Notice: This put up is a condensed model of a chapter from half three of the forthcoming ebook, Deep Studying and Scientific Computing with R torch. Half three is devoted to scientific computation past deep studying. All through the ebook, I concentrate on the underlying ideas, striving to elucidate them in as “verbal” a means as I can. This doesn’t imply skipping the equations; it means taking care to elucidate why they’re the best way they’re.
How do you compute linear least-squares regression? In R, utilizing lm(); in torch, there may be linalg_lstsq().
The place R, typically, hides complexity from the person, high-performance computation frameworks like torch are likely to ask for a bit extra effort up entrance, be it cautious studying of documentation, or taking part in round some, or each. For instance, right here is the central piece of documentation for linalg_lstsq(), elaborating on the driver parameter to the perform:
`driver` chooses the LAPACK/MAGMA perform that shall be used.
For CPU inputs the legitimate values are 'gels', 'gelsy', 'gelsd, 'gelss'.
For CUDA enter, the one legitimate driver is 'gels', which assumes that A is full-rank.
To decide on one of the best driver on CPU take into account:
- If A is well-conditioned (its situation quantity just isn't too massive), or you don't thoughts some precision loss:
- For a basic matrix: 'gelsy' (QR with pivoting) (default)
- If A is full-rank: 'gels' (QR)
- If A just isn't well-conditioned:
- 'gelsd' (tridiagonal discount and SVD)
- However if you happen to run into reminiscence points: 'gelss' (full SVD).Whether or not you’ll must know this may depend upon the issue you’re fixing. However if you happen to do, it definitely will assist to have an thought of what’s alluded to there, if solely in a high-level means.
In our instance downside under, we’re going to be fortunate. All drivers will return the identical consequence – however solely as soon as we’ll have utilized a “trick”, of types. The ebook analyzes why that works; I gained’t do this right here, to maintain the put up fairly brief. What we’ll do as a substitute is dig deeper into the assorted strategies utilized by linalg_lstsq(), in addition to a number of others of widespread use.
The plan
The way in which we’ll set up this exploration is by fixing a least-squares downside from scratch, making use of assorted matrix factorizations. Concretely, we’ll method the duty:
-
By way of the so-called regular equations, probably the most direct means, within the sense that it instantly outcomes from a mathematical assertion of the issue.
-
Once more, ranging from the conventional equations, however making use of Cholesky factorization in fixing them.
-
But once more, taking the conventional equations for a degree of departure, however continuing by way of LU decomposition.
-
Subsequent, using one other sort of factorization – QR – that, along with the ultimate one, accounts for the overwhelming majority of decompositions utilized “in the actual world”. With QR decomposition, the answer algorithm doesn’t begin from the conventional equations.
-
And, lastly, making use of Singular Worth Decomposition (SVD). Right here, too, the conventional equations aren’t wanted.
Regression for climate prediction
The dataset we’ll use is obtainable from the UCI Machine Studying Repository.
Rows: 7,588
Columns: 25
$ station 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,…
$ Date 2013-06-30, 2013-06-30,…
$ Present_Tmax 28.7, 31.9, 31.6, 32.0, 31.4, 31.9,…
$ Present_Tmin 21.4, 21.6, 23.3, 23.4, 21.9, 23.5,…
$ LDAPS_RHmin 58.25569, 52.26340, 48.69048,…
$ LDAPS_RHmax 91.11636, 90.60472, 83.97359,…
$ LDAPS_Tmax_lapse 28.07410, 29.85069, 30.09129,…
$ LDAPS_Tmin_lapse 23.00694, 24.03501, 24.56563,…
$ LDAPS_WS 6.818887, 5.691890, 6.138224,…
$ LDAPS_LH 69.45181, 51.93745, 20.57305,…
$ LDAPS_CC1 0.2339475, 0.2255082, 0.2093437,…
$ LDAPS_CC2 0.2038957, 0.2517714, 0.2574694,…
$ LDAPS_CC3 0.1616969, 0.1594441, 0.2040915,…
$ LDAPS_CC4 0.1309282, 0.1277273, 0.1421253,…
$ LDAPS_PPT1 0.0000000, 0.0000000, 0.0000000,…
$ LDAPS_PPT2 0.000000, 0.000000, 0.000000,…
$ LDAPS_PPT3 0.0000000, 0.0000000, 0.0000000,…
$ LDAPS_PPT4 0.0000000, 0.0000000, 0.0000000,…
$ lat 37.6046, 37.6046, 37.5776, 37.6450,…
$ lon 126.991, 127.032, 127.058, 127.022,…
$ DEM 212.3350, 44.7624, 33.3068, 45.7160,…
$ Slope 2.7850, 0.5141, 0.2661, 2.5348,…
$ `Photo voltaic radiation` 5992.896, 5869.312, 5863.556,…
$ Next_Tmax 29.1, 30.5, 31.1, 31.7, 31.2, 31.5,…
$ Next_Tmin 21.2, 22.5, 23.9, 24.3, 22.5, 24.0,… The way in which we’re framing the duty, almost every little thing within the dataset serves as a predictor. As a goal, we’ll use Next_Tmax, the maximal temperature reached on the next day. This implies we have to take away Next_Tmin from the set of predictors, as it might make for too highly effective of a clue. We’ll do the identical for station, the climate station id, and Date. This leaves us with twenty-one predictors, together with measurements of precise temperature (Present_Tmax, Present_Tmin), mannequin forecasts of assorted variables (LDAPS_*), and auxiliary data (lat, lon, and `Photo voltaic radiation`, amongst others).
Notice how, above, I’ve added a line to standardize the predictors. That is the “trick” I used to be alluding to above. To see what occurs with out standardization, please take a look at the ebook. (The underside line is: You would need to name linalg_lstsq() with non-default arguments.)
For torch, we cut up up the information into two tensors: a matrix A, containing all predictors, and a vector b that holds the goal.
[1] 7588 21Now, first let’s decide the anticipated output.
Setting expectations with lm()
If there’s a least squares implementation we “consider in”, it certainly have to be lm().
Name:
lm(formulation = Next_Tmax ~ ., information = weather_df)
Residuals:
Min 1Q Median 3Q Max
-1.94439 -0.27097 0.01407 0.28931 2.04015
Coefficients:
Estimate Std. Error t worth Pr(>|t|)
(Intercept) 2.605e-15 5.390e-03 0.000 1.000000
Present_Tmax 1.456e-01 9.049e-03 16.089 With an defined variance of 78%, the forecast is working fairly effectively. That is the baseline we wish to test all different strategies in opposition to. To that goal, we’ll retailer respective predictions and prediction errors (the latter being operationalized as root imply squared error, RMSE). For now, we simply have entries for lm():
rmse perform(y_true, y_pred) {
(y_true - y_pred)^2 %>%
sum() %>%
sqrt()
}
all_preds information.body(
b = weather_df$Next_Tmax,
lm = match$fitted.values
)
all_errs information.body(lm = rmse(all_preds$b, all_preds$lm))
all_errs lm
1 40.8369Utilizing torch, the short means: linalg_lstsq()
Now, for a second let’s assume this was not about exploring totally different approaches, however getting a fast consequence. In torch, we now have linalg_lstsq(), a perform devoted particularly to fixing least-squares issues. (That is the perform whose documentation I used to be citing, above.) Similar to we did with lm(), we’d most likely simply go forward and name it, making use of the default settings:
b lm lstsq
7583 -1.1380931 -1.3544620 -1.3544616
7584 -0.8488721 -0.9040997 -0.9040993
7585 -0.7203294 -0.9675286 -0.9675281
7586 -0.6239224 -0.9044044 -0.9044040
7587 -0.5275154 -0.8738639 -0.8738635
7588 -0.7846007 -0.8725795 -0.8725792Predictions resemble these of lm() very carefully – so carefully, in actual fact, that we might guess these tiny variations are simply attributable to numerical errors surfacing from deep down the respective name stacks. RMSE, thus, must be equal as effectively:
lm lstsq
1 40.8369 40.8369It’s; and it is a satisfying final result. Nonetheless, it solely actually took place attributable to that “trick”: normalization. (Once more, I’ve to ask you to seek the advice of the ebook for particulars.)
Now, let’s discover what we are able to do with out utilizing linalg_lstsq().
Least squares (I): The traditional equations
We begin by stating the aim. Given a matrix, (mathbf{A}), that holds options in its columns and observations in its rows, and a vector of noticed outcomes, (mathbf{b}), we wish to discover regression coefficients, one for every characteristic, that permit us to approximate (mathbf{b}) in addition to attainable. Name the vector of regression coefficients (mathbf{x}). To acquire it, we have to clear up a simultaneous system of equations, that in matrix notation seems as
[
mathbf{Ax} = mathbf{b}
]
If (mathbf{A}) have been a sq., invertible matrix, the answer may instantly be computed as (mathbf{x} = mathbf{A}^{-1}mathbf{b}). It will infrequently be attainable, although; we’ll (hopefully) all the time have extra observations than predictors. One other method is required. It instantly begins from the issue assertion.
After we use the columns of (mathbf{A}) for (mathbf{Ax}) to approximate (mathbf{b}), that approximation essentially is within the column house of (mathbf{A}). (mathbf{b}), however, usually gained’t be. We would like these two to be as shut as attainable. In different phrases, we wish to reduce the gap between them. Selecting the 2-norm for the gap, this yields the target
[
minimize ||mathbf{Ax}-mathbf{b}||^2
]
This distance is the (squared) size of the vector of prediction errors. That vector essentially is orthogonal to (mathbf{A}) itself. That’s, after we multiply it with (mathbf{A}), we get the zero vector:
[
mathbf{A}^T(mathbf{Ax} – mathbf{b}) = mathbf{0}
]
A rearrangement of this equation yields the so-called regular equations:
[
mathbf{A}^T mathbf{A} mathbf{x} = mathbf{A}^T mathbf{b}
]
These could also be solved for (mathbf{x}), computing the inverse of (mathbf{A}^Tmathbf{A}):
[
mathbf{x} = (mathbf{A}^T mathbf{A})^{-1} mathbf{A}^T mathbf{b}
]
(mathbf{A}^Tmathbf{A}) is a sq. matrix. It nonetheless may not be invertible, by which case the so-called pseudoinverse could be computed as a substitute. In our case, this won’t be wanted; we already know (mathbf{A}) has full rank, and so does (mathbf{A}^Tmathbf{A}).
Thus, from the conventional equations we now have derived a recipe for computing (mathbf{b}). Let’s put it to make use of, and evaluate with what we obtained from lm() and linalg_lstsq().
AtA A$t()$matmul(A)
Atb A$t()$matmul(b)
inv linalg_inv(AtA)
x inv$matmul(Atb)
all_preds$neq as.matrix(A$matmul(x))
all_errs$neq rmse(all_preds$b, all_preds$neq)
all_errs lm lstsq neq
1 40.8369 40.8369 40.8369Having confirmed that the direct means works, we might permit ourselves some sophistication. 4 totally different matrix factorizations will make their look: Cholesky, LU, QR, and Singular Worth Decomposition. The aim, in each case, is to keep away from the costly computation of the (pseudo-) inverse. That’s what all strategies have in widespread. Nonetheless, they don’t differ “simply” in the best way the matrix is factorized, but additionally, in which matrix is. This has to do with the constraints the assorted strategies impose. Roughly talking, the order they’re listed in above displays a falling slope of preconditions, or put in another way, a rising slope of generality. As a result of constraints concerned, the primary two (Cholesky, in addition to LU decomposition) shall be carried out on (mathbf{A}^Tmathbf{A}), whereas the latter two (QR and SVD) function on (mathbf{A}) instantly. With them, there by no means is a must compute (mathbf{A}^Tmathbf{A}).
Least squares (II): Cholesky decomposition
In Cholesky decomposition, a matrix is factored into two triangular matrices of the identical measurement, with one being the transpose of the opposite. This generally is written both
[
mathbf{A} = mathbf{L} mathbf{L}^T
] or
[
mathbf{A} = mathbf{R}^Tmathbf{R}
]
Right here symbols (mathbf{L}) and (mathbf{R}) denote lower-triangular and upper-triangular matrices, respectively.
For Cholesky decomposition to be attainable, a matrix needs to be each symmetric and optimistic particular. These are fairly robust circumstances, ones that won’t usually be fulfilled in observe. In our case, (mathbf{A}) just isn’t symmetric. This instantly implies we now have to function on (mathbf{A}^Tmathbf{A}) as a substitute. And since (mathbf{A}) already is optimistic particular, we all know that (mathbf{A}^Tmathbf{A}) is, as effectively.
In torch, we receive the Cholesky decomposition of a matrix utilizing linalg_cholesky(). By default, this name will return (mathbf{L}), a lower-triangular matrix.
# AtA = L L_t
AtA A$t()$matmul(A)
L linalg_cholesky(AtA)Let’s test that we are able to reconstruct (mathbf{A}) from (mathbf{L}):
LLt L$matmul(L$t())
diff LLt - AtA
linalg_norm(diff, ord = "fro")torch_tensor
0.00258896
[ CPUFloatType{} ]Right here, I’ve computed the Frobenius norm of the distinction between the unique matrix and its reconstruction. The Frobenius norm individually sums up all matrix entries, and returns the sq. root. In principle, we’d prefer to see zero right here; however within the presence of numerical errors, the result’s ample to point that the factorization labored wonderful.
Now that we now have (mathbf{L}mathbf{L}^T) as a substitute of (mathbf{A}^Tmathbf{A}), how does that assist us? It’s right here that the magic occurs, and also you’ll discover the identical sort of magic at work within the remaining three strategies. The thought is that attributable to some decomposition, a extra performant means arises of fixing the system of equations that represent a given process.
With (mathbf{L}mathbf{L}^T), the purpose is that (mathbf{L}) is triangular, and when that’s the case the linear system could be solved by easy substitution. That’s finest seen with a tiny instance:
[
begin{bmatrix}
1 & 0 & 0
2 & 3 & 0
3 & 4 & 1
end{bmatrix}
begin{bmatrix}
x1
x2
x3
end{bmatrix}
=
begin{bmatrix}
1
11
15
end{bmatrix}
]
Beginning within the high row, we instantly see that (x1) equals (1); and as soon as we all know that it’s easy to calculate, from row two, that (x2) have to be (3). The final row then tells us that (x3) have to be (0).
In code, torch_triangular_solve() is used to effectively compute the answer to a linear system of equations the place the matrix of predictors is lower- or upper-triangular. A further requirement is for the matrix to be symmetric – however that situation we already needed to fulfill so as to have the ability to use Cholesky factorization.
By default, torch_triangular_solve() expects the matrix to be upper- (not lower-) triangular; however there’s a perform parameter, higher, that lets us appropriate that expectation. The return worth is an inventory, and its first merchandise incorporates the specified resolution. As an example, right here is torch_triangular_solve(), utilized to the toy instance we manually solved above:
torch_tensor
1
3
0
[ CPUFloatType{3,1} ]Returning to our working instance, the conventional equations now seem like this:
[
mathbf{L}mathbf{L}^T mathbf{x} = mathbf{A}^T mathbf{b}
]
We introduce a brand new variable, (mathbf{y}), to face for (mathbf{L}^T mathbf{x}),
[
mathbf{L}mathbf{y} = mathbf{A}^T mathbf{b}
]
and compute the answer to this system:
Atb A$t()$matmul(b)
y torch_triangular_solve(
Atb$unsqueeze(2),
L,
higher = FALSE
)[[1]]Now that we now have (y), we glance again at the way it was outlined:
[
mathbf{y} = mathbf{L}^T mathbf{x}
]
To find out (mathbf{x}), we are able to thus once more use torch_triangular_solve():
x torch_triangular_solve(y, L$t())[[1]]And there we’re.
As regular, we compute the prediction error:
all_preds$chol as.matrix(A$matmul(x))
all_errs$chol rmse(all_preds$b, all_preds$chol)
all_errs lm lstsq neq chol
1 40.8369 40.8369 40.8369 40.8369Now that you just’ve seen the rationale behind Cholesky factorization – and, as already urged, the concept carries over to all different decompositions – you would possibly like to avoid wasting your self some work making use of a devoted comfort perform, torch_cholesky_solve(). It will render out of date the 2 calls to torch_triangular_solve().
The next strains yield the identical output because the code above – however, after all, they do cover the underlying magic.
L linalg_cholesky(AtA)
x torch_cholesky_solve(Atb$unsqueeze(2), L)
all_preds$chol2 as.matrix(A$matmul(x))
all_errs$chol2 rmse(all_preds$b, all_preds$chol2)
all_errs lm lstsq neq chol chol2
1 40.8369 40.8369 40.8369 40.8369 40.8369Let’s transfer on to the subsequent technique – equivalently, to the subsequent factorization.
Least squares (III): LU factorization
LU factorization is known as after the 2 components it introduces: a lower-triangular matrix, (mathbf{L}), in addition to an upper-triangular one, (mathbf{U}). In principle, there are not any restrictions on LU decomposition: Supplied we permit for row exchanges, successfully turning (mathbf{A} = mathbf{L}mathbf{U}) into (mathbf{A} = mathbf{P}mathbf{L}mathbf{U}) (the place (mathbf{P}) is a permutation matrix), we are able to factorize any matrix.
In observe, although, if we wish to make use of torch_triangular_solve() , the enter matrix needs to be symmetric. Subsequently, right here too we now have to work with (mathbf{A}^Tmathbf{A}), not (mathbf{A}) instantly. (And that’s why I’m exhibiting LU decomposition proper after Cholesky – they’re comparable in what they make us do, although in no way comparable in spirit.)
Working with (mathbf{A}^Tmathbf{A}) means we’re once more ranging from the conventional equations. We factorize (mathbf{A}^Tmathbf{A}), then clear up two triangular techniques to reach on the closing resolution. Listed here are the steps, together with the not-always-needed permutation matrix (mathbf{P}):
[
begin{aligned}
mathbf{A}^T mathbf{A} mathbf{x} &= mathbf{A}^T mathbf{b}
mathbf{P} mathbf{L}mathbf{U} mathbf{x} &= mathbf{A}^T mathbf{b}
mathbf{L} mathbf{y} &= mathbf{P}^T mathbf{A}^T mathbf{b}
mathbf{y} &= mathbf{U} mathbf{x}
end{aligned}
]
We see that when (mathbf{P}) is wanted, there may be an extra computation: Following the identical technique as we did with Cholesky, we wish to transfer (mathbf{P}) from the left to the best. Fortunately, what might look costly – computing the inverse – just isn’t: For a permutation matrix, its transpose reverses the operation.
Code-wise, we’re already conversant in most of what we have to do. The one lacking piece is torch_lu(). torch_lu() returns an inventory of two tensors, the primary a compressed illustration of the three matrices (mathbf{P}), (mathbf{L}), and (mathbf{U}). We are able to uncompress it utilizing torch_lu_unpack() :
lu torch_lu(AtA)
c(P, L, U) % torch_lu_unpack(lu[[1]], lu[[2]])We transfer (mathbf{P}) to the opposite aspect:
All that continues to be to be achieved is clear up two triangular techniques, and we’re achieved:
y torch_triangular_solve(
Atb$unsqueeze(2),
L,
higher = FALSE
)[[1]]
x torch_triangular_solve(y, U)[[1]]
all_preds$lu as.matrix(A$matmul(x))
all_errs$lu rmse(all_preds$b, all_preds$lu)
all_errs[1, -5] lm lstsq neq chol lu
1 40.8369 40.8369 40.8369 40.8369 40.8369As with Cholesky decomposition, we are able to save ourselves the difficulty of calling torch_triangular_solve() twice. torch_lu_solve() takes the decomposition, and instantly returns the ultimate resolution:
lu torch_lu(AtA)
x torch_lu_solve(Atb$unsqueeze(2), lu[[1]], lu[[2]])
all_preds$lu2 as.matrix(A$matmul(x))
all_errs$lu2 rmse(all_preds$b, all_preds$lu2)
all_errs[1, -5] lm lstsq neq chol lu lu
1 40.8369 40.8369 40.8369 40.8369 40.8369 40.8369Now, we take a look at the 2 strategies that don’t require computation of (mathbf{A}^Tmathbf{A}).
Least squares (IV): QR factorization
Any matrix could be decomposed into an orthogonal matrix, (mathbf{Q}), and an upper-triangular matrix, (mathbf{R}). QR factorization might be the most well-liked method to fixing least-squares issues; it’s, in actual fact, the tactic utilized by R’s lm(). In what methods, then, does it simplify the duty?
As to (mathbf{R}), we already know the way it’s helpful: By advantage of being triangular, it defines a system of equations that may be solved step-by-step, by way of mere substitution. (mathbf{Q}) is even higher. An orthogonal matrix is one whose columns are orthogonal – which means, mutual dot merchandise are all zero – and have unit norm; and the good factor about such a matrix is that its inverse equals its transpose. Basically, the inverse is tough to compute; the transpose, nevertheless, is straightforward. Seeing how computation of an inverse – fixing (mathbf{x}=mathbf{A}^{-1}mathbf{b}) – is simply the central process in least squares, it’s instantly clear how important that is.
In comparison with our regular scheme, this results in a barely shortened recipe. There isn’t a “dummy” variable (mathbf{y}) anymore. As an alternative, we instantly transfer (mathbf{Q}) to the opposite aspect, computing the transpose (which is the inverse). All that continues to be, then, is back-substitution. Additionally, since each matrix has a QR decomposition, we now instantly begin from (mathbf{A}) as a substitute of (mathbf{A}^Tmathbf{A}):
[
begin{aligned}
mathbf{A}mathbf{x} &= mathbf{b}
mathbf{Q}mathbf{R}mathbf{x} &= mathbf{b}
mathbf{R}mathbf{x} &= mathbf{Q}^Tmathbf{b}
end{aligned}
]
In torch, linalg_qr() provides us the matrices (mathbf{Q}) and (mathbf{R}).
On the best aspect, we used to have a “comfort variable” holding (mathbf{A}^Tmathbf{b}) ; right here, we skip that step, and as a substitute, do one thing “instantly helpful”: transfer (mathbf{Q}) to the opposite aspect.
The one remaining step now could be to resolve the remaining triangular system.
lm lstsq neq chol lu qr
1 40.8369 40.8369 40.8369 40.8369 40.8369 40.8369By now, you’ll expect for me to finish this part saying “there may be additionally a devoted solver in torch/torch_linalg, specifically …”). Properly, not actually, no; however successfully, sure. For those who name linalg_lstsq() passing driver = "gels", QR factorization shall be used.
Least squares (V): Singular Worth Decomposition (SVD)
In true climactic order, the final factorization technique we talk about is probably the most versatile, most diversely relevant, most semantically significant one: Singular Worth Decomposition (SVD). The third side, fascinating although it’s, doesn’t relate to our present process, so I gained’t go into it right here. Right here, it’s common applicability that issues: Each matrix could be composed into parts SVD-style.
Singular Worth Decomposition components an enter (mathbf{A}) into two orthogonal matrices, referred to as (mathbf{U}) and (mathbf{V}^T), and a diagonal one, named (mathbf{Sigma}), such that (mathbf{A} = mathbf{U} mathbf{Sigma} mathbf{V}^T). Right here (mathbf{U}) and (mathbf{V}^T) are the left and proper singular vectors, and (mathbf{Sigma}) holds the singular values.
[
begin{aligned}
mathbf{A}mathbf{x} &= mathbf{b}
mathbf{U}mathbf{Sigma}mathbf{V}^Tmathbf{x} &= mathbf{b}
mathbf{Sigma}mathbf{V}^Tmathbf{x} &= mathbf{U}^Tmathbf{b}
mathbf{V}^Tmathbf{x} &= mathbf{y}
end{aligned}
]
We begin by acquiring the factorization, utilizing linalg_svd(). The argument full_matrices = FALSE tells torch that we would like a (mathbf{U}) of dimensionality similar as (mathbf{A}), not expanded to 7588 x 7588.
[1] 7588 21
[1] 21
[1] 21 21We transfer (mathbf{U}) to the opposite aspect – an affordable operation, because of (mathbf{U}) being orthogonal.
With each (mathbf{U}^Tmathbf{b}) and (mathbf{Sigma}) being same-length vectors, we are able to use element-wise multiplication to do the identical for (mathbf{Sigma}). We introduce a brief variable, y, to carry the consequence.
Now left with the ultimate system to resolve, (mathbf{mathbf{V}^Tmathbf{x} = mathbf{y}}), we once more revenue from orthogonality – this time, of the matrix (mathbf{V}^T).
Wrapping up, let’s calculate predictions and prediction error:
lm lstsq neq chol lu qr svd
1 40.8369 40.8369 40.8369 40.8369 40.8369 40.8369 40.8369That concludes our tour of necessary least-squares algorithms. Subsequent time, I’ll current excerpts from the chapter on the Discrete Fourier Remodel (DFT), once more reflecting the concentrate on understanding what it’s all about. Thanks for studying!
Photograph by Pearse O’Halloran on Unsplash
