corners. Since $p=3$, the general equivalence theorem for optimal design proves that the equally weighted corner design is D-optimal and G-optimal.

For the augmentation, all $m$ added observations share the regressor $f=(1,x_1,x_2{)}^T$. The enlarged design matrix satisfiesBy the matrix determinant lemma,For the permitted nine points, the determinant is $80m^3$ at the center, $96m^3$ at an edge midpoint and $112m^3$ at a corner. ThereforeAll four corners tie for the best mandated single-point augmentation. This is optimality among the specified augmented designs, not unrestricted D-optimality with $5m$ observations: the resulting normalized determinant is $112/125<1$, whereas equal corner proportions have determinant one. The restriction that the extra $m$ runs all occur at one point is what prevents recovering equal proportions.

Code each factor level by $x_j\in \{-1,1\}$. A nonconstant factorial contrast indexed by a nonempty factor set $A$ has sign column $g_A(x)={\prod }_{j\in A}{x}_j$. Flipping any coordinate in $A$ reverses that sign and pairs the treatment combinations, so exactly $2^{m-1}$ have each sign. Allocating the two signs to two experimental blocks therefore divides the full factorial design into equal experimental blocks. The contrast is confounded with experimental blocks because its column is constant within each experimental block: changing its coefficient and compensating by the two experimental block intercepts leaves every fitted value unchanged. It has no information separate from experimental block differences. Randomize the allocation of the two groups to days and the run order within days.

Write $A,B,C$ for temperature, pressure and catalyst, respectively, with low levels and catalyst one coded minus. In the supplied allocation, $ABC=-1$ on day one and $ABC=1$ on day two. Thus the three-factor interaction $ABC$ is confounded with days. Every other nonconstant factorial column is orthogonal to the day contrast, because its product with $ABC$ is a nonconstant sign column on the full cube and consequently sums to zero.

If all interactions are negligible, the fitted model contains an intercept, one day contrast and the three main effects. The corrected total statistical degrees of freedom partition asThe three residual statistical degrees of freedom are the unused two-factor interaction columns $AB,AC,BC$, pooled as error under the stated negligible-interaction assumption. There is no independent pure-error replication of treatment combinations.

For four days with two runs each, all three main effects can be estimated. One regular allocation pairs each treatment combination with its complete opposite:The block-generating columns can be $AB$ and $AC$; their product $BC$ is also constant in experimental blocks. Thus $AB,AC,BC$ span the three experimental block contrasts, while $A,B,C$ and $ABC$ vary within experimental blocks. The main effects remain mutually orthogonal and separate from the experimental block effects; with an additive main-effect model there is one residual degree of freedom, represented by $ABC$.

For the further request about $AC$, the distinction between regular confounding and unrestricted pairing is important. A regular four-block allocation cannot retain all three main effects and $AC$. Its three confounded columns form a rank-two defining subgroup. To preserve main effects, these columns must lie in $\{AB,AC,BC,ABC\}$. If the subgroup includes $ABC$ and a two-factor column, their product is a main-effect column, which is forbidden. Hence the only permissible nonidentity subgroup is $\{AB,AC,BC\}$, and it necessarily confounds $AC$. This proves impossibility in the usual regular factorial-blocking framework, rather than relying on a count of available statistical degrees of freedom.

The literal request does not explicitly restrict arrangements to regular experimental blocks. If arbitrary pairings are permitted, a nonregular construction can estimate $A,B,C,AC$ together, assuming the other interactions are omitted from the model. An explicit allocation isEvery treatment combination occurs exactly once. Fit one unrestricted intercept per day and the four coefficients ${\beta }_A,{\beta }_B,{\beta }_C,{\beta }_{AC}$. By estimability from within-block differences, subtracting the second run from the first in each experimental block removes all day effects and yields the coefficient matrixThus all four coefficients are uniquely determined from the four day differences. In this broader design class the answer is yes, under the reduced four-effect model. The resulting fit is saturated: four experimental block intercepts plus four treatment coefficients use all eight observations, leaving no residual estimate of error variance. Replication or external error information would be needed for inference. This does not contradict the impossibility for regular orthogonal confounding, and it does not claim simultaneous unbiased estimation when arbitrary additional interactions are retained.

A randomized complete block design groups similar experimental units into experimental blocks and places every treatment once in each experimental block, independently randomizing treatment allocation within experimental blocks. Blocking removes background between-block variation from within-block treatment comparisons. A balanced incomplete block design has $v$ treatments and $b$ experimental blocks of size $k<v$, with each treatment occurring in $r$ experimental blocks and every distinct treatment pair together in $\lambda$ experimental blocks. Counting treatment occurrences and co-occurrences gives $vr=bk$ and $r(k-1)=\lambda (v-1)$.

A Latin square of order $q$ places each of $q$ treatment symbols once in every row and every column of a $q\times q$ array. This controls two nuisance directions. A Graeco-Latin square superposes two mutually orthogonal Latin squares: each square is Latin, and each ordered pair of symbols from the two squares occurs exactly once. The two symbol factors can then be fitted together with row and column effects. Random permutations of row labels, column labels and treatment symbols provide a suitable randomized allocation without destroying these balances.

For the operator experiment, fit the additive normal linear modelHere $r_i,c_j$ account for the two fabric directions, and ${\tau }_o$ is the operator effect. Additivity, independent equal-variance errors and the normal assumption supply the exact F-test; there is only one observation per cell, so an unrestricted interaction model is not available.

In the analysis of variance for a Latin square, the three centered factor spaces each have dimension $4-1=3$. Together with the intercept the model has rank ten, leaving $16-10=6$ residual statistical degrees of freedom. The corrected total has $16-1=15$. The completed calculation isUnder the null hypothesis ${\tau }_1=\cdots ={\tau }_4=0$, the operator F-statistic isThe upper five-percent critical value for $F_{3,6}$ is $4.76$, so the operator effect is not significant at five percent in this model.

The decision is independent of the order of fitting rows, columns and operators. To prove the required orthogonality, let $u_i$ be any centered row coefficients and $w_o$ any centered operator coefficients. Their observation-space inner product is ${\sum }_{i,j}{u}_i{w}_{o(i,j)}={\sum }_i{u}_i{\sum }_o{w}_o=0$, since every operator occurs once in each row. The same calculation applies to columns and operators, and all row-column pairs occur once as well. Thus the three centered spaces are mutually orthogonal; their projections commute and sequential sums of squares equal adjusted sums of squares in every fitting order.

With machines included, the layout is a Graeco-Latin square: the machine symbols are Latin, and all sixteen operator-machine pairs occur exactly once. Fitin addition to the earlier constraints. The machine space contributes three new statistical degrees of freedom and is orthogonal to the previous factor spaces, so the operator sum of squares remains $18.25$. The residual sum of squares and statistical degrees of freedom becomeHenceThis exceeds $F_{3,3}(0.05)=9.28$ but not $F_{3,3}(0.01)=29.46$. Therefore the operator effect is now significant at five percent, but not at one percent. Adjustment for machines has removed substantial residual variation without altering the operator numerator. This illustrates why a balanced nuisance factor can change precision and significance even when it does not change the estimated treatment contrasts.

The calculation is principal component analysis on a correlation matrix, rather than on an unscaled covariance matrix. For each of the eleven rating variables, subtract its column mean and divide by its column standard deviation. If $Z$ is the resulting $43\times 11$ matrix, use the divisor convention of the fitted object to writeHere the scaling standard deviations use divisor $43$, consistently with this convention. Using sample standard deviations with divisor $42$ gives the same correlation matrix and directions, with the corresponding score scale. Every diagonal entry of $R$ is one, so ${\sum }_k{\lambda }_k=\mathrm{tr}R=11$.

A principal component is the linear combination of the standardized ratings along $a_k$. Its principal component score for judge $i$ is $t_{ik}=z_i^T{a}_k$, and its variance is ${\lambda }_k$. The displayed component standard deviations are $\sqrt{{\lambda }_k}$; the explained variance of a principal component is ${\lambda }_k/11$. In particular,The first component explains about $92.122\%$ of the standardized variation, and the first two jointly explain $95.982\%$. Components three, four and five add about $2.321\%$, $0.830\%$ and $0.339\%$. The cumulative line adds these proportions successively; it is not the proportion explained by that component alone. Retaining two components discards about $4.018\%$ of the total standardized squared variation.

The displayed principal component loadings are the unit eigenvector coefficients $a_{jk}$. The first direction has positive, broadly similar weights for all ratings. It measures an overall favorable-rating tendency: a judge high on most ratings has a high first score. The second direction compares integrity and demeanour, with large positive coefficients, against such attributes as decision promptness, case-flow management and physical ability, with negative coefficients. Thus among similarly rated judges, a positive second score indicates relatively stronger integrity/demeanour compared with those operational ratings. This is a description of a data direction, not a causal interpretation. Multiplying a whole eigenvector and its scores by minus one reflects the display but changes nothing substantive.

These principal component loadings are not themselves variable-component correlations. With standardized variables,For example, the integrity correlation with component one is approximately $3.1833\times 0.289\simeq 0.920$. Blank loadings in the printed second column indicate suppressed small entries, not exact zeros. The complete unrounded loading vectors in the fitted object must be used to calculate scores.

A two-dimensional representation plots the 43 pairswith each point labeled by its judge. In the fitted software object these are the first two columns of the score matrix. Nearby points have similar projections of their standardized ratings. The rank-two reconstruction ${\hat{z}}_i=t_{i1}{a}_1+t_{i2}{a}_2$ is the orthogonal projection onto the leading principal plane and minimizes the total squared reconstruction error over all two-dimensional linear subspaces. To see this, for any orthonormal pair of directions the captured variance is the sum of its two Rayleigh quotients, at most ${\lambda }_1+{\lambda }_2$ by diagonalizing $R$; the remaining variance is the total minus that captured sum. Distances in the plot omit the smaller components, so a close projected pair need not be equally close in every original rating. The actual 43 coordinates cannot be recovered from the summary alone; the supplied data matrix or fitted score matrix is needed.

The printed descriptive list omits INTG, although the loading table includes it. INTG denotes integrity and supplies the eleventh variable. The standard data set also contains a contacts variable; the stated eleven-rating analysis uses the ratings, not contacts.