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Let $\hat{\beta }=(X^{T}X{)}^{-1}{X}^{T}Y$, $H=X(X^{T}X{)}^{-1}{X}^T$, ${\hat{y}}_i=x_i^T\hat{\beta }$, and $e_i=y_i-{\hat{y}}_i$. The horizontal axis of the residual-versus-fitted plot is ${\hat{y}}_i$, the fitted values, and the vertical axis is $e_i$, the regression residual, in millimetres. Under the normal linear model, $\mathrm{Var}(e_i)={\sigma }^2(1-h_{ii})$, where $h_{ii}$ is the regression leverage.

In the quantile-quantile plot, the vertical values are the ordered standardized regression residualsand the horizontal values are corresponding theoretical quantiles ${\Phi }^{-1}(p_k)$ of the standard normal distribution, with plotting positions such as $p_k=(k-1/2)/n$. The exact plotting-position convention has little practical effect here. Under Gaussian errors these points should approximately follow a straight line; they are not independent because fitting induces residual correlations.

The main visible concern is curvature in the conditional mean. The red smooth in the residual-versus-fitted plot descends from positive residuals at low fitted values, becomes negative in the middle, and rises again at high fitted values. This suggests the strictly linear time effects may be inadequate. There is no clear monotone widening of the residual scatter, so strong heteroscedasticity is not apparent. The quantile-quantile plot has modest tail deviations and a few labelled observations, but does not show a dramatic departure from normality. These plots cannot establish independence across days or laboratories; check residuals against day, laboratory, and sampling order as well. A large coefficient of determination does not remove the visible mean-model concern.

Check that the chosen mean structure adequately describes the transformed response. Examine the residual-versus-fitted plot and residuals versus each predictor: systematic curvature suggests omitted nonlinear effects or an interaction term. Plot residual spread against fitted values, for example using a scale-location plot, to assess homoscedasticity. Compare standardized residuals with a normal distribution using a quantile-quantile plot, especially for finite-sample Student's t-distribution and F-test inference.

Also check independence using the sampling design and residuals versus time, collection site, or other groups; spatially related photovoltaic systems may have correlated errors that are invisible in a residual-versus-fitted plot. Investigate large standardized regression residuals, high regression leverage, and influential observations using Cook's distance. Check that the design matrix has full rank and that severe multicollinearity is not making estimates unstable. Reassess whether the response transformation and retained predictors improve these diagnostics and prediction; a coefficient p-value alone cannot establish model adequacy. If observations are independent only conditional on site effects, use an appropriate dependence model rather than treating correlated systems as independent replicates.

For the residual-versus-fitted plot, compute fitted values ${\hat{y}}_i={\hat{\beta }}_0+{\hat{\beta }}_1{x}_{i1}+{\hat{\beta }}_2{x}_{i2}$ and regression residuals $e_i=y_i-{\hat{y}}_i$; the plotted point is $({\hat{y}}_i,e_i)$.

For the normal Q-Q plot, let $H=X(X^{T}X{)}^{-1}{X}^T$ be the hat matrix for the full three-column design matrix and let $s^2=\mathrm{RSS}/(n-3)$. Form the standardized regression residualsand order them as $r_{(1)}\le \cdots \le r_{(n)}$. R plots $r_{(j)}$ against ${\Phi }^{-1}(p_j)$, where $p_j$ are its plotting positions. For $n>10$, $p_j=(j-1/2)/n$; $\Phi$ is the standard normal distribution function. This standardization is specified in R's diagnostic-plot documentation.

The original PDF shows a residual cloud around zero without compelling systematic curvature or a clear funnel, and a Q-Q pattern approximately following the reference line. Observations 81 and 83 are conspicuous positive tail points, with a smaller lower-tail departure. The plots show no decisive violation of linearity, constant variance or normality, but the extreme residuals merit inspection. They do not test independence across observations and cannot establish that all model assumptions hold.

A quantile-quantile plot compares ordered observations with quantiles of a reference distribution, or with quantiles of another sample. Approximate linearity supports agreement up to location and scale. A regression normal Q-Q plot compares ordered standardized regression residuals with standard normal distribution quantiles.

A scale-location plot compares the square root of the absolute standardized regression residual with the fitted values. Systematic changes in level or spread suggest heteroscedasticity. It assesses the error scale, while a residual-versus-fitted plot primarily reveals mean patterns as well as changing spread.